ch05 by Salim Kasap

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

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.

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.

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%

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%

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

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.

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.

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.

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.

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.

Chapter 5 Portfolio theory

Figure p.68: Risk / Return Annual return

Annual risk

Risk premium

Sharpe ratio

Treasury

6.19%

‘0%’ (0.20%)

n/a

BigBroth er Inc.

12.18%

17.62%

5.99%

0.34

Spec LLP

15.04%

50.16%

8.85%

0.18

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%

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

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’

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.

Chapter 5 Portfolio theory

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

20 30 Number of stocks

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%

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%

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.

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.

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 )