1. Théorie Financière Risk and expected returns (2)
Professeur André Farber

2. Risk and return Objectives for this session: 1. Review: 2 risky assets
2. Many risky assets
3. Beta
4. Optimal portfolio
5. Equilibrium: CAPM
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3. Review: The efficient set for two assets
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4. Formulas Returns: normal distribution Expected return: Variance:
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5. Choosing portfolios from many stocks
Porfolio composition :
(X1, X2, ... , Xi, ... , XN)
X1 + X Xi XN = 1
Expected return:
Risk:
Note:
N terms for variances
N(N-1) terms for covariances
Covariances dominate
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6. Using matrices
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7. Some intuition
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8. Example
Consider the risk of an equally weighted portfolio of N "identical«  stocks:
Equally weighted:
Variance of portfolio:
If we increase the number of securities ?:
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9. Diversification
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10. Unsystematic or diversifiable risk
Conclusion
1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated
2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks
The variance of a security's return can be broken down in the following way:
The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio:
Portfolio risk
Total risk of individual security
Unsystematic or diversifiable risk
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11. The efficient set for many securities
Portfolio choice: choose an efficient portfolio
Efficient portfolios maximise expected return for a given risk
They are located on the upper boundary of the shaded region (each point in this region correspond to a given portfolio)
Expected Return           
Risk
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12. Optimal portofolio with borrowing and lending
Optimal portfolio: maximize Sharpe ratio
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13. Capital asset pricing model (CAPM)
Sharpe (1964) Lintner (1965)
Assumptions
Perfect capital markets
Homogeneous expectations
Main conclusions: Everyone picks the same optimal portfolio
Main implications:
1. M is the market portfolio : a market value weighted portfolio of all stocks
2. The risk of a security is the beta of the security:
Beta measures the sensitivity of the return of an individual security to the return of the market portfolio
The average beta across all securities, weighted by the proportion of each security's market value to that of the market is 1
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14. Market equilibrium: illustration
Wealth
Risk free asset
Market
Portfolio
Firm 1
Firm 2
Firm 3
Optimal portfolio
100%
20%
50%
30%
Alan 10
-10 20 4 6
Ben -5 25 5
12.5
7.5
Clara 30 15 3
4.5 60 12 18
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15. Capital Asset Pricing Model
Expected return RM Rj
Risk free interest rate βj 1
Beta
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16. Beta
Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

17. Measuring the risk of an individual asset
The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification.
The standard deviation is not an correct measure for the risk of an individual security in a portfolio.
The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification.
Remember: the optimal portfolio is the market portfolio.
The risk of an individual asset is measured by beta.
The definition of beta is:
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18. Beta Several interpretations of beta are possible:
(1) Beta is the responsiveness coefficient of Ri to the market
(2) Beta is the relative contribution of stock i to the variance of the market portfolio
(3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified
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19. Beta as a slope
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20. A measure of systematic risk : beta
Consider the following linear model
Rt Realized return on a security during period t
 A constant : a return that the stock will realize in any period
RMt Realized return on the market as a whole during period t
 A measure of the response of the return on the security to the return on the market
ut A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0
Partition of yearly return into:
Market related part ß RMt
Company specific part a + ut
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21. Beta - illustration Suppose Rt = 2% + 1.2 RMt + ut If RMt = 10%
The expected return on the security given the return on the market
E[Rt |RMt] = 2% x 10% = 14%
If Rt = 17%, ut = 17%-14% = 3%
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22. Measuring Beta Data: past returns for the security and for the market
Do linear regression : slope of regression = estimated beta
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23. Decomposing of the variance of a portfolio
How much does each asset contribute to the risk of a portfolio?
The variance of the portfolio with 2 risky assets
can be written as
The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio.
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24. Example
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25. Beta and the decomposition of the variance
The variance of the market portfolio can be expressed as:
To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio
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26. Marginal contribution to risk: some math
Consider portfolio M. What happens if the fraction invested in stock I changes?
Consider a fraction X invested in stock i
Take first derivative with respect to X for X = 0
Risk of portfolio increase if and only if:
The marginal contribution of stock i to the risk is
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27. Marginal contribution to risk: illustration
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28. Beta and marginal contribution to risk
Increase (sightly) the weight of i:
The risk of the portfolio increases if:
The risk of the portfolio is unchanged if:
The risk of the portfolio decreases if:
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29. Inside beta
Remember the relationship between the correlation coefficient and the covariance:
Beta can be written as:
Two determinants of beta
the correlation of the security return with the market
the volatility of the security relative to the volatility of the market
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30. Properties of beta Two importants properties of beta to remember
(1) The weighted average beta across all securities is 1
(2) The beta of a portfolio is the weighted average beta of the securities
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31. the excess market return (the market risk premium)
Risk premium and beta
3. The expected return on a security is positively related to its beta
Capital-Asset Pricing Model (CAPM) :
The expected return on a security equals:
the risk-free rate
plus
the excess market return (the market risk premium)
times
Beta of the security
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32. CAPM - Illustration
Expected Return 1
Beta
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33. CAPM - Example Assume: Risk-free rate = 6% Market risk premium = 8.5%
Beta Expected Return (%)
American Express
BankAmerica
Chrysler
Digital Equipement
Walt Disney
Du Pont
AT&T
General Mills
Gillette
Southern California Edison
Gold Bullion
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34. CAPM – two formulations
Consider a future uncertain cash flow C to be received in 1 year.
PV calculation based on CAPM:
The standard approach to calculate the present value of a risky future cash flow is to discount the expected cash flow at a risk-adjusted discount rate.
But that there another method. The expected cash flows are adjusted to obtain certainty equivalents. These certainty equivalent cash flows are discounted at the risk-free interest rate.
Here is an example.
You want to calculate the present value of an expected cash flow of 100 to be received in one year. The standard deviation of this cash flow is 20. The correlation with the market portfolio is 0.5. The risk-free interest rate is 5%, the risk premium on the market portfolio is 8% and the standard deviation of the market is 20%. Lambda, the price of covariance risk is .08 / (.20)² = 2
cov(C,rM) = Correlation(C,rM) σC σM = (0.5)(20)(.20) = 2
The certainty equivalent cash flow is CEQ = E(C) – λ cov(C,rM) = 100 – 2 × 2 = 96
V = 96 / 1.05 = 91.43
See Brealey and Myers 9.6 for additional details.
See Brealey and Myers Chap 9
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35. Risk-adjusted expected cash flow
Using risk-adjusted discount rates is OK if you know beta.
The adjusted risk-adjusted discount rate does not work for OPTIONS or projects with unknown betas.
To understand how to proceed in that case, we need to go deeper into valuation theory.
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36. Example (see Introduction)
You observe the following data:
Value
Up market (u)
Proba = 0.75
Down market (d)
Proba = 0.25
Expected return
Bond
95.24
105 5%
Market Portfolio
100
1.2
0.80
10%
What is the value of the following asset? What are its expected returns?
We now want to show that the uncertainty equivalent can be interpreted as an expected value in a risk neutral world.
Do to this, we will represent uncertainty by states of the world. In our presentation we limit ourselves to a story with 2 states (boom and recession) and two assets: a bond and a stock whose market prices can be observed.
We now wish to price a new asset. How to proceed?
NewAsset ?
200
100
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37. Valuation of project with CAPM
Step 1: calculate statistics for the market portfolio:
Up mkt Proba = .75
Down mkt Proba = .25
Return
20%
-20%
Expected return:
Market risk premium:
Variance:
Price of covariance:
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38. Valuation of project with CAPM (2)
Step 2: Calculate statistics for the project
Expected cash flow:
Covariance with market portfolio:
(Reminder: )
Step 3: Value the project
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39. Valuation of project with CAPM (3)
Once the value of the project is known, the beta can be calculated.
Value
Up mkt Proba = .75
Down mkt Proba = .25
Cash flow
154.76
200
100
Returns
29.23%
-35.38%
Expected return:
Beta:
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