Download X - ULB

Corporate Finance
Risk and Return
Prof. André Farber
SOLVAY BUSINESS SCHOOL
UNIVERSITÉ LIBRE DE BRUXELLES
Risk and return
• Objectives for this session:
• 1. Review
• 2. Efficient set
• 3. Optimal portfolio
• 4. CAPM
A.Farber Vietnam 2004
|2
Review : Risk and expected returns for
porfolios
• In order to better understand the driving force explaining the benefits
from diversification, let us consider a portfolio of two stocks (A,B)
• Characteristics:
– Expected returns :
– Standard deviations :
– Covariance :
RA , RB
 A , B
 AB   AB A B
• Portfolio: defined by fractions invested in each stock XA , XB
XA+ XB= 1
• Expected return on portfolio:
• Variance of the portfolio's return:
RP  X A RA  X B RB
 P2  X A2 A2  2 X A X B AB  X B2 B2
A.Farber Vietnam 2004
|3
The efficient set for two assets: correlation
=0
A.Farber Vietnam 2004
|4
Example
Riskless rate
A
B
Correlation
Exp.Return
5
15
20
0
Prop.
0.50
0.50
A
B
Sigma
0
20
30
Variance
0
400
900
Variance-covariance
400
0
0
900
Cov(Ri,Rp)
X
200.00
0.50
Variance
S t.de v.
Exp.Ret. Rp
450.00
0.50
325.00
18.03
17.50
A.Farber Vietnam 2004
|5
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
 P2  (1  X ) 2  M2  2 X (1  X ) iM  X 2 i2
• Take first derivative with respect to X for X = 0
d P2
 2( iM   M2 )
dX X 0
• Risk of portfolio increase if and only if:
 iM  
2
M
• The marginal contribution of stock i to the risk is
 iM
A.Farber Vietnam 2004
|6
Marginal contribution to risk: illustration
35.00
30.00
Risk of portfolio
25.00
20.00
15.00
10.00
5.00
0.00
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Fraction in B
Cor = 0
Cor = 0.25
Cor = 0.50
Cor = 0.75
Cor = 1.0
A.Farber Vietnam 2004
|7
1
Choosing portfolios from many stocks
• Porfolio composition :
 (X1, X2, ... , Xi, ... , XN)
 X1 + X2 + ... + Xi + ... + XN = 1
RP  X 1 R1  X 2 R2  ...  X N RN
• Expected return:
 P2   X 2j  2j   X i X j ij   X i X j ij
• Risk:
j
i
j i
i
j
• Note:
 N terms for variances
 N(N-1) terms for covariances
 Covariances dominate
A.Farber Vietnam 2004
|8
Some intuition
Var
Cov
Cov
Cov
Cov
Cov
Var
Cov
Cov
Cov
Cov
Cov
Var
Cov
Cov
Cov
Cov
Cov
Var
Cov
Cov
Cov
Cov
Cov
Var
A.Farber Vietnam 2004
|9
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
A.Farber Vietnam 2004
|10
Choosing between 2 risky assets
•
Choose the asset with the highest
ratio of excess expected return to
risk:
Sharpe ratio 
•
Exp.Return
Expected return
Ri  RF
B
i
A
Example: RF = 6%
•
A
Risk
•
A
9%
10%
•
B
15%
20%
•
Asset Sharpe ratio
•
A (9-6)/10 = 0.30
•
B (15-6)/20 = 0.45 **
Risk
A.Farber Vietnam 2004
|11
Optimal portofolio with borrowing and
lending
Optimal portfolio:
maximize Sharpe ratio
A.Farber Vietnam 2004
|12
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
A.Farber Vietnam 2004
|13
Optimal portfolio: property
RM  R F
Slope =
M
M
xj
RF
Slope =
R j  RM

2
jM   M
 M
A.Farber Vietnam 2004
|14
Risk premium and beta
• 3. The expected return on a security is positively related to its beta
• Capital-Asset Pricing Model (CAPM) :
R  R F  ( RM  R F )  
• 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
A.Farber Vietnam 2004
|15
CAPM - Illustration
Expected Return
RM
RF
1
Beta
A.Farber Vietnam 2004
|16
CAPM - Example
•
Assume:
Risk-free rate = 6%
•
Market risk premium = 8.5%
Beta
Expected Return (%)
•
American Express
1.5
18.75
•
BankAmerica
1.4
17.9
•
Chrysler
1.4
17.9
•
Digital Equipement
1.1
15.35
•
Walt Disney
0.9
13.65
•
Du Pont
1.0
14.5
•
AT&T
0.76
12.46
•
General Mills
0.5
10.25
•
Gillette
0.6
11.1
•
Southern California Edison 0.5
10.25
•
Gold Bullion
5.40
-0.07
A.Farber Vietnam 2004
|17
Pratical implications
• Efficient market hypothesis + CAPM: passive investment
 Buy index fund
 Choose asset allocation
A.Farber Vietnam 2004
|18