Aenorm 70

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vol. 18 dec. ‘10

This edition:

Visualising Internal Migration

Day-to-day Use of Risk Models Cointegration of Natural Gas Markets and Valuation of Transportation Options Pension Fund Decision Making: An Altering AOW Retirement Age

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Colofon Chief editor Ewout Schotanus Editorial Board Ewout Schotanus Editorial Staff Tara Douma Daniella Brals Chen Yeh Annelieke Baller Dianne Kaptein Jan Nooren

How to Spend Your Life

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

Advertisers DNB KPMG NIBC RiskQuest SNS Reaal TNO TNT Towers Watson Information about advertising can be obtained from Daan Oosterbaan at: commercielezaken@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, E2.02 1018 WB Amsterdam tel. 020-5254134 Kraket De Boelenlaan 1105, 1A-19 1081 HV Amsterdam tel. 020-5986015

by: Professor J.S. Cramer

From what I have seen I like the articles in AENORM. They often illustrate down-to-earth applications of fairly simple techniques, far removed of the rarefied atmosphere of the classroom. Some of these techniques have arisen in quite different fields. Econometrics, epidemiology (or medical statistics), and other applied sciences such as biology, engineering, meteorology and archaeology all make use of much the same basic statistical and quantitative devices, adapted to the subject matter under review, and sooner or later they all make the same discoveries. At the same time, they all wish to be recognized as separate and distinct disciplines with specific procedures and a terminology of their own, jealously guarding their borders. When it comes to cooperation across disciplines, most scientists are narrow-minded and wish to have nothing to do with outsiders from other fields which they generally regard with contempt. At times this phenomenon even occurs within a discipline, when some strong and ambitious individual establishes a school with a new paradigm, new procedures and a terminology of its own – take Freud’s analytical approach in psychology (admittedly an extreme case), or, in econometrics, the Cowles Commission’s approach to simultaneous equations or Nelder’s advocacy of General Linear Models. On a much smaller and quite innocent scale, there may well be some distinct traits that separate the work of the Rotterdam econometrics group from that of their Amsterdam colleagues.

There are two sides to this phenomenon. In the world of meta-science – the people who devise science policy and hand out grants and subventions - it is generally held that such closed schools of thought are a bad thing. If you apply for a subsidy it will help a lot if you give your project an interdisciplinary slant. This sounds good, but I sometimes doubt whether its supposed merits are supported by empirical evidence. When horses drew carts and carriages they were made to wear blinkers so that they would not be distracted from the directions of the driver, and this made them go faster. In the same way, a narrow view of scientific progress may be very productive, because the way forward is clear. It also fosters team spirit among a dedicated group working together for years and years.

For the individual and his or her intellectual development the lesson is however the opposite: she or he definitely should not stay too long in the same environment. If Amsterdam and Rotterdam econometrics differ (and all the others even more) it is a good idea to leave Amsterdam at once when you have completed your studies and to go to work in a faraway place where there are no other Amsterdam graduates. If you go for a Ph. D. and must stay in academe, at least try to go to some other university. You will learn lots of new things.

In a wider perspective, I would strongly recommend to continue with fairly frequent changes of environment and, possibly, of occupation during your lifetime. Life is too short to spend it all in one place, doing one thing. There are indeed psychological theories to the effect that radical changes in lifestyle will lengthen your perceived life. Changes of occupation, residence and living conditions will enrich your life, even if they turn out to be thoroughly unpleasant. The world of today offers more opportunities for career change than ever before. So if you think of doing finance, ultimately setting up a hedge fund of your own, it may be a good idea to begin by spending a few years breeding pigs.

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00 70 Visualising Internal Migration

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by: Edwin de Jonge This article focuses on visualising internal migration. Many statistical datasets can be modelled as a weighted digraph. In this article we use migration in 2006 as a working case.

Migration is an important demographic for local policymakers. It is typically presented in detailed tables describing the in- and outflow of inhabitants per region. This local regional view is useful, but gives a fragmented picture of overall migration. To reveal and show patterns migration is modelled as a large weighted digraph. One major problem in this respect is that these of graphs are rendered poorly and cluttered with mainstream graph visualisation software. This article shows and discusses several visualisations of migration and their problems, including cartographic flow map and graph visualisation.

Pension Fund Decision Making: an Altering AOW Retirement Age

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by: Lars van den Berg This article focuses on the impact of a change in the AOW retirement age for pension funds. The Dutch system of social security and pensions is internationally recognised for its level of reliability and services provided to participants (OECD, 2010). Economic and demographic developments have put this system under pressure. In 2009 the Dutch government decided that changes were needed to keep the system sustainable in the future.

One of this measures is the increase of the Dutch General Old Age Act (AOW) retirement age.

There is More Behind Government Spending than You Think

14

by: Chen Yeh The aftermath of the recent credit crisis is starting to show its signs: after a bunch of bailouts and fiscal investments, many governments over the world are forced to cut their spending.

However some governments seem to cut down their expense budgets more drastically than others. While the magnitude of these economic measures differ from country to country, economists wonder which countriesâ&#x20AC;&#x2122; spending patterns are similar and in particular why they are similar. In traditional macroeconomics one can find models that partially address these questions, however Persson and Tabellini (American Economic Review, 2004) adopt a different approach by combining views from political science and economics.

Their empirical paper is based on sophisticated arguments and addresses determinants of government expenditure that are not straightforward.

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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

Day-to-day Use of Risk Models

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by: Hans Heintz, Frank Pardoel This article focuses on the role of risk models. Much has been written about the subprime mortgage crisis and the subsequent credit crisis. Several parties have been appointed as the source of the (current) financial climate: banks, governments, rating agencies, regulators, risk managers and even founders of complex derivative products. The most extreme opinions even imply that one man can be held responsible for the crisis. Besides aforementioned, risk models have also been criticized and held responsible. The key question to be answered reads: Is accusing mathematical models as a cause for the crisis not similar as holding a firearm responsible for a shooting fatality?

Cointegration of Natural Gas Markets and Valuation of Transportation Options

26

by: Ieva Jeglinskaite This research focuses on the dynamics of natural gas prices in the U.S. and U.K. natural gas markets. In contrast to previous assumptions about separated gas markets, recent research attempts to jointly model the markets, and this research extends the relevant cointegration analysis. More specifically, the question of stationarity of natural gas prices is carried out through an ADF test, which incorporates deterministic seasonality. The test results suggest that prices behave as non-stationary processes. The long-run relationship of these processes is analyzed under two frameworks, namely the vector error correction and the unobserved components. The results provide evidence in favor of the existence of the long-run relationship. Finally, different models for the dynamics of natural gas prices are compared in terms of valuation of gas transportation with the destination flexibility option.

Puzzle

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Facultive

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Operations Research and Management

Visualising Internal Migration by: Edwin de Jonge

Many statistical datasets can be modelled as a weighted digraph. In this article we use migration in 2006 as a working case. Migration is an important demographic for local policymakers. It is typically presented in detailed tables describing the in- and outflow of inhabitants per region. This local regional view is useful, but gives a fragmented picture of overall migration. To reveal and show patterns we model migration as a large weighted digraph. One major problem in this respect is that these of graphs are rendered poorly and cluttered with mainstream graph visualisation software. We show and discuss several visualisations of migration and their problems, including cartographic flow map and graph visualisation.

Introduction

Figure 1. Internal migration (2006) in Google Earth.

Migration can be described as a matrix with elements mij that describe the migration from region i to region j. In 2006 there were 459 municipalities in the Netherlands.

This means that there are potentially 210,2221 flows between the municipalities. Many of these flows are nonexistent. In 2006 the number of migration flows between municipalities was 60,073 (28.5% of the potential flows).

De maximum flow size in the flow data is 2,888, the median is 2 and the mean 10.86. Figure 1 shows the distribution of the flow size: there are some large migration flows, but the majority are small. This is one of the aspects that make it difficult to find a comprehensive visualisation of these migration statistics.

Weighted digraphs in statistics A weighted digraph is a mathematical graph consisting of nodes (or vertices) and directed weighted edges (or ties), i.e. a connection (edge) between nodes has a direction and a weight. Many official statistics can be modelled as weighted digraphs, for example trade, migration and

commuting, where numbers of goods or people flow from place A to B, but also balance systems, such as the System of National Accounts or a national energy balance sheet, where money or energy carriers flow from node A to node B. A more formal notation of a digraph is a set of nodes N, a set of edges E and an accompanying set of weights w.

N {n1 , n2 ,}, nn }

E {e1i , j , e2k ,l , }, ema ,b } w {w1i , j , w2k ,l , }, wma ,b }

Edwin de Jonge Edwin de Jonge is a methodologist working at Statistics Netherlands. His expertise is prototyping complex statistical software and applying data visualisation to statistical data.

A weighted digraph can be also represented using an adjacency matrix A where cell aij represents the weight of the edges. A weight of zero means that there is no edge.

Applied to our internal migration dataset, this means that aij is the number of people moving from municipality i to municipality j.

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The number of potential flows is nâ&#x20AC;&#x2030;(nâ&#x20AC;&#x2030;-â&#x20AC;&#x2030;1).

Zie jij groeikansen als de postvolumes dalen?

YEP!

Het Young Executive Programme van TNT Post H He zijn spannende tijden bij TNT Post. Dat is niet altijd even makkelijk. Maar dat is niet erg, Het want je wil tot het uiterste gaan. Bij TNT Post vind je pas echt een uitdaging. Door toenemende wa concurrentie en digitalisering dalen de postvolumes. Dat vraagt om creatieve en intelligente co oplossingen. Dan is er veel mogelijk. Je mag direct meedoen en leert de organisatie vanuit alle op hoeken kennen. Daarom kiezen echte talenten voor het YEP van TNT Post. Kijk voor meer informatie op www.werkenbijtnt.nl. Als zekerheden verdwijnen, komen de kansen.

Operations Research and Management

Figure 2. Internal net migration (2006) in Google Earth.

Figure 3. Migration Amsterdam (2006) in Google Earth.

Visualising migration

persons, a proportional linear width is not an option. In our visualisation we chose a log scale. Tobler’s (2003) Z-order technique of placing more important flows in front of less important ones is also applicable in our case.

It underlines the importance of flows with respect to less important flows. We chose to implement the Z Order as height of the flow using the 3D functionality of Google Earth. To reduce this clutter we use a technique that is commonly used in graph visualisations: we make less important flows more transparent than important flows.

This technique has a nice side effect: crowded areas are coloured darker and less transparent as a result of the overlapping flow arrows. This makes it easier to detect these crowded spots.

The direction of the flow is also an important property to be encoded in the visualisation. It serves all visualisation goals. Without flow direction the visualisation only shows the gross flow between two regions. Our visualisation contains a large number of flows. The direction is encoded by an arrow head. Although this is a useful way to encode direction for a small number of flows, for large numbers of flow arrows we came to the conclusion that the direction of the flows became less clear. As in- and outflows have the same colour it is also difficult to see the net migration flow.

To reduce visual clutter in the visualisation we could reduce the number of flow arrows. One option is to combine in- and outflow into one net flow arrow: the net migration flow between regions. This visualisation would contain half the number of flow arrows of an inand outflow visualisation. Furthermore it shows the net result of all these migrations more clearly than the two arrow approach.

It does not show the volume of the migration that took place in that year, however. The migration volume could be encoded in the colour of the flow arrow. For example, a larger volume could be encoded in a more salient colour. However, we find that such encoding is difficult to interpret: in our opinion colour is not a good way to encode migration volume.

Most graph visualisation research focuses on algorithms for positioning nodes also known as graph layout. These algorithms try to minimise the crossing of edges, thereby generating a clear overview for most graphs. Most of these algorithms do not function well for weighted digraphs. First of all because direction is not taken into account: in a digraph the direction of an edge is important and there may also be an edge in the opposite direction.

Furthermore weight is not taken into account when positioning the nodes.

A successful visualisation of the migration dataset should: •

• • •

give an impression of the overall migration flow and volume; show clearly the important migration flows; show clearly the important net migration flows; show all in- and outflow per region.

As designing an integrated visualisation that serves all these goals it is quite ambitious, we break down the visualisation into components for each goal. These components can be combined by a user by switching them on and off.

The properties of the data make it difficult to meet the goals. The number of flows is so large that the naïve implementation of drawing every flow gives a cluttered result which is difficult to interpret. This problem is aggravated because (almost) every flow mij has a reverse flow mji. We addressed these problems with visual encoding and specialised views.

Migration flow size is a very important property that must be encoded in the visualisation. It is a measure for how important the migration flow is compared with other flows and is therefore responsible for fulfilling at least two of the visualisation goals. It gives an impression of the migration volume and it emphasises large flows. An important aspect is that the distribution of flow sizes is skewed; but encoding should deal with that. Flow size is typically encoded by the width of the flow arrow.

As the migration flows range from 1 person to 2,888

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Discussion Flow mapping is a long existing practice in cartography; the design principles are given and discussed in various cartographic textbooks such as Dent (1999), Slocum (1999) and Ormeling & Kraak (1997).

Many effective flow maps are designed by hand because manually routing the flows gives a clear result.

The principles underlying such a flow layout have only recently been used in computer generated flow layouts.

The flow map layout algorithm described in Ormeling & Kraak (1997) gives an attractive example of computer generated unidirectional migration visualisation. These flow maps typically contain 100 flows at the most. Most flow maps found in the literature are unidirectional: the flows share one single starting or ending point.

The research on bidirectional migration maps (Slater (1981)) we found mostly dates from around 1990, when interactive 3-dimensional cartographic systems where not widely available.

Our carefully designed visualisation is interactive and uncovers several data patterns. First significant non-local migrations can be seen: e.g. Groningen to Amsterdam.

The net migration picture reveals migration hubs: e.g.

Nijmegen attracts migrants from nearby provinces but loses migrants to Amsterdam and Utrecht.

Slocum, Terry A.. Thematic cartography and visualization.

New Jersey: Prentice Hall, 1999.

Tobler, W.. “Experiments in migration mapping by computer.” American Cartographer (1987).

References Google Earth, available at http://earth.google.com/ Masser, I.. “The design of spatial systems for internal migration systems.” Regional Studies 10 (1976): 39-5.

Slater, P.. “Comparisons of aggregation procedures for interaction data.” Socio-Economic Planning Sciences 15 (1981): 1-8.

Tobler, W.. “Movement mapping.” http://www.csiss.org/ clearinghouse/FlowMapper/MovementMapping.pdf (2003).

StatLine. The statistical database Netherlands. http://statline.cbs.nl/

of

Statistics

The Google Census project. http://gecensus.stanford. edu/gcensus Phan, Doantam, Ling Xiao, Ron Yeh, Path Hanrahan & Terry Winograd. “Flow Map Layout.” InfoVis (2005).

Dent, Borden D.. Cartography: Thematic map design.

New York: McGraph-Hill, 1999.

Ormeling, F. & M.J. Kraak. Cartography, visualisation of spatial data. Harlow Longman, 1997. 2nd edition.

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Actuarial Science

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Actuarial Science

Pension Fund Decision Making: an Altering AOW Retirement Age by: Lars van den Berg The Dutch system of social security and pensions is internationally recognised for its level of reliability and services provided to participants (OECD, 2010). Economic and demographic developments have put this system under pressure. In 2009 the Dutch government decided that changes were needed to keep the system sustainable in the future. One of these measures is the increase of the Dutch General Old Age Act (AOW) retirement age. For this article, research has been performed on the impact of a change in the AOW retirement age for pension funds.

Introduction A pension fund has different possibilities to anticipate on a change in the AOW retirement age. This article focuses on the financial results of these decisions for different scenarios in the change of the pensionable age. The first paragraph of this article explains the demographic and economic developments that threaten the sustainability of the pension system. The interdependencies between the first and second pillar pension are stated in the second paragraph. In the third paragraph an overview is given of the possible actions a pension fund can take as a response to a change of the AOW retirement age. This is followed by a quantitative analysis of these decisions and the conclusion. This article is a summary of a thesis published in June 2010 (Berg, 2010). At that time the new government and its current law proposals concerning the AOW were not yet known. The scenarios analysed in the thesis are based on the suggestions published in the different party programmes of the major Dutch political parties. Since the differences between the current law proposal on the AOW and some of the ideas in these programmes are fractional, the results from this article can be used as a bench-mark for the current bill on the AOW.

Demographic and economic developments influencing the pension system The income of a Dutch pensioner consists of three elements: the pay-as-you-go AOW pension, the capital funded supplementary occupational pension and individual savings. This article concentrates on the first two types of pension. These sources of pension income have come under pressure through several reasons. The largest problem for the AOW is the change in demography

that is taking place in the Netherlands. The number of people of 65 years and older is expected to grow until the year 2040, while an opposite effect is expected for the working population. This development is displayed in the population pyramids of figure 1 and 2 (CBS, 2010).

As the AOW benefits for retirees are paid by the labour force this development causes problems for the AOW. For the coming years a growing burden of AOW benefits will have to be borne by a smaller group of people.

Another trend that is observed amongst the population is the increase in the life expectancy. People are living older because of improved living standards and medical care. The mortality projections that have been published in the past have always been underestimating the actual life probabilities. The recent publications of cohort tables by the Dutch Actuarial Association (2010) and the CBS (2010) show that the life expectancy of people aged 65 has increased compared to previous cohort tables. As a result, pension funds have to pay out benefits to pensioners for a longer period of time than expected on beforehand. The premiums that have been paid by current retirees to cover their benefits have been too low, which causes a deficit for pension funds. Furthermore, the actuarial required

Lars van den Berg Lars van den Berg obtained his master degree in Actuarial Sciences and Mathematical Finance at the University of Amsterdam in June 2010.

He wrote his thesis under the supervision of prof.dr. Jan B. KunĂŠ from the University of Amsterdam and drs. Hans de Mik AAG and drs. Jochem Fischer AAG of Ernst & Young Actuarial Services. After his graduation Lars started to work as a consultant at Ernst & Young Actuarial Services.

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Actuarial Science

Figure 1. Population pyramids Netherlands 2010.

Figure 2. Population pyramids Netherlands 2040.

premiums have become unaffordable for the people who are currently accruing pension rights and want to retire at 65. Figure 3 displays the remaining life expectancy at age 65 for the population forecast of Statistics Netherlands (CBS). The life expectancies in this figure are presented in two different ways. The period life expectancy is the average number of years a person would life if he experienced the particular mortality rates from that period throughout the rest of his life. This life expectancy does not take changes in mortality rates over time into

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Figure 3. Remaining life expectancy at age 65.

account. As mortality rates are generally expected to decrease, this number is an underestimation of the actual life expectancy. The second type of life expectancy is called the cohort life expectancy. This value is calculated using mortality rates that do allow for known or projected changes in mortality rates for later years. The mortality rates for a particular age used in a cohort table change over time while the mortality rates within a period life table remain the same.

Another development that has had impact on the pension sector is the decrease in the interest rate used for discounting pension obligations. The latter caused the obligations of pension funds to increase. This came together with the credit crunch that had a negative influence on the balances of the government and the pension funds. The asset portfolios of most pension funds decreased in value because of declines at the stock exchange. The combination of these two effects put the coverage ratio of a large group of pension funds below the required minimum level.

These developments made the government think about the sustainability of the pension system. One of the ideas was to increase the AOW retirement age. This is a complex measure because the current retirement age of 65 is seen as an acquired right by a lot of Dutch citizens. This is not surprising since this retirement age has not changed since the introduction of the AOW in 1957. The government decided however that an increase in the AOW retirement age was inevitable. The law proposal for the increase of the AOW age of retirement to 67 was submitted in 2009 (Min SZW, 2009). Due to the fall of the government in February 2010 new elections were held in June 2010.

From these elections a new government was formed and an alternative bill for the AOW was submitted.

Interdependencies AOW and second pillar pension Pension funds apply a retirement age of 65 years in their pension scheme. This pensionable age is derived from the statutory AOW age. The relationship between the AOW retirement age and the age of retirement within the second pillar pension is arranged in the so

Actuarial Science

Implementing modified AOW retirement age within 2nd pillar pension scheme The decision of the pension board will affect the accrual of future pension rights of active participants and the level of the future premiums. Accrued pension rights, also from retirees and inactive participants, will remain unchanged. A pension fund has several options how to implement the increased AOW retirement age within the pension scheme. These choices are defined below: • Decision A: The pension fund can choose to increase the retirement age within the pension scheme parallel to the AOW pensionable age. For the active participants this will have an effect on the accrual of future pension rights. The annual accrual of pension rights will be calculated with the modified accrual percentages. In addition to this the active participants will work for a longer period of time in which they accrue rights.

For the participants already retired or left the pension scheme nothing changes. A fund can give the active participants the opportunity to postpone the accrued pension benefits for the retirement age of 65 to the modified retirement age which will result in higher benefits.

• Decision B: The pension fund and employer can also decide to retain the retirement age within the pension

Figure 4. Development AOW retirement age for different scenarios political parties.

67,5 67 AOW rretirement age

called Witteveen-framework. Pension funds follow the Witteveen-framework for the fiscal accrual of pension rights. This framework has set rules to which a pension fund must comply with respect to the accrual of pension rights for its participants. One of the basic principles of the Witteveen-framework is, that as a result of a change in the retirement age it must be possible to accrue a pension that is at least equal to the pension that would be reached under the previous pensionable age (Min Fin, 2010). The framework further states maximum accrual percentages that apply to the supplementary occupational pension.

These percentages are determined in such way that a pension income equal to 70 percent of the last earned salary is achievable. This pension income also includes the AOW benefits received from the government. One of the assumptions used in the calculation of the accrual percentages is a length of 40 service years for an average worker. This duration is based on a retirement age of 65 years. A change in the retirement age will also affect the number of pensionable service years, since the increase of the AOW retirement age results in extra labour years.

Annual accrual rates can therefore be decreased to achieve the same level of pension benefits at retirement age.

Lower annual accrual will result in lower yearly actuarial required premiums. It is up to the social partners to decide how to anticipate on the change in the Witteveenframework. They are not obliged to follow the increase in the retirement age as long as they comply with the Witteveen-framework and adjust the accrual percentages accordingly.

66,5 66 65,5 65 64,5 2010

2015

2020

2025

2030

2035

2040

2045

2050

2055

2060

Year Scenario PVDA Scenario VVD Scenario PVV and SP

Scenario CDA Scenario GroenLinks

scheme at 65 years and to compensate the missing AOW benefits between the age of 65 and the new AOW retirement age. This can be financed through higher premium payments or lower second pillar benefits from 65 years onwards. The accrual of future pension rights should be based on the modified percentages as mentioned in the Witteveen-framework.

• Decision C: A third possibility for the pension fund is to keep the retirement age at 65. The participants will have to decide for themselves how to compensate for the missing AOW benefits between the age of 65 and the increased AOW retirement age.

Scenarios for the change of the AOW retirement age During the elections of June 2010 the increase of the AOW pensionable age was an important topic. All the political parties suggested a different plan for the change of the AOW retirement age. The scenarios for the development of the retirement age are displayed in figure 4. This figure shows which retirement age is applicable for a person who reaches the age of 65 in a particular year. This figure shows that the ideas from the political parties PVV and SP were more conservative, because they proposed to retain the AOW retirement age at 65. Other parties agreed that changes were needed and all suggested an increase of the AOW retirement age. However, each party had a different year of implementation and annual increase of the retirement age into mind. The VVD wanted to increase the AOW retirement age the earliest. They suggested to increase the retirement age from 2011 with two months per year until 2022. However the final retirement age of 67 was first reached in the plan of the CDA. The next paragraph provides an analysis on the financial impact of the two plans for a pension fund and it’s participants. The PVDA had the same annual increase as the CDA in mind, but only wanted to implement it 5 years later in 2020.

The suggestion of GroenLinks deviated from the other ideas since they wanted to increase the AOW retirement age to 67 for everybody born after January 1st 1972. The effective implementation of this higher retirement age would therefore take place from 2037 onwards.

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Actuarial Science

Table 1. Results pension fund ABC decision A (left of the diagonal) and decision B (right of the diagonal).

Additional costs scenarios on the right compared to scenarios below (%)

Basic scenario Scenario PvdA Scenario CDA Scenario VVD Scenario GroenLinks

Basic scenario

Scenario PVDA

Scenario CDA

Scenario VVD

x 2.7 4.5 6.8 −1.1

12.8 x 1.8 4.4 −3.7

15.8 2.5 x 2.5 −5.5

16.4 3.0 0.5 x −7.8

Quantitative analysis fictive pension fund ABC The scenarios from the previous paragraph are analysed for the fictive pension fund ABC. This fund consists of 80 persons who are born between 1946 and 1985. Each birth year is represented by two persons of whom one is a man and the other a woman. The pension fund was founded on the 1st of January 2010. For all participants is assumed that no values transfers of pension rights have taken place. It is a closed pension fund which means that no inflow or outflow of participants occurs. All participants receive an annual wage increase which is age dependent.

This consists of inflation and an incidental wage increase which is age dependent. To analyse the results a model has been designed that projects the future accrual of pension rights for each participant in an average pay pension system. For the calculation methods used in the model I would like to refer to my thesis (Berg, 2010).

The active participants pay an annual pension premium which is equal to a percentage of the salary in a particular year. This percentage is calculated by dividing the required premium by the total salaries of the participants in a particular year. The results for the different scenarios can be compared by taking a look at the variable which represents the present value of all future premiums. This variable gives an indication of the costs of a particular scenario. As was discussed in an earlier paragraph, pension fund ABC has several possibilities to anticipate on a change in the AOW retirement age. For this article the results of decisions A and B will be discussed.

If pension fund ABC would like to set the retirement age within the pension scheme equal to the AOW retirement age, option A is chosen. Table 1 shows the results for this decision as a ratio of the indication variable for the different scenarios. The results for option A are presented to the left of the diagonal in the shaded part of the table. In the basic scenario the retirement age retains at 65. The table shows that the premium costs of pension fund ABC are lower as compared to the basic scenario for the scenarios PVDA, CDA and VVD. The increase of the retirement age is accompanied by lower accrual percentages which results in lower annual premiums.

The present value of all future premiums is the lowest in the scenario of VVD. The costs for scenario CDA are 2.5

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Scenario GroenLinks

5.6 −6.9 −9.7 −10.2 x

percent higher. This means that an immediate increase of the retirement age with small annual steps results in lower premium costs than the scenario in which the retirement age is increased at a later stage with higher annual steps. Another interesting result appears for the scenario GroenLinks. The premium costs in this scenario are higher compared to the basic scenario. These higher costs are caused by the participants who become 65 after 2037. These participants earn a high pensionable salary which results in high premiums for the accrual of pension rights. The effect of the increase of the retirement age to 67 and resulting lower accrual percentages is outweighed by the combination of high pensionable salaries and two additional years of pension accrual from these participants.

Pension fund ABC can also choose to retain the retirement age within the pension scheme at 65 and to compensate the missing AOW benefits for its participants.

The results of choice B are presented to the right of the diagonal in table 1. This option results in large additional costs for the different scenarios. The most expensive scenario for pension fund ABC is VVD. This scenario implies a direct increase of the AOW retirement age. This means that the pension fund directly has to compensate participants for missed AOW benefits. To finance this, large premiums are required from the participants and the employer. As compared to the results for decision A, the order of the costs saving scenarios has reversed. The scenario of GroenLinks is the least expensive since in this scenario the higher AOW retirement age will only be effective from 2037 onwards. So only a small group of participants will receive AOW compensation benefits.

Since the premiums for the basic scenario have already reached the affordable maximum limit it would not be advisable for pension fund ABC to choose for option B.

Conclusion Economic and demographic developments threaten the Dutch pension system. Retirees are living longer than expected and the number of pensioners is expected to increase in the coming decades. In addition to this the current economic climate does not allow for compensation of these effects. To keep the pension system sustainable in the future, the government decided to increase the AOW

Actuarial Science

age of retirement. This has an impact on the second pillar pension system through the Witteveen-framework.

A pension fund has several options to anticipate on an increase in the retirement age. A fund can choose to retain the pensionable age within the pension scheme at 65 and to compensate for the missed AOW benefits of its participants. Results in this article show that this is an expensive option. The current premiums for the annual accrual of pension rights have already reached the maximum. Therefore this does not seem like a good measure for a pension fund board to implement in its pension scheme. A pension fund can also choose to change the pensionable age in the pension scheme parallel to the AOW retirement age. From the results of the different scenarios in this article can be concluded that this leads to lower required premium payments for the accrual of pension rights. The downside of this option is the lower annual accrual of pension rights. Since the premiums have reached the maximum it is advisable for pension fund boards to change the pension age within their pension schemes equally to the AOW retirement age.

As we are all living longer than we originally expected, this is a suggestion we should be able to live with.

References Berg, L van den. “Wijziging van de AOW en het aanvullende pensioenstelsel: Een analyse naar het effect van de wijziging van de AOW op het aanvullende pensioenstelsel.” MA thesis University of Amsterdam, 2010.

Dutch Actuarial Association (AG). “AG Prognosetafel 2010-2060.” 24 Aug. 2010.

Ministry of Finance (Min Fin). “Wet op de Loonbelasting 1964.” Ministry of Finance 24 Feb. 2010.

Ministry of Social Affairs and Employment (Min SZW). “Wetsvoorstel tot wijziging van de Algemene Ouderdomswet, de Wet inkomstenbelasting 2001 en de Wet op de Loonbelasting 1964 i.v.m. de verhoging van de leeftijd waarop het recht op ouderdomspensioen ontstaat.” Ministry of Social Affairs and Employment 2 Dec. 2009.

Organisation for economic co-operation and development (OECD). “ELS Pensions.” OECD: Stat Extracts 15 Feb. 2010.

Statistics Netherlands (CBS). “Bevolkingspiramide.”

Statistics Netherlands. CBS, 20 Dec. 2010, Web. 20 Dec. 2010 Statistics Netherlands (CBS). “Kerncijfers van de bevolkingsprognose, 2010-2060.” CBS.Statline CBS 17 Dec. 2010.

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Econometrics

There is More Behind Government Spending Than You Think by: Chen Yeh The aftermath of the recent credit crisis is starting to show its signs: after a bunch of bailouts and fiscal investments, many governments over the world are forced to cut their spending. However some governments seem to cut down their budgets more drastically than others. While the magnitude of these economic measures differ from country to country, economists wonder which countries’ spending patterns are similar and in particular why they are similar. In traditional macroeconomics one can find models that partially address these questions, however Persson and Tabellini (American Economic Review, 2004) adopt a different approach by combining views from political science and economics. Their empirical paper is based on sophisticated arguments and addresses determinants of government expenditure that are not straightforward.

Introduction Traditional models of government expenditure often involve economic variables, however the field of political economy also considers institutional factors. Persson and Tabellini (2004, henceforth P&T) consider the effect of constitutional rules on fiscal policy outcomes. These constitutional rules consist of electoral rules and forms of government. In most democracies, the electoral rule is either proportional or majoritarian. When a country adopts a proportional voting rule, the distribution of votes closely matches the distribution of seats in the representative democracy, whereas the majoritarian rule is equivalent to “the winner takes all” system.1 For forms of government, P&T consider presidential and parliamentary ones. In their definition of presidential countries, the chief executive or cabinet (regarding issues of economic policy) is not accountable to the legislature through a vote of confidence, i.e. its power is independent of any political institution. As a result, a country is called parliamentary where this vote of confidence is actually required.

In their paper, P&T are trying to address a simple question, namely how do electoral rules and forms of government influence fiscal policy? The authors empirically test a series of claims, hypothesized by both political scientists and economists. Their basic setup is the simple, yet trustworthy, technique of ordinary least squares (OLS). They also relax the necessary OLS assumptions and perform some robustness tests to check whether their empirical results are reliable. It seems that standard OLS regressions confirm earlier theoretical work, however their other results are not that strong, but do give interesting insights regarding constitutional rules and fiscal policy outcomes.

The interaction between constitutional rules and fiscal policy outcomes: some theory Although P&T’s contribution is mainly empirical, the foundation of their econometric framework is firmly motivated by theoretical models. A stream of economic models predict that proportional electoral systems and

The proportional voting rule is for example used in the Netherlands for parliamentary elections, whereas the majoritarian voting rule is adopted in the United States during their elections to vote for a president.

1

In this issue of AENORM, we continue to present a series of articles. These series contain summaries of articles which have been of great importance in economics or have caused considerable attention, be it in a positive sense or a controversial way.

Reading papers from scientific journals can be quite a demanding task for the beginning economist or econometrician. By summarizing the selected articles in an understanding way, the AENORM sets its goal to reach these students in particular and introduce them into the world of economic academics.

For questions or criticism, feel free to contact the AENORM editorial board at aenorm@vsae.nl

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Econometrics

parliamentary regimes should be associated with more public goods, a bigger and broader program of welfare spending and a larger overall size of government.2 While these models are all somewhat different in nature, they do reach a similar conclusion: under proportional elections, the composition of public spending is more biased towards large groups in the population. To satisfy the needs of these large groups, a higher level of government spending is induced when compared to the majoritarian voting rule. The main arguments are twofold.

The first reason is district magnitude (how large a share of the legislature is elected in a typical district): political parties have strong incentives to seek support of broad and large parts of the population under proportional elections as legislatures are elected in mainly large districts. The other argument is based on the electoral formula (how vote shares are converted to seat shares in the legislature).

In a majoritarian system, a political candidate usually only needs 50 percent of the district vote. Moreover the politician would actually suffice with 50 percent of those particular votes, which implies that theoretically this candidate could win the entire election with just 25 percent of the total amount of votes. Under a proportional system, 50 percent of the national vote is needed to win the elections. Thus when facing a proportional voting rule, politicians are once again induced to target larger segments of the population.

Even though the effect of forms of government on fiscal policy has not received much attention, there are a few formal studies. As mentioned earlier, the key difference between presidential and parliamentary regimes is the required confidence vote. Diermeier and Feddersen (1998) argue that confidence requirements induce more legislative cohesion: when the chief executive receives the support of a stable majority of legislators, they vote together on legislation, pursuing the joint interest of its represented voters. This means that government spending is providing benefits to a majority of voters. Moreover, these voters become the residual claimant on additional revenue and therefore prefer high taxes and spending.3 By contrast, in presidential regimes there are often powerful minorities (typically the constituency of powerful officeholders) and none of them are residual claimants, which implies that they do pay (relatively) high costs but do not reap the benefits. Thus they resist high government spending.

Econometric methodology: the average treatment effect and assumptions Constitutional reforms have been very rare, i.e. time

variation is minimal and thus little can be gained by estimating panel data. Hence their conclusions about the effect of constitutions on policy outcomes must be identified from cross-country regressions. P&T therefore use a cross-section of 80 democracies in the 1990â&#x20AC;&#x2122;s. The main problem however is that constitution selection is not random: countries with different constitutions also differ in many other aspects. Thus how are the effects of constitutional rules on fiscal policy outcomes isolated from other observable and non-observable policy determinants?

P&Tâ&#x20AC;&#x2122;s econometric model consists of two parts. The first deals with the constitution selection of country i Si; thus what electoral rule (Si = 1 when majoritarian and 0 otherwise) or form of government (Si = 1 when presidential and 0 otherwise) a country i uses. To simplify their model, these two aspects of constitutional rules are estimated separately with the following equation:

Si

­1 if G ( X i ) ei ! 0 Ž o/w ¯0

where Xi is a vector of observables, including colonial origin and geographic location.4 Countries have changed their constitutional rules very few times in the last 40 years (coined constitutional inertia) and political scientists suggest that this information on constitutional history can be exploited to explain cross-country variation. Thus three time dummies are also included in Xi that indicate the origin of the current constitution.5 The second part of their framework determines the fiscal policy outcome Yi, given by6 Yi F ( Si , Z i ) ui Thus P&Tâ&#x20AC;&#x2122;s goal is to estimate the effect on fiscal policy of a hypothetical shift from Si = 0 to Si = 1, the so-called average treatment effect of constitutional reform. When estimating this framework with OLS, two assumptions are imposed. Conditional independence implies that the error terms in both equations (i.e. ei and ui) are uncorrelated and linearity demands that F(.) is linear with constant coefficients. However these assumptions might be restrictive and not realistic. Conditional independence is a strong assumption given the non-random distribution of constitution since historical variables determining constitution could also influence policy outcomes.

Naturally we do not have a problem when all common historical determinants of government policy are included in the constitution selection equation. However how do

Persson and Tabellini (1999), Milesi-Ferreti et al. (2002) and Persson et al. (2000) are recent examples of the broad theoretical literature about the link between constitutional rules and fiscal policy.

3 A residual claimant is entitled to the net benefits, i.e. the amount of surplus that is left after deducting costs from income.

4 P&T also rely on other cultural and geographic variables such as: distance from the equator, percentage of population with English or a European language as the mother tongue, ethno-linguistic fractionalization and population size.

5 These dummies include the periods of pre 1920, 1921 â&#x20AC;&#x201C; 1950 and 1951 â&#x20AC;&#x201C; 1980. The period after 1980 is taken as the default, i.e. equals zero when a countryâ&#x20AC;&#x2122;s constitution originates from this period.

6 Several regressions are estimated, each using a different measure for size of government. These include central government spending (cgexp), central government revenues (cgrev) and government deficit (dft). All these are ratios of GDP. 2

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December 2010

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Econometrics

Table 1. Results from the OLS regressions. *, ** and *** indicate significancy at the 10, 5 and 1 percent level respectively (Source: Persson and Tabellini, 2004).

Dependent variable

cgexp

pres

-5.18 (1.93)*** -6.32 (2.11)***

maj propres majpar pajpres pres_newdem

cgexp

-6.56 (3.01)*** -6.96 (3.72)*** -10.37 (3.03)***

cgrev

dft

cgexp

cgexp

cgexp

-5.00 (2.47)** -3.68 (2.15)*

0.16 (1.15) -3.15 (0.87)***

-2.65 (2.70) -1.45 (2.32)

-7.75 (2.70)*** -7.94 (3.74)**

-6.46 (2.98)** -6.33 (2.48)**

3.50 (2.72) 3.58 (4.03) -4.08 (2.23)*

maj_newdem newdem pres_baddem maj_baddem baddem F-test (pres) F-test (maj) Sample Observations R²

0.43 1990's 80 0.71

1990's 80 0.70

1990's 76 0.68

we know that we have dealt with this omitted variables bias? P&T justify their methodology by using a proper set of control variables, however there are always other factors that can be overlooked. Therefore the authors use Heckman correction which very loosely can be described as estimating the original two-part model with no adjustments at all and the corresponding bias.7 Finally the earlier estimates are then corrected with the estimated bias and should lead to unbiased results. Another option involves the classical Instrumental Variables (IV).

Lastly, the linearity assumption can be relaxed.

Linearity is taken as a convenient local approximation of a more general model. However the constitutional effect on policy outcomes may be stronger in older or better democracies. As these features differ systematically across constitutional groups, the local approximation may no longer be tenable and the linear estimates are biased.

Their solution consists of using matching methods.

More weight is given to the comparisons of similar countries to reduce the effect of any nonlinearities. The basic idea is that we should compare the performance of similar countries, because their selection into different constitutions is largely random. Thus for each country with a particular constitutional rule, we try to find its “twin” or a “set of close relatives” to it with the alternative constitutional rule.

7

1990's 72 0.50

4.01** 3.18* 1960-1973 1990's 42 80 0.79 0.72

2.42 (4.16) 2.06 (5.97) -5.73 (3.46) 1.40 0.66 1990's 80 0.70

Econometrics at work: results of the regressions In their basic OLS setup, the results are in line with the earlier mentioned theoretical frameworks. A switch from proportional to majoritarian elections in a country chosen at random reduces total government spending by about 6 percent of GDP, which can be seen in the first column of table 1.

Similarly the estimates for form of government indicate that presidentialism reduces the overall size of government by roughly 5 percent of GDP. The effects of the two constitutional features also appear to be additive.

A F-test does not reject the null hypothesis that the estimated coefficient of majpres in column 2 of table 1 equals the sum of the estimated coefficients of propres and majpar. Thus these estimates seem to indicate that introducing a presidential form of government and majoritarian electoral rules in a proportionalparliamentary country would reduce central government spending by a significant amount of 10 percent of GDP. When other measures of size of government are used, most of the results remain intact. While the effects are smaller, the signs are preserved in the government revenues setup (column 3 of table 1): a hypothetical switch from presidential to parliamentary government would increase both spending and revenues by the same

This is feasible due to some identification assumptions which we will not discuss in this article.

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Econometrics

Table 2. Results from the Heckman specifications, IV estimations and matching methods. *, ** and *** indicate significancy at the 10, 5 and 1 percent level respectively (Source: Persson and Tabellini, 2004).

Dependent variable

cgexp

cgexp

cgexp

cgexp

cgexp

cgexp

cgexp

pres

-5.29 (2.18)**

-11.52 (4.54)**

-6.51 (3.71)*

-4.22 (3.99)

-5.86 (4.53)

-2.54 (2.26)

-7.30 (2.36)***

-4.18 (3.17) col_uka, laam 1990's pres maj 2SLS

-4.86 (3.57)

-6.59 (3.40)*

-5.76 (2.59)**

1990's pres maj Stratification

1990's pres maj Nearest neighbor

1990's pres maj Kernel

65(pres) 67(maj)

65(pres) 67(maj)

65(pres) 67(maj)

maj

-6.21 (2.82)** Conts & Cols Yes Sample 1990's Endogenous selection maj Estimations Rho Chi-2 Adjusted R² Observations

-6.77 (1.98)*** Yes 1990's pres

-4.83 (3.19) col_uka 1990's pres maj Heckman Heckman 2SLS ML ML 0.05 0.62 (0.29) (0.33) 3.29 0.59 75 75 75

amount of 5 percent. However the effect of majoritarian elections is cut in half, resulting in a reduction of about 3 percent of GDP. This basic setup is robust to the specification of the control vector Zi. When less influential controls are dropped or other controls such as income inequality, ethnic and linguistic fractionalization and whether the country was a former social country are added, the results do not change much as the estimated coefficient of majoritarian elections is always negative and (almost) always significant. This also holds true when P&T focus on the age of the constitution and the quality of democracy: constitutional effects appear stronger in old democracies and the effect of presidential regimes is stronger in higher quality democracies, whereas the effect of majoritarian elections remains stable.

In table 2 the results are shown when the standard OLS assumptions of conditional independence and linearity are relaxed. The first two columns show the Heckman adjustments and it seems that the results are strengthened: the signs of the coefficients are in line with theoretical results and significant. Furthermore the magnitude of these coefficients are larger in absolute value than under the previous setup which indicates that the OLS regressions were upwards biased. However the other two estimation strategies do not seem to be that successful.

The instrumental variables estimations (columns 3 and 4) lead to coefficients that are not significant. Moreover when P&T test for the validity of their instruments it does not lead to promising results. A similar story holds for the matching methods: both stratification and nearest neighbor techniques lead to insignificant coefficients.

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2.23 0.59 75

Conclusion In political science and economics, scientists have hypothesized about the effects of electoral rules and forms of government on fiscal policy outcomes in the form of theoretical models. In their paper, P&T empirically test the predictions of these frameworks in a cross-country data set of 80 democracies. As predicted a hypothetical shift in constitutional rules, coined the average treatment effect, does affect government spending.

Majoritarian elections lead to smaller governments when compared to proportional elections: it reduces total government spending by about 6 percent of GDP. The data also seems to support hypothetical changes concerning forms of government. Presidentialism roughly cuts the size of government by about 5 percent of GDP. While these results are robust regarding control variables variation, the basic OLS framework does seem to rely on strong assumptions. However when these assumptions are relaxed, the results still stand, although they do seem to be less firm.

P&Tâ&#x20AC;&#x2122;s econometric framework does incorporate another important implicit assumption: constitution affects policy outcomes directly. However political scientists have doubts about this particular mechanism.

They argue that constitutional rules have an impact on factors such as party structures, types of government and occurrence of elections or government crises.

These in turn may affect fiscal policy outcomes, which implies that constitutional rules only indirectly affect government spending. However this is not reflected in P&Tâ&#x20AC;&#x2122;s empirical model. Acemoglu (2005) argues that

Econometrics

this is crucial and criticizes P&T’s methodology heavily: he considers their results to be cases of robust correlation rather than causal effects. Nonetheless, P&T’s work is a first step in empirically identifying relationships between constitutional rules and fiscal policy and may make you wonder that there is more behind government spending than you think.

References Diermeier, D. and T. Feddersen. “Cohesion in Legislatures and the Vote of Confidence Procedure.” American Political Science Review 92.3 (1998): 611 – 621.

Milesi-Ferretti, G.M., R. Perotti and M. Rostagno.

“Electoral Systems and Public Spending.” Quarterly Journal of Economics 117.2 (2002): 609 – 657.

Persson, T. and G. Tabellini. “The Size and Scope of Government: Comparative Politics with Rational Politicians.” European Economic Review 43.4 - 6 (1999): 699 – 735.

Persson, T., G. Roland and G. Tabellini. “Comparative Politics and Public Finance.” Journal of Political Economy 108.6 (2000): 1121 – 1161.

Persson, T. and G. Tabellini. “Constitutional Rules and Fiscal Policy Outcomes.” American Economic Review 94.1 (2004): 25 – 45.

Acemoglu, D.. “Constitutions, Politics and Economic Growth: A Review Essay on Persson and Tabellini’s ‘The Economic Effects of Constitutions’.” Journal of Economic Literature 43 (2005): 1025 – 1048.

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Econometrics

Day-to-Day Use of Risk Models by: Hans Heintz and Frank Pardoel

Much has been written about the subprime mortgage crisis and the subsequent credit crisis. Several parties have been appointed as the source of the (current) financial climate: banks, governments, rating agencies, regulators, risk managers and even founders of complex derivative products. The most extreme opinions even imply that one man can be held responsible for the crisis: Lewis S. Ranieri1). Others argue that the former president of the United States (U.S.) George W. Bush2) is to be held responsible (see the text box for a background story).

Besides aforementioned, risk models have also been criticized and held responsible. This article focuses on the role of risk models. The key question to be answered reads: Is accusing mathematical models as a cause for the crisis not similar as holding a firearm responsible for a shooting fatality?

Introduction The Black-Scholes-Merton options pricing model, the Capital Asset Pricing Model (CAPM), the Vasicek Model, all examples of great mathematical models which are correct if and only if several assumptions are satisfied. By assuming these underlying assumptions, models enable the users to obtain outcomes with high (theoretical) power in a relative simplified way. Despite these (non-realistic) assumptions, theoretical models are an important analysis tool and are still used by many financial institutions (i.e. banks, insurance companies, rating agencies). Examples of underlying assumptions which correspond to often applied models are: normality of the underlying process, independency between model variables, perfect and symmetric information and the constant volatility assumption.

Without these assumptions, no sensible models can be applied. However, the user must always apply common sense in interpreting the results. Consider for example the period prior to the subprime mortgage crisis. This period was characterized by seemingly unbounded stock prices, excessive returns and sky-high profits. Simultaneously, the model users started to rely more and more on the Ranieri is a former bond trader and is considered as the pioneer behind securitization. In the mid-seventies, he introduced the securitization of mortgages, better known as mortgage-backed securities (MBS).

2 Speaker of the House of Representatives Nancy Pelosi: “Consider the terrible consequences of the ‘anything goes’ Bush Administration, whose irresponsible non-regulation of financial institutions has led to this crisis.”

1

model outcomes and perceive the model outcomes as the single truth. This diminishing awareness of the shortcomings of mathematical models combined with an economic exuberance, proofed to be a dangerous combination. Preaching precaution in times of economic euphoria is a difficult and an ungrateful task, however it

Hans Heintz After graduating in business econometrics in 1995, Hans joined ING Barings to work within the Trading Risk department. In 2000, Hans became account manager large corporate relations at Deutsche Bank Corporate Finance. In 2006 Hans moved back to ING, where he helped to establish a new model validation department.

In 2008 Hans together with three other partners founded RiskQuest, an Amsterdam based consultancy firm specialized in mathematical models for financial institutions.

Frank Pardoel Frank graduated in Econometrics in 2009 at the University of Amsterdam, track Mathematical Economics. Frank started his career within Hewitt Associates in 2007, working as an actuarial consultant and member of the Asset Liability Management team. In 2010 he decided to join RiskQuest. Frank currently joins the postgraduate program Risk Management for Financial Institutions at the VU Graduate School of Economics and Business. Frank is an associate at RiskQuest.

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Econometrics

Table 1. In case of Gaussian VaR, the number of standard deviations corresponds with a certain probability and an expected number of times by which the realization will exceed the VaR measure. # Standard deviations

Probability exceeding VaR

E[number of days until event]

1 2 3 4 5 6

15.8655254% 2.2750132% 0.1349898% 0.0031671% 0.0000287% 0.0000001%

1 in 6 1 in 44 1 in 741 1 in 31,574 1 in 3,488,556 1 in 1,013,594,635

appeared not to be a superfluous task.

Theory versus reality An illustration of the relation between theory and practice involves the Gaussian Value at Risk (VaR), widely used by many financial institutions. The VaR is a measure of the maximum loss not exceeded with a given probability defined as the confidence level, over a given period of time. Mathematically described as: VaRĮ

inf{z | P[ Z ! z ] (1 Į )}

The confidence interval is constructed using the properties of the normal distribution. This implies for example: the probability of exceeding the VaR measure in case of using the significance level corresponding with three standard deviations is approximately 0.135% or 1 event in 741 days; see table 1 for more information.

Mandelbrot (2004) showed the discrepancy between the assumed probability under Gaussian VaR and Dow Jones realizations during the period 1916 and 2003 (prior to the crisis). On the basis of Gaussian VaR a daily movement of more than 3.4% should occur 58 days while the Dow Jones showed such a move 1,001 times for this period. Even less accurate is the expectation corresponding to a more than 7% move. Gaussian VaR predicts an occurrence once every 300,000 years while the realization counts 48 days. The above example does not

Table 2. The table illustrates the predicted and realized 3-year default probability of U.S. mortgage-backed securities issued in the period 2005 to 2007.

Rating

Predicted PD (as of June 2006)

Realized PD (as of July 2009)

AAA AA+ AA AAA+ A ABBB+ BBB BBBBBB BBB-

0.008% 0.014% 0.042% 0.053% 0.061% 0.088% 0.118% 0.340% 0.488% 0.881% 0.488% 0.881%

0.10% 1.68% 8.16% 12.03% 20.96% 29.21% 36.65% 48.73% 56.10% 66.67% 56.10% 66.67%

imply that VaR models cannot be used to estimate risks; it just addresses the importance of taking into account the shortcomings of models due to their assumptions. By underestimating or even ignoring these model restrictions, underestimation of the risks will eventually occur.

Another practical example involves the credit ratings published by the U.S. rating agencies (i.e. Moody’s and S&P). To those ratings certain probabilities of default (PDs) are attached based on historical observations.

Banks use those ratings to estimate the risk and return corresponding to their rated counterparties.

Unconditionally applying the credit ratings and PDs can lead to inaccurate estimations, as demonstrated in table 2.

The realized defaults in the period until July 2009 differ significantly from the predicted defaults published in June 2006. The large differences are primarily due to the crisis. This observation immediately indicates the weakness of the model, which is not appropriate for adverse movements. The text box provides background information on the subprime mortgage crisis and it considers the main flaws which eventually led to the credit crisis.

Fannie Mae and Freddie Mac The Federal National Mortgage Association (hereafter: Fannie Mae) has been established in 1938 – in the aftermath of the Great Depression – in order to increase homeownership by lending federal money to banks against a relative low rate. Fannie Mae was part of Roosevelt’s New Deal. Thirty years later in 1968, the U.S. government privatized Fannie Mae in order to reduce government expenditures; thereby raising the budget for the Vietnam War. In 1970, a second mortgage association was established also known as Freddie Mac.

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The two players, Fannie and Freddie form the heart of the (secondary) mortgage market. Their business in a nutshell: they purchase mortgages in the primary market thereby providing cash for new mortgages. The purchased mortgages are then securitized – mortgagebacked securities – and sold to numerous investors, see figure 1.

After the deregulation of the secondary mortgage market the government did not own Fannie Mae or Freddie Mac. Nevertheless a majority of investors

Econometrics

Figure 1. A simplified illustration of the U.S. mortgage market.

Payments minus fee

Investor II

Investor III Payments minus fee

Mortgage Bank A Payments

P a y m e n t s

Fannie Mae

Payments

Payments

Mortgage Bank B

Payments

H O M E

Payments

ts men Pay

Payments minus fee

P o o l e d

Pa ym e nt s

Investor I

Pa ym ents

Freddie Mac

Mortgage Bank C

Payments

Mortgage Bank D

Payments

Payments

O W N E R S

Payments

Tertiary Market (MBS)

Secondary Market

was unaware of the exact relationship and issued loans and MBS did not receive the appropriate rating. Furthermore, the former relationship with the government provided several benefits: lower capital requirements, lower bond yields, less debt restrictions and legal regulations.

In return, both parties supported the – on December 16, 2003 by President Bush signed – American Dream Downpayment. The objective of the act was increasing the number of low income mortgages. This is the reason some consider President Bush as a cause for the crisis.

The overall consequences were a loosened risk framework and – due to their large involvement in the financial market – a significant increase in systemic risk. On top of the previous points, the boards of Fannie Mae and Freddie Mac extended their business rapidly by issuing more and even less transparent derivative products without taking appropriate (operational) risk management actions.3 They underestimated prepayment risk4 and even kept unsold complex securitized products on their own balances, thereby increasing their risk exposure.5 Together with these risk management flaws, senior executives of Fannie Mae manipulated accounting figures in order to report stable and constant growth rates thereby receiving undeserved bonuses.6 The Office of

Primary Market

Federal Housing Enterprise Oversight (OFHEO) had investigated the matter and their director James B. Lockhart stated: “The image of Fannie Mae as one of the lowest-risk and “best in class” institutions was a facade”.

In 2007 when (low income) homeowners defaulted on their monthly payments the several risk management flaws caused a domino effect. The high exposure to credit risk was not sufficiently captured by the risk management department and insufficient economic capital was reserved. The VaR measures of Fannie and Freddie were too conservative – lacked additional stress testing and scenario analysis – which resulted in huge unforeseen looses. Furthermore, complex securitized derivatives seemed much riskier than on forehand and huge liquidity problems occurred. Fannie Mae and Freddie Mac faced the inability to meet their obligations.

Systemic risk – referring to the collapse of the entire financial system – seemed a reasonable threat due to the large interconnection between banks in the primary and secondary market and other financial institutions. In the Q4 2008, the U.S. government supported Fannie Mae and Freddie Mac with billions of dollars thereby preventing bankruptcy.

In case of Freddie Mac, the notional amount of derivatives contracts increased gradually from $72 billion to $1.7 trillion without appropriately adjusting the operational department budget (Cunningham (2004)).

4 Prepayment occurs in the event a house is sold, death, resignation but mostly when interest rates decline. If prepayment occurs, duration of mortgage portfolios decrease and affects the match of risk exposure between assets and liabilities.

5 Freddie Mac bought Interest Rate Swaps (IRS) to hedge the additional risk. However the hedge was less efficient than presumed due to high variety of interest rates and durations and insufficient stress testing.

6 Office of Federal Housing Enterprise Oversight (OFHEO) report, 20 September 2004.

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Conclusion Like the econometrician Henri Theil said: “You should not believe the model, you should use the model”. It is ruled out that mathematical models are guilty to the current crisis. They were used incorrectly (in some cases even on purpose). It therefore seems more appropriate to accuse the model user instead of the model itself. Nobel Prize winner Robert Merton supports this statement: “The cause of this crisis is not attributable to models that were too complicated or that too much trust was placed in them. Rather, there was too much trust in the people who used the models. (..) The solution is not: the models are too complicated to understand for bank managers and supervisors, and hence, the models must go. No, the bank managers and supervisors who do not understand the models must go!”

The current complex financial world requires mathematical models in order to value, monitor and estimate risks. Nevertheless, there should be sufficient caution and restraint regarding the outcomes and conclusions. Even the most advanced, complete and innovative mathematical model applied to the real world will still rely on assumptions. Therefore it is crucial to be aware of the shortcomings of the model and the importance of the underlying assumptions. Risk models might not predict or prevent a future crisis, they will at least– if used appropriately – support to understand and clarify the risks.

Furthermore, models will become more and more important due to the immense amount of data available now and in the future, the rising complexity, increasing competition in the financial market and the technological improvements. RiskQuest supports the vision of Henri Theil and focuses on the development, validation and calibration of risk models especially for banks, insurance companies and pension funds.

References Avgouleas, Emilios. “The Global Credit Crisis, Behavioural Finance, and Financial Regulation, In Search of a New Orthodoxy.” Journal of Corporate Law Studies 9.1 (2009): 23-59.

Cleveland, Paul A.. “Freddie Mac: A Mercantilist Enterprise.” Mises Daily 14 Mar. 2005.

Cuomo, Andrew M.. “No Rhyme or Reason: The Heads I Win, Tails You Lose, Bank Bonus Culture.” 2009.

Duffie, Darrell & Erin Yurday. “Risk at Freddie Mac.” Stanford Graduate School of Business 12 Dec. 2004.

Hudson, R. L. & B.B. Mandelbrot. “The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward.”

Basic Books Aug. 2004.

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The Professional Risk Managers International Association (PRMIA), “Fannie Mae & Freddie Mac.” PRMIA at www.prmia.org, 2009.

For more information: www.riskquest.com.

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Econometrics

Cointegration of Natural Gas Markets and Valuation of Transportation Options by: Ieva Jeglinskaite This research focuses on the dynamics of natural gas prices in the U.S. and U.K. natural gas markets. In contrast to previous assumptions about separated gas markets, recent research attempts to jointly model the markets, and this research extends the relevant cointegration analysis. More specifically, the question of stationarity of natural gas prices is carried out through an ADF test, which incorporates deterministic seasonality. The test results suggest that prices behave as non-stationary processes. The long-run relationship of these processes is analyzed under two frameworks, namely the vector error correction and the unobserved components. The results provide evidence in favor of the existence of the long-run relationship. Finally, different models for the dynamics of natural gas prices are compared in terms of valuation of gas transportation with the destination flexibility option.

Introduction

markets, the principal objectives of this research are:

For a long time, integration of global gas market was limited due to the lack of maturity of transportation infrastructure. In addition, gas transportation contracts had destination clauses forbidding exporters or importers to divert natural gas cargoes. Consequently, three main segmented gas markets, in Europe, North America and Asia, were developed with limited competition in between.

Nevertheless, the technological development and decline in transportation costs lead to growing investments into Liquefied Natural Gas (LNG) transportation. As a result, gas is becoming, although very slowly, an internationally traded commodity providing a missing mechanism for the global market integration.

Furthermore, due to technological improvements gas producers can send cargoes over longer distances. The option to choose among alternative markets raises the need to value investments in gas transportation.

With the main focus on the U.K. and U.S. natural gas

•

•

Ieva Jeglinskaite Ieva Jeglinskaite obtained her master degree of Econometrics at the University of Amsterdam in August 2010. This article is a summary of her master thesis, which was written under supervision of dr. Noud P.A. van Giersbergen during internship at KYOS Energy Consulting.

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to extend cointegration analysis between the two traditional separated markets; to analyze different models for valuation of investments in gas transportation.

In specific, this research employs advanced econometric methodologies in order to address the following research questions: • •

•

Gas prices: mean reverting or random walk?

What is the long-run relationship between the U.K. and U.S. natural gas markets?

To what extent do different models for the dynamics of natural gas prices lead to different values of transportation investments?

More specifically, this research applies Augmented Dickey-Fuller (ADF) test, developed by Dickey and Fuller (1981), to investigate the presence of a unit root in gas prices. In addition, the test is extended in order to incorporate deterministic seasonality. Seasonality is modeled using trigonometric functions. Moreover, this research utilizes Johansen (1995) cointegration approach and unobserved components method, developed by Morley (2007), in order to examine the long-run relationship between natural gas markets.

The last part of the research considers the illustrative application of the valuation of gas transportation. Not only correlation, but also cointegration, has to be taken into account while modeling the joint dynamics of the prices.

Econometrics

Econometric Methodology The Augmented Dickey-Fuller Test In the spirit of Enders and Lee (2006), this research considers a case of ADF test, allowing for a single frequency deterministic (seasonal) component: Çťyt

§ 2Ę&#x152;kt ¡ ČĄyt 1 ÄŽ ÄŽk sin ¨ ¸ Š T š § 2Ę&#x152;kt ¡ p Č&#x2022;k cos ¨ ¸ ÂŚ Č&#x2122;i Çťyt i Ä°t Š T š i1

(1)

where T represents the number of trading days in a year and k represents a particular frequency of cycles in one year. The unit root test is performed by testing the null hypothesis H0: Ď = 0 versus an alternative H1: Ď < 1.

The regression is estimated using Ordinary Least Squares (OLS) in order to obtain the t-statistic for Ď . Critical values for the t-statistic can be obtained by Monte Carlo simulation from a nonstandard distribution.

Johansen Cointegration Method and VECM Johansen method starts with an estimation of an adequate VAR model, which is further rewritten into vector error correction form. Then, cointegration rank tests, based on VECM and likelihood ratio test statistics, have to be carried out.

The Vector Error Correction representation of the VAR(p) model is of the following form ÇťX t

p 1

Č&#x2020;X t 1 ... ÂŚ ÄŤi ÇťX t i Ä­Dt Ä°t , t 1,..., T

(2)

i 1

where Î = â&#x2C6;&#x2019;â&#x20AC;&#x2030;(Î i+1â&#x20AC;&#x2030;+â&#x20AC;&#x2030;...â&#x20AC;&#x2030;+â&#x20AC;&#x2030;Î p), Î&#x201C;i = â&#x2C6;&#x2019;â&#x20AC;&#x2030;(1â&#x20AC;&#x2030;â&#x2C6;&#x2019;â&#x20AC;&#x2030;Î 1â&#x20AC;&#x2030;â&#x2C6;&#x2019;â&#x20AC;&#x2030;...â&#x20AC;&#x2030;â&#x2C6;&#x2019;â&#x20AC;&#x2030;Î p) for i = 1,â&#x20AC;&#x2030;...,â&#x20AC;&#x2030;pâ&#x20AC;&#x2030;â&#x2C6;&#x2019;â&#x20AC;&#x2030;1, and Îľt is n-dimensional Gaussian process, so that Îľt ~ NIDâ&#x20AC;&#x2030;(0,â&#x20AC;&#x2030;ÎŁ). The term Î Xt-1 is referred to as the long-run or the error correction term of the model.

Johansen cointegration test is based on the rank of the matrix Î . Three cases, provided in (Johansen and Juselius, 1990), have to be examined: â&#x20AC;˘

â&#x20AC;˘

â&#x20AC;˘

Correlated Unobserved Components Model The CUCM was originally proposed by Morley (2007), who applied it for the analysis of aggregate consumption and permanent income. The model written in a statespace representation and the parameters are estimated through a Kalman filter.

The multivariate CUCM for the logarithms of two price series, which are cointegrated, is defined as

The matrix Î has rank zero. That is, all series are non stationary, have no common stochastic trend and there is no linear combination of the variables that is stationary; The matrix Î is of full rank, i.e., rank(â&#x20AC;&#x2030;Î â&#x20AC;&#x2030;) = n. Then the process Xt is stationary and has no common stochastic trend as well; The matrix Î has reduced rank, i.e., rank(â&#x20AC;&#x2030;Î â&#x20AC;&#x2030;) = r, 0 < r < n. Then there exist n Ă&#x2014; r matrices Îą and β with rankâ&#x20AC;&#x2030;(Îą) = rankâ&#x20AC;&#x2030;(β) = r such that Î = ιβâ&#x20AC;&#x2122;.

Johansen (1995) formulates the null hypothesis of at most r distinct stationary cointegration vectors as Hâ&#x20AC;&#x2030;(r): Î = ιβâ&#x20AC;&#x2122; and provides two likelihood ratio test statistics, LRtrace and LRmax, for testing this hypothesis.

Xtus = Ď&#x201E;t + u1tâ&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;,

(3)

Xtus = Îą + ÎłĎ&#x201E;t+ut2â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;,

(4)

Ď&#x201E;t = Îź + Ď&#x201E;t-1 + θtâ&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;,

(5)

Ď&#x2020;1(L)u = Îľ â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;,

(6)

Ď&#x2020;2(L)ut2 = Îľt2â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;,

(7)

1 t

1 t

where θt ~ NID(0,â&#x20AC;&#x2030;Ď&#x192;θ2â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;), Îľt1â&#x20AC;&#x2030;â&#x20AC;&#x2030;~ NID(0,â&#x20AC;&#x2030;Ď&#x192;12â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;), Îľt2â&#x20AC;&#x2030;â&#x20AC;&#x2030;~ NID(0,â&#x20AC;&#x2030;Ď&#x192;22â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;â&#x20AC;&#x2030;), Ď&#x201E;t is the common stochastic trend, whereas ut1 and ut2 are the transitory (cyclical) components and follow unobservable finite-order autoregressive AR(p) processes.

The parameter Îą represents a level shift.

Furthermore, the model allows the innovations of the unobserved components to be correlated. Morley et al.

(2003) and Morley (2007) show that, given sufficient autoregressive dynamics in the transitory components, an UC model with correlated innovations can be identified.

Data Natural gas is traded through spot or long-term contracts in the market at special locations, called hubs. Henry Hub (HH) is the most important natural gas hub in North America, whereas the National Balancing Point (NBP) is a national hub in the U.K. For the further analysis, the NBP and HH one-month ahead daily forward prices are used as a proxy for natural gas prices of the U.K. and U.S. gas markets respectively.

The examined period covers working days from January 2000 until February 2010 (2645 observations).

The data is split into two sample periods: the in-sample period (1825 observations), which is used to analyze and estimate the models of interest, and the out-of-sample period (820 observations), which is used to compare the models in terms of their forecasting performance.

To ensure the compatibility, the U.S. gas prices from US$/MBtu have been converted into pence/therm using the conversion factor (1 MBtu = 10 therms) and daily exchange rates acquired from the European Central Bank.

Numerical Results ADF Test with Deterministic Seasonality This section performs the ADF test for the null hypothesis that the U.K. and U.S. gas prices contain a unit root. The regression (1) is estimated. The T is equal to 260 and

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A N D E R H A L F U U R V O O R D E E I N D B E S P R E K I N G VA N D E J A A R R E K E N I N G VA N E E N G R O O T R E C L A M E B U R E A U

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

Econometrics

Table 1. The LR trace and maximum eigenvalue statistics together with the critical values.

H0

Table 3. Parameter estimates of CUCM-AR(2) model.

Standard errors reported in parentheses.

Eigenvalue Lrtrace 0.05 CVtrace LRmax 0.05 CVmax

r = 0

0.0097

23.87

19.96

17.82

15.67

r ≤ 0

0.0033

6.05

9.24

6.05

9.24

U.S

U.K

φ11 0.9188 (0.854) φ12 −0.0274 (0.947) σ1 0.0001 (0.001)

α 0.286 (0.137) γ 0.854 (0.050) φ21 1.011 (0.019) φ22 −0.0234 (0.019) σ2 0.049 (0.001)

Table 2. Normalized cointegrating vector and parameters of speed of adjustment.

U.S. Cointegrating vector

Speed of adjustment

U.K.

β1 1.000

constant β2 −0.928 −0.452 (−6.811) (−0.999) α1 α2 −0.0035 0.0122 (−1.21) (−3.821)

represents the number of working days in a year. Similarly as in Enders and Lee (2006), Δyt is regressed on α(t) for each integer frequency values in the interval (1,12) in order to find which frequency minimizes the corresponding sum of squared residuals. Frequency 1 represents the annual seasonal pattern, frequency 2 represents the semiannual seasonal pattern and frequency 12 represents the monthly seasonal pattern. The frequency k = 1 is chosen for the U.K. gas prices as it gives the smallest value of the sum of squared residuals. Analogously, the frequency of 6 is chosen for the U.S. gas prices.

The estimated ADF regressions provide evidence that the U.K. and U.S. natural gas prices contain one unit root.

Testing for the Rank of Cointegration Table 1 shows the results of the Johansen LR trace and the maximum eigenvalue statistics for the rank of cointegration.

The intercept is assumed to enter the cointegration relationship and the seasonal components are included in Dt as exogenous variables. Considering both statistics, the hypothesis of no cointegration can be rejected at 5% significance level. Furthermore, the hypothesis of at most one cointegration relationship cannot be rejected at 5% significance level, which indicates that two price series might be related in the long-run.

Table 2 shows the estimated (normalized) cointegrating vector and the corresponding parameters of speed of adjustment.

The β parameters represent the long-run coefficients and the α parameters represent the speed of adjustment to disequilibrium. In the case of the U.S. gas prices, the estimated speed of adjustment coefficient α1 is −0.0035 with t-statistic of −1.21, indicating that it is not significant.

Common Stochastic Trend

μ 0.0005 (0.001) σθ 0.038 (0.001)

Correlations

ρθ1 0.995 (0.001) ρθ2 −0.529 (0.039) ρ12 −0.531 (0.024)

In contrast, the corresponding coefficient α2 for the U.K. prices is 0.0122 and it is significant (the t-statistic is 3.821). The estimated coefficient is very low, indicating that the adjustment towards equilibrium is very slow.

Correlated Unobserved Components Model In the first step, the deseasonalized prices are constructed, while in the second step, the CUCM is estimated using a Kalman filter on the deseasonalized prices. It is assumed that the transitory (short-term) components follow an AR(2) process to ensure the identification of the model .

Table 3 reports the maximum likelihood estimates.

The autoregressive coefficients reported in Table 3 represent the dynamics of the transitory components.

In the case of the U.S. price series, the estimated autoregressive coefficients are statistically insignificant, suggesting that the U.S. prices do not have rich enough dynamics for periodic cycles. Furthermore, they exhibit almost zero variance σ1 and high correlation with the permanent component (equal to 0.99). These values support the possibility that the U.S. price series do not have transitory component. Thus, the U.S. gas prices represent a common factor, which is, actually, a random walk. Moreover, the estimated CUCM implies that the cointegration relationship of the U.S. and U.K. gas prices is (−0.854, 1) or, alternatively, (1, −1.17) if normalized with respect to the U.S. Model Comparison In order to compare the VECM and CUCM, the rootmean-square error (RMSE) is calculated for the insample fitted values. In short, there is little difference among the residuals, although VECM seems to provide slightly better fit to the observed data.

In addition, we calculate RMSE for the 1-step ahead,

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Econometrics

Table 4. RMSE for the forecasts with respect to forecasting steps r.

RMSE(r)

VECM

CUCM

1-step ahead

0.03979

0.04119

5-step ahead

0.07124

0.07240

60-step ahead

0.2022

0.2143

Johansen, S. and K. Juselius. “Maximum likelihood estimation and inference on cointegration - with application to the demand for money.” Oxford Bulletin of Economics and Statistics 52.2 (1990): 169–210.

5-steps ahead and 60-steps ahead forecasts. Table 4 represents the corresponding results, which indicate that the VECM slightly outperforms the CUCM.

Morley, J. “The slow adjustment of aggregate consumption to permanent income.” Journal of Money, Credit, and Banking 39 (2007): 615–638.

Valuation of Investments into Natural Gas Transport The valuation problem is carried out by Monte Carlo simulation for the following four models: mean-reverting (MR), random walk (RW) with the drift, VECM and CUCM. The experimental results show that the simple MR model provides the lowest expected value of gas transportation.

However, if prices are assumed to be integrated of order one, the standard MR model is inadequate. An alternative model, RW, could be used in that case. However, simple RW model with the neglected cointegration leads to a much wider distribution of the expected values and time fractions of option exercise. Consequently, the VECM and the CUCM can be used as reasonable alternatives to model mean-reverting prices, the mean being the stochastic long-run equilibrium level.

Morley, J., C. Nelson and C. Zivot. “Why are the Beveridge-Nelson and unobserved components decompositions of GDP so different?” Review of Economics and Statistics 85.2 (2003): 235–243.

Conclusion This research focuses on the dynamics of natural gas prices and extends the cointegration analysis between the U.K. and the U.S. gas markets. In addition, a comparison study of different models for gas prices is carried out through valuation of gas transportation.

In term of cointegration methods, the CUCM appears to be an interesting alternative model to VECM in cointegration analysis. The advantage of the VECM lies in its efficiency in estimating the model. Johansen cointegration test is implemented in most of the nowadays econometric software, while CUCM, however, requires Kalman filtering and the estimation procedure is sensitive to the given set of initial values. Nevertheless, it would be interesting to compare the methods not only in the bivariate, but also in a multivariate, framework.

References Dickey, D. and W. Fuller. “Likelihood ratio statistics for autoregressive time series with a unit root.”

Econometrica 46.4 (1981): 1057–1072.

Enders, W. and J. Lee. “Testing for a unit root with a nonlinear Fourier function.” Working papers University of Alabama (2006).

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Johansen, S. “Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.” Oxford University Press: Advanced Texts in Econometrics, 1995.

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vol. 18 (70)

December 2010

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 “Crossing the Bridge” The group of four can cross the bridge in two similar ways. First the two people who are the fastest cross the bridge with the torch. This takes 2 minutes. Then one of them goes back with the torch, let’s say the one who crosses the bridge in 1 minute. Then he hands over the torch to the two people who were left behind and they cross the bridge together in 10 minutes. Then the person who was already across the bridge goes back to pick up the last one which takes 2 times 2 minutes. All in all they crossed the bridge in 2 + 1 + 10 + 2 + 2 = 17 minutes.

Answer to “Marathon” Let’s say Pete’s starting number is s. The sum of the numbers which are lower than s is equal to s (s - 1) / 2 and the sum of the numbers which are higher is equal to n (n + 1) / 2 - s (s + 1) / 2. Now we need to find a solution to the equation s (s - 1) / 2 = n (n + 1) / 2 - s (s + 1) / 2 or equivalent s2 = n (n + 1) / 2. This means n (n + 1) / 2 is the square of an integer. Because n and n + 1 do not have common dividers, and one of them is an odd number which can not be divided by 2, we need to find an odd number that is a square of an integer and in addition, this odd number plus or minus 1 should be two times the square of an integer. Between 10 and 100, we have three odd squares, namely 25, 49 and 81. 25 and 81 are not the right answer because 12, 13, 40 and 41 are not a square of an integer. However, (49 + 1) / 2 = 25 which is a square of an integer. So n = 49 and s is the square root of 49 * 50 / 2 = 1225 which is 35.

the edge, is exactly 8% of the total number of pieces in the jigsaw. Determine the possible number of pieces the jigsaw could contain. Hint: there are three solutions.

Shopping Jack is going shopping at the supermarket. Every time he picks up a product, he types the price in on his calculator. However, in stead of pushing the ‘add’-button every time he picks up another product he accidentally uses the ‘multiply’-button. At the counter, where all the products are added up, he has to pay € 7.35. Jack does this without protesting, because his calculator shows the same price, i.e. € 7.35. Determine the prices of Jack’s products, knowing that he only bought three things and all the prices are above € 1.-.

Cucumbers A salesman has a pile of cucumbers which weights 200 pounds. These cucumbers consist for 99% of water. The salesman lays the cucumbers in the sun for the whole day.

At the end of the day a lot of water has evaporated and the pile of cucumbers consist for only 98% of water. How many pounds does the pile of cucumbers weight now?

Jigsaw Pete has a Jigsaw that is m by n pieces. It is given that the number of pieces on the border, including the pieces on

Solutions Solutions to the two puzzles above can be submitted up to March 1st 2010. You can hand them in at the VSAE room (E2.02/04), mail them to aenorm@vsae.nl or send them to VSAE, for the attention of Aenorm puzzle 70, Roeterstraat 11, 1018 WB Amsterdam, Holland. Among the correct submissions, one book token will be won.

Solutions can be both in English and Dutch.

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vol. 18 (70)

December 2010

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In the past two months there was a lot of activity in the VSAE. At the end of November we went with 52 students to Berlin for the weekend, which was a lot of fun. We visited the Reichstag, an icehockey game and on the last day a beer brewery.

In December we had the Actuarial Congress in a beautiful old church named ‘De Duif’ with a lot of interesting lectures about Longevity Risk including a panel discussion about this subject.

There was also a last drink in 2010 with Cocktail as dresscode for which both members as alumni were invited. Although the weather stopped a lot of people from coming, it had been snowing all day, it was still a great evening.

In addition, a new board was formed during the last General Members Meeting and they will be in charge starting from February 2011. Our year as a board is coming to an end. It was a great year with a lot of great events and activities and we sure had fun. But now it is time to give another board the opportunity to run the VSAE and fulfil their great plans for the association.

Agenda

Agenda •

29 January

• 10 February Monthly Drink

•

3 February

• 15 February National Econometricians Day (LED)

• 15 February National Econometricians Day (LED)

•

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The last months of 2010 were a great success for Kraket.

In November we had our annual ‘Casedag’ in the Victoria hotel. The day started with an interesting talk by the CPB about how they analyze policy proposals of political parties in the 2010 Dutch general elections. After that a mix of Econometric- and Math students participated in cases presented by a number of companies.

The “Sinterklaasborrel” was also a lot of fun. The board prepared poems and had presents for all the members.

At the time of writing, a group of our students is in Tokyo for our study trip. From 8 January to 22 January, sixteen of our best students and two Professors are in Japan to visit Japanese companies, universities and to get to know the culture.

In the following month we have planned a lot of activities to look forward to. The 29th of January we will visit the New Year’s Party hosted by Getronics Consulting.

The Winter Gala, which will be held in cooperation with Anquilla and Salus is coming up at the 3rd of February and of course there is the LED in Rotterdam on February the 15th.

The board hopes that 2011 will bring even more success for Kraket and we hope to see you at one of the upcoming activities.

February 8 General Members Meeting

•

17-18 February

•

12-14 April

Risk Intelligence Competition Econometric Game 2011

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vol. 18 (70)

December 2010

New Year’s party with Getronics Consulting ‘Winter Gala’

• 24 February General Members Meeting

LED 2011

Landelijke Econometristendag

Date:

February 15th

Location:

Beurs-WTC

Rotterdam

Subscribe at www.leditbeyourday.nl

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