Finance and Derivatives: Theory and Practice Sébastien Bossu and Philippe Henrotte

Chapter 5 Portfolio theory

Chapter 5 Portfolio theory

1. Summary of portfolio valuation 

Under the usual assumptions of absence of arbitrage  and infinite liquidity, the arbitrage price of a portfolio  of N assets is equal to the sum of asset prices pk  multiplied by their respective quantities qk: N

P = ∑ pk qk = p1q1 + p2 q2 + L pN q N k =1

This valuation method for portfolios is known as  ‘mark­to­market’. When it comes to buying or selling  the portfolio, the transaction should take place at that  price to be fair.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

1. Summary of portfolio valuation (2) 

Market agents  usually  have  the  right  to short  sell.  In  this case the portfolio quantities can be negative. In  practitioners’  jargon  a  positive  quantity  is  called  a  long  position  and  a  negative  quantity  a  short  position.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2. Risk and return 2.1. Risk and return of an asset 

Consider three assets: 

the stock  of  BigBrother  Inc.,  a  large  multinational  IT  company; a  Treasury  bond  issued  by  the  government  of  a  developed  country; and a share in the Spec LLP hedge fund.

With annual compound returns:   

BigBrother Inc. Treasury Spec LLP

12.18%  6.19% 15.04%

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

Figure p.67: Monthly returns  BigBrother Inc.

Treasury

Spec LLP

January

-3.01%

0.43%

-13.47%

February

1.31%

0.44%

18.30%

March

-2.87%

0.52%

8.55%

April

6.64%

0.47%

-18.45%

May

3.03%

0.61%

3.56%

June

7.32%

0.43%

26.75%

July

-4.86%

0.45%

-7.52%

August

-2.07%

0.52%

2.79%

Sept.

10.35%

0.53%

-8.19%

Oct.

-4.13%

0.52%

5.87%

Nov. Dec.

-2.54% 3.77%

0.55% 0.55%

-12.43% 19.53%

Average

1.08%

0.50%

2.11%

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.1. Risk and return of an asset (2) 

Based on  these  calculations  an  unsophisticated  investor would probably decide to put his fortune into  the asset that gives the highest return: Spec LLP. Such  an  investor  fails  to  think  over  why  company  stocks  or  hedge  funds  give  higher  returns  than  Treasury bonds. The answer is that these three assets do not carry  the same risk  Bonds

Stocks

Hedge Fund Risk

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.1. Risk and return of an asset (3) 

The intuitive  perception  by  rational  investors  of  the  risk level of an asset will typically be reflected by the  volatility of its returns. The higher the risk, the more  volatile the returns. Stock  returns  are  usually  more  volatile  than  bond  returns, which is consistent with the intuitive idea that  stocks are subject to many more economic risks than  bonds.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.1. Risk and return of an asset (3) 

This is  why  in  finance  risk  is  synonymous  with  volatility,  which  is  universally  measured  as  the  annualized standard deviation of asset returns:

σ periodic =

1 N 2 ( r − r ) ∑ t N − 1 t =1

σ annual = σ periodic × Number of periods per year

r where     is the average periodic return.  Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.1. Risk and return of an asset (4) 

Volatility for the three assets:   

BigBrother Inc. Treasury Spec LLP

17.62%  0.20% 50.16%

These numbers reflect the intuitive distribution of risk  levels between stocks, bonds and hedge funds.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.2. Risk­free asset; Sharpe ratio 

The asset  with  zero  volatility  is  called  the  risk­free  asset and its return is called the risk­free rate rf.  

In our previous example the Treasury bond, whose volatility of  0.20% is very close to zero, would be a suitable proxy for the  risk­free  asset,  in  which  case  the  risk­free  return  would  be  6.19%.

The risk­free  rate  is  the  minimum  return  an  investor  should expect from other risky assets. The difference rA  – rf  between the expected return of a  given  risky  asset  A  and  the  risk­free  rate  is  called  the  risk premium of A.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.2. Risk­free asset; Sharpe ratio (2) 

Investors should demand a higher risk premium when  the risk is higher. As such, the return performance of an asset must be  compared to the risk incurred. This is exactly what the Sharpe ratio does:

Premium rA − rf SharpeA = = Risk σA

The Sharpe  ratio  is  the  premium  per  unit  of  risk  incurred.  The  ratio  is  higher  if  the  risk  premium  is  higher and the risk is lower.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

Figure p.68: Risk / Return Annual return

Annual risk

Sharpe ratio

Treasury

6.19%

‘0%’ (0.20%)

0

n/a

BigBroth er Inc.

12.18%

17.62%

5.99%

0.34

Spec LLP

15.04%

50.16%

8.85%

0.18

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.3. Risk and return of a portfolio 

Let P be a portfolio of N assets in proportions w1, w2  …, wN summing to 100% with returns R1, R2 …, RN,  respectively. 

Example: 75% BigBrother Inc., 12.5% Treasury, 12.5% Spec LLP.

The portfolio return is then: N RP = ∑ wk Rk = w1 R1 + w2 R2 + L + wN RN k =1

Example: 75% x 12.18% + 12.5% x 6.19% + 12.5% x 15.04% = 11.79%

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.3. Risk and return of a portfolio (2) 

The portfolio volatility is NOT the weighted average  of volatilities: 

Example:

σP = 15.22% (mark­to­market)     < 75% x 17.62% + 12.5% x 0.20% + 12.5% x 50.16%

This is because the asset returns are correlated:  For 2 assets: σ P = Var ( w1 R1 + w2 R2 )

= w12σ 12 + w22σ 22 + 2w1w2σ 1σ 2 ρ1,2 1 44 2 4 43 1 44 2 4 43 sum of variances

< w1σ 1 + w2σ 2

covariance term

iff ρ1,2 < 1

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

2.3. Risk and return of a portfolio (3) 

Example portfolio:   

Return: 11.79% Risk: 15.22% Sharpe: (11.79% ­ 5.99%) / 15.22% = 0.37  

BigBrother Inc: 0.34 Spec LLP: 0.18

Here, correlation results in an improved risk­return  profile This effect is called ‘gains of diversification’

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

3. Gains of diversification; portfolio  optimization 

‘Diversification’ is the scholarly term for not putting  all of one’s eggs in one basket.  By investing in an equally weighted portfolio of 10  assets rather than a single one, we can dramatically  reduce our risk.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

Figure p.71: Diversification (Dow Jones  EuroStoxx50 index) Portfolio Risk 40% 35% 30% 25% 20% 15% 0

10

20 30 Number of stocks

40

50

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

3. Gains of diversification; portfolio  optimization (2) 

Example: risk­return profiles of various portfolios  invested in BigBrother Inc. and Spec LLP. Weight Spec LLP

Risk

Return

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

17.62% 17.33% 18.49% 20.84% 24.04% 27.80% 31.92% 36.28% 40.81% 45.44% 50.16%

12.18% 12.47% 12.75% 13.04% 13.32% 13.61% 13.90% 14.18% 14.47% 14.75% 15.04%

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

Figure p.72: Risk­Return profiles  Return

Spec LLP

15.5% 14.5%

Lower Risk and Higher Return than BigBrother Inc.

13.5% 12.5%

BigBrother Inc.

11.5% 15%

25%

35%

45%

55%

Risk Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

4. The Capital Asset Pricing Model  (CAPM) 

2 fundamental principles of portfolio theory:  

Higher risk implies higher expected return. More diversification implies lower risk.

These two principles are not always consistent: 

rA = rB = 10% ;

rP = w rA + (1 – w)rB = 10% for all w

 

But there exists an optimal weight w* minimizing the risk. Thus, the first principle seems to be violated: the risk of A or  B is higher than the risk of P but it is mathematically  impossible to get compensation by higher returns.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

4. The Capital Asset Pricing Model (2) 

To resolve  this  paradox,  the  Capital  Asset  Pricing  Model  proposes a distinction between two types of risk: Market risk (or systematic risk):   

Common to all risky assets Reflects general market trends Cannot be eliminated by diversification and must be rewarded with  higher returns.

Specific risk (or idiosyncratic risk):  

Specific to each asset Corresponds  to  price  fluctuations  stemming  from  the  asset’s  own  characteristics Can  be  eliminated  by  diversification  and  therefore  is  generally  not  rewarded by the market.

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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Chapter 5 Portfolio theory

4. The Capital Asset Pricing Model (3) 

With further  assumptions,  the  conclusion  of  the  CAPM is that the expected return of an asset A is the  function of only 3 parameters: 

Risk­free rate rf

Market risk premium rM – rf

Sensitivity of A to market movement βA

Specifically:

rA = rf + β A (rM − rf )

Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

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ch05

Finance and Derivatives: Theory and Practice Finance and Derivatives: Theory and Practice Sébastien Bossu and Philippe Henrotte