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Finance and Derivatives: Theory and PracticeFinance and Derivatives: Theory and PracticeSébastien Bossu and Philippe Henrotte

Chapter 5Portfolio theory

2Sébastien 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

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:

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

1 1 2 21

N

k k N Nk

P p q p q p q p q



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.



2. Risk and return2.1. 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. 12.18% Treasury 6.19% Spec LLP 15.04%



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. -2.54% 0.55% -12.43%Dec. 3.77% 0.55% 19.53%

Average 1.08% 0.50% 2.11%



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



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.



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:

where is the average periodic return.

2periodic

1

annual periodic

1 ( )1

Number of periods per year

N

tt

r rN

r



2.1. Risk and return of an asset (4) Volatility for the three assets:

BigBrother Inc. 17.62% Treasury 0.20% Spec LLP 50.16%

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



2.2. Risk-free asset; Sharpe ratio The asset with zero volatility is called the risk-free

asset and its return is called the risk-free 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 risk-free asset, in which case the risk-free return would be 6.19%.

The risk-free 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 risk-free rate is called the risk premium of A.



2.2. Risk-free 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:

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.

PremiumRisk

A fA

A

r rSharpe



Figure p.68: Risk / Return Annual return Annual

riskRisk premium

Sharpe ratio

Treasury 6.19% ‘0%’ (0.20%) 0 n/a

BigBrother Inc. 12.18% 17.62% 5.99% 0.34

Spec LLP 15.04% 50.16% 8.85% 0.18



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:

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

= 11.79%

1 1 2 21

N

P k k N Nk

R w R w R w R w R



2.3. Risk and return of a portfolio (2) The portfolio volatility is NOT the weighted

average of volatilities: Example: σP = 15.22% (mark-to-market)

< 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:

1 1 2 2

2 2 2 21 1 2 2 1 2 1 2 1,2

sum of variances covariance term

1 1 2 2 1,2

( )

2

iff 1

P Var w R w R

w w w w

w w



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 risk-return profile

This effect is called ‘gains of diversification’



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.



Figure p.71: Diversification (Dow Jones EuroStoxx50 index)

15%

20%

25%

30%

35%

40%

0 10 20 30 40 50Number of stocks

Portfolio Risk



3. Gains of diversification; portfolio optimization (2) Example: risk-return profiles of various

portfolios invested in BigBrother Inc. and Spec LLP.Weight Spec LLP Risk Return

0% 17.62% 12.18% 10% 17.33% 12.47% 20% 18.49% 12.75% 30% 20.84% 13.04% 40% 24.04% 13.32% 50% 27.80% 13.61% 60% 31.92% 13.90% 70% 36.28% 14.18% 80% 40.81% 14.47% 90% 45.44% 14.75%

100% 50.16% 15.04%



Figure p.72: Risk-Return profiles

11.5%

12.5%

13.5%

14.5%

15.5%

15% 25% 35% 45% 55%Risk

BigBrother Inc.

Spec LLPReturn

Lower Risk and Higher Returnthan BigBrother Inc.



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.



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.



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:

Risk-free rate rf Market risk premium rM – rf Sensitivity of A to market movement βA

Specifically:( )A f A M fr r r r

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