Aenorm 67

67

vol. 18 june ‘10

This edition:

Modelling the Scores of Premier League Football Matches

And:

Valuation and risk of Koopgarant Mortality Forecasting for Small Populations: The SAINT Framework Cartel Harm with Input Substitutability: Theory and an Application

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Eerlijk over werken bij AEGON AEGON biedt financiële oplossingen voor ieder moment in het leven.

Verzekeringen, pensioenen, hypotheken, spaarproducten en beleggingsproducten.

In Nederland werken wij met circa 3200 mensen vanuit vier kantoren om mensen te helpen aan een betere financiële toekomst. Deze kantoren staan in Den Haag, Leeuwarden, Nieuwegein en Groningen. Wereldwijd werken zo’n 31.500 mensen voor AEGON. Wij zijn actief in de Verenigde Staten, Europa, Canada en het Verre Oosten. Daarmee is AEGON één van ’s werelds grootste beursgenoteerde verzekeraars.

Werken bij AEGON Werken kun je bij meer bedrijven. Waarom AEGON? Het antwoord ligt uiteindelijk bij jezelf. We kunnen je wel helpen met wat feiten. AEGON is een standvastige organisatie die weet wat hij wil. Het doel is de beste en grootste pensioen- en inkomensverzekeraar te zijn. En dat is meer dan pensioenen en schadeverzekeringen. Het is behoud van inkomen en bezit. Nu en later.

Bij de grootste moet je niet meteen denken in getallen en euro’s. De grootste en beste willen zijn betekent toonaangevend willen zijn in het nemen van verantwoordelijkheid.

Dat we baanbrekend willen zijn met integere, transparante producten en diensten.

De toekomst kunnen we niet voorspellen. Wat we wel kunnen: ons steeds aanpassen. Elke dag weer vernieuwen en verbeteren. Met die instelling zijn we klaar voor morgen.

Kom maar op.

Mogelijkheden Als je met plezier naar je werk gaat, presteer je beter.

Dat is goed voor jou. En als je bij AEGON werkt, is dat ook goed voor AEGON. Dat plezier in je werk is natuurlijk persoonlijk.

Toch kunnen we er iets over zeggen dat vrijwel iedereen aanspreekt die bij AEGON werkt: Je wilt het verschil maken.

Je wilt de ruimte om te ondernemen.

Je wilt vast en zeker je ambities kunnen uitleven.

Wat is jouw doel?

We zijn op zoek naar zelfbewuste en ondernemende mensen die verantwoordelijkheid nemen. We kijken naar je sociale vaardigheden en je realisatiekracht.

Krijg je een goed gevoel bij begrippen als zelfstandigheid, ambitie en ondernemen? Dan willen we graag weten wat jouw doel is. We kunnen je nog genoeg vertellen. Maar wij zijn vooral ook benieuwd naar wat jij te vertellen hebt.

Voor concrete startfuncties of stages bij AEGON kijk je op onze website: www.aegon.nl/werk Meer weten? Bel dan met Ineke ten Cate op 070 344 5372, of stuur een mailtje naar Campus@aegon.nl

Colofon Chief editor Ewout Schotanus Editorial Board Ewout Schotanus Editorial Staff Daniëlla Brals Lennart Dek Winnie van Dijk Annelies Langelaar Chen Yeh Ron Stoop

Is it Possible to Prevent a Crisis?

Design United Creations © 2009 Lay-out Taek Bijman Cover design ©iStockphoto.com /RTimages (edit by Michael Groen) Circulation 2000 A free subscription can be obtained at www.aenorm.eu.

Advertisers AEGON All Options DNB Flow Traders KPMG TNO Towers Watson SNS REAAL Information about advertising can be obtained from Axel Augustinus at info@vsae.nl Insertion of an article does not mean that the opinion of the board of the VSAE, the board of Kraket or the redactional staff is verbalized. Nothing from this magazine can be duplicated without permission of VSAE or Kraket. No rights can be taken from the content of this magazine.

ISSN 1568-2188 Editorial Staff adresses VSAE Roetersstraat 11, C6.06 1018 WB Amsterdam tel. 020-5254134 Kraket De Boelenlaan 1105 1018 HV Amsterdam tel. 020-5986015

by: Ewout Schotanus For the last two years the financial crisis has been a big issue around the world. Banks all over the world have collapsed, governments have put billions into the world’s financial system to avoid even greater catastrophes and several hundred thousand people have been fired worldwide. Committees were brought into life to investigate the cause of the financial crisis and to find out who is to blame. The question which has been asked and has been debated for so many times is: “How can we prevent another financial crisis in the future?”

Although this is of course an important issue that has to be looked into, I personally think the question ‘how to survive the next financial crisis’ is much more interesting and realistic.

One problem with a crisis is it is unfair. Most people still got their job, their house and their accrued pension benefit. Therefore, the total overall loss due to the crisis is only distributed among a small percentage of the population. As a consequence, a large percentage of these people lost everything they had, their job, their house and their savings.

The people not really affected by the crisis will soon forget about it and the ones that were affected will probably be more focused on surviving another crisis in their own life.

I think the same holds for companies. The ones that still exist will just try to save themselves and within a few years they will only try to maximize their own profits. In addition, most people and companies are not able to prevent or provoke another crisis. The ones that have this kind of power, what incentive do they have to use this power in order to safe others in stead of just doing what is best for themselves.

Therfore, I think it is up to governmental organizations to prevent another crisis. This process has already been set in motion. In a broad perspective it is known what caused the collapse of the system and who or what is to blame. However, the precise chain of reactions leading to the crisis will never be known. Furthermore, the first steps leading to the most recent financial crisis were already taken after the crisis in 2001 by lowering the rate of borrowing money, which was initiated to stimulate the economy. Decisions with a positive effect in the short run can have negative effects in the long run. At last, if the next crisis will be caused by something we did not encounter before it is impossible to prevent this crisis. For example, fast growing economies like India and China are slowly changing the financial world of which the effects in the long run are hard to predict. It can also be possible another volcano in Iceland will erupt lasting for months instead of weeks. What would be the consequences for a country like the Netherlands that depends such a great deal on the Schiphol airport?

To come to a conclusion, I think the most important thing is to keep people aware of the fact another crisis is inevitable, because people tend to forget quite easily during prosperous times. If everyone is convinced it will happen again some day, people and companies will tend to take less risks which might be our best chance in preventing another crisis.

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© 2009 VSAE/ AENORM

vol. 18 (67)

June 2010

1

00 67 Modelling the Scores of Premier League Football Matches

04

by: Daan van Gemert

This paper estimates the fulltime scores of Premier League football matches using a statistical model which accounts for dependence between the number of goals scored by the home and away team. For the marginal distributions of the number of home and away goals, the censored zero inflated Poisson distribution and the censored Negative Binomial distribution are compared. Also, the profitability of these models against the bookmakers is investigated.

Cartel Harm with Input Substitutability: Theory and an Application

09

by: Daan in 't Veld This article examines the substitution effects of cartels. After a group of input producers has fixed a high price, its customers may change the relative demand of their inputs. To model this, the elasticity of substitution is used. Surprisingly, it is found that both with close substitutes and close complements, forming a cartel may not be profitable.

Mortality Forecasting for Small Populations: The SAINT Framework

15

by: Esben Masotti Kryger In recent years mortality forecasting has become a hot topic in demographics and life and pension insurance. The main reason for this renewed and augmented interest in the dynamics of mortality is a growing recognition of longevity: the fact that human beings (in rich countries) live still longer. In essence, there seems to be a struggle between deteriorations such as inactivity and obesity on one side, and improved treatment and other scientific advances on the other side.

Valuation and Risk of Koopgarant by: Bert Kramer

19

One of the biggest problems on the Dutch housing market is the large gap between the rental and the owner-occupied market. Since the latter is characterized by high prices due to limited supply, first-time buyers often face an affordability issue. Dutch housing corporations have developed several hybrid forms of sale over time. The most popular hybrid form of sale is called Koopgarant. Koopgarant combines a discount on the initial sales price with a number of conditions on the sales contract.

On the Effectiveness of Leniency Programs in Cartel Enforcement

23

by: H. Houba, E. Motchenova and Q. Wen The aim of introducing a leniency program is to reduce the maximal cartel price below the one under traditional antitrust regulation. However, we will argue that, given the current setup of leniency programs that do not allow for rewards, this is impossible. To the contrary, if the leniency program is wrongly designed, it even may have an adverse effect.

We provide conditions under which such adverse effects are eliminated: Individual fine reductions in case of multiple reporting firms should be moderate.

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vol. 18 00 june m. y.

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BSc - Recommended for readers of Bachelor-level MSc - Recommended for readers of Master-level PhD - Recommended for readers of PhD-level

Matching and the Allocation of Indivisible Objects via Deferred-Acceptance under Responsive Priorities

29

by: Bettina Klaus It is well-known that economics is the science of allocating scarce resources. Sometimes, there is more to the allocation mechanism than simple price setting and taking. There are economic problems in which using money and prices to match resources and consumers is not usually done or is considered immoral or even illegal. If money cannot or should not be used to determine who gets what, how else can we decide on matching resources to consumers?

Who Should Have the Control Right in the Organization?

33

by: Te Bao This article provides two application of incomplete contract theory to real life problems.

In the one on the privatization of public service sectors, we showed that the coexistence of private and public ownership is better than either privatizing all firms or let all of them to stay un-privatized. In the one on the divisional structure in an organization, we show why the coexistence of two divisions performing the same task is less likely to be seen than the coexistence of two firms producing the same goods in the market.

Approximation of Economic Capital for mortality and longevity risk

36

by: Erik Beckers Mortality and longevity risk form one of the primary sources of risk in many products of a life insurer. Life insurers wish to determine the amount of capital that should be available to cover these risks precisely. This capital is referred to as the Economic Capital, which may deviate in magnitude from the regulatory required capital. A few conventional methods are available to determine the Economic Capital for mortality and longevity risk, but these methods either lack stochastic scenarios for the mortality rates or seem to fail in their practical implementation because of extensive simulation time.

Complex Dynamics in an Asset Pricing Model with Updating Wealth

43

by: S. Brianzoni, C. Mammana, and E. Michetti We consider an asset pricing model with wealth dynamics and heterogeneous agents. By assuming that all agents belonging to the same group agree to share their wealth whenever an agent gets in the group (or leaves it), we develop an adaptive model which characterizes the evolution of the wealth distribution when agents switch between different trading strategies.

Two groups with heterogeneous beliefs are considered: fundamentalists and chartists.

Counting the Number of Sudoku’s by Importance Sampling Simulation

47

by: Ad Ridder Stochastic simulation can be applied to estimate the number of feasible solutions in a combinatorial problem. This idea will be illustrated to count the number of possible Sudoku grids. It will be argued why this becomes a rare-event simulation, and how an importance sampling algorithm resolves this difficulty.

Puzzle

51

Facultive

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Modelling the Scores of Premier League Football Matches by: Daan van Gemert The aim of this thesis is to develop a model for estimating the probabilities of premier league football outcomes, with the potential to form the basis for a profitable betting strategy against online bookmakers. Bets with a positive expected value can be made when the online bookies misprice their odds and when the model is accurate enough to detect these profitable betting opportunities. This thesis describes the development of the model and investigates the accuracy of the forecasts of this model compared to the predictions of online bookmakers. The model developed in this thesis estimates the final score of a football match, from which the predictions for the match result and the margin of victory can easily be calculated.

Data The data described in this section contain information about all Premier League matches only. The championship games are not included for the analysis of the data, since the final purpose of this paper is to find an appropriate model which serves as a basis to place bets on matches played in the premier league competition. When estimating the model, information about all championship games from the past five seasons will also be included. This extra data gives more information about teams who played only one season in the premier league or are totally new to the premier league.

The dataset consists of 1900 matches played during the seasons 2004-2005 to 2008-2009. All games end after 90 minutes of play plus some injury time determined by the referees, so there is no extra time or penalty shootout involved. The result of a football game is expected to be influenced by a variety of explanatory variables. The model in this paper only makes use of the full time match results from the past. All other factors such as injuries and suspensions are ignored.

Over the past five seasons, an average number of 2,523 goals were scored in a premier league football match, with a variance equal to 2,640. The teams playing

Daan van Gemert I obtained a BSc in Actuarial Sciences in june 2008 at the University of Amsterdam.

This article is a summary of my master thesis written under the supervision of Dr. J.C.M. van Ophem in order to obtain the master degree in Econometrics at the same University. The subject of my thesis is mainly chosen because of my great interests in football and gambling.

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at home scored an average number of 1,468 goals, with a variance of 1,617. For the away teams, this average was 1,055 with a variance of 1,158. The observations are slightly overdispersed; the variance exceeds the mean for both the home and away goals.

Marginal Distributions for home and away goals The number of goals scored by the home team and the number of goals scored by the away team are treated independently at first. The two simple count variables Y1 and Y2, the number of home and away goals respectively, with observations y1g and y2g (g = 1,...,1900) are considered.

This section compares the censored versions of the Poisson, zero inflated Poisson and Negative Binomial distributions to analyse these count data variables.

Censoring takes place after five goals. The results of the fits of these distributions to the observed data are shown in tables 1 and 2.

The Negative Binomial distributions give a closer fit to the data then the Poisson distributions for both the home and away goals. This is not surprising, since the Poisson distribution is a special case of the Negative Binomial distribution (the Negative Binomial distribution converges to the Poisson distribution as the parameter r goes to infinity). The Negative Binomial distribution accounts for overdispersion by introducing the extra parameter r.

Looking at the Poisson models for the home and away goals, it seems that the Poisson distribution overestimates the number of matches where the home and away teams score one or two goals, at the cost of the games where the home and away teams do not score any goals. So the data contain more zeros than the Poisson distribution predicts. The zero inflated Poisson distribution includes a proportion (1- π) of extra zeros and therefore accounts for the problem of excess of zeros.

Econometrics

Table 1. Expected frequencies for the number of home goals, based on the censored Poisson, censored zero inflated Poisson and censored Negative Binomial distributions. Maximum likelihood parameter estimates are given at the bottom of the table, with their standard errors between parentheses.

Table 2. Expected frequencies for the number of away goals, based on the censored Poisson, censored zero inflated Poisson and censored Negative Binomial distributions. Maximum likelihood parameter estimates are given at the bottom of the table, with their standard errors between parentheses.

Home Goals, censored distributions

Away Goals, censored distributions

# Home Goals

Observed

Poisson

ZIP

Neg. Binomial

# Away Goals

Observed

Poisson

ZIP

Neg. Binomial

0 1 2 3 4 ≥5 Total

469 621 456 217 100 37 1900

439.9 643.6 470.8 229.6 84.0 32.1 1900.0

469.0 610.2 462.7 233.9 88.7 35.5 1900.0

466.1 628.7 448.9 225.5 89.4 41.4 1900.0

0 1 2 3 4 ≥5 Total

692 680 335 131 51 11 1900

661.7 698.0 368.1 129.4 34.1 8.7 1900.0

692.0 657.1 366.4 136.2 38.0 10.3 1900.0

695.8 665.8 349.6 133.3 41.2 14.3 1900.0

1.463 (0.028)

0.965 (0.016) 1.516 (0.038)

1.055 (0.024)

0.946 (0.024) 1.115 (0.037)

π2 = Parameters

λ2 = r2 =

π2 = 1.465 (0.029) 17.067 (7.27)

The validity of the different models for the marginal distributions of home and away goals are tested by performing chi-square goodness-of-fit tests. The test results are presented in table 3. According to these tests, the censored Negative Binomial distribution gives the best fit to both the home and away goals data.

Assuming that the marginal distributions for home and away goals are independent from each other is quite strong. Since teams compete each other in a football match, it is likely that there exists some correlation between the number of goals scored by the home team and the number of goals scored by the away team. In the next chapter, a model is presented which accounts for a dependence structure between the marginal distributions of home and away goals. Both the censored Negative Binomial as well as the censored zero inflated Poisson distributions are taken as the marginal distributions in the correlated model. The estimation results of the two different models are then compared.

The Model The model presented in this section takes the following aspects into account (see also Dixon and Coles (1997)): • The different abilities of teams in a match.

• Teams playing in their home stadium have a ‘home ground advantage’ over the away teams.

• A team’s total ability is determined by its ability to

Parameters

λ2 =

1.055 (0.025) 10.260 (3.79)

r2 =

attack and its ability to defend.

• The correlation of the scoring performances of two competing teams in a match.

The Goal Model: the marginal distributions Consider a game between home team i and away team j.

Let n be the number of different teams in the dataset. Let Y1ij denote the number of goals scored by the home team and Y2ij the number of goals scored by the away team.

It is assumed that these two count variables are either drawn from a censored zero inflated Poisson distribution or from a censored Negative Binomial distribution. In both cases the parameters λ1ij and λ2ij are defined by λ1ij = exp {γ + αi – βj}, λ2ij = exp {αj – βi}.

The parameter γ represents the advantage of the home team when playing at its home stadium. This home ground advantage parameter is assumed to be equal for every team1. The set of parameters αi and βi, i =1,…,n, measure team i’s attack and defence strength. In words; in a match between home team i and away team j, the number goals scored by the home team depends on (home) team i’s ability to attack, team j’s ability to defend and the home ground advantage. The number of goals scored by the away team depends on team j’s ability to attack and team

In early stages of the development of the model, I have estimated different home effects for every team. This has led to insignificant parameter estimates.

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Econometrics

i’s ability to defend.

(

The Goal Model: the dependence structure To model the relation between the two scoring processes, the method introduced by van Ophem (1999) is followed.

Here, the bivariate cumulative normal distribution function is used to relate any two discrete random processes. If it is assumed that the two count variables Y1 and Y2 are marginally censored Negative Binomial distributed with unknown parameters (λ1ij, r1) and (λ2ij, r2), then the method to formalize the dependence structure between these two possibly correlated discrete stochastic variables is as follows: • Set the numbers η1,...,η5 and γ1,...,γ5 equal to:  κ −1  ηκ = Φ −1  ∑ p NB ( k ; λ1ij , r1 )  ,  k =0   −1   γ  = Φ −1  ∑ p NB ( m; λ2 ij , r2 )  m 0 =  

and define η6 = γ6 = +∞. pNB Denotes the density function of the censored NB distribution and Ф–1 is the standard normal quantile function.

• Maximize the following log likelihood function with respect to αi ,βi, γ, r1, r2 and ρ: log L ( αi , βi , γ , r1 , r2 , ρ; i = 1,..., n ) =

∑ log  B ( η N

g =1

y1 g +1

(

) ( ; ρ) + B (η , γ

, γ y2 g +1 ; ρ − B η y1 g , γ y2 g +1 ; ρ

− B η y1 g +1 , γ y2 g

y1 g

y2 g

is obtained after setting Pr(Y1g ≤ y1g, Y2g ≤ y2g) = B η y1 g +1 , γ y2 g +1 ; ρ . So the multivariate joint distribution function is constructed by specifying the dependence structure as a bivariate normal distribution on the transformed marginal distributions of Y1 and Y2, i.e. the Gaussian copula is used.

)

(1)

)

;ρ  

where B(•,•;ρ) is the bivariate normal cumulative distribution function with expectations 0, variances 1 and correlation ρ.

The log likelihood function from the second step

)

Results Estimation of the model yields an estimated dependence parameter of ρˆ = 0.074 (0.018) for the model with Negative Binomial marginal distributions and a dependence parameter estimate of ρˆ = 0.070 (0.017) for the zero inflated Poisson model. So the number of goals scored by the home team and the number of goals scored by the away team are positively correlated. For both models the dependence parameter ρ is highly significant.

The estimation results of the ability parameters for the NB model, including their standard errors, can be found in table 4. The ZIP model gives comparable estimates.

The teams are ranked according to the total ability; the sum of the estimated attacking and defending strength parameters. The standard errors of the estimated ability parameters are typically about 0.05 – 0.07.

The home ground advantage parameter γ is estimated to be equal to 0.30 (0.018) for the NB model and 0.31 (0.026) for the ZIP model. Thus; home teams have a highly significant home ground advantage over the away teams.

The parameter estimates for π1 and π2 in the ZIP model are equal to πˆ 1 = 1.040 (0.010) and πˆ 2 =1.049 (0.016).

These estimates are significantly bigger than one with p-values of 0.007% and 0.160%. This implies a reduction in the probability of zero goals, which is not what we expected since we observed an excess of zeros in the data for home and away goals. This result may be caused by the fact that in the correlated model, the dependence parameter already accounts for an extra proportion of the score 0-0. Estimation of the NB model yields dispersion parameter estimates of and rˆ1 = 2,675,100 and rˆ2 = 1,138,100. These extreme high estimates suggest that using marginal censored Poisson distributions for home and away goals might be equally appropriate as using the

Table 3. Results of chi square goodness of fit tests for the number of home goals and the number of away goals. The degrees of freedom for these tests are 4, 3 and 3 respectively.

Null-hypothesis

H0: H0: -

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Number of home goals follows a… censored Poisson distribution censored Zero - Inflated Poisson distribution censored Negative Binomial distribution Number of away goals follows a… censored Poisson distribution censored Zero - Inflated Poisson distribution censored Negative Binomial distribution

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p-value

Conclusion

10.39% 38.89% 52.09%

do not reject H0 at 10% level do not reject H0 at 10% level do not reject H0 at 10% level

0.80% 4.21% 25.64%

reject H0 at 1% level reject H0 at 5% level do not reject H0 at 10% level

Econometrics

Table 4. Attacking and defending parameter estimates and standard errors from the NB model for all teams included in the dataset. Teams are ranked according to their total strength. The estimates are obtained with maximum likelihood and each match has equal weight in determining the parameter estimates.

Team

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Chelsea Man United Liverpool Arsenal Everton Tottenham Aston Villa Man City Blackburn Bolton Reading Middlesbrough Wigan Portsmouth Newcastle Fulham West Ham Birmingham Sunderland West Brom Sheffield United Swansea Wolves Preston Crystal Palace Stoke Charlton Ipswich

se( )

se( )

Team

se( )

se( )

1.60 1.66 1.51 1.66 1.26 1.38 1.33 1.19 1.21 1.19 1.19 1.16 1.10 1.14 1.17 1.16 1.14 1.00 1.06 1.15 0.99

0.06 0.06 0.06 0.06 0.07 0.06 0.06 0.07 0.07 0.07 0.06 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.06 0.06 0.06

1.95 1.76 1.66 1.52 1.38 1.19 1.16 1.24 1.17 1.18 1.15 1.11 1.16 1.12 1.08 1.07 1.07 1.15 1.02 0.92 1.04

0.10 0.09 0.08 0.08 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.07 0.06 0.06 0.06

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

Watford Cardiff Bristol City Hull Burnley Southampton Sheffield Weds Colchester Plymouth Leeds Derby Blackpool Leicester QPR Norwich Coventry Doncaster Millwall Luton Barnsley Nott Forest

1.01 0.91 0.87 0.90 0.85 0.95 0.84 1.06 0.81 0.77 0.92 0.85 0.72 0.82 0.91 0.85 0.63 0.62 0.95 0.79 0.72

0.06 0.06 0.10 0.07 0.06 0.06 0.07 0.09 0.06 0.08 0.06 0.10 0.07 0.06 0.06 0.06 0.15 0.11 0.09 0.08 0.10

0.85 0.91 0.94 0.88 0.92 0.81 0.85 0.63 0.88 0.91 0.76 0.80 0.93 0.81 0.72 0.74 0.95 0.92 0.59 0.67 0.73

0.06 0.06 0.10 0.07 0.06 0.06 0.07 0.09 0.06 0.08 0.06 0.09 0.07 0.06 0.06 0.06 0.14 0.10 0.08 0.07 0.09

1.03 1.00 0.98 0.96 0.89 1.00 1.05

0.12 0.06 0.06 0.06 0.06 0.07 0.06

0.99 0.96 0.98 0.99 1.02 0.90 0.83

0.14 0.06 0.06 0.06 0.06 0.06 0.06

50 51 52 53 54 55

Crewe Gillingham Scunthorpe Southend Brighton Rotherham

1.00 0.68 0.70 0.72 0.55 0.43

0.09 0.15 0.15 0.14 0.11 0.17

0.43 0.71 0.68 0.53 0.68 0.67

0.08 0.12 0.12 0.11 0.09 0.12

marginal censored Negative Binomial. Apparently, the censored Negative Binomial distribution is not a very suitable choice for the marginal distributions in correlated model specified in the previous chapter. Taking censored Poisson distributions as the marginal distributions in the correlated model produces the same estimation results for all the parameters and standard errors (at least up to two decimals).

Betting Results The data used in this section contain all matches played from the beginning of season 2009/2010 up to the 7th of February 2010 (a total of 242 games), including information about the odds from more than 30 bookmakers. Match odds, total goals odds and Asian handicap odds are all available in the data2. Similar to Kuypers (2000), the following betting strategy is applied: place â‚Ź1 on outcome i of a particular match j if

Match odds are the odds for betting on the match result (home win, draw or away win). Total goal odds are the odds for betting on the total number of goals scored in the match. Only two outcomes are separated; over or under than 2.5 goals. Asian handicap is a form of betting on the match result, where one of the two teams receives a virtual head start. The team who scores the most with the handicap applied is deemed the winner.

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1 + r1 < Probability Ratio Predicted Probability from Model = < 1 + r2 1/ Oij

where Oij are the best available odds on outcome i for match j.

Overall, the results of the Poisson model seem to be better than the results of the ZIP model. For betting on match results, the Poisson model would have made a maximum profit of €9.71. This profit was generated for r1 = 6% and r2 = 13%, placing a total number of 62 bets, 24 of which were winning. So a total return of 15.66% was made. For betting on the total score of the match, most betting strategies would have resulted in a loss, and both models do not seem to be capable of generating a profit against the bookmakers. For Asian handicap betting however, profits are made for almost every strategy, and both models seem to perform better against the bookmaker on this form of betting. For the betting strategy based on the Poisson model with r1 = 2% and r2 = 15% a profit of €12.09 was made, generated by placing 76 bets (out of which 37 bets were winning and 17 bets were voided). This equals a total return of 15.91%.

The maximum returns of 15.66% and 15.91% for betting on match results and Asian handicap betting can be compared to the returns realized by using a random betting strategy: placing a €1 bet on one outcome at random in 62/ 76 randomly chosen matches. For betting on 62 randomly chosen match results, a simulation of 100,000 returns resulted in an average profit of - €1.98 (-3.20%) with an estimated standard deviation of 13.34.

For Asian Handicap betting, the estimated expected profit of 76 random bets equals - €3.17 (-4.17%) with an estimated standard deviation of 9.22. So a random betting strategy obviously results in negative returns, but the total profit after 62 or 76 random bets show high standard deviations, and the positive returns of €9.71 and €12.09 are only 0.9 and 1.7 standard deviations above the returns obtained from random betting. Although some of the betting strategies seem to be profitable, this still does not say much about the capability of the model to make profit over the bookmakers. Because of the high standard deviations it is also not clear which model gives the better results.

Concluding remarks This paper developed a model for estimating the scores of Premier League matches. Previous literature was extended by allowing for correlation between the scoring performances of the home and away team. A correlation coefficient of about 7% was found, suggesting that the number of goals scored by the home and away team are positively correlated. A structural shortcoming of the model presented in this paper is that every match has an equal weight in determining the ability parameters αi and βi. So it is assumed that teams have a constant attacking

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and defending strength over the whole period. This is very unrealistic, since a team’s strength varies over time and tends to be dynamic. The stochastic development of the ability parameters should be specified to improve the model on this topic. It was found that profitable betting opportunities based on the model estimations do exist, comparing the odds from more than 30 bookmakers.

However, due to the high variance of the realized returns the betting results have to be interpreted with caution.

References Dixon, M. J. and S. G. Coles. “Modelling association football scores and inefficiencies in the football betting market.” Applied Statistics 46.2 (1997):265 – 280.

Kuypers, T. “Information and efficiency: an empirical study of a fixed odds betting market.” Applied Economics 32 (2000):1353 – 1363.

Ophem, H.v. “A general method to estimate correlated discrete random variables.” Econometric Theory 15 (1999):228 – 237.

Mathematical Economics

Cartel Harm with Input Substitutability: Theory and an Application by: Daan in 't Veld This article examines the substitution effects of cartels. After a group of input producers has fixed a high price, its customers may change the relative demand of their inputs. To model this, the elasticity of substitution is used.

Surprisingly, it is found that both with close substitutes and close complements, forming a cartel may not be profitable.

Introduction A cartel is an agreement between firms to fix prices above competitive levels. While higher prices typically lead to a transfer in wealth from customers towards cartel members, the main concern for antitrust authorities is the accompanying loss in social welfare. For this reason, antitrust law is assigned to ex ante deter cartel agreements, and ex post punish the offenders and return the wealth to the rightful owner.

Simple as this may sound, the analysis of cartels has turned out to be very complex. Unlike other deprivations (for example, a bank robbery), a cartel often provokes a cycle of reactions from many relevant agents in the economy; wealth is redistributed among all these agents.

To model a number of vertical cartel effects, Han, Schinkel and Tuinstra (2008) use a fixed proportions production chain with an arbitrary number of layers. My purpose is to investigate horizontal or substitution effects, connected with firms outside the cartel, but within the same production layer.

Covering the behavioural changes of the different agents, game theory is a useful tool to explain and describe cartel effects. In this article I will compare two equilibria attached to situations before and after the cartel.

Surprisingly, it will be found that forming a cartel is not always profitable. This rather paradoxical proposition was for the first time set forth in a famous paper of Salant, Switzer and Reynolds (1983) in the context of mergers.

Intuitively, the possibility of a disadvantageous cartel arises because other firms in the same production layer react to the new strategy of the cartel members.

Model of production and assumptions We will look at the three-layer industry presented in Figure 1. There are n downstream firms producing consumer

goods, using two inputs from m1 and m2 upstream suppliers. Downstream quantity and price is denoted by (Q, p); upstream market quantities and prices by (Z1, w1) and (Z2, w2), respectively. The separation of the upstream firms is the major deviation from the mainstream cartel literature. I will explore the implications of price-fixing agreements by the m1 primary input producers under the existence of secondary input market.

The industry will be subject to the following assumptions. First, the relation between the two inputs and the output is specified. We use a CES (constant elasticity of substitution) production function: 1 σ −1 1 σ −1 σ Q( Z1 , Z 2 ) = ( Z 1 σ + Z 2 σ ) σ −1 , σ ≥ 0 2 2

(1)

The CES production function is constructed in such a way that σ equals the elasticity of substitution, defined as −

∂[ Z1 / Z 2 ] w1 / w2 ∂[ w1 / w2 ] Z1 / Z 2

Daan in 't Veld Daan in ‘t Veld completed the mathematical economics track of the MSc in Econometrics (cum laude) at the University of Amsterdam.

This article is a summary of his master thesis written under the supervision of dr. Jan Tuinstra.

Currently, Daan stays at the University St. Petersburg to fully master the Russian language and attempt to understand the economic situation in Russia.

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Figure 1. The three-layer industry with two groups of input producers.

individual profits (indicated with *), and the cartel state in which the m1 firms collude (with C). Game-theoretically, in the latter state there remains only one decision-making entity in the primary input market.

Equilibrium analysis Given the demand curve p(Q) and prices w1 and w2, every downstream firm i maximises individual profits πi: max π i = max( p (Q ) − mc ( w1 , w2 )) qi , qi

(2)

qi

where mc(w1, w2) are the constant marginal costs of the downstream firms, and can be derived from (1). This is a quite simple optimization problem in one dimension, where only the produced quantities of the n − 1 firms other than i are unknown. Using the fact that all firms are identical, the first order condition for (2) provides the competitive Cournot equilibrium quantity: In words, the elasticity of substitution measures how the demand ratio of inputs Z1/Z2 reacts to a change in the price ratio w1/w2. The higher σ, the more elastic the input substitution, and, for example, the more the demand ratio Z1/Z2 will fall after a price increase ∆w1.

More insight in the effect of σ can be obtained by looking at two limit cases. For σ↓0 there are no substitution possibilities for firms, and the inputs, here called perfect complements, are used in equal proportions. Indeed, it is possible to derive that limσ↓0Q(Z1,Z2) = min(Z1,Z2).

Alternatively, when σ approaches infinity, Z1/Z2 becomes very sensitive to the input price ratio, and every downstream firm will tend more and more towards the input with the lowest price. Thus inputs become perfect substitutes. It is easy to see that (1) now becomes linear: Q(Z1,Z2 ) = 12 Z1 + 12 Z 2 .

The typical assumption is that firms compete in quantities, and that prices are determined by market forces. In other words, all three markets are Cournot oligopolies. Naturally, upstream production precedes downstream production; downstream production precedes consumption. We use backward induction for our sequential game with two stages. Therefore, we will start with the downstream market equilibrium for given input prices w1 and w2. Then we will derive the optimal choices for the upstream firms that will lead to an equilibrium with certain (w1, w2), using the knowledge of the strategies of the downstream firms.

Finally, there are two technical assumptions to simplify the analysis. Marginal costs in the input markets are set constant and equal to d and e, respectively. Next, consumer demand will be expressed by a linear demand curve p(Q) = a − bQ. As our point of interest in this industry is cartel effects, we compare the market outcomes in two states: the competitive state in which all firms are maximising

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qi ( w1 , w2 ) =

1 ( a − mc ( w1 , w2 )). ( n + 1)b

(3)

For the two upstream markets some complications arise. Consider the primary input market. While it is assumed that upstream firms recognise the optimal downstream strategy given by (3), they also have to reckon with the simultaneous production of the secondary input firms, Z2. When all m1 firms behave competitive, they are maximizing profits as follows: max π1 j = max( w1 ( Z1 , Z 2 ) − d ) ∗ z1 j , z1 j

z1 j

(4)

where w1(Z1, Z2) is an inverse demand function that determines the market-clearing price as a function of Z1 and Z2. For the cartel, the counterpart problem of (4) is: max Π1 ( Z1 ) = max( w1 ( Z1 , Z 2 ) − d ) ∗ z1 .

Z1

(5)

Z1

In both the competitive and cartel state, the optimization problem results in a first order condition of the form FOC1(Z1, Z2) = 0. This first order condition can be rewritten as an aggregate reaction function: R1(Z2) ≡ {Z1 | FOC1(Z1, Z2) = 0}

(6)

Note that this is a rather peculiar reaction function, because it describes the optimal reaction of the complete primary input market to the production of the secondary market. This notion of aggregate reaction functions was earlier used by Salant, Switzer and Reynolds (1983). Due to the symmetry, we can collect all individual reactions together in one reaction.

TNO

Mijn fascinatie Doelgericht innoveren. Nieuwe producten, nieuwe diensten,

nieuwe mogelijkheden creĂŤren. Creatieve antwoorden vinden op de vragen die de samenleving stelt. Werken aan betere oplossingen: sneller, veiliger, slimmer, efficiĂŤnter. Dat is mijn fascinatie.

Mathematical Economics

Figure 2. Aggregate reaction functions and the equilibrium for σ→1.

Figure 3. The merger paradox for σ→∞.

The equilibria are found in the intersection points of R1(Z2) and R2(Z1), where R1(Z2) belongs to firms either competing or colluding. In Figure 2, the reaction functions and the equilibrium are drawn in the {Z1, Z2}–plane.

Clearly, in the intersection point no firm of any market can raise higher profits by changing his production level.

Although it is difficult to derive the reaction functions explicitly for general σ, the two limit cases can be worked out. This will enable us to understand two possible situations in which forming a cartel is not profitable: the merger paradox and the reinforced race-to-the-bottom.

function in the cartel state, it suffices now in (7) to replace m1 by 1.

The equilibria in the two states for perfect substitutes are represented in Figure 3. By forming a cartel, the reaction function of the primary input firms shifts proportionally towards the vertical axis. The cartel profits are maximised by R1C (Z2), and therefore Π1(R1C (Z2)) > m1 n 2( a − 2 d ) Π1(R1* (Z theZ 2cartel equilibrium, ( Z22))) for (Z2. However, in − ) = every n + 1 produces b 1 + 1 market the secondaryminput more, and therefore it is possible that the cartel makes fewer profits than the sum of the competitive primary firms.

This result has become known as the merger paradox since the famous paper of Salant, Switzer and Reynolds (1983). The merger paradox, as I will also call it in the present context of cartels, occurs as a special case in our model. Salant, Switzer and Reynolds gave a necessary condition for the advantageousness of the cartel if d=e. For us, it is enough to know that there exist some upperbound σ , given the other parameters in the model, where for σ>σ forming a cartel is not profitable because of the merger paradox.

The merger paradox We start with perfect substitutes (σ→ ∞). Now downstream firms only choose the input with the lowest price, and have a linear production function Q = 12 Z1 + 12 Z 2 . This means that downstream marginal costs equal mc(w1, w2) = 2min(w1, w2). Only when prices are equal, both inputs are produced. Combining these results with (3), it is not difficult to derive that under w1 = w2, w1 ( Z1 , Z 2 ) =

The reinforced race-to-the-bottom

1 1 ( n + 1)b (a − ( Z1 + Z 2 )).

2 2 n

With this relatively simple formula for w1, the first order condition of (4) can be solved to the aggregate reaction of competing firms R1* ( Z 2 ) =

m1 n 2( a − 2 d ) − Z2 ) ( m1 + 1 n + 1 b

(7)

and we see that the reaction function is linear and downward-sloping. In order to obtain the reaction

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For close complements, as well as for close substitutes, it will turn out that there exist disadvantageous cartels.

In the limit case of perfect complements, our model reproduces a puzzling result of Sonnenschein (1968).

He showed that firms producing perfect complements optimally choose their quantity just below the choice of the opponent, hereby creating a shortage of their own product, and an accompanying high price. This leads to a so-called race-to-the-bottom until Z1 = Z2 = 0.

The race-to-the-bottom holds even when there exist perfect substitutes for every complement, i.e. as in our situation where m1, m2 > 1, which was also shown by Dari-

Mathematical Economics

Figure 4. The reinforced race-to-the-bottom for σ=0.25.

relatively well documented. The application underlined the relevance for taking substitution effects into account.

References Dari-Mattiaci, G. and F. Parisi. “Substituting complements.” Journal of Competition Law and Economics 2.3 (2006):333-347.

Han, M.A., M.P. Schinkel. and J. Tuinstra. “The overcharge as measure for antitrust damages.”

Amsterdam center for law and economics working paper series 2008-08.

Salant, S.W., S. Switzer. and R.J. Reynolds. “Losses from horizontal merger: the effects of an exogenous change in industry structure on Cournot-Nash equilibrium.”

Quarterly Journal of Economics 48 (1983):185-200.

Mattiaci and Parisi (2006). This means that for perfect complements, the Cournot equilibrium constitutes to zero production in the entire industry. In accordance with Sonnenschein, a cartel is unable to resolve this situation.

It may seem trivial that a cartel cannot succeed in raising profits in a paralysed industry. However, a cartel may also create such a situation itself. Any cartel will reduce the primary input production Z1, in order to profit from the excess demand. This means that for σ close to 0, the cartel actually reinforces the race to the bottom, and therefore even fewer profits are made. Again the strong reaction of the secondary input producers is critical for the loss of profits. This reinforced race-to-the-bottom is illustrated in Figure 4.

Even when in the cartel equilibrium both inputs are produced with a positive quantity, it is possible that the rigorous restriction of the production, initiated by the cartel formation, leads to fewer profits. This means that, depending on the remaining parameters, there exists a underbound σ , for which σ < σ the (reinforced) race-tothe-bottom occurs.

Sonnenschein, H. “The dual of duopoly is complementary monopoly; or, two of Cournot’s theories are one.”

Journal of Political Economy 76 (1968):316-318.

Conclusions We have seen that cartels may be disadvantageous if there is either a close substitute or close complement available.

Given the remaining parameters, a cartel is profitable only if σ > σ > σ . The merger paradox and race-to-the-bottom, known for limit cases, were previously not connected and interpreted in terms of the σ-spectrum.

Other conclusions in the thesis were related to the ex post estimation of cartel harm using the common overcharge measure, and the possible compensation to the secondary input producers. Additionaly, I applied the model to the lysine conspiracy of the 1990’s. This cartel between four global producers of feed supplements is one of the most harmful ever discovered, and, consequently,

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Actuarial Sciences

Mortality Forecasting for Small Populations: The SAINT Framework by: Esben Masotti Kryger In recent years mortality forecasting has become a hot topic in demographics and life and pension insurance. The main reason for this renewed and augmented interest in the dynamics of mortality is a growing recognition of longevity: the fact that human beings (in rich countries) live still longer. For instance, life expectancy has increased by roughly a year every fourth year in the US for most of the 20th and 21st centuries! And this development has not stopped even at present life expectancies of 81 and 76 years for women and men respectively. At the same time there is an ongoing debate about the prospects for human life length. One side argues that improvements cannot be expected to continue at the same pace as hitherto, so there will be no drastic increase in, say, life expectancy in this century. Others maintain that there is no reason to suspect that there exists a definite upper limit to human life span, and that improvements could continue at unchanged speed for some time still. In essence, there seems to be a struggle between deteriorations such as inactivity and obesity on one side, and improved treatment and other scientific advances on the other side.

Background Demographers and life and pension insurers are highly interested in the¸ future development of mortality.

Demographers typically project the size and composition of national or regional populations, while insurers merely estimate the future number of survivors (or deceased) within their respective populations. To the latter end the central quantity is the so-called time- and age-dependent force of mortality, μ(t, x), which can be interpreted as the (annualized) probability to die at time t and age x. This quantity is not observable, but under standard assumptions it can be approximated consistently by the death rate, m(t, x), which can be calculated directly from population data.

Typically death rates are published for integer-valued t and x, e.g. representing a whole calender year and an “age last birthday”, and with the implicit assumption that the underlying force of mortality is constant on an appropriate region, e.g. [t, t+1) × [x, x+1). The observed death rate turns out to be rather erratic in small populations, and much smoother in larger populations, c.f. Figure 1, which compares the death rates for 70-year old women in a pool of Western countries and Denmark since 1950. The figure shows that although the trend is similar (save the period 1980-1995) the annual fluctuations are much larger in the mortality of the smaller population. This irregularity is caused by the random nature of death; that is, even if one knows the correct distribution of fatalities, these may occur at any future time. This concept is often referred to as unsystematic variability or diversifiable mortality risk as it vanishes in larger populations due to the Law of Large Numbers. However, as the true underlying future

mortality, i.e. the distribution of fatalities is unknown, there will be an additional discrepancy between forecasts and realizations - even in populations of unbounded size. This remainder is denoted systematic variability or non-diversifiable mortality risk. There is good reason to suspect that smaller populations exhibit larger systematic and unsystematic variability.

For the remaining text we shall study 70-year old Danish women only, but rest assured that the conclusions apply to both genders, many nationalities and all relevant age groups. Data for this study originates from the Human Mortality Database (www.mortality.org), which offers free access to updated records on death counts and population sizes for a long list of countries. The database is maintained by University of California, Berkeley, United States and Max Planck Institute for Demographics Research, Germany. The aforementioned pool collects mortality data from 19 Western countries, including USA and Japan, and is about 100 times as large as Denmark.

Esben Masotti Kryger Esben Masotti Kryger is an industrial ph.d.student (actuarial science) at ATP (the Danish Labour Market Supplementary Pension Scheme) and the University of Copenhagen.

He undertakes research in the areas of pension scheme design, portfolio selection, and mortality modelling. Previous research has been published in the Scandinavian Actuarial Journal, The Geneva Risk and Insurance Review.

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1960

1970

1980

1990

2000

0.020 0.010

0.020 0.010

0.005

1950

Death rate

Pool Denmark

Figure 2. Mortality forecasts and 95% two-sided margnal confidence intervals from the Lee-Carter model for 70year old Danish women. The lines stem from a forecast based on the estimation period 1950:2005, while the points represent the data window 1970:2005.

0.005

0.035 0.020 0.025 0.015

Death rate

0.015

0.020 0.025

0.035

Figure 1. Mortality development for 70-year old Western and Danish women.

Year

2010

Members of the pool will be denoted Western.

Standard forecasting methods The most widespread mortality is the seminal model by Lee and Carter (1992), which models the death rates as log m(t, x) = a(x) + b(x)κ(t) + ε(t, x), with the normalizations ∑xb(x) = 1, ∑tκ(t) = 0. The fitted values of κ are treated as observed and subsequently modelled as a random walk with drift. A modern version of the model assumes that the number of deaths in each year and each group follows independent Poisson distributions. This text shall not go deeper into the technicalities of the Lee-Carter model, only mention that it typically works well in large populations due to stability of data. It performs correspondingly poorly in small populations for the same reason. As an example, see Figure 2, which compares forecasts of the mortality of 70-year old Danish women based on two different estimation periods. The forecasts are in total disagreement about the trend, and - to a lesser extent - the width of the associated uncertainty. In comparison to most Life and Pension insurers the Danish population is in fact rather large with cohort sizes exceeding 20,000 persons at age 70.

For the sake of calculating the technical provisions for life insurance and pensions it is essential that mortality forecasts are trustworthy, which means they should be plausible and robust - and preferably have little uncertainty. This task is particularly difficult for long horizons, as short term variability is relatively small and of little importance to the technical provisions. As can be seen from Figure 2 these essential criteria are not satisfied when the Lee- Carter model is applied to Denmark, and this phenomenon is general for small populations when using random-walk models. Determination of the trend mortality is harder for smaller populations. This is because the data window used for the estimation becomes more central the larger the fluctuations in data. That is, the models are less robust when applied to small populations.

Also, the uncertainty of the forecasts seems unreasonably large in the light of historic data. The reason is that these

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models translate the observed annual fluctuations into correspondingly large forecast uncertainties. Overall, the difference between basing an insurance business on one or the other projection in Figure 2 is enormous.

When applying the Lee-Carter model to the pooled data the trend determination is far more robust, and the confidence intervals of the projections are much narrower, c.f. Figure 3.

Figure 4 shows an example of diverging mortality projections. According to those forecasts, the Danish trend is far outside the confidence interval for the West, and vice versa. Altogether, this set of projections seems unrealistic. Rather than divergence, data suggests that future mortalities for the two countries should develop similarly. In addition to this data-based explanation intuition also supports the idea of converging mortality for countries, which are broadly similar with respect to socio-economic factors, lifestyle, technology etc. For further insight into this topic, see e.g. Tuljapurkar, Li and Boe (2000), Wilson (2001), or Wilmoth (1998).

A robust approach to forecasting In order to forecast the mortality of a small populations in a robust manner it is a good idea to use the aforementioned plausible forecasts for large populations. To this end one should find a large reference population, the future mortality of which can be assumed (or tested) to be similar to that of the population of interest. The SAINT (Spread Adjusted InterNational Trend) concept does exactly this by modelling the deviations between a trend mortality and the mortality of the population of interest. First, a mortality trend forecast for the reference population, μref(t, x) is taken as given for a number of years, t, and ages, x. This estimate could have come from anywhere, e.g. a model or an expert. It is then assumed that the ageand time-specific death count of the subpopulation is Poisson distributed with force of mortality μ(t, x) = μref(t, x) exp (r(x)´y(t)), where r(x) is an n-dimensional set of fixed regressors,

Actuarial Sciences

2010

2020

2030

2040

2050

0.020 0.010 0.005

0.010

Death rate

0.020

Figure 4. Mortality forecasts and 95% two-sided marginal confidence intervals from the Lee-Carter model for 70year old Western (circles) and Danish (lines) women. The data window is 1970:2005.

0.005

0.020 0.010 0.005

0.010 0.005

Death rate

0.020

Figure 3. Mortality forecasts and 95% two-sided marginal confidence intervals from the Lee-Carter model for 70year old Western women. The lines stem from a forecast based on the estimation period 1950:2005, while the points represent the data window 1970:2005.

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Year

and y(t) is an n-dimensional time series, which is to be estimated and forecasted. Assuming that y(t) is stationary will ensure that the median mortality forecast does not diverge from the trend. In the following illustration we shall assume that lim E[ y (t )] = 0, t →∞

such that the trend mortalities converge: lim E[log  (t , x ) − log ref (t , x )] = 0, t →∞

for all ages, x. It is also possible, however, to work with a non-zero limiting assumption on E[y], which can be relevant for industry-wide pension funds for instance, the members of which may have a suspected mortality pattern that is substantially different from that of the larger, say national, population.

An example As an application of the SAINT framework we shall demonstrate how to construct a robust forecast for Danish women. This example has n = 3 and r(x) = (1, ax + b, cx2 + dx + e) with a, b, c, d, e chosen such that the regressors are orthogonal, such that the second coordinate has values ±1 at the lowest and highest ages included in the forecast respectively, and such that the third regressor has values +1 at both the highest and the lowest age considered. This construction ensures that the regressors can be interpreted as a level, a slope, and a curvature effect respectively. For the time-series part we choose the model y(t) = Ay(t – 1) + ε(t), with A being a stationary n × n matrix and ε being i.i.d.

Gaussian with mean zero and covariance matrix ­Ω, i.e. a Gaussian VAR(1)-model. Notice that the aforementioned non-zero limiting spread can be obtained by including a fixed term on the right-hand side of the equation, which can be pre-specified or estimated - depending on the

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application in mind. The sketched structure gives the following 95% two-sided marginal confidence intervals for the future Danish mortality in year T + h at age x:  (T + h, x ) ∈ ref (t + h, x ) exp( Ah r ( x )′y (T ) ± 1.96 r ( x )′(∑ i = 0 Ai ( Ai )′)r ( x ), h =1

where T is the last observation year. It is notable that these bounds increase very slowly with time - in contrast to those of random-walk models.

Figure 5 shows the results of this technique applied to Denmark with a pooled international data set consisting of most Western countries as the reference population.

It is particularly noteworthy that the long-run trend is reasonably stable across all three estimation periods, and thus much more believable than above. This will result by construction, since the mean forecast for Danish mortality converges towards the reference trend, which varies only little across estimation periods due to robustness.

However, the confidence intervals are also far narrower than in Figure 2, and they increase much slower.

The SAINT approach can obviously be applied to other stationary spread dynamics as well as different models for the reference population, be they deterministic or stochastic. Further, the method can be used to ensure coherent projections between genders within the same population, or within a whole group of populations.

The method is described in detail and implemented with a so-called frailty mortality model (which is wellsuited to forecast mortality among the oldest old) for the reference populations in the paper “Modelling Adult Mortality in Small Populations: The Saint Model” by Søren Fiig Jarner and Esben Masotti Kryger.

References Jarner, Søren Fiig and Esben Kryger. “Modelling adult mortality in small populations: The SAINT model.”

Pensions Institute Discussion Paper PI-0902. 2008. (http://www.pensions-institute.org/workingpapers/ wp0902.pdf).

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Figure 5. Mortality forecasts and 95% two-sided marginal confidence intervals from the SAINT model for 70-year old Danish women. The lines mark the Danish data and projections, while the circles represent an international data set, which is used as the reference population. The upper panel is based on the data window 1950:2005, and the lower panel on 1970:2005.

Lee, Ronald D. and Lawrence R. Carter. “Modeling and forecasting of U.S. mortality.” Journal of the American Statistical Association 87 (1992):659-675.

Tuljapurkar, Shripad, Nan Li and Carl Boe. “A universal pattern of mortality decline in the G7 countries.”

Nature 405 (2000):789-792.

Wilmoth, John R. “Is the pace of Japanese mortality decline converging toward international trends?”

Population and Development Review 24 (1998):593600.

Wilson, Chris. “On the scale of global demographic convergence 1950-2000.” Population and Development Review 27 (2001):155-171.

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Econometrics

Valuation and Risk of Koopgarant by: Bert Kramer One of the biggest problems on the Dutch housing market is the large gap between the rental and the owneroccupied market. Since the latter is characterized by high prices due to limited supply, first-time buyers often face an affordability issue. To promote home ownership and greater freedom of choice for their customers, Dutch housing corporations have developed several hybrid forms of sale over time. The most popular hybrid form of sale is called Koopgarant. Koopgarant combines a discount on the initial sales price with a number of conditions on the sales contract. The most important condition is that the future profit (or loss) is shared at turnover1. Until the end of 2009, around 15,000 Koopgarant houses have been sold. And its popularity is increasing: in 2009 5,000 to 6,000 houses were sold, compared to fewer than 3,000 in 2008.

Not much literature exists on the valuation and risk of Koopgarant from a housing corporation perspective. In contrast to a regular sale where the role of the housing corporation ends after the purchase has been made, Koopgarant involves a lasting commitment of both parties. That is, housing corporations have the obligation to buy back the house at turnover. For internal steering from an economical perspective, housing corporations need to have insight into the economic value of these options and obligations. The first objective of this paper is therefore to propose valuation formulas to calculate economic values for houses sold as Koopgarant.

Furthermore, housing corporations are not only interested in the current economic value, but also in the potential financial (cash flow) risk. The solvency and, especially, the liquidity of housing corporations have deteriorated dramatically since 2008. This deterioration has been caused by developments on the housing markets and by government interventions like the introduction of corporation tax as of 1 January 2008, the maximization of the annual rent increase at price inflation, and the obligation to deposit â&#x201A;Ź 75 million a year in a private investment fund over the next 10 years. This fund will be used to invest in urban renewal projects in 40 problem neighborhoods throughout the country. The second objective of this paper is, therefore, to establish the risk profile of Koopgarant. Thus, should housing corporations hold an additional risk buffer for their Koopgarant portfolio, or can they use up all sales revenues?

Koopgarant Koopgarant is the successor of the socially-bound ownership (Maatschappelijk Gebonden Eigendom) concept, first introduced in the 1970s. Tenants buy 1

The future moment when the original buyer sells his home.

the house at a discount of the free market value on the condition that they will later resell it to the landlord at the same discount and subsequently share profit or loss. The housing corporation is obliged to buy back the house.

Koopgarant is quite successful with around 100,000 houses offered to the tenants, and around 15,000 houses actually sold up to the end of 2009. Housing corporations have to buy a license to apply this concept. As of 1 January 2010, 160 housing corporations have a license.

The share of the housing corporation and the purchaser in the value increase (or decrease) of the house depends on the discount on the market value at the initial purchase.

The relationship between initial discount and profit share is based on a fair value calculation. That is, the fair value profit share is calculated from the view point of the purchaser, taking into account the conditions of the sales contract. See Conijn & Schweitzer (2000) for the fair value calculations. The parameters differ between existing and new built dwellings. The reason is, that there are differences in the savings on the additional costs (like transfer tax) related to the discount. The discounts and profit shares are presented in table 1.

The minimum profit and loss share for the purchaser is 50%. Value increases due to improvements by the

Bert Kramer Bert Kramer is Senior Researcher at Ortec Finance bv in Rotterdam, the Netherlands. He holds a Master degree in Econometrics and a Ph.D. in Management from the University of Groningen, both passed cum laude. He has held an Assistant Professor of Finance position at Tilburg University, and was a lecturer at the Amsterdam School of Real Estate. Within Ortec Finance his fields of expertise are ALM for non-life insurance companies, housing corporations and property investors.

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Table 1. Discounts and profit shares with Koopgarant Share of housing corporation in value increase or decrease Discount for buyer

15% 20% 25% 30% 35%

Existing houses

30% 22.5% 40% 30% 50% 37.5% 50% 45% 50% 50% Source: Netherlands Ministry of Housing, Spatial Planning and the Environment, MG2006-06, 10 July 2006.

purchaser are entirely for the purchaser. For example: Example

A

B

Initial market value Discount Sales price (A) Market value at turnover Value increase due to improvements (B) Value increase due to market development (C) Buy back price = A+B+0.5*C

160,000 25% 120,000 200,000 10,000

160,000 25% 120,000 150,000 10,000

30,000

-20,000

145,000

120,000

After buyback, the housing corporation is free to decide what to do with the house. For a more thorough description, the reader is referred to Noordenne & Vos (2006).

Gruis et al (2005) discuss a number of risk factors for socially-bound ownership, amongst others: 1. A drop in house prices after the start of the program means that the housing corporation has to share losses with buyers who want to sell. Moreover, people are less likely to buy a house in a declining housing market.

This might lead to an increased number of houses let and a decreased number of houses sold, leading to an additional loss.

2. The program involves a lasting commitment, which could end with the obligation to buy back many houses at the end of their lifespan.

Gruis et al (2005) conclude that the main financial risk lies in using up too much of the revenues too early.

Accordingly, risk management must become a crucial part of strategic policy.

Valuation of Koopgarant We will discuss potential valuation models for 2

20

Please contact the author for the deduction of the formulas.

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Koopgarant. In all valuation formulas, the initial sales revenue is excluded from the value at time 0. Sales and buyback costs are also not included. The formulas can easily be extended to include these factors. We calculate economic values for two options after buyback: A. Resell against market value; B. Continue Koopgarant; For both options, the buyback price at time t, P(t), is equal to: P(t ) = (1 − d ) ⋅ MV (0) + s ⋅ ( MV (t ) − MV (0)),

(1)

with d the initial discount and s the profit share for the customer. In case t lies in the future, MV(t) is uncertain and should be replaced by the product of the current market value and the expected indexation until the time of buyback.

For option A (resell against market value), the present value is based on the difference between the resale price and the buy back price and can be calculated as follows2: −1  p   1+ r  PV (t ) =  − 1  ⋅  ( d + s − 1) ⋅ MV (0) ⋅   1 − p   1 − p 

  1+ r + (1 − s ) ⋅ MV (t ) ⋅  − 1 (1 − ) ⋅ (1 + ( )) p E m  

−1

 , 

(2)

with discount rate r, the expected yearly development of the house price index E(m) and p the turnover rate For option B (continue as Koopgarant) we distinguish two cases: infinite and finite economic life of the house.

The assumption of an infinite economic life of the house is mainly relevant for single family homes. The underlying assumption is that purchasers will keep their property well maintained, thus infinitely expanding economic

Econometrics

Figure 1. Value of Koopgarant model A (resell against market value)

Figure 2. Value of Koopgarant model B (continue Koopgarant) Figure 2: Value of Koopgarant model B (continue Koopgarant)

Figure 1: Value of Koopgarant model A (resell against market value) 80.000

60.000 50.000

Economic value

50.000-60.000 40.000-50.000 30.000-40.000 20.000-30.000

4

3

Exp. house price increase

2

1

0

{

{

E(N )  + (1 − d − s ) ⋅ MV (t ) ⋅  1 −  1+1E+(rm )  

}

−1

 − 1 , 

5

E(N )

}

4

3

Exp. house price increase

−1

(3)

with E(N) the expected time to resale.

The, more complex, formula with finite economic life is available from the author on request, as is a valuation formula for a third option, letting the house after buyback.

Note that the potential costs and risks around the end of the economic life of a house can have a large impact on the value of a Koopgarant house (see Gruis et al (2005)).

Housing corporations should therefore pay careful attention to this issue.

For the option B, using the probability distribution of the time to buyback is complex. No closed form formulas exist. An alternative would be to use Monte Carlo simulation for valuation purposes, in which case a large number of random drawings from the probability distribution of the time to buyback would be used to value the contract.

As an example, we calculate the value of Koopgarant for one single house with the following characteristics: • • • •

50.000-75.000 25.000-50.000 0-25.000

50 15

life. For multi family homes this assumption might be too strong. With infinite economic life, we can derive a formula when E(m) < r. That is: PV (t ) = (1 + r ) − E ( N ) ⋅ ( d + s − 1) ⋅ MV (0) ⋅ 1 −  1+1E+(rm ) 

75.000-100.000

25.000

Exp. habitation period

25

100.000-125.000

125.000 75.000

30.000

5

125.000-150.000

150.000

50.000

520.000

6

150.000-175.000

175.000

100.000

Economic value

40.000

15

175.000-200.000

200.000

60.000-70.000

70.000

200.000-225.000

225.000

70.000-80.000

Free market value at time of initial sale: € 200,000; Discount: 25%; 50-50 profit / loss sharing; Discount rate: 6%;

In figures 1 and 2, we show economic values as a function of the expected habitation period and the expected house price increase.

Most housing corporations currently keep the value of the buyback obligation off the balance sheet. From these figures, we can conclude that the buyback obligation only has a value of zero in case we assume that house prices remain stable and that the housing corporation will resell the dwelling as Koopgarant after each buyback. In all

2

1

0

Exp. habitation period

25

other cases, the buyback obligation has a positive value.

Thus, when house prices are expected to rise, keeping the buyback obligation off balance is too conservative.

Furthermore, we can see that for low expected house price increases, the economic value of reselling against market value is higher than the value of continuing Koopgarant.

However, for high expected house price increases, continuing Koopgarant leads to higher economic values.

The breakeven point lies between 3% and 4%. Finally, for expected house price increases above 4%, the economic value of Koopgarant lies above the free market value of € 200,000.

Financial risk of Koopgarant In this section we evaluate the effects of Koopgarant in a dynamic stochastic sense. Economic values are based on many assumptions and housing corporations run risk(s) as realizations can substantially differ from these assumptions. Important risk factors are, for instance: a. Development of house prices; b. Changes in the expected habitation period.

Changes in these risk factors will lead to variation in the actual cash flows (direct return) and to changes in the economic value (indirect return). Higher sensitivity to changes in these risk factors means larger risk buffers for housing corporations.

We analyze the sensitivity of the economic values for changes in these external parameters by means of Monte Carlo simulation. The stochastic scenarios of house price changes and price inflation are generated with a Vector AutoRegressive (VAR) model estimated on annual time series data. Volatilities, correlations and dynamics (i.e. auto- and cross-correlations) are in accordance with the historical statistics. The expected values for the different variables are overruled, based on current market and forward looking information. For each of the stochastic scenarios the economic value is calculated. In this way, a “cloud” of possible outcomes is generated. The spread of this cloud represents the sensitivity for economic parameters.

Like Conijn & Schweitzer (2000), we use 8 and 14 years for the expected habitation period. These are the

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Figure 3. Probability distributions buyback obligation 8 year habitation period, expected house price increase 3%

Figure 4. Probability distributions buyback obligation 14 year habitation period, expected house price increase 3%

average number of years Dutch households stay in their owner-occupied houses (14 years for single family homes, 8 years for multi-family homes). For the expected house price increase we use 3%. To show the sensitivity of Koopgarant to abovementioned risk factors, we use the same example as in the previous section, extended with the option to let after buyback. The results are presented in figures 3 and 4.

When we assume continuation of Koopgarant after buyback, the mode of the distribution is zero for both habitation periods. The expected value is, however, positive given the extremely fat tail on the right side of the distribution. The downside risk is relatively limited as long as the house is resold after buyback, either as Koopgarant or against market value. The maximum loss is around € 25,000 in that case, but the likelihood of a loss is still around 20% when continuation of Koopgarant is assumed. When it is assumed that the housing corporation will let the house after buyback, there is only a very small probability that a profit will be made. And losses of over € 100,000 are not unlikely. Finally, the potential spread in outcomes is very large, especially for longer expected habitation periods.

References

Conclusions In this paper we have discussed valuation and risk issues related to Koopgarant. With Koopgarant a house is sold to the tenant at a discount. Koopgarant is the most popular hybrid form of sale offered by housing corporations. Its loss sharing part can be interesting for potential buyers in the current difficult housing market. However, it also involves a lasting commitment of both the housing corporation and the buyer. Determining the economic value, expected return and financial risk is therefore more complex than with a regular sale. Up till now, housing corporations do not have sufficient insight into value, return and risk. We have shown that the current reporting approach does not resemble the economic value. For internal steering the economic value should be used.

Finally, we also conclude that for housing corporations Koopgarant leads to relatively high financial risks.

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Conijn, Johan & Mark Schweitzer. Fair Value bij Verzekerd Kopen. Amsterdam: Rigo Research en Advies, 2000. Available at www.rigo.nl.

Gruis, Vincent, Marja Elsinga, Arjan Wolters & Hugo Priemus. “Tenant Empowerment Through Innovative Tenures: An Analysis of Woonbron-Maasoevers’ Client’s Choice Programme.” Housing Studies 20.1 (2005):127-147.

Kramer, Bert & Ton van Welie. “An Asset Liability Management Model for Housing Associations.”

Journal of Property Investment & Finance 19.6 (2001): 453-471.

Noordenne, Maurice van & Maarten Vos. Atlas koopvarianten. Rotterdam: SEV, 2006. Available at www.sev.nl.

Mathematical Economics

On the Effectiveness of Leniency Programs in Cartel Enforcement by: Harold Houba, Evgenia Motchenkova and Quan Wen Recently, antitrust policies in the US and the EC have undergone substantial reforms and currently include leniency programs as a key ingredient, see US Department of Justice (1993) and EC (2006). Leniency programs grant total or partial immunity from fines to cartel members collaborating with the antitrust authority (AA) by revealing information about the cartel. Leniency programs are based upon the economic principle that firms, which broke the law, might report their illegal activities if given proper incentives. Effective leniency programs might dissolve existing cartels or, even better, a priori deter such illegal activities. US Department of Justice claims big success of these programs in practice. Also the recent conviction of the Dutch chemical company Akzo-Nobel was the result of an application for leniency several years ago. Despite claimed successes, we raise some theoretical reasons for concern.

The aim of introducing a leniency program is to reduce the maximal cartel price below the one under traditional antitrust regulation. At face value, introducing the possibility to apply for leniency makes it harder for cartel members to sustain cartel prices. However, we will argue that, given the current setup of leniency programs that do not allow for rewards, this is impossible. To the contrary, if the leniency program is wrongly designed, it even may have an adverse effect by enhancing systematic collusion and reporting that increases the cartel price. We provide conditions under which such adverse effects are eliminated: Individual fine reductions in case of multiple reporting firms should be moderate.

Our arguments are derived from a game-theoretic analysis in which competition among firms and application for leniency is modeled as an infinitelyrepeated sequential game. Our discussion fits well the growing literature on evaluating the effectiveness of antitrust policies and the design of leniency programs for cartel enforcement, see e.g. Motta and Polo (2003), Rey (2003), Spagnolo (2008), Harrington (2004, 2008), Motchenkova (2004), and Chen and Harrington (2007).

In our analysis, we focus on a new perspective based upon the maximal cartel price, i.e., the highest cartel price for which the conditions for sustainability hold, which replaces the profit-maximizing cartel price. Since the maximal cartel price reflects the consumers' worst situation of market abuse, this new perspective is more natural and appealing. In this article, we consider a simple version of the model in Houba et al. (2009) to illustrate our results.

The Model There are n ≥ 2 symmetric firms that compete in prices over infinitely many periods in the presence of a leniency

program. Figure 1 describes the situation as an infinitelyrepeated sequential game with a common discount factor δ ∈(0, 1) per period. We concentrate on symmetric outcomes. Each period consists of the following twostage game: First all firms simultaneously set their prices, and after prices are observed and profits π(p1,...,pn) ≡ π(p) are realized, all firms simultaneously decide whether to report, where R stands for reporting and N for refraining from reporting. If no firm reports, the firms will be found guilty of collusion with probability β(p) and if the firms are found guilty of sustaining cartel price p ∈(pN, pM], each firm will have to pay an one-time fine k(p)π(p). If some firms report, the leniency program is executed. We distinguish between the fine α(p, N)π(p) for the single reporting firm in a silently operating cartel and the fine α(p, N)π(p) for each firm in case the cartel agreement includes application for leniency in every period. So, our model is rich enough to investigate improperly designed leniency programs that give incentives for a cartel to systematically report if it can reduce each member's expected fine from β(p)k(p)π(p) if no firm reports to α(p,R)π(p) if all firms report. We assume that all functions are well-behaved.

H. Houba, E. Motchenkova and Q. Wen Harold Houba is associate professor at the Department of Econometrics of the VU University. His specialization is bargaining theory with applications to labor and environmental economics.

Evgenia Motchenkova is assistant professor at the Department of Economics of the VU University. Her specialization is the economics of antitrust regulation, in particular leniency programs.

Quan Wen is professor at the Department of Economics of the Vanderbilt University in Nashville. His specializations are bargaining theory, repeated games and their applications to economics.

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Figure 1. A repeated game with price competition and application for leniency.

The profit function allows for many standard static oligopoly models. We denote the static competitive (Nash) equilibrium price and the monopoly price in the absence of antitrust policy by pN and pM, respectively, and normalize π(pN) to zero. We focus on grim-trigger strategies to sustain cartel prices, in which any unilateral deviation leads to the competitive equilibrium price in every period thereafter.

Stylized facts from the OECD countries on antitrust policies, see OECD (2002), suggest an expected penalty roughly between 30% to 50% of illegal gains, or 2 ≤ β ( p ) k ( p ) ≤ 12 . This is captured by the more general 7 1 range, 720 ≤≤ β ( p ) k ( p ) <≤ 1; 2 in our discussion. This implies that the expected fine at any price above the competitive price is lower than the (per period) cartel profit and, therefore, any cartel is tempted to set its price above the competitive (Nash) equilibrium price.

The rules of the current leniency guidelines in most OECD countries allow full amnesty for a single reporting firm and only partial fine reductions in case of multiple reporting firms. This ranking is captured by the more general assumption 0 ≤ α(p, N) ≤ α(p, R) ≤ k(p). The US leniency program is captured by α(p, N) = 0 and the EC leniency program corresponds to 0 ≤ α(p, N) ≤ k(p), where both α(p, N) and k(p) are constant and α(p, N) can be reinterpreted as a constant percentage of fine reduction.

All (expected) fines are increasing in p and this conforms to the legal reasoning that more severe violations, i.e. higher cartel prices, are punished harder.

A major issue is whether the introduction of a leniency program reduces the maximum cartel price. For that reason, we first investigate the benchmark case in which only a traditional antitrust policy without a leniency program is present.

The Maximal Cartel Price under Traditional Policies All firms follow standard grim-trigger strategies to sustain a cartel price of p > pN. Unilateral deviations by cartel members dissolve the cartel, but detection does not.

The latter describes the situation of notorious cartels that continue illegal business as usual after each detection. Let V(p) be the present value of a firm's expected profit if the cartel sets price p ∈[pN, pM] in every period. V(p) is equal to the current illegal gains π(p), minus the expected fine 1

24

Figure 2. pA ≤ pC is the largest p such that λ(p) ≥ Λ(p).

β(p)k(p)π(p), plus the expected continuation profit δV(p), since detection does not dissolve the cartel. Solving for V(p) yields that V ( p) =

1− β ( p)k ( p) 1− δ

π ( p ).

Note that V(pN) = 0 due to π(pN) = 0, and otherwise V(p) >0 because β(p)k(p) < 1.

Given grim-trigger strategies, the profit from a unilateral deviation is equal to the short term net gain πopt(p) = maxp' π(p', p,..., p) in the current period, minus an expected fine of zero (no prosecution), plus the infinite stream of competitive profits, π(pN) = 0. So, the condition to support cartel price p is V(p) ≥ πopt(p). We obtain 1− β ( p)k ( p) 1− δ

π ( p ) ≥ π opt ( p ) .

Let λ(p) be the relative size of a firm's cartel profit to the gains under the best unilateral deviation:  πopt( p ) , λ ( p ) =  π ( p)  1,

for p ∈ ( p N , p M  , for p = p N ,

where we denote λ = lim ε → 0 λ ( p N + ε ) ≤ 1.1 Then, (1) can be rewritten as +

λ ( p) ≥ Λ ( p) ≡

1− δ , 1− β ( p)k ( p)

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(2)

where λ(·) represents the degree of cartel stability in a sector. The right-hand side of (2) is decreasing in δ,

λ ( p because + ε ) ≤ 1.the classical Bertrand competition implies a discontinuity at p = pN. Introducing λ ¸=islim necessary, ε → 0+ N

(1)

Mathematical Economics

pA =

max

p∈  p N , p M   

p,

s.t. ( 2 ) .

(3)

Figure 2 illustrates how to find pA as the intersection of the downward sloping curve λ(p) and the upward sloping curve Λ(p). In this figure, pC is the highest cartel price under no regulation, i.e. β(p) = k(p) = 0, for which Λ(p)=1−δ. We observe graphically that regulation may reduce the highest cartel price: pN ≤ pA ≤ pC ≤ pM. This graphical method deserves a place in future textbooks!

Similar to partial equilibrium analysis, shifts in AA policies do not affect the λ(p) curve and only translate into shifts of the Λ(·) curve.

The standard approach is to derive the lowest δ for which the monopoly price can be sustained. From (2), it follows that cartel price p can be sustained by the cartel if and only if δ ≥ δ (p) , where δ (p) = 1−λ(p) [1−β(p) k(p)]<1. In particular, for pM and δ ∈[ δ (pM), 1), the antitrust policy is not effective to deter the cartel from setting its monopoly price. To destabilize cartels from sustaining pM the right-hand side of this expression must be greater than 1, i.e. β(pM)k(pM) > 1 . In general, since all OECD antitrust policies satisfy 0 < β(p)k(p) < 1, cartels cannot be eradicated.

The main message of this section is a mixed blessing for antitrust regulation. On the one hand, we can identify non-empty sets of parameter values for which antitrust regulation is effective in reducing the highest cartel price.

On the other hand, as long as the legal system obeys condition 0 < β(p)k(p) < 1, there always will be a nonempty set of parameter values for which pA = pM, meaning the antitrust policy is totally ineffective on this set.

The Maximal Cartel Price under Leniency There are two fundamentally different ways the cartel may respond to a leniency program. Either, the cartel operates silently, i.e. no reporting, or the cartel seeks to exploit the leniency program by systematically colluding and reporting.2 For each case, one can provide grimtrigger strategies, where we assume that reporting in case of a silent cartel would terminate the cartel and, in case of systematic reporting, does not dissolve the cartel.

The maximum cartel price under leniency (L) is given by

2

pL =

max

p∈  p N , p M   

s.t. either λ ( p ) ≥ Λ ( p ) and or λ ( p ) ≥

p, 1 ≥ Λ ( p ) (4) 1 − α ( p, N )

1− δ .

1 − α ( p, R )

(5)

The constraints listed under (4) are the equilibrium conditions under which a silent cartel can sustain the cartel price p. We denote the highest of such prices pS. Similar, the constraint (5) is the equilibrium condition under which systematically colluding and reporting (R) can sustain the cartel price p. We denote as pR the highest of such prices.

A silent cartel that deliberates to sustain cartel price p must make sure that no firm has an incentive to undercut this price, i.e. (2), and that no firm has an incentive to report, be fined α(p, N)π(p) and receives the infinite stream of competitive profits instead of continuing the cartel, i.e., face the expected fine of β(p)k(p)π(p) followed by δV(p). The second condition can be rewritten more conveniently as 1/(1−α(p, N)) ≥ Λ(p). At face value, leniency adds another constraint for silently operating cartels and seems to make life for such cartels harder. It can be shown, however, that for any function α(p,N)≥0, the second condition always holds in case the first condition holds. The underlying economic intuition is simple: By reporting, a single reporting firm blows up the cartel and looses an infinite stream of positive expected profits from collusion and, by reporting this firm gets −α(p, N)π(p) + δ · 0 ≤ 0. Obviously, the firm has no incentive to apply for leniency, unless sufficiently large rewards would be given (which in turn might invoke systematic collusion and reporting). So, we conclude that pS = pA, where pA as in (3).

The implication pS = pA is that if the cartel can sustain the cartel price p ∈[pN, pA] under traditional regulation, then introducing an leniency program without rewards allows the cartel to maintain its illegal activity with the same cartel price by operating silently.

Characterizing the maximal cartel price pL has become a relatively easy task. The conditions (4) and (5) can be translated into two intervals of sustainable cartel prices, i.e., either p ∈[pN, pS] or p ∈[pN, pR] and therefore these constraints are equivalent to [pN, pS] [pN, pR] = [pN,max{pS, pR}]. Since also pS = pA, the following result immediately follows. ∩

meaning that an increase in δ would relax (2). As firms care more about the future, it becomes easier to sustain a cartel price. More importantly, an overall increase in detection probabilities β(p) or fines k(p) would make collusion harder to sustain.

The highest cartel price supported by grim-trigger strategies under a traditional antitrust policy (A) is given by

Theorem 1 Under any leniency program with α(p,N) ≥ 0, we have pL = max{pA, pR} ≥ pA. The implication of Theorem 1 is negative: Given current antitrust rules, i.e. β(p)k(p) < 1, it is impossible

This is for explanatory reasons, and we discuss this issue in the concluding remarks.

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Mathematical Economics

Figure 3. The maximal cartel price pL is the maximum of pA and pR.

to find policy functions α(p,N) ≥ 0 and α(p,R) ≥ 0 that will reduce the range of sustainable cartel prices under traditional AA regulation. To the contrary, introducing a leniency program to existing antitrust regulation may even enhance cartels by allowing them to sustain higher prices as Figure 3 illustrates. At best, the leniency program achieves pL = pA and, at worst, it achieves pL >pA. If pL = pA, then the maximal cartel price can be supported by strategies in which the cartel operates silently and the leniency program fails to break this price and the silence.

Otherwise, i.e. pL > pA, the maximal cartel price cannot be sustained under traditional regulation, but it can be sustained by strategies in which the cartel systematically colludes and reports to the AA. In this case, observing reporting is evidence of an increased range of sustainable cartel prices!

Figure 3 illustrates a robust simple case in which pR > A p is associated with a unique crossing of the curves Λ(·) and λ(·) at a lower cartel price than the unique crossing at some higher price of the curves 1−1α−( ⋅δ, R ) and λ(·). In that case, the curve Λ(·) at p = pA lies above the curve 1− δ A L R A 1− α ( ⋅, R ) at p = p and this suffices for p = p > p . Thus, 1− δ A L A 1 − if α(p ,R)< λ ( p ) , then p > p and the maximal cartel price increases due to leniency.

In words, if the fine reductions for multiple reporting firms are too generous, introducing a leniency program increases the maximal cartel price instead of reducing it. In order to avoid this type of adverse effects, fine reductions should be limited such that α(pA,R) ≥ 1 − λ1(−pδ ) .

So, moderate fine reductions in case of multiple reporting firms would not only allow to avoid adverse effects in the form of systematic collusion and reporting, but also prevent that introduction of leniency will increase the maximal cartel price.

A

A

A Characterization of Leniency Programs In this section, we classify all possible leniency programs applicable for the case of multiple reporting α(p, R) in the (p, α)-space. In this classification, two issues stand out: By what strategies can we sustain the cartel price p (if

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Figure 4. Classification of leniency programs α (∙, R).

sustainable) and whether the program is exploitable. The leniency program is exploitable at cartel price p when both strategies support p as an equilibrium price and systematically colluding and reporting yields a higher profitt than a silently operating cartel, i.e. 1− α1(−pδ, R ) π(p) > V(p).

Combining all results implies that the (p, α)-space is divided by the vertical line p = pA and the curves 1 − Λ1−(δ⋅ ) and 1 − 1λ−(⋅δ) that intersect at p = pA. Figure 4 illustrates these curves. Below we discuss the cases when p and α(p, R) belong to each of these regions separately: A1: Since p ≤ pA and α(p, R) ≤ 1 − λ1(−pδ), both strategies sustain p as an equilibrium price. Since also α(p, R) > 1 − λ1(−pδ) , the leniency program is not exploitable at p, and the cartel prefers to operate silently.

A2: As in A1, both strategies sustain p as an equilibrium price. Since also α(p, R) < 1 − Λ1−( pδ ) , the leniency program is exploitable at p, and the cartel prefers to systematically report.

A3: Since p > pA and α(p, R) ≤ 1 − λ1(−pδ), only the systematic collusion and reporting strategies form an equilibrium.

Even though the traditional antitrust policy would eradicate this cartel price, the leniency program annihilates this positive effect by enhancing collusion on this cartel price.

B1: Since p ≤ pA and α(p, R) > 1 − λ1(−pδ), only the strategies in which the cartel operates silently form an equilibrium.

B2: Since p > pA and α(p, R) > 1 − λ1(−pδ), none of the strategies form an equilibrium. Hence, the cartel price p cannot be sustained, firms prefer to deviate, and no cartel formation takes place.

Hence, introducing leniency program α(p, R) in either of the regions A1, B1 or B2 does not change cartel practices, and the cartel either operates silently to sustain p or does not form at all. However, adverse effects do arise for α(p, R) in regions A2 and A3 due to the leniency program.

In region A2, the leniency program does not eradicate a

Mathematical Economics

previously sustainable cartel price p, but it is exploitable and might induce systematic reporting. This can be seen as a first type of adverse effects of leniency programs. In region A3, introduction of the leniency program makes it possible to sustain a previously unsustainable cartel price p implying that leniency programs in this region are harmful. This can be identified as a second type of adverse effects. For the class of non-decreasing policy functions α(·, R), regions A2 and A3 can be avoided by allowing only moderate (or no) fine reductions in case of multiple reporters, i.e. α(pA,R) > 1 − λ1(−pδ ) .

The above discussion implies that introduction of wrongly designed leniency programs (e.g. leniency programs that give too generous fine reductions α(p, R) to all reporting firms in case of multiple reporting) can lead to an improvement in cartel stability with higher prices sustained by the cartel. At the same time, under silently operating cartels, in case only one firm reports, our results imply that leniency programs without rewards to a single reporting firm, i.e. α(p, R), do not have any impact on the range of sustainable cartel prices.

A

Conclusion In our discussion, a simplifed version of the general model in Houba et al. (2009) is outlined. All results presented here extend to the general case. Two issues deserve some attention here. First, we discussed the case that firms may apply as often for leniency as they seem fit. In many OECD countries, however, past offenders are excluded from leniency programs. The latter case would not qualitatively change our main conclusions, but require more elaborated arguments. A second issue is that leniency in this paper has to be understood as that the firms may apply for leniency only before the antitrust agency launches an investigation in their sector and that the firms loose their right to apply for leniency once the investigation has started. In many OECD countries, the leniency program also allows firms to apply for leniency in case an investigation has started, but then the structure of the fines changes accordingly. The latter is left for future research and might yield different policy implications.

Harrington, J.. “Cartel pricing dynamics in the presence of an antitrust authority.” The Rand Journal of Economics 35 (2004):651-673.

Harrington, J.. “Optimal corporate leniency programs.”

The Journal of Industrial Economics LVI(2) (2008):215-246.

Houba, H., E. Motchenkova, and Q. Wen. “The effects of leniency on maximal cartel pricing.” Tinbergen Institute Discussion Paper (2009): 09-081/1.

Motchenkova, E.. “Effects of leniency programs on cartel stability.” CentER Discussion Papers Series 2004-98, Tilburg University, Tilburg.

Motta, M. and M. Polo. “Leniency programs and cartel prosecution.” International Journal of Industrial Organization 21 (2003):347-379.

OECD. “Fighting hard-core cartels: Harm, e®ective sanctions and leniency programs.” OECD Report 2002, OECD, Paris, France, http://www.SourceOECD.org.

Rey, P.. “Towards a theory of competition policy.” In M. Dewatripont, L. Hansen, and S. Turnovsky (Eds.), Advances in Economics and Econometrics: Theory and Applications. Cambridge University Press, 2003.

Spagnolo, G.. “Leniency and whistleblowers in antitrust.”

In P. Buccirossi (Ed.), Handbook of Antitrust Economics.

MIT Press, 2008.

References Chen, J. and J. Harrington. “The impact of the corporate leniency program on cartel formation and the cartel price path.” In V. Ghosal and J. Stennek (Eds.), The Political Economy of Antitrust. Elsevier (2007).

D.O.J.. “Us corporate leniency policy.” (1993). http:// www.usdoj.gov/atr/public/guidelines/0091.htm.

EC. “Commission notice on immunity from fines and reduction of fines in cartel cases.” Official Journal of the European Union (2006/C 298/11). Brussels (2006).http:// europa.eu.int/comm/competition/antitrust/leniency/.

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Wat als haar bankbiljet DNB van 50 euro niet tegen 60 graden kan?

Als ze gaat stappen, heeft ze nooit een portemonnee bij zich. Ze heeft geen zin om met een tas te zeulen en zo’n dikke bult in je achterzak ziet er niet uit. Dan maar gewoon wat losse briefjes mee. Maar wat als ze dat de volgende dag even is vergeten? En haar laatste beetje studiefinanciering in de bonte was verdwijnt?

Daarom zorgt de Nederlandsche Bank (DNB) voor een goede kwaliteit van onze bankbiljetten. Maar eeuwig gaan ze niet mee. Voor een zo soepel en veilig mogelijk betalingsverkeer onderscheppen we dagelijks duizenden vuile en beschadigde bankbiljetten. Het betalingsverkeer in goede banen leiden, is niet de enige taak van DNB. We houden ook toezicht op de financiële instellingen en dragen – als onderdeel van het Europese Stelsel van Centrale Banken – bij aan een solide monetair beleid. Zo maken we ons sterk voor de financiële stabiliteit van Nederland. Want vertrouwen in ons financiële stelsel is de voorwaarde voor welvaart en een gezonde economie. Wil jij daaraan meewerken? Kijk dan op www.werkenbijdnb.nl. | Economen | Juristen

Werken aan vertrouwen.

Econometrics

Matching and the Allocation of Indivisible Objects via DeferredAcceptance under Responsive Priorities by: Bettina Klaus It is well-known that economics is the science of allocating scarce resources. Often this is done using money as our daily shopping routines confirm. Sometimes, there is more to the allocation mechanism than simple price setting and taking, but even auction mechanisms as encountered on e-bay are now well accepted rationing mechanisms.

However, there are economic problems in which using money and prices to match resources and consumers is not usually done or is considered immoral or even illegal. Examples for this type of problems are the assignments of students to public schools or universities or the assignment of transplant organs to patients who urgently need these transplants. If money cannot or should not be used to determine who gets what, how else can we decide on matching resources to consumers1?

Matching Practice: the National Resident Matching Program (NRMP)2 A by now famous and classic example of two-sided matching is the National Resident Matching Program (NRMP): a centralized clearing house to assign medical students to so-called intern or residency positions after their M.D. Degree. As with many entry-level jobs, work conditions and salaries are not very flexible and therefore do not play any role in the matching process.

The NRMP was established in 1952 because of persisting problems in the assignment of residents to hospitals. First, between 1900 and 1945, the medical resident market was decentralized and experienced unraveling of appointment dates – in an attempt to compete for the best students, hospitals offered residencies to medical students earlier and earlier up to a point when in 1945 positions were offered two years in advance of graduation. This led to some inefficiency in the matching. Second,when medical schools tried to control unraveling by not releasing information about candidates before a specified date that they all agreed on, the practice of exploding offers (extremely short decision times to accept offers) again destabilized the market A recent survey on these types of markets is Sönmez and Ünver (2009).

2 For a detailed discussion of the American medical interns/ residents market I refer to Roth (1984,2003).

3 I strongly recommend Gale and Shapley's (1962) seminal paper – simply beautiful!

1

(through missed as well as broken agreements). The original NRMP matching algorithm introduced in 1952 as well as the redesigned NRMP matching algorithm introduced in 1998 are based on the so-called deferred acceptance algorithm that I will explain next.

Matching Theory and Deferred Acceptance3 In 1962 two game theorists/mathematicians - David Gale and Lloyd Shapley - wrote a now famous paper about “College Admissions and the Stability of Marriage”. The college admissions model exactly captures the situation for medical residents (but the college admissions model was formulated by Gale and Shapley, 1962, independently of the NRMP): a set of students (medical residents) has to be assigned to a set of colleges (hospitals with residency programs). The ingredients of a college admissions

Bettina Klaus Bettina Klaus is full professor of mircoeconomic theory at the University of Lausanne (Switzerland).

She has obtained her PhD from Maastricht University in 1998 and has held academic positions at the University of Nebraska-Lincoln (USA), the University Autonoma de Barcelona (Spain), Maastricht University (Netherlands), and Harvard Business School (USA). Bettina Klaus currently works in the fields of market design (matching) and social choice (normative resource allocation).

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problem are as follows:

students if too many propose).

• a set of students, • a set of colleges each with a quota, i.e., the maximal number of students it can admit, • students strict preferences over colleges (including the option to not go to college), • and colleges' responsive preferences over sets of students.

Continue this procedure until all students are either tentatively assigned to a college or have exhausted their list of acceptable colleges. This algorithm produces a stable matching!

There is also a deferred acceptance algorithm where the colleges propose to students, but here I exclusively focus on the student proposing deferred acceptance algorithm, which has the following nice properties.

Responsiveness means that colleges' preferences over sets of students are based on a strict ranking of the students that the college has (e.g., based on an entrance exam). For simplicity, I assume that all students are acceptable (they pass the minimal entrance requirements for all colleges).

Then, a college's preference relation is responsive if it would like to add a student if its quota is not exhausted and if it would prefer to exchange one of its students for a better student in case the quota is already met.

A solution to a colleges admissions problem is a matching of students to colleges such that each student gets assigned at most one college and such that colleges' quotas are respected. A matching is stable, if no student is assigned to a college (s)he does not want to go to (individual rationality) and there is no blocking by a student-college pair such that the student prefers the college to the current match and the college also would like to add the student (either because the quota is not exhausted or because the college would then not admit a less preferred student). Stability is not only a theoretically appealing property, it plays an important role in entrylevel job-matching since an unstable matching is not likely to persist (it is too easy to check if there is a winwin improvement for both sides of the market – you would definitely make a couple of phone calls to check if the employers you would prefer by chance would also prefer you before final hiring decisions are made).

However, given a college assignment or similar problem, do stable matchings always exist? And if they do, how do we find one? Gale and Shapley (1962) provided a simple and fast algorithm, called the student proposing deferred acceptance algorithm, to compute a stable matching: Step 1 for students: each student proposes to her\his favorite college.

Step 1 for colleges: each college tentatively assigns the best students who proposed to it without exceeding its quota (and rejects some students if too many propose).

Step 2 for students: each student currently not tentatively assigned proposes to her/his favorite college among those who have not yet rejected her/him.

Steps 2 for colleges: each college tentatively assigns the best students who proposed to it and who were tentatively assigned without exceeding its quota (and rejects some 4

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A recent survey on school choice is Klijn (2008).

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• it finds a stable matching (Gale and Shapley, 1962); • it finds the best/optimal stable matching for all students (Gale and Shapley, 1962).

• it is a weakly dominant strategy for all students to state their true preferences (Roth, 1985). In other words, a student cannot obtain a better match by lying about her/his preferences.

School Choice and Deferred Acceptance4 Some cities in the US operate so-called school choice programs in which students submit preferences about different district schools and based on “priorities” students are matched with schools. The school choice problem is very similar to the college admissions problem with the difference that schools' preferences over sets of students are replaced by priorities, i.e., rankings of students that might reflect objective priority criteria determined by the school district. In New York City, priorities at schools are determined by exam scores and in Boston aspects such as walking distance and siblings in the same school are taken into account. Both school systems exhibited signs of market failure and were redesigned using the student proposing deferred acceptance algorithm as main building block (see Abdulkadiroğlu, Pathak, and Roth, 2005 a,b).

Allocating Indivisible Objects using Deferred-Acceptance So far we have argued that deferred acceptance has nice theoretical properties for college admissions and school choice (note that preferences of colleges or priorities of schools are given). Furthermore, deferred acceptance emerged “naturally” as the mechanism of the NRMP clearinghouse and it was implemented in Boston and New York City’s school choice programs. Finally, I will demonstrate that deferred acceptance also has very nice and robust properties for allocation models without preferences or fixed priorities.

In Ehlers and Klaus (2009) we study the allocation of indivisible objects to a set of agents. The assignment of students to schools could be considered an example as well as the assignment of dormitory rooms to students. We assume that agents (e.g., students) in these situations have

Econometrics

strict preferences over the (object) types (e.g., dormitory rooms of a certain type or in a certain building) and that (object) types might come with a capacity constraint (the maximal number of dormitory rooms of the same type).

We have various results for this general class of allocation problems, but here – due to space limitations - I will only present one of our results for the class of problems where exactly one object of each type is available.

Our approach is to first define desirable properties that an allocation rule should satisfy and then see which class of rules satisfies all these properties.

We consider situations where resources may change, i.e., it could be that additional objects are available. When the change of the environment is exogenous, it would be unfair if the agents who were not responsible for this change were treated unequally. We apply this idea of solidarity and require that if additional resources become available, then all agents (weakly) gain. This requirement is called resource-monotonicity.

Next, we impose the mild efficiency requirement of weak non-wastefulness5 as well as the very basic and intuitive properties of individual rationality6 and unavailable object invariance7.

We also impose the invariance property truncation invariance8. Our last property is the well-known strategic robustness condition of strategy-proofness (which we mentioned before when stating that it is a weakly dominant strategy for students to state their true preferences when the student proposing deferred acceptance algorithm is used).

Assuming that an allocation rule satisfies all six elementary and intuitive properties described, we construct a priority structure of objects over agents. In other words, if an allocation rule satisfies all properties mentioned above, then any object is just like a college (with quota one due to our assumption that we have one object of each type) and has a priority ranking over agents. Thus, objects can be endowed with responsive preferences over agents and we can apply the agent proposing deferred acceptance algorithm to obtain an assignment or matching of objects to agents. We call a rule that is based on the agents-proposing deferredacceptance algorithm with responsive priorities a responsive DA-rule. Not only can we construct a priority structure of objects over agents based on our properties, but the only allocation rule satisfying the properties is the DA-rule based on the constructed priority structure.

To summarize, we characterize the class of responsive DA-rules by a set of basic and intuitive properties, namely, unavailable object invariance, individual rationality, weak non-wastefulness, resource-monotonicity, truncation

invariance, and strategy-proofness (Ehlers and Klaus, 2009, Theorem 1). For further characterizations along this line (and technical details of the model, the properties etc.) we refer to Ehlers and Klaus (2009).

The main conclusion of Ehlers and Klaus (2009) is, that deferred acceptance also plays an important role for allocation problems that might not come in the form of a college admissions problem (where colleges have responsive preferences) or a school choice problem (where schools have fixed priorities over students).

When allocating objects to agents, it might be necessary to assign pseudo (responsive) preferences or priorities to objects in order to guarantee the desirable properties mentioned above.

References Abdulkadiroğlu, Atila, Parag A. Pathak, and Alvin E. Roth. “The New York City High School Match.”

American Economic Review, Papers and Proceedings 95 (2005a, May):364-367.

Abdulkadiroğlu, Atila, Parag A. Pathak, Alvin E. Roth, and Tayfun Sönmez. “The Boston Public School Match.” American Economic Review, Papers and Proceedings 95 (2005b, May):368-371.

Ehlers, Lars, and Bettina Klaus. “Allocation via Deferred Acceptance.” CIREC Cahier 17-2009 (http://www. cireq.umontreal.ca/publications/17-2009-cah.pdf).

Gale, David, and Lloyd Shapley. “College Admissions and the Stability of Marriage.” American Mathematical Monthly 69 (1962):9-15.

Klijn, Flip. “Mechanism Design in School Choice: Some Lessons in a Nutshell.” Boletín de la Sociedad de Estadística e Investigación Operativa 24 (2008):1122 (http://www.seio.es/BEIO/files/BEIOv24n3_IO_ FKlijn.pdf).

Roth, Alvin E.. “The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory.” Journal of Political Economy 92 (1984): 991-1016.

Roth, Alvin E.. “The College Admissions Problem is not Equivalent to the Marriage Problem.” Journal of Economic Theory 36 (1985):277-288.

No agent who does not receive any object would prefer to obtain an object that is not assigned.

Each agent weakly prefers an object she\he is assigned to not receiving any object.

7 The rule only depends on the set of available objects and not on objects that are not currently available.

8 If an agent reduces his set of acceptable objects without changing the preference between any two real objects and his assigned object remains acceptable under the new preference, then the rule should choose the same allocation for the initial profile and the new one.

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Roth, Alvin E.. “The Origins, History, and Design of the Resident Match.” Journal of the American Medical Association 289 (2003, February):909-912.

Roth, Alvin E.. “Deferred Acceptance Algorithms: History, Theory, Practice, and Open Questions,” International Journal of Game Theory 36, Special Issue in Honor of David Gale on his 85th birthday: (2008):537-569.

Sönmez, Tayfun, and M. Utku Ünver. “Matching, Allocation, and Exchange of Discrete Resources.”

Handbook of Social Economics (2009) Alberto Bisin, Jess Benhabib, and Matthew Jackson, eds., Elsevier (http://ssrn.com/abstract=1311517).

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Mathematical Economics

Who Should Have the Control Right in the Organization? by: Te Bao This article provides two application of incomplete contract theory to real life problems. In the privatization of public service sectors, we show that the coexistence of private and public ownership is better than either privatizing all firms or let all of them to stay un-privatized, because private and public ownership usually provide different price-quality combination of services, and the coexistence makes full use of the competition and enriched choice set. In the divisional structure in an organization, we show why the coexistence of two divisions performing the same task is less likely to be seen than the coexistence of two firms producing the same goods in the market. In one organization, it is always better for the principal (the top manager or headquarter of the organization) to integrate these divisions and assign the control right to the agent whose interest is more in line with her own. This is not the case only when there is negative externality between the efforts of the agents, or when the principal wants to encourage the agent of lower interest congruence to search more in order to get more efficiency-enhancing bargaining.

Introduction Contract theory, or the theory of incentives is a very important tool for modern economic analysis and the design of economic institutions. There are two major approaches of contract theory: the complete contract approach and the incomplete contract approach. The complete contract approach assumes people can write every future contingency into a contract, so the most important thing is to find an optimal contract that maximizes the profit of the principal. The incomplete contract approach assumes that people may have limited ability to contract on everything, because it evolves too much complexity and cost. Therefore, the important thing is to find the optimal allocation of control right, meaning who should have the right to make decisions under certain circumstance. This paper shows two applications of the incomplete contract theory to real life situations. The first one is about the debate on privatization of public service sectors, and the second one is about the separation and integration of divisions inside an organization.

Ownership Structures in the Public Service Sectors The privatization of former public sectors is a very important trend in the post-war Europe, especially the UK in the 1970s and the transition economies during the 1980s and 1990s. The privatization in the private good sectors usually makes large success, while in the public service sectors incurs continual questions and heated debate. On the one hand, privatization usually leads to high profit to the owners of firms in both sectors. On the other hand, the quality of the service provided by

the privatized public service firms may go substantially poorer, such as the deteriorated level of security and treatment to prisoners in the privatized American prisons reported by AFSME (1985), Donahue (1988,1989), Logan (1990,1992) and analyzed in Hart et al. (1997).

Wang and Bao (2008) surveyed the literature on the optimal ownership structure for the provision of public service, and proposed a proposition (which is later proved in Wang and Xu, 2007) that the coexistence of public and private ownership dominates each single form of ownership in all cases.

The reasoning is mainly in line with the famous work by Hart et al. (1997): consider there is a politician who can hire a manager to operate a facility of public ownership for the provision of public service, or make the manager the owner of the facility, and purchase his service, what is the difference between these two cases?

The most important one is that when the manager does not have the ownership rights, the politician can always fire the manager so that the manager can not get any benefit from the innovation he makes on the facility, while the

Te Bao Te Bao is currently a PhD student at CeNDEF, University of Amsterdam and Tinbergen Institute.

He received his bachelor degree in economics from Fudan University, Shanghai, China and his Msc in Economics from the Hong Kong University of Science and Technology. His research interests include behavioral finance, experimental economics and contract theory. This article is based on his coauthored papers on contract theory with Prof. Yongqin Wang at Fudan University.

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politician simply cannot do so with an owner of a private firm. With this potential firing threat, the politician can always ask the manager to leave part of the future profit of innovation to the firm when the manager proposes such innovation, which harms the incentive of the manager to make innovations.

The effect of this lower incentive can be either positive or negative: when the innovation is to enhance both the profit of the firm and the quality of service to the public, such as new technology that promotes efficiency, private ownership is better than public ownership. When the innovation is to raise the profit of the firm at the expense of the welfare of the consumers, such as that the doctors prescribe too expensive medicine when they have cheaper options with the same effect, or the contractors build cheap but unsafe dams, the public owner ship may be better because the manager at least has no incentive to adopt the profitable but not socially desirable technology.

In sum, the firms with public ownership may be less efficient but more reliable, while it is vice versa for the private ownership.

The idea of our model is that the public and private ownership arrangements can provide different pricequality bundles of public service. So the coexistence of two types of ownership is better than single one because: (1) when the consumers are heterogeneous, it provides more options, so that it is easier for each type of consumers to give his optimal bundle; (2) when the consumers are homogeneous, it is easier for the consumers to reduce information asymmetry in the cost and quality of the service by evaluating the relative performance of the two arrangements.

Divisional Structures in Organizations In one of the founding papers of incomplete contract theory, Grossman and Hart (1986) studied the optimal conditions on whether two firms should be integrated or separated, and when they should be integrated, which party should be granted the control right over the big firm. This inspires us to think about a similar problem between two divisions within one organization.

The difference between the two problems is that the integration and separation decision between two firms is often decided by negotiation between the firms, while between two divisions performing the same task in one organization this decision is made by the head of the whole organization. In the terminology of contract theory, that means the ownership structure between two firms involves two equal parties, while in the case of two divisions it often involves one principal P (the head of the organization) and two agents A1, A2 (the manager of the two original divisions).

In Bao and Wang (2009), we studied this problem following the framework by Aghion and Tirole (1997).

The agents search for projects. The payoff from one project to each of them is randomly distributed. The correlation coefficients between the payoffs to each two

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Figure 1. from left to right: separation, integration under A1 control and integration under A2 control.

parties are called the coefficients of interest congruence between them. The higher the coefficient is, the higher the chance that the payoff to one party is high when the payoff to the other party is high.

There are three candidate divisional structures: separation, integration under A1 control, and integration under A2 control, as shown in Figure 1.

In the separation case each agent controls one division of his own. Each one manager searches for projects for their own division. With a probability that is increasing with his searching effort he knows the profit profile for all projects. Then he can choose his “favorite” project, the project which brings himself the highest private benefit, and some benefit to others depending on the coefficient of interest congruence. When one manager fails to know the profit information in his own division, the other one can take his role if the other is informed. In the case of integration, one agent is made the chief of both divisions, and the other is his subordinate. When both the chief and the subordinate are informed about the profit profile of the projects, the chief can overrule the subordinate.

When the chief is not informed and the subordinate is, the subordinate can choose. No project is chosen when neither of them is informed.

One interesting thing here is that the chief in the integration case may not want to overrule all the time.

For instance, if the chief is A1 and the subordinate is A2, when the favorite project of A1 is chosen, A1 gets a private benefit b and A2 gets γ1b , while when the favorite project of A2 is chosen, A1 gets γ2b and A2 gets b, γ1,γ2 ∈ [0,1].

When γ1< γ2, A1 can make both parties better off if he agrees the favorite project of A2 is chosen, and ask A2 to give him a private transfer so that the final payoff lies in [b, b + (γ2 − γ1)b] for him and [γ1b, γ2b] for A2. In Figure 2 this is shown as the point is moving from point (b, γ1b) to the direction of north east.

So there are mainly three forces that influence the manager’s incentive to search: (1) the more divisions assigned under his control, the more likely that his favorite project can be guaranteed, the more he likes to search; (2) the poorer the interest congruence he has with the other manager (the less he gets when the favorite of the other is chosen), the more he likes to search in order to avoid this case; (3) the larger the share he can get from bargaining, the more he likes to search, because the uninformed party can not bargain with the other.

Mathematical Economics

Figure 2. A1 can make both parties better by moving from his own favorite project to the favorite of A2, and grab a part from the cooperative surplus.

References: American Federation of State, County and Municipal Employees (AFSCME). Does Crime Pay? An Examination of Prisons for Profit. Washington, DC: AFSCME, 1985, Print.

Aghion, P., and J. Tirole. “Formal and Real Authority in Organizations.” Journal of Political Economy 105.1 (Feb 1997):1-29.

Bao, Te and Yongqin Wang. “Incomplete Contract and Divisional Structures” Tinbergen Institute Discussion Papers 2009.

Donahue, John. Prisons for Profit: Public Justice, Private Interests. Washington, DC: Economic Policy Institute, 1988, Print.

Conclusion Above are stories about how the agents interact between each other. From the point of the principal, the trade off seems even simpler. On the one hand, she would like both parties to search as much as possible, so that she can avoid the case no project is chosen, and everyone gets 0.

On the other hand, if the interest congruence between her and the A1 and A2 are different, she prefers the favorite project of the one with higher interest congruence with her to be chosen. We solved the optimization problem for the agents with respect to their effort and for the principal. The result turns out to be very intuitive when there is no spill-over effect between the agents’ effort: it is always the best strategy for the principal to integrate the divisions, and assign the control right to the manager who have higher interest congruence with her.

However, we still need to find an explanation why separation of divisions of the same function sometimes still appears in reality: the universities may have more than one research center in more or less one general research field, the big companies may own more than one brand in more or less one type of goods of similar price and quality. One simple and practical reason might be: when the organization is too big, the attention of one chief can not meet the need to operate a very large group.

There are also several theoretical reasons we found in our model: (1) when there is dissonance effect between the efforts of two managers, meaning the managers work less efficiently when they are together than if they work alone, it is better to have separation; (2) when the bargaining promotes the interest congruence between the principal and the agents, the principal should use separation to encourage the agent whose interest congruence is originally lower to search, because this agent will search very little when the divisions are integrated under the other’s control.

Donahue, John. The Privatization Decision: Public Ends, Private Means. New York: BasicBooks, 1989, Print.

Grossman, Sanford J., and Oliver D. Hart. “The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration.” Journal of Political Economy XCIV (1986):691-719.

Oliver Hart, Andrei Shleifer and Robert W. Vishny.

“The Proper Scope of Government: Theory and an Application to Prisons.” Quarterly Journal of Economics 112.4 (Nov 1997):1127-1161 Logan, Charles H.. Private Prisons: Cons and Pros. New York: Oxford University Press, 1990, Print.

Logan, Charles H.. “Well Kept: Comparing Quality of Confinement in Private and Public Prisons.” Journal of Criminal Law and Criminology LXXIII (1992): 577-613.

Wang, Yongqin and Te Bao. “The Ownership Arrangement for Public Service: What do We Know?” World Economic Papers 2008 (3):57-75.

Wang, Yongqin and Haibo Xu. “Social Heterogeneity and Optimal Mix Between Public and Private Provision of Public Goods.” Journal of Chinese Political Science 12.3 (December 2007):1874-6357.

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Approximation of Economic Capital for Mortality and Longevity Risk by: Erik Beckers Mortality and longevity risk form one of the primary sources of risk in many products of a life insurer (Cairns, Blake and Dowd, 2006a). Longevity risk defines the risk of the mortality rates of a group of insured being lower than expected, while mortality risk defines the risk of the mortality rates of the same group being higher than expected. Life insurers wish to determine the amount of capital that should be available to cover these risks precisely, since setting too much capital aside is costly and will damage the competitive position and setting too little capital aside will damage the financial position. This capital is referred to as the Economic Capital, which may deviate in magnitude from the regulatory required capital. A few conventional methods are available to determine the Economic Capital for mortality and longevity risk, but these methods either lack stochastic scenarios for the mortality rates or seem to fail in their practical implementation because of extensive simulation time.

Introduction This article addresses whether the calculation process of determining the Economic Capital for mortality and longevity risk can be significantly simplified through a new method with duration and convexity concepts for mortality. This method should of course provide an accurate approximation of the Economic Capital. We examine this by selecting twenty stochastic mortality scenarios from a simulation of a stochastic mortality model and compare the impact on the Best Estimate Liability (BEL) of these scenarios in cash flow models with the approximation of the new method.

An annuity and a term life insurance cash flow model are constructed for this purpose. These products differ both in their mortality and longevity risk and therefore form a good example of the different risks a life insurer is facing. Annuities deliver a policyholder a periodical benefit as long as the policyholder is still alive, which makes longevity risk one of the main risk drivers of

Erik Beckers Erik Beckers recently obtained his Master degree in Actuarial Science and Mathematic Finance at the University of Amsterdam. This article is a summary of his Master Thesis, which was written at Towers Watson under supervisory of M.P.M. Visser and J. Borst. The supervisors at the University of Amsterdam were W.J. Willemse and H.J. Plat.

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annuities. Conversely, term life insurance policies deliver a policyholder a single benefit when the policyholder dies, which makes mortality risk one of the main risk drivers of term life insurance policies. The characteristics of the policyholders in these cash flow models are furthermore constructed to be typical for similar products of a common Dutch life insurer. This article therefore serves as a nice example to Dutch life insurers.

Economic Capital Stochastic mortality adds uncertainty to the expectations of the future mortality rates. A life insurer may use stochastic mortality scenarios to construct confidence intervals for future mortality rates and eventually for the Economic Capital.

The first conventional method to determine this Economic Capital is therefore to implement a set of stochastic mortality scenarios in the cash flow models of the different products of the insurer. The liabilities are then calculated for every scenario in every cash flow model, from which the Economic Capital is derived as a fixed percentage of the worst outcomes of the change in the BEL. Through the different stochastic scenarios, the BEL is estimated for all of the products on a great variety of mortality rates, which provides a more or less accurate representation of the Economic Capital. However, since the cash flow models have to be run over and over again for all scenarios to determine the liabilities, this involves a great deal of simulation time. Determining the liabilities of a single cash flow model for 10,000 scenarios may already take days to complete.

To overcome the extensive simulation time, most Dutch

Actuarial Sciences

life insurers currently use a different method to estimate the Economic Capital for mortality and longevity risk. In this method, a shock parameter is specified and applied to the mortality rates in the cash flow models. This shock may be a fixed percentage increase to the mortality rates to determine the mortality risk, or a fixed percentage decrease to the mortality rates to determine the longevity risk. Determining the Economic Capital through such a shock parameter is in line with the advice of CEIOPS (2009) on Solvency II. The advantage of a mortality shock is that only a few runs of the cash flow models are needed to determine the Economic Capital, which therefore forms a great improvement to the simulation time. However, an equal mortality shock is applied to all ages and calendar years with this method, while mortality rates usually do not improve equally along different ages and years and may differ from the expected rates. This can be covered by stochastic mortality scenarios.

New Method Since the conventional methods for determining the Economic Capital for mortality and longevity risk all have their disadvantages, a new method is proposed here.

First a set of stochastic scenarios is simulated for future mortality rates and thereafter a parameter describing the change in mortality is determined for every scenario.

The “mortality duration” and “mortality convexity” are determined at the same time for every cash flow model, i.e. for every product of the life insurer. These three parameters are thereafter combined in a Taylor series expansion, from which the Economic Capital is determined relatively straightforward.

Since this method requires only three runs of the cash flow models, it involves much less simulation time than the method for implementing stochastic scenarios in existing cash flow models and does not lack the use of stochastic mortality scenarios. The new method is described more thoroughly in the following sections.

Mortality Duration and Convexity Wang et al (2009) introduce mortality duration and convexity in their paper and state that these are best measured by effective duration and convexity along the same lines as for the interest rate risk of the liabilities of a life insurer. When determining Macaulay duration and modified duration, improvements in future mortality rates will not be incorporated, while through effective duration future mortality dynamics are incorporated more precisely.

Mortality duration (Dμ) and convexity (Cμ) are thus measured through formulas (1) and (2).

PV - - PV + 2 ⋅ PV ⋅ ∆

(1)

PV − − 2 ⋅ PV + PV + PV ⋅ ( ∆  ) 2

(2)

D = C =

Where, • PV- is the present value of the liabilities when all future mortality rates have decreased with ∆μ1; • PV+ is the present value of the liabilities when all future mortality rates have increased with ∆μ; • PV is the present value of the liabilities when all future mortality rates are equal to the assumptions; and • ∆μ represents the difference between the increased mortality rate and the mortality rate from the assumptions (thus ∆μ = μ+ − μ).

Only three runs of a cash flow model with three different scenarios for the mortality rates are therefore necessary to determine the mortality duration and convexity of a life insurance product.

Stochastic mortality model To simulate stochastic scenarios of future mortality rates, a stochastic mortality model should be used. Following the development in stochastic interest rate models, many of these stochastic mortality models have been introduced during the past twenty years. The interested reader may refer to Cairns, Blake and Dowd (2006a) for an overview of these models.

The Cairns, Blake and Dowd (2006b) model is selected to simulate stochastic scenarios for the future mortality rates. This model relies on the linear pattern in the logit of the mortality rates for ages 40 and older, which can be observed in the formula of the model, formula (3).

A disadvantage of the Cairns, Blake and Dowd (CBD) model is that it is only really accurate for ages 40 and older. Big advantages are however that the model is quick and easy in simulations and that the model combines two stochastic factors which provides a greater variety in shape and movements of the term structure of mortality rates compared to an one-factor model.  q (t , x )  (1) (2) logit q (t , x ) = ln   = κt + κt ( x - x ) (3) 1 q ( t , x )  

Where, • q(t,x) represents the initial mortality rate, i.e. the probability of a person aged x at time 0 dieing before time t; • κ1 and κ2 are two stochastic factors, following a bivariate random walk with drift; and

This μ should not be confused with the force of mortality μx which is the usual definition of μ. Here μ is defined as a fixed shock applied to the mortality rates qx.

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Table 1. Comparison of change in Best Estimate Liability for term life insurance and annuities (Values in EUR) TERM

ANNUITIES

SCENARIO

CHANGE OF BEL MOSES

CHANGE OF BEL TAYLOR SERIES

% DIFFERENCE

A B C D E F G H I J Q R S T U V W X Y Z

131,160 148,444 204,412 119,842 133,917 121,091 163,708 163,682 60,062 123,948 (207,729) (204,614) (188,617) (237,787) (195,142) (271,390) (262,898) (269,006) (255,008) (290,877)

128,399 146,992 205,717 118,586 134,944 125,220 165,200 167,462 59,830 120,189 (203,525) (201,441) (184,370) (234,779) (190,842) (267,662) (261,630) (264,052) (254,339) (288,009)

-2.1% -1.0% 0.6% -1.0% 0.8% 3.4% 0.9% 2.3% -0.4% -3.0% -2.0% -1.6% -2.3% -1.3% -2.2% -1.4% -0.5% -1.8% -0.3% -1.0%

• x = n −1 ∑ i xi is the mean age in the sample range, with n the number of different ages in the sample range.

Taylor series expansion After selecting the CBD model we are now able to simulate a set of stochastic scenarios for future mortality rates for a range of ages from past mortality data. A set of 1,000 scenarios is simulated in this example from CBS2 mortality data. A parameter representing the change in the mortality rates weighted on the claim amounts of the individual policyholders is then obtained for every stochastic scenario. For an explanation on how to obtain this parameter, one may turn to Beckers (2009).

To determine the effect of mortality duration and convexity and the stochastic scenarios on the value of the BEL, a Taylor series expansion is adopted represented by formula (4).

CHANGE OF CHANGE OF BEL BEL MOSES TAYLOR SERIES

(461,798) (477,027) (571,241) (384,130) (362,897) (404,290) (393,536) (431,735) (244,629) (230,654) 539,495 560,637 516,801 608,741 528,860 636,412 685,027 607,448 692,913 702,315

(440,764) (460,558) (562,095) (370,831) (359,112) (392,036) (396,160) (429,863) (230,969) (239,496) 542,100 560,565 515,876 614,744 529,200 650,024 693,569 622,794 698,939 718,769

 PV = - D  ⋅ PV ⋅   +

% DIFFERENCE

-4.6% -3.5% -1.6% -3.5% -1.0% -3.0% 0.7% -0.4% -5.6% 3.8% 0.5% 0.0% -0.2% 1.0% 0.1% 2.1% 1.2% 2.5% 0.9% 2.3%

1 ⋅ C  ⋅ PV ⋅ (  ) 2 2

(4)

Where, • Dμ and Cμ are effective mortality duration, respectively effective mortality convexity; • PV represents the value of the BEL at t = 0 of the best estimate scenario; and • ∆μ3 represents the parameter describing the change in mortality rates for every scenario as described above.

The best estimate scenario is generated by fitting the kappas of the CBD model as a random walk and then projecting these kappas forward without any stochastic innovations. The mortality duration and convexity are determined through this same best estimate scenario, which is represented by PV in (1) and (2) and shocked two times for respectively the – and the + scenario.

The Taylor series expansion in (4) can then be applied

See http://statline.cbs.nl for the complete CBS database Again, μ should not be confused with the force of mortality μx. In this formula, μ is defined as the difference in mortality rates of a stochastic scenario compared to the best estimate scenario.

2 3

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w w w.g a a a n . n U

Š 2010 KPMG N.V., alle rechten voorbehouden.

KPMG

T i e n m i n U T e n v o o r d e p r e s e n TaT i e va n e e n p r o p o s a l a a n d e C F o

a u d i t J ta x J a dv i s o r y

Actuarial Sciences

Table 2. Comparison of change in Best Estimate Liability for complete portfolio of life insurer (Values in EUR) COMPLETE PORTFOLIO SCENARIO

A B C D E F G H I J Q R S T U V W X Y Z

CHANGE CHANGE OF OF BEL BEL MOSES TAYLOR SERIES

(330,639) (328,582) (366,830) (264,288) (228,979) (283,200) (229,828) (268,053) (184,567) (106,706) 331,766 356,023 328,184 370,955 333,719 365,022 422,129 338,442 437,905 411,437

(312,365) (313,566) (356,377) (252,245) (224,168) (266,816) (230,960) (262,401) (171,139) (119,307) 338,575 359,124 331,506 379,965 338,357 382,362 431,939 358,742 444,600 430,761

% DIFFERENCE

-5.5% -4.6% -2.8% -4.6% -2.1% -5.8% 0.5% -2.1% -7.3% 11.8% 2.1% 0.9% 1.0% 2.4% 1.4% 4.8% 2.3% 6.0% 1.5% 4.7%

to quickly estimate the change of the BEL (â&#x2C6;&#x2020;PV) for the 1,000 stochastic scenarios.

Results To examine the accuracy of the method with the Taylor series expansion, twenty scenarios are selected and the change in the BEL is determined for these scenarios by both the Taylor series expansion and the cash flow projection modelling program MoSes4.

These impacts on the BEL should be approximately equal to each other, but one should keep in mind that the determination of the Economic Capital for mortality and longevity risk already involves a lot of uncertainty (i.e. choice of stochastic model, method of simulating, choice of parameters and data used). The approximation of a scenario may therefore deviate a few percentage points from the actual impact on the BEL, but this should not be too substantial.

Since the Economic Capital is based on scenarios with the largest increases or decreases in mortality rates, 4

40

these scenarios are most important to be approximated accurately by the new method. The twenty selected scenarios are therefore selected as the ten scenarios with the largest increases in mortality rates and the ten scenarios with the largest decreases in mortality rates compared to the best estimate scenario.

Table 1 provides the results of the twenty scenarios for a term life insurance products and an annuity product (both having 500 insured). Note both products having opposite capital needs, which show the difference in longevity and mortality risk. The impact on the BEL of the twenty scenarios is approximated relatively well by the Taylor series for both products. The maximum difference between the Taylor series and the cash flow model is 5.6%, but most scenarios have significantly lower differences.

Since the Economic Capital is based on the complete portfolio of a life insurer, both products are combined and we obtain the results in table 2. Here again the impact on the BEL of the twenty scenarios is approximated relatively well by the Taylor series for most scenarios, except for the scenarios where the impact is small and negative.

A test has therefore been performed on scenarios with a small and negative impact on the BEL, which shows the same substantial deviations for scenarios with a small and negative impact on the BEL but not for others. Since the Economic Capital is not distracted from scenarios with negative or small impacts to the BEL, the approximation for the Economic Capital is however relatively well for the complete portfolio.

The approximations of the Taylor series expansions should be accurate for different yield curves as well and therefore the calculations of above are repeated for an inverted yield curve and a high and a low fixed interest rate. Since the new method approximates the results of the cash flow models relatively well under all of these different interest rate environments, it is robust to different scenarios of the interest rates.

One may increase the prudence of the new method by constructing a range around the parameter describing the change in mortality in the Taylor series expansion, from which then a range of possible impacts on the BEL is determined for every stochastic mortality scenario. It is furthermore interesting to examine the accuracy of the new method for a different stochastic mortality model.

This is left for future research.

Conclusion A new method for determining the Economic Capital for mortality and longevity risk of a life insurer is introduced in this article. This method combines the advantages and overcomes the shortcomings of conventional and desired methods. With a Taylor series expansion, which combines effective mortality duration and convexity, the impact on

More information on MoSes is available on http://www.towerswatson.com/services/Financial-Modeling-Software

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the Best Estimate Liability of a set of stochastic scenarios for the mortality rates is directly derived. The Economic Capital for mortality and longevity risk is then derived straightforward from these Taylor series. The new method approximates this Economic Capital relatively well, even under different interest rate environments.

References Beckers, Erik T. “Approximation of Economic Capital for mortality and longevity risk using duration and convexity concepts.” Master Thesis University of Amsterdam (2009), see: http://download265.mediafire. com/stnb9jjm1d2g/2nxzdg0hm2n/Master_Thesis_ ET_Beckers.pdf Cairns, Andrew J.G., David Blake and Kevin Dowd.

“Pricing death: Frameworks for the valuation and securitization of mortality risk.” ASTIN Bulletin 36.1 (2006a):79-120.

Cairns, Andrew J.G., David Blake and Kevin Dowd.

“A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration.”

Journal of Risk and Insurance 73.4 (2006b):687-718.

CEIOPS (Committee of European Insurance and Occupational Pension Supervisors). “CEIOPS’ Advice for Level 2 Implementing Measures on Solvency II: Standard formula SCR - Article 109 c Life underwriting risk.” Consultation Paper 49, October 2009.

Wang, Jennifer L., H.C. Huang, Sharon S. Yang and Jeffrey T. Tsai. “An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach.” Journal of Risk and Insurance (2009), Forthcoming.

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Mathematical Economics

Complex Dynamics in an Asset Pricing Model with Updating Wealth by: Serena Brianzoni, Cristiana Mammana and Elisabetta Michetti In recent years, several models have focused on the study of the asset price dynamics and wealth distribution when the economy is populated by boundedly rational heterogeneous agents with Constant Relative Risk Aversion (CRRA) preferences. Chiarella and He (2001) study an asset pricing model with heterogeneous agents and fixed population fractions. In order to obtain a more appealing framework, Chiarella and He (2002) allow agents to switch between different trading strategies. More recently, Chiarella et al. (2006) consider a market maker model of asset pricing and wealth dynamics with fixed proportions of agents. A large part of contributions to the development and analysis of financial models with heterogeneous agents and CRRA utility do not consider that agents can switch between different predictors. Moreover, the models which allow agents to switch between different trading strategies make the following simplified assumption: when agents switch from an old strategy to a new strategy, they agree to accept the average wealth level of agents using the new strategy.

Motivated by such considerations, we develop a new model based upon a more realistic assumption. In fact, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). The most important fact is that the wealth of the new group takes into account the wealth realized in the group of origin, whenever agents switch between different trading strategies. This leads the final system to a particular form, in which the average wealth of agents is defined by a continuous piecewise function and the phase space is divided into two regions. Nevertheless, our final dynamical system is three dimensional and we can find all the equilibria. We will prove that it admits two kinds of steady states, fundamental steady states (with the price being at the fundamental value) and non fundamental steady states. Several numerical simulations supplement the analysis.

Wh ,t +1 = (1 − zh ,t ) wh ,t R + zh ,t wh ,t (1 + ρt +1 ) = wh ,t [ R + zh ,t ( ρt +1 − r )]

where zh,t is the fraction of wealth that agent-type h invests in the risky asset, ρt = (pt + yt – pt–1)/pt–1 is the return on the risky asset at period t and wh,t is the average wealth of agent type h at time t defined as the total wealth of group h in the fraction of agents belonging to this group.

The individual demand function zh,t derives from the maximization problem of the expected utility of Wh,t+1, i.e. z h ,t = max zh ,t Eh ,t [u h (Wh ,t +1 )] where Eh,t is the belief of investor-type h about the conditional expectation, based on the available information set of past prices and dividends. Following Chiarella and He (2001), the optimal (approximated) solution is given by:

The model Consider an economy composed of one risky asset paying a random dividend yt at time t and one risk free asset with constant risk free rate r = R – 1 > 0. We denote by pt the price (ex dividend) per share of the risky asset at time t.

In order to describe the wealth’s dynamics, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). Hence, the dynamics of the wealth of investor h is described by the following equation:

S. Brianzoni, C. Mammana and E. Michetti Serena Brianzoni has a PostDoc Position at the Department of Financial and Economic Institutions of the University of Macerata (Italy).

Cristiana Mammana is Professor in Mathematics at the Faculty of Economics and Director of the Department of Financial and Economic Institutions of the University of Macerata (Italy).

Elisabetta Michetti is a Lecturer at the Department of Financial and Economic Institutions of the University of Macerata (Italy).

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Mathematical Economics

zh ,t =

zh ,t =

Eh , t [ ρt +1 − r ] λh σ

Eh , t [ ρtwhere +1 − r ] λ is the relative risk aversion coefficient and h λh σ h2 =Varh,t[ρt+1–r] is the belief of investor h about the conditional variance of excess returns.

In our model, different types of agents have different beliefs about future variables and prediction selection is based upon a performance measure φh,t. Let nh,t be the fraction of agents using strategy h at time t. Hence, as in Brock and Hommes (1998), the adaptation of beliefs is given by:

nh ,t +1 =

exp[ β (φh,t − Ch )] Z t +1

Wɶ h,t = nh ,t wh ,t , h = 1, 2

2 h

, Z t +1 = ∑ exp[ β (φh ,t − Ch )] h

which we call weighted average wealth of group h.

Hence, the weighted average wealth of group 1 at time t+1 is given by:  n1,t+1W2,t +1 + n1,tW1,t +1 =  ɶ W1,t +1 =  = n1,t (W1,t +1 − W2,t +1 ) + n1,t +1W2,t +1 if mt +1 ≥ mt .  n W if m < m t +1 t  1,t +1 1,t +1

In a similar way we can derive the weighted average wealth of group 2:  n2,t +1W2,t +1 if mt +1 ≥ mt Wɶ 2,t +1 =   n2,t (W2,t +1 − W1,t +1 ) + n2,t +1W1,t +1 if mt +1 < mt .

where the parameter β is the intensity of choice measuring how fast agents choose between different predictors and Ch ≥ 0 are the costs for strategy h. We observe that at Finally, we define wh,t as the weighted average wealth of time t+1 agent h measures his realized performance and group h in the total weighted average wealth, i.e. wh,t=Wɶ h,t = nh ,t wh ,t , h = ɶ(1 −=zn=,t )1,ww2h ,t R,and + hzhh=,t=1,2. w + ρthe then chooses whether to stay in group h or to switch to /∑hWɶ h,t W =where 1,h2,t (1 In hn, th+,t1w= t +1 ) following, hW , t ,h , t h hn h ,t h , t h,t another one. Hence, we measure the past performance as we will consider dynamics of the state variable = whthe ,t [ R + z h ,t ( ρt +1 − r )] the personal wealth coming from the investment in the wt:=w1,t– w2,t, i.e. the difference in the relative weighted − zh ,t ) wh ,t :R + zh ,t wh ,t (1 + average ρt +1 ) risky asset with respect to the W average wealth wealths. As a consequence, the dynamics of h , t +1 = (1 the state variable wt can be described by the following = wh ,t [ R + zh ,t ( ρt +1 − r )] φh,t = zh,t(ρt+1 – r). system: In this work, we focus on the case of a market populated by two groups of agents, i.e. h = 1,2, and assume that at any time agents can move from group i to group j, with i,j = 1,2 and i ≠ j, while both movements are not simultaneously possible. We define Δnh,t+1= nh,t+1– nh,t as the difference in the fraction of agents of type h from time t to time t +1. Notice that in a market with two groups of agents it follows that Δn1,t+1 = –Δn2,t+1. As a consequence, we can have two different cases: 1) Δn1,t+1 ≥ 0, if Δn1,t+1 fraction of agents moves from group 2 to group 1 at time t+1, 2) Δn1,t+1 < 0, if Δn1,t+1 fraction of agents moves from group 1 to group 2 at time t+1.

Following Brock and Hommes (1998), we define the difference in fractions at time t, i.e. mt = n1,t – n2,t, so that n1,t = (1+mt)/2 and n2,t = (1–mt)/2, with: β mt +1 = tanh[ (φ1,t − φ2,t − C1 + C2 ] 2

Afterwards, conditions Δn1,t+1 ≥ 0 and Δn1,t+1 < 0 can be replaced by mt+1 ≥ mt and mt+1 < mt, respectively. In order to describe the wealth dynamics of each group, we define shareh of produced by group h to the Wɶ h,t =asnhthe = 1,wealth 2 ,t wh , t , total wealth:

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wt +1

 FG1 + 1 if = F 2  G − 1 if

mt +1 ≥ mt mt +1 < mt

where: F1 = −2 F2 = 2

1− mt +1 1− mt

1+ mt +1 1+ mt

(1 − wt )[ R + z2,t ( ρt +1 − r )],

(1 + wt )[ R + z1,t ( ρt +1 − r )],

G = (1 − wt )[ R + z2,t ( ρt +1 − r )] + (1 + wt )[ R + z1,t ( ρt +1 − r )].

In this work we assume that price adjustments are operated by a market maker who knows the fundamental price, see Chiarella et al. (2006). After assuming an i.i.d. dividend process and zero supply, we obtain: 1 + wt 1 − wt pt +1 − pt = α ( z1,t + z2,t ).

2 2 pt

We analyze the case in which agents of type 1 are fundamentalists, believing that prices return to their fundamental value, while traders of type 2 are chartists, who do not take into account the fundamental value but their prediction selection is based upon a simple linear trading rule. In other words, we assume that E1,t(pt+1)=p* and E2,t(pt+1) = apt with a > 0. Therefore the demand

Mathematical Economics

Figure 1. State variable wt versus time for an i.c. x0 = 0.8, m0 =1 and w0 = 1 and parameter values α = 0.5, λ = 1, σ2 = 1, r =0.02, C = 0.5, β = 1. (a) a = 0.1. Periodical fluctuations. (b) a = 0.5. Convergence. (a)

(b)

w

wt

1

t

1

Figure. 2. Bifurcation diagrams of the state variables w.r.t. a. Initial condition and parameter values as in Figure 1.

−1

0

60

t

−1

0

60

t

functions are given by: z1,t =

1 (1 + r )( xt − 1), λσ 2

z2,t =

1 [ a − 1 + r ( xt − 1)], λσ 2

where xt = p*/pt is the fundamental price ratio.

The final nonlinear dynamical system T is written in terms of the state variables xt, mt and wt: x1,t = f1 ( xt , wt ) =

α 2 λσ 2

xt ( xt − 2 + a + 2r ( xt − 1) + ( xt − a ) wt ) + 1

mt +1 = f 2 ( xt , wt ) = tanh{ β2 [ λσ1 2 ( xt − a )[ 2 λσα 2 ( xt − 2 + a +2r ( xt − 1) + ( xt − a ) wt ) + r ( xt − 1)] − C ]},

wt +1

F1 = F2 =

 FG1 + 1 if = f 3 ( xt , mt , wt ) =  F 2  G − 1 if

mt +1 ≥ mt mt +1 < mt

1 [ a −1+ r ( x −1)][ α ( x − 2 + a + 2 r ( x −1) + ( x − a ) w ) + r ( x −1)]} t t t t t t λσ 2 2 λσ 2 (1− mt )[exp{ β [ 12 ( xt − a )[ α 2 ( xt − 2 + a + 2 r ( xt −1) + ( xt − a ) wt ) + r ( xt −1)] − C ]}+1] λσ 2 λσ

4(1+ wt ){ R +

1 λσ 2

1. Let a ≠ 1, then: • for a < 1+r there exist two steady states: the fundamental equilibrium

,

1 [(1+ r )( x −1)][ α ( x − 2 + a + 2 r ( x −1) + ( x − a ) w ) + r ( x −1)]} t t t t t t λσ 2 2 λσ 2 1 ( x − a )[ α ( x − 2 + a + 2 r ( x −1) + ( x − a ) w ) + r ( x −1)] − C ]} +1] t t t t t t 2 λσ 2 λσ 2

(1+ mt )[exp{ − β [

G = 2R +

Lemma 1. The number of the steady states of the system T depends on the parameter a:

,

−4(1− wt ){ R +

Enf becomes a fundamental steady state. More precisely, every point E = (1,Wtanh{−(Cβ)/2, is +a zfundamental h , t +1 = (1 − z h ,t ) wh}, t R h , t wh , t (1 + ρt +1 ) equilibrium, i.e. the long-run at = wh ,t [wealth R + zh ,t (distribution ρt +1 − r )] a fundamental steady state is given by any constant w ∈ [ −1,1] . In other words, a continuum of steady states exists: they are located in a one-dimensional subset (a straight line) of the phase space. In this case the steady state wealth distribution which is reached in the long run by the system strictly depends on the initial condition.

Summarizing, the following lemma deals with the existence of the steady states:

,

[ xt − 2 + a + 2r ( xt − 1) + ( xt − a) wt ] ⋅

[ 2 λσα 2 ( xt − 2 + a + 2r ( xt − 1) + ( xt − a ) wt ) + r ( xt − 1)]

Results Our model admits two types of steady states: - fundamental steady states characterized by x = 1, i.e. by the price being at the fundamental value, - non fundamental steady states for which x ≠ 1.

More precisely, for a ≠ 1 the fundamental steady state Ef of the system is such that wf = 1 and there exists a non fundamental steady state Enf such that wnf = −1, xnf = (1−a)/r +1. Observe that at the fundamental (nonfundamental) equilibrium the total wealth is owned by fundamentalists (chartists). When a = 1 the fixed point

Cβ }, w f = 1) 2 and the non-fundamental equilibrium E f = ( x f = 1, m f = tanh{−

1− a β 1 1+ r + 1, tanh{ [ 2 (1 − a ) 2 − C ]},1) Enf = ( r 2 λσ r

• for a ≥ 1 + r the fundamental steady state Ef is unique 2. Let a = 1, then: Cβ • Every point E = (1; tanh{− 2 }, w ) is a fundamental equilibrium.

Now we move to the study of the asymptotic dynamics of the system by using numerical simulations. We first consider the extreme case in which, at the initial time, all agents are fundamentalists and they own the total wealth. Moreover, we focus on the more interesting a's

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Figure. 3. Two dimensional bifurcation diagram in parameters' plane (a, β) for i.c x0 = 0.8, m0 = w0 = 1. The other parameter values as in Figure 1.

range of values, i.e. a < 1 + r. In Figure 1 (a) we present the trajectory of the state variable wt while assuming that at the initial time m0 = w0 = 1 and the price is above its fundamental value. We choose two different values of a and observe that if a is low enough the system fluctuates (a 12-period cycle is observed) while, as a increases the long run dynamics becomes simpler (wt → 1 after a long transient).

Numerical simulations show the existence of a value aɶ such that for all aw ∈ [(−aɶ1,1] , 1+r) the system converges to Ef , while more complex dynamics is exhibited if a< aɶ , i.e. for a low enough (see Figure 2). Notice that the system preserves similar features, i.e. the stabilizing effect of a in the long run, also in the other extreme case w0 = m0 = −1, i.e. at the initial time the market is dominated by chartists who own all the wealth. In Figure 2 we observe a succession of periodic windows which occur at border collision bifurcations (see Hommes and Nusse, 1991 and Nusse and Yorke, 1995). These periodic windows have the following properties: (i) the width of the periodic window reduces monotonically as the parameter a increases; (ii) the windows are characterized by the same periodicity; (iii) for a > aɶ , the periodic windows terminate in convergence.

In order to consider also the role of the parameter β, in Figure 3 we present a two-dimensional bifurcation diagram in the parameters' plane (a, β) in the extreme case w0 = m0 = 1 and the price x0 above the fundamental value. Rich dynamics is exhibited and the final behaviour increases in complexity as β increases (according to what happens in asset pricing models with heterogeneous agents and adaptiveness). Other numerical simulations confirm the evidence that the systems is more complicated for high values of β and low values of a.

In order to focus on the role of the initial condition, we fix a low enough. In Figure 4 we present the basins of attraction in the plane (m0, w0), being the initial price below the fundamental. Observe that if we move

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Figure 4. Basins of attraction in the plane (m0,w0) for x0 = 1.3, a = 0.1 and the other parameter values as in Figure 1. (a) β = 1.8, (b) β = 2.

parameter β, the structure of the basin also undergoes to a change. More precisely, it seems to become more complex as β increases, confirming the evidence previously obtained. In fact the structure of the basin becomes fractal providing the strong dependence of the final dynamics w.r.t. the initial conditions (in panel (b) ouside the stability region the system may converge to a 10-period cycle or to a more complex attractor).

In summmary, our model characterizes the evolution of the distribution of wealth in a framework with heterogeneous beliefs and adaptiveness. Analytical and numerical results show coexistence of attractors, sensitive dependence on initial conditions and complicated dynamics. Complexity is mainly due to border collision bifurcations which are involved by the wealth dynamics.

References Brock, W.A. and C.H. Hommes. “Heterogeneous beliefs and routes to chaos in a simple asset pricing model.”

Journal of Economic Dynamics and Control 22 (1998):1235-1274.

Chiarella, C. and X. He. “Asset price and wealth dynamics under heterogeneous expectations.” Quantitative Finance 1.5 (2001):509-526.

Chiarella, C. and X. He. “An adaptive model on asset pricing and wealth dynamics with heterogeneous trading strategies.” Working paper no. 84, School of Finance and Economics UTS, (2002).

Chiarella, C. R. Dieci, and L. Gardini. “Asset price and wealth dynamics in a financial market with heterogeneous agents.” Journal of Economic Dynamics and Control 30 (2006):1755-1786.

Hommes, C.H. and H. E. Nusse. “Period three to period two bifurcation for piecewise linear model.” Journal of Economics 54 (1991):157-169.

Nusse, H.E. and J. A. Yorke. “Border-collision Bifurcations for Piecewise Smooth One-dimensional Maps.” International Journal of Bifurcation and Chaos 5 (1995):189-207.

Econometrics

Counting the Number of Sudoku’s by Importance Sampling Simulation by: Ad Ridder Stochastic simulation can be applied to estimate the number of feasible solutions in a combinatorial problem. This idea will be illustrated to count the number of possible Sudoku grids. It will be argued why this becomes a rareevent simulation, and how an importance sampling algorithm resolves this difficulty.

The Sudoku puzzle was introduced in the Netherlands as late as 2005, but became quickly very popular. Nowadays most newspapers publish daily versions of various difficulties. The idea of the puzzle is extremely simple as can be seen from the following figures. Consider a 9×9 grid, divided into nine 3×3 blocks, where some of the boxes contain a digit in the range 1,2,...,9, as shown in Figure 1.

As can be seen, there are no duplicates in any row, column or block. The problem is to complete the grid by filling the remaining boxes with 1–9 digits, while keeping the property of no duplicates in any row, column, or block. The solution is a Sudoku grid, see Figure 2.

In this paper we are interested in the number of possible Sudoku grids. This is a combinatorial problem that might be solved by complete enumeration (Felgenhauer and Javis, 2005). However, we shall show that a simple stochastic simulation approach gives an extreme good approximation, within a few percent of the exact number.

The simulation is based on importance sampling and this idea can be used also for counting problems where one does not know the exact number. Counting problems are important in certain areas of computer science (Meer, 2000), also from a theoretical point of view of algorithmic complexity. Many counting problems have no exact polynomial-time algorithm (Jerrum, 2001), and then one searches for polynomial-time approximation algorithms which are fast and give accurate approximations. A popular technique is to get estimates by some Markov chain Monte Carlo method (Mitzenmacher and Upfal, 2005). Recently, counting problems have been considered as rare-event problems [1,2,11], and then one might apply importance sampling techniques for getting fast and accurate estimates. Our paper is in this spirit.

Counting by simulation

interest, defined by some property of its elements. Denote by |Ω| and |A| their sizes. We assume the following.

Assumption 2.1.

1. The population size |Ω| is known (but large), the size |A| of the set of interest is unknown.

2. It is easy to generate samples ω ∈ Ω from the uniform probability distribution u on Ω.

3. It is easy to test whether a sample ω ∈ Ωi s an element of the target set A or not.

Here ‘easy’ means that the associated algorithms run in polynomial time. We consider (Ω, u) as a finite probability space, and let X : Ω → Ω be the identity map with induced probability P. Thus for any set B ⊂ Ω P ( X ∈ B) = u (B) =

B 

Specifically for the set of interest we get A = P ( X ∈ A)  = u ( A) 

Ad Ridder Ad Ridder is an associate professor at Department of Econometrics of the Vrije Universiteit Amsterdam, where he teaches courses in simulation, numerical methods, and stochastic models. His main research interests are in the area of applied probability, rare event simulation, and performance evaluation of stochastic systems.

Let Ω be a finite population of objects and A ⊂ Ω a set of

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Figure 1. Example of an unsolved sudoku grid

Figure 2. Solution of the sudoku grid in figure 1

Now we can see a simulation approach for estimating |A|: generate X1,...,Xn as n i.i.d. copies of X, and n d 1 1{ X i ∈ A}as an unbiased estimator of P(X ∈ A). n ∑ i =1 More importantly,

For our Sudoku problem we consider the population to be all 9 × 9 grids for which each row is a permutation of the digits 1–9. We call such grids 9-permutation grids, and it is easy to see that there are

Y (n) ≐ 

1 n ∑ 1{ X i ∈ A} n i =1

(1)

is an unbiased estimator of |A|, that is commonly known as the direct, or the crude Monte Carlo estimator. The performance of this estimator is measured through the sample size n required to obtain 5% relative error with 95% confidence (Bucklew, 2004), i.e., P(| Y (n) − |A| | < 0.05|A|) ≥ 0.95.

(2)

Assuming n large enough to approximate Y(n) ~ N(|A|,Var[Y(n)]) with Var Y ( n )  = 

2 1 n

u ( A ) (1 − u ( A ) ) = A 

1 n

(1 − u ( A ) )

and letting z1−α/2 = 1.96 ≈ 2 to be the 95% two-sided critical value (or percentile) of the standard normal distribution, we get that (2) holds if and only if 2  2   2 Var Y ( n )  ≤ 0.05 A ⇔ n ≥  (1 − u ( A ) )   0.05  A

= 1600

1 − u ( A)

(3)

u ( A)

Equivalently, the performance criterion (2) holds if and only if the relative error RE[Y(n)] ≤ 2.5%, where a relative error of an estimator is defined to be the ratio of its standard deviation to its mean.

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|Ω| = (9!)9 ≈ 1.091 · 1050 9-permutation grids, an example given in Figure 3.

Notice that also the items 2. and 3. of Assumption 2.1 are satisfied. However, the direct simulation algorithm fails to work: an experiment that generated 100 million 9-permutation grids, did not found any Sudoku grid among them. The reason is that the set A of Sudoku grids is a rare event in the ‘world’ of 9-permutation grids, i.e., the probability that a randomly generated 9-permutation grid turns out to be a Sudoku grid, is extremely small.

We shall see later that this probability is u(A) = |A|/|Ω| ≈ 6.240 · 10−29. Hence, the required sample size (3) is n≈

1600 ≈ 2.5 ⋅ 103131 ~ 2.5 · 10 u ( A)

which would take about 8·1018 years on my current home PC (105 random 9-permutation grids are generated per second). Notice that the total number of calls of the random number generator would be about 2.5 ·1031 · 72 = 1.8 · 1033 which is nowadays no problem (L’Ecuyer, 2006).

Importance sampling Suppose the simulation is executed by generating random 9-permutation grids according to a probability q on 1 n ∈ Aits 1{ X ibe ( n ) ≐ u(ω)/q(ω) } Ω. For each ω ∈ Ω, let Y L(ω) ∑ n i =1 associated likelihood ratio. Then it is easy to see that the corresponding (unbiased) importance sampling estimator of |A| becomes

Econometrics

Figure 3. Example of a 9-permutation grid

Yq ( n ) ≐ 

1 n ∑ 1{ X i ∈ A} L ( X i ). n i =1

Figure 4: Generated sudoku grid up to box (3,5) using importance sampling algorithm

(4)

1 EFF  γɵ q ( n )  ≐   Var  γɵ n  × CPU  γɵ n   q ( )  q ( )

What we do actually is to estimating the rare-event probability P(X ∈ A,) by importance sampling. Recently, where CPU stands for the computation time. Better there is much interest in techniques and methods performance is obtained by smaller RE, larger RAT, and concerning rare events and how to estimate their higher EFF. probabilities, both from a theoretical point of view, and from a practical vue point, see Buckley (2004), Rubino Importance sampling for generating Suand Tuffin (2009) and the more popular Taleb (2007). doku grids The idea of importance sampling is to let the rare event to occur more often in such a way that the likelihood We propose the following change of measure for our ratio does not blow up. The main issue, therefore, is to Sudoku problem. find the new probability (or change of measure) q such that the importance sampling estimator is efficient or 1. Start with an empty grid. optimal. Clearly, an estimator is optimal if its variance is 2. Simulate the grid row-by-row from top to bottom. zero. In that case a single sample suffices! However, in 3. Simulate a row box-by-box from left to right.

2 ɵ Suppose practice this is not realizable. More realistic is to achieve log En 4. that box (i, j) (row i, column j) has to get a 1  γ q ( n )  ɵ   ≐ ∈  1 Y n X A ≐ γ n RAT ≈ 2that it lies in the k-th 3 × 3-block (numbered ( ) { } an efficient importance sampling estimator. Let q (n) digit, and ( ) ∑ i ɵ ( n ) n q Yq(n)/|Ω| be the associated estimator of the probability logi =E1  γleft-to-right and top-to-bottom). Then, eliminate from  P(X ∈ A), then we say that it is efficient if {1,...,9} all digits that have already been generated (i) in the boxes of row i (i.e., left of j), (ii) in the boxes of column j (i.e., above i), and (iii) in the boxes of the 2 log E  γɵ q ( n )  (5) k-th 3 × 3-block (i.e., in the rows above i). Let Rij be   ɵ RAT  γ q ( n )  ≐ ≈2   the remaining digits. log E  γɵ q ( n )    5. If Rij is empty, generate in all remaining boxes random digits 1–9 similarly as under the uniform measure.

Basically it means that the required sample size to obtain The grid will not be a Sudoku grid.

(3) grows polynomially, when the estimated probability 6. If Rij is non-empty, box (i, j) gets a digit generated decays exponentially fast to zero (L’Ecuyer et al., 2008). uniformly from Rij. Repeat from item 4. with the next It is easy to show that the crude Monte Carlo estimator box until all boxes are done has RAT = 1, and that always RAT ≤ 2.

Comparison of the performances of the crude Monte Example 3.1.

Carlo and the importance sampling estimators will be Suppose that the grid is generated up to box (i, j) = (3, 5) given by considering their relative errors (RE, defined as presented in Figure 4. above) and their efficiency EFF defined by Under the original uniform measure, box (3, 5) would get a digit uniformly drawn from {1, 2, 5, 8, 9}. However,

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under the change of measure, box (3, 5) gets a digit from {2, 8} (uniformly).

The likelihood ratio L(ω) = u(ω)/q(ω) is a product of the likelihood ratios of the boxes: L (ω ) =

∏ ( )

i, j i =1,...,9 j =1,...,9

ℓ ij ,

References

where the box likelihood ratio is ℓ ij =

 1 / (10 − j ) =  1 / Rij qij 1 

u ij

Bayati, M., J. Kim, J. and A. Saberi. “A sequential algorithm for generating random graphs.” To appear in Algorithmica, Springer (2009). Available at http:// dx.doi.org/10.1007/s00453-009-9340-1

if Rij ≠ 0 otherwise

While executing the importance sampling algorithm, we calculate the likelihood ratio by updating the product after each newly generated digit. For instance in Example 3.1, the current likelihood ratio is (clearly, the box likelihood ratios of the first row are all equal to 1):   6 5 4   4 3 2   3 2 1    3 2 1   3           9 8 7   6  .  9 8 7   6 5 4   3 2 1           row 2

row 3

Box (3, 5) will get either 2 or 8, both with likelihood ratio ℓ35 = 2/5 , which is going to be multiplied with the expression above.

In this way we generated 106 grids in about 15 seconds resulting in the following performance (we repeated this experiment 100 times and took the averages). estimate

RE

RAT

EFF

6.662 · 1021

3.051%

1.895

1.941 · 1058

The average relative error of these hundred estimates to the actual value was 2.1%. The standard estimator (1) would have estimated performance after 106 simulated grids estimate

RE

-

1.266· 10 % 13

RAT

EFF

1.0

1.041· 1033

Conclusion We have shown how importance sampling simulation is applicable to approximate counting problems in combinatorial problems, and we gave an illustration for counting the number of possible 3×3 Sudoku grids.

The performance of the estimator indicates that the

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importance sampling algorithm is efficient. However, when applied to larger Sudoku grids, this algorithm seems to be susceptible to the same difficulty that it does not generate quickly feasible Sudoku grids. Currently, we investigate importance sampling algorithms based on splitting the sample space and apply Gibbs sampling for generating samples.

June 2010

Blanchet, J., and D. Rudoy. “Rare event simulation and counting problems.” Chapter 8 in Rare event simulation using Monte Carlo methods, eds. Rubino G. and Tuffin, B., Wiley, (2009):171-192.

Bucklew, J.A.. Introduction to Rare Event Simulation.

Springer, New York, 2004.

L’Ecuyer, P. 2006. “Random Number Generation.”

Chapter 3 in Elsevier Handbooks in Operations Research and Management Science: Simulation, eds.

S. G. Henderson and B. L. Nelson, Elsevier Science, Amsterdam, (2006):55-81.

L’Ecuyer, P., J.H. Blanchet, B. Tuffin, and P.W. Glynn.

“Asymptotic robustness of estimators in rare-event simulation.” ACM Transactions on Modeling and Computer Simulation To appear (2008).

Felgenhauer, B., and F. Javis. Enumerating possible Sudoku grids. 2005. Available at http://www.afjarvis. staff.shef.ac.uk/sudoku/sudoku.pdf Jerrum, M.. Counting, sampling and integrating: algorithms and complexity. Birkhäuser, 2001.

Meer, K.. “Counting problems over the reals.” Theoretical Computer Science 242 (2000):41-58.

Mitzenmacher, M. and E. Upfal. Probability and computing: randomized algorithms and probabilistic analysis. Cambridge University Press, 2005.

Rubino G. and B. Tuffin (Eds.). Rare event simulation using Monte Carlo methods. Wiley, 2009.

Rubinstein, R.Y. 2007. “How many needles are in the haystack, or how to solve fast #P-complete counting problems.” Methodology and Computing in Applied Probability 11 (2007):5-49.

Taleb, N.N.. The black swan: the impact of the highly improbable. Random House, 2007.

Puzzle On this page you find a few challenging puzzles. Try to solve them and compete for a price! But first we will provide you with the answers to the puzzles of last edition.

Answer to “the real mathematical student”

Let be the amount of hours that has past since midnight. According to the student this amount has to be exactly equal to a quarter of the time from midnight till now plus half the time from now until midnight. Putting this in a formula we get x = x/4 + (24 - x)/2. This equation is solved for x = 9.6 , which means it is now 9:36 am.

Answer to “shooting contest”

The shooter can get the following points for his shot, namely 2, 3, 5, 10, 20, 25 and 50. After six shots he managed to get 96 points. Three of those six shots were doublets, which means only three different areas on the shooting target have been hit and each of those three areas is hit twice. This means we have to find out how the shooter can get 48 points with three shots. The only way he could do this is by shooting a bullet through 3, 20 and 25. To get 96 points and three doublets he has to hit does areas twice.

Here you find the puzzles of this edition

Shuffle the cards

Solutions Solutions to the two puzzles above can be submitted up to July 1st 2010. You can hand them in at the VSAE room (C6.06), mail them to aenorm@vsae.nl or send them to VSAE, for the attention of Aenorm puzzle 67, Roeterstraat 11, 1018 WB Amsterdam, Holland. Among the correct submissions, one book token will be won.

Solutions can be both in English and Dutch.

A machine to shuffle the cards changes the order of the cards inserted always in the same way. When the 13 playing cards are inserted in the following order: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, the cards will always come out in the following order: Jack, 9, 7, 10, 6, Queen, 2, 8, 5, King, Ace, 3, 4. How many times does the machine have to shuffle so the order of the cards is exactly the same as the original order?

Cycling or walking?

Jack is going to his sports club by foot. He could also have chosen to use his racing bicycle, which would take him to his sports club 5.5 times as fast as walking, but Jack chooses to go walking in stead. After 900 metres he is at a point it does not matter whether he goes back home to get his bicycle or just keeps on walking to the sports club. It will take him exactly the same amount of time.

What is the distance between his home and the sports club?

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Since february 1st there has been a new VSAE board and already a lot has happened since then. On the 11th of February we went with a group of 20 students to TNO, one of our main sponsors, to solve a case about the mobile network capacity during one of the first stages of the Giro d’Italia in the Netherlands.

On the 8th of March the National Econometrians Day was organized by the VSAE and Kraket for all the students in Econometrics in the Netherlands. The event took place in the RAI and there were over 400 students en almost 30 companies.

On the 12th, 13th and 14th of April the VSAE organized the Econometric Game. 25 universities from all over the world competed in this year’s game. The case was about the HIV/Aids epidemic in Sub-Saharan Africa and the students had to investigate people’s behaviour towards participating in a voluntary blood test.

For the next couple of months we don’t have any big events on our agenda, but the summer brings a totally different challenge. If everything goes according to plan, we are going to move to a much bigger room in the UvA building.

With the summer holiday coming up fast, the Kraket board is planning their last activities before the new board takes over. The board is very satisfied with the activities in the last couple of months. We started the year with a nice dinner, accompanying Towers Watson. The winter sport in January was an even bigger success than expected. Everyone had a great week and nobody has broken anything. Other activities were, for example, a hall soccer tournament, a gala, paintball activity and of course the soccer tournament together with VSAE. All these activities were a big success.

The biggest event was the LED. Kraket and VSAE organized a fantastic day in the RAI in Amsterdam. All the students that were present had the chance to meet several interesting companies. With a lot of participants, an interesting day program and a great party at the end, this day was very successful.

In the last months of this college year, some other activities are planned. In June the traditional Kraket weekend will take place. As every year the location will be secret until arrival. With 48 participants this weekend is fully posted.

The closing off activity will take place on June 24, after a sportive activity we will watch the match between Holland and Cameroon. The board hopes that the last activities will be just as successful as the previous activities.

Agenda

Agenda

• 28-29 May Year Closing

• 2 June Inhouse day Optiver

• 18 June Alumni Drink

• 11-13 June Kraket weekend

• 25 June Monthly Drink

• 15 June ALV • 24 June Closing off activity

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