FINANCE
10. Risk and expected returns
Professor André Farber
Solvay Business School
Université Libre de Bruxelles
Fall 2006
Measuring the risk of an individual asset
• La mesure du risque d’un titre dans un portefeuille doit tenir compte de
l’impact de la diversification.
• L’écart type n’est donc pas la bonne mesure.
• Le risque se mesure par la contribution du titre au risque du portefeuille.
• Remember: the optimal portfolio is the market portfolio.
• The risk of an individual asset is measured by beta.
• The definition of beta is:
i 
Cov( Ri , RM )
 ( RM )
2
 iM
 2
M
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Beta
• Plusieurs interprétations du beta:
• (1) Beta mesure la sensibilité de Ri par rapport au marché
• (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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Beta as a slope
30
20, 27.5
25
15, 25
20
15
15, 15
Return on asset
10
Slope = Beta = 1.5
5
0
-15
-10
-5
0
-5, -5
5
10
15
20
25
-5
-10
-5, -15
-15
-10, -17.5
-20
Return on market
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A measure of systematic risk : beta
• Consider the following linear model
Rt      RMt  ut
•
•
•
•
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
– Company specific part
ß RMt
 + ut
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Measuring Beta
• Data: past returns for the security and for the market
• Do linear regression : slope of regression = estimated beta
C
B
A
1 Beta Calculation - monthly data
Market
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Mean
StDev
Correl
R²
Beta
Intercept
Data
Date
1
2
3
4
5
6
7
8
9
10
11
12
A
E
D
G
H
I
B
1
0
0.00%
4.33%
78.19%
61.13%
0.63
-1.32%
4.55%
10.46%
71.54%
51.18%
1.40
1.64%
Rm
5.68%
-4.07%
3.77%
5.22%
4.25%
0.98%
1.09%
-6.50%
-4.19%
5.07%
13.08%
0.62%
RA
0.81%
-4.46%
-1.85%
-1.94%
3.49%
3.44%
-4.27%
-2.70%
-4.29%
3.75%
9.71%
-1.67%
RB
20.43%
-7.03%
-10.14%
6.91%
4.65%
7.64%
8.41%
-1.25%
-11.19%
13.18%
19.22%
3.77%
2.08%
5.36%
F
D3. =AVERAGE(D12:D23)
D4. =STDEV(D12:D23)
D5. =CORREL(D12:D23,$B$12:$B$23)
D6. =D5^2
D7. =SLOPE(D12:D23,$B$12:$B$23)
D8. =INTERCEPT(D12:D23,$B$12:$B$23)
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Beta and the decomposition of the variance
• The variance of the market portfolio can be expressed as:
2
M
 X11M  X 2 2M  ...  X i iM  ...  X n nM
• To calculate the contribution of each security to the overall risk, divide
each term by the variance of the portfolio
 iM
 nM
 1M
 2M
X 1 2  X 2 2  ...  X i 2  ...  X n 2  1
M
M
M
M
or
X 11M  X 2  2 M  ...  X i  iM  ...  X n  nM  1
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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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Inside beta
• Remember the relationship between the correlation coefficient and the
covariance:
 iM
 iM 
 i M
• Beta can be written as:
 iM
 iM
i
 2   iM
M
M
• 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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Properties of beta
• Two importants properties of beta to remember
• (1) The weighted average beta across all securities is 1
X 11M  X 2  2 M  ...  X i  iM  ...  X n  nM  1
• (2) The beta of a portfolio is the weighted average beta of the securities
 P  X 1P 1M  X 2 P  2M  ...  X iP  iM  ...  X nP  nM
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Risk premium and beta
• 3. The expected return on a security is positively related to its beta
• Capital-Asset Pricing Model (CAPM) :
R  RF  ( RM  RF )  
• 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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CAPM - Illustration
Expected Return
RM
RF
1
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Beta
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CAPM - Example
•
•
•
•
•
•
•
•
•
•
•
•
•
Assume:
Risk-free rate = 6%
Beta
American Express
1.5
BankAmerica
1.4
Chrysler
1.4
Digital Equipement
1.1
Walt Disney
Du Pont
1.0
AT&T
0.76
General Mills
0.5
Gillette
0.6
Southern California Edison 0.5
Gold Bullion
-0.07
Market risk premium = 8.5%
Expected Return (%)
18.75
17.9
17.9
15.35
0.9
13.65
14.5
12.46
10.25
11.1
10.25
5.40
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Pratical implications
• Efficient market hypothesis + CAPM: passive investment
• Buy index fund
• Choose asset allocation
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Arbitrage Pricing Model
Professeur André Farber
Market Model
• Consider one factor model for stock returns:
Rj  Rj   j F  j
• Rj = realized return on stock j
• R j= expected return on stock j
• F = factor – a random variable E(F) = 0
• εj = unexpected return on stock j – a random variable
• E(εj) = 0
Mean 0
• cov(εj ,F) = 0 Uncorrelated with common factor
• cov(εj ,εk) = 0 Not correlated with other stocks
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Diversification
• Suppose there exist many stocks with the same βj.
• Build a diversified portfolio of such stocks.
R j Rj   jF
• The only remaining source of risk is the common factor.
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Created riskless portfolio
• Combine two diversified portfolio i and j.
• Weights: xi and xj with xi+xj =1
• Return:
R x R x R
P
i
i
j
j
 ( xi Ri  x j R j )  ( xi  i  x j  j ) F
• Eliminate the impact of common factor  riskless portfolio
xi i  xi  j  0
• Solution:
xi 
 j
i   j
xj 
i
i   j
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Equilibrium
• No arbitrage condition:
• The expected return on a riskless portfolio is equal to the risk-free rate.
 j
i   j
Ri 
i
i   j
R j  RF
At equilibrium:
Ri  RF
i

R j  RF
j
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
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Risk – expected return relation
Linear relation between expected return and beta
R j  RF  
j
For market portfolio, β = 1
RM  RF  
Back to CAPM formula:
R j  RF  ( RM  RF ) j
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