Return and Risk: The Capital Asset Pricing Model
(Text reference: Chapter 10)
Topics
general notation
single security statistics
covariance and correlation
return and risk for a portfolio
diversification
efficient set with two assets
diversification with many assets
efficient set with many assets
the capital market line
the capital asset pricing model
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 1
General Notation
E(R j ) = expected return on a security/portfolio j
σ 2j = variance of some security/portfolio j
σAB = covariance between two variables A and B
ρAB = correlation between two variables A and B
Ω = the number of possible future states of the economy
ωi = a possible future state of the economy, i ∈ {1, . . . , Ω}
pi = probability of occurrence of state ωi , i ∈ {1, . . . , Ω}
Ri, j = return of security j in state ωi , i ∈ {1, . . . , Ω}
Rt, j = return of security j in period t, t ∈ {1, . . . , T }
β j = beta of some security/portfolio j
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 2
Single Security Statistics
expected return
historical sample: R̄ j =
1
T
T
Rt, j
∑t=1
population: E(R j ) = ∑Ω
i=1 pi × Ri, j
example:
ωi
pi
RA - auto stock
RB - gold stock
recession
0.25
20%
normal
0.50
−8%
5%
3%
boom
0.25
18%
−20%
calculate the expected return for each stock:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 3
Cont’d
variance and standard deviation
T
1
(Rt, j − R¯ j )2
historical sample: σ 2j = T −1
∑t=1
2
population: σ 2j = ∑Ω
i=1 pi × (Ri, j − E(R j ))
standard deviation: s.d.(R j ) = σ j
a measure of risk: a probability weighted average of
squared deviations of a security’s return from its
expected return
calculate the variances and standard deviations for A
and B
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 4
Covariance and Correlation
variance (s.d.) measures variability of a single variable
(e.g. stock)
correlation and covariance measure the statistical
relationship between two variables (e.g. 2 stocks)
covariance (σAB ) - direction of the relationship
correlation (ρAB ) - strength and direction of
relationship (−1 ≤ ρAB ≤ 1)
examples:
negative covariance: interest rates and bond prices,
housing starts and interest rates, exchange rates
and exports, etc.
positive covariance: profits and stock prices,
exchange rates and imports, dividends and stock
prices, etc.
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 5
Cont’d
positive correlation/covariance
two variables X and Y tend to move together
when X is above (below) its mean, Y tends to be above (below) its mean
Perfect Positive Correlation
120
Perfect Positive Correlation
90
110
85
100
80
90
75
Y
Value
X
Y
80
70
70
65
60
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
60
90
1
Correlation = 0.85
120
95
100
105
X
110
115
120
110
115
120
Correlation = 0.85
90
85
100
80
90
75
Y
Value
X
Y
110
80
70
70
65
60
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
AFM 271 - Return and Risk: The Capital Asset Pricing Model
0.7
0.8
0.9
1
60
90
95
100
105
X
Slide 6
Cont’d
negative correlation/covariance
two variables X and Y tend to move in opposite directions
when X is above (below) its mean, Y tends to be below (above) its mean
Perfect Negative Correlation
120
Perfect Negative Correlation
90
110
85
100
80
90
75
Y
Value
X
Y
80
70
70
65
60
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
60
90
1
Correlation = −0.85
120
95
100
105
X
110
115
120
110
115
120
Correlation = −0.85
90
85
100
80
90
75
Y
Value
X
Y
110
80
70
70
65
60
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
0.7
0.8
0.9
1
60
90
95
100
105
X
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 7
Cont’d
zero correlation/covariance
two variables X and Y are not (linearly) related
Zero Correlation
120
Zero Correlation
90
110
85
100
80
90
75
Y
Value
X
Y
80
70
70
65
60
0
0.1
0.2
0.3
0.4
0.5
Time
0.6
AFM 271 - Return and Risk: The Capital Asset Pricing Model
0.7
0.8
0.9
1
60
90
95
100
105
X
110
115
120
Slide 8
Cont’d
covariance formulas
historical sample:
1 T
σAB = cov(RA , RB ) =
∑ [(Rt,A − R̄A ) × (Rt,B − R̄B)]
T − 1 t=1
population:
Ω
σAB = cov(RA , RB ) = ∑ [pi × (Ri,A − E(RA )) × (Ri,B − E(RB ))]
i=1
calculate the covariance between stock A and stock B:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 9
Cont’d
correlation formula
historical sample/population:
ρAB = corr(RA , RB ) =
σAB
cov(RA , RB )
=
σA × σ B
σA × σ B
calculate the correlation between stock A and stock B:
graphically:
RA
Ri, j
20%
20%
10%
10%
0%
R
N
B
state
−10%
−20%−10%
−10%
−20%
−20%
AFM 271 - Return and Risk: The Capital Asset Pricing Model
10% 20%
RB
Slide 10
Cont’d
correlation and covariance - further observations
σAB = σBA
ρAB = ρBA
even if two variables are actually uncorrelated, a
historical sample will not yield a zero covariance due
to measurement errors, sampling error, etc.
covariance σAB can take any value, and is in squared
deviation units
indicates direction of (linear) relationship
correlation coefficient ρAB can take on values
between -1 and 1 (inclusive)
indicates direction and strength of (linear)
relationship
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 11
Return and Risk for a Portfolio
let Y, Z be random variables, and a, b be real numbers, recall
E(aY + bZ) = aE(Y ) + bE(Z)
var(aY + bZ) = a2 var(Y ) + b2 var(Z) + 2abcov(Y, Z)
the expected return of a portfolio P is a weighted average of the
expected returns of the individual securities:
E(RP ) =
N
∑ X j E(R j )
j=1
where N is the number of securities in P, X j is the percentage
of funds invested in security j (∑Nj=1 X j = 1), and E(R j ) is the
expected return of security j
e.g. find the expected return on a portfolio consisting of 75% in
stock A and 25% in stock B:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 12
Cont’d
to see why the E(RP ) formula holds, consider the N = 2 case
and let ni be the number of shares held of stock i (i = 1, 2)
letting Si (t) be the price of stock i at time t, the portfolio value at
time t = 0 is V (0) = n1 S1 (0) + n2 S2 (0)
note that X1 = n1 S1 (0)/V (0), X2 = n2 S2 (0)/V (0)
since Si (1) = Si (0)(1 + Ri ), we have
V (1) = n1 S1 (1) + n2 S2 (1) = n1 S1 (0)(1 + R1 ) + n2 S2 (0)(1 + R2 )
⇒
V (1)
n1 S1 (0)(1 + R1 ) + n2 S2 (0)(1 + R2 )
− 1 = Rp =
−1
V (0)
n1 S1 (0) + n2 S2 (0)
= X1 (1 + R1 ) + X2 (1 + R2 ) − 1
= X1 R1 + X2 R2
⇒ E(R p ) = X1 E(R1 ) + X2 E(R2 )
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 13
Cont’d
variance of a portfolio P: σP2 = ∑Nj=1 ∑Nk=1 X j Xk σ jk where j
and k each represent a security, σ jk = cov(R j , Rk ), and
σ j j = σ 2j = var(R j )
for N = 2
σP2
2
=
∑ ∑ X j Xk σ jk
j=1 k=1
2
=
2
∑
j=1
X j X1 σ j1 + X j X2 σ j2
= X1 X1 σ11 + X1 X2 σ12 + X2 X1 σ21 + X2 X2 σ22
= X12 σ12 + X22 σ22 + 2X1 X2 σ12
= X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2 ρ12
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 14
Cont’d
s.d. of a portfolio P: σP =
√
varP
e.g. find the variance and standard deviation of a
portfolio 75% in stock A and 25% in stock B
observations:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 15
Diversification
combining stocks into a portfolio reduces risk: how does this
happen?
claim: as long as ρAB < 1, we get diversification (i.e. the s.d. of
the portfolio of 2 securities is less than the weighted average of
the s.d.’s of the individual securities). Proof:
σP2 = X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2 ρ12
< X12 σ12 + X22 σ22 + 2X1 X2 σ1 σ2
(if ρ12 < 1)
= (X1 σ1 + X2 σ2 )2
⇒ σP < X1 σ1 + X2 σ2
thus portfolio risk is not a weighted average of the risks of the
stocks in the portfolio; we get diversification whenever
correlation is less than +1
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 16
Efficient Set With Two Assets
suppose E(RA ) = 0.20, E(RB ) = 0.15, σA = 0.3098, σB = 0.0775, ρAB = −0.5
some possible risk/return combinations:
XA
0.0
0.2
0.4
0.6
0.8
1.0
E(RP )
0.15
0.16
0.17
0.18
0.19
0.20
σP
0.0775
0.0620
0.1084
0.1725
0.2405
0.3098
Feasible set with −0.5 correlation
0.22
0.21
0.2
E(Rp)
0.19
0.18
0.17
0.16
0.15
0.14
0.13
0
0.05
0.1
0.15
s.d.
0.2
0.25
0.3
0.35
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 17
Cont’d
opportunity set: (a.k.a. feasible set) set of all attainable
portfolios (attained via various combinations of two
stocks)
domination: P1 dominates P2 if
E(RP1 ) = E(RP2 ) and σP1 < σP2 , or
σP1 = σP2 and E(RP1 ) > E(RP2 )
efficient set: set of attainable portfolios which result in
maximum expected return for a given s.d. (or
alternatively, minimum s.d. for a given expected return)
note that the efficient set does not contain any portfolios
which are dominated
minimum variance portfolio: portfolio having the lowest
s.d. of all the portfolios in the feasible set
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 18
Cont’d
finding the minimum variance portfolio:
σP2 = XA2 σA2 + (1 − XA )2 σB2 + 2XA (1 − XA )σAB
this is a differentiable concave function, so a minimum
exists. To find it, differentiate w.r.t. XA and set the result to
zero:
so
d σP2
= 2XA σA2 + 2(1 − XA )(−1)σB2 + 2(1 − 2XA )σAB
dXA
= 2 XA σA2 − (1 − XA )σB2 + (1 − 2XA )σAB
= 2 XA (σA2 + σB2 − 2σAB ) + (σAB − σB2 )
σ 2 − σAB
d σP2
= 0 ⇒ XA∗ = 2 B 2
dXA
σA + σB − 2σAB
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 19
Cont’d
what happens for different values of ρAB ?
Portfolio Standard Deviation σP
XA
E(RP )
ρAB = 1.0
ρAB = 0.5
ρAB = 0.0
0.0
15%
0.0775
0.0775
0.0775
0.2
16%
0.1240
0.1074
0.4
17%
0.1704
0.6
18%
0.8
19%
1.0
20%
ρAB = −0.5
ρAB = −1.0
0.0877
0.0620
0.0000
0.1526
0.1324
0.1084
0.0774
0.2169
0.2032
0.1884
0.1725
0.1549
0.2633
0.2559
0.2483
0.2405
0.2323
0.3098
0.3098
0.3098
0.3098
0.3098
0.0775
0.0775
Feasible set with varying correlations
0.21
0.2
0.19
E(Rp)
0.18
0.17
0.16
0.15
0
AFM 271 - Return and Risk: The Capital Asset Pricing Model
0.05
0.1
0.15
s.d.
0.2
0.25
0.3
Slide 20
Cont’d
observations:
lower ρAB :
more bend in the curve
lower s.d. (see table)
greater diversification
if ρAB = −1, we can find a risk free portfolio (i.e.
having σP = 0)
recall that this is extremely unlikely with two
stocks, but can happen in a portfolio consisting of
a stock and a derivative (e.g. stock and a put on
that stock) over short periods
in reality, only one curve is possible, since ρAB is
unique; other curves in our graph are hypothetical
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 21
Diversification With Many Assets
the risk an asset adds to a portfolio must be measured with
reference to its relationship to other securities in the portfolio
the variance of the return on a portfolio with many securities is
more dependent on the covariances between the individual
securities than on the variances of the individual securities
proof:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 22
Cont’d
proof cont’d:
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 23
Cont’d
σP2 = total risk = diversifiable risk + undiversifiable risk
σP2
N
about 30 stocks required for optimal diversification
diversifiable risk: risk relating to uncorrelated events
that gets eliminated when we hold many securities
undiversifiable risk: risk relating to correlated events
that is not eliminated by owning many securities
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 24
Efficient Set With Many Assets
feasible set becomes an area
efficient frontier is still a curve
E(RP )
σP
how do we find the efficient frontier?
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 25
Summarizing to Here
investors are risk-averse; try to increase expected return and
reduce risk of their portfolios
investors can reduce risk by choosing stocks which are not
perfectly correlated
the risk added by a security to a portfolio has to be measured
with reference to its relationship to other securities in the
portfolio (variances and covariances determine risk)
a rational risk-averse investor chooses investments that are
efficient
the optimal investment lies on the efficient frontier
investors are not compensated for bearing diversifiable risk,
and only undiversifiable risk is priced in the market
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 26
The Capital Market Line
introduce risk-free borrowing/lending asset f with return R f :
σ f = 0 by definition
σA f = 0 by definition for any risky asset A
portfolio combinations of risky portfolio B and risk free asset f :
E(RP ) = (1 − XB )R f + XB E(RB )
1/2
= XB σB
σP = (1 − XB )2 σ 2f + XB2 σB2 + 2(1 − XB )XB σ f σB ρB f
σP
⇒ XB =
σ
B
σP
σP
R f + E(RB )
⇒ E(RP ) = 1 −
σB
σB
E(RB ) − R f
= Rf +
· σP
σB
AFM 271 - Return and Risk: The Capital Asset Pricing Model
E(RP )
Slide 27
Cont’d
σP
choose point of tangency A to get capital market line (CML)
separation principle:
1. determine point A, independent of personal preferences;
and
2. based on degree of personal risk aversion, pick a point on
CML as your investment point
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 28
The Capital Asset Pricing Model
so far we have considered decisions of one investor
now consider many investors:
homogeneous expectations assumption: all
individuals have same expectations regarding
returns, variances, and covariances
under homogeneous expectations, all investors
choose the same portfolio of risky assets
represented by A
in equilibrium, A must be the market portfolio - a
value-weighted portfolio of all existing securities
(often proxied by indexes (e.g. S&P/TSX))
one implication: index investing (pick a broad stock
index such as S&P/TSX, then divide your investment
between this index and the risk free asset)
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 29
Cont’d
β measures the sensitivity of the change in return of an
individual security to the change in return of the market
portfolio M:
cov(R j , RM )
βj =
2
σM
sign of β depends on sign of covariance/correlation
example: βC = 2
E(RC )
E(RM )
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 30
Cont’d
β can be estimated by linear regression:
Rt, j = α j + β j × Rt,M + εt, j ,
where α j = intercept, β j = security β , and
εt, j = regression error term
characteristics of β
βM = 1
βR f = 0
β j and E(R j ) have a positive, linear relationship
β j > 1 ⇒ unusually sensitive to market movements
β j < 1 ⇒ unusually insensitive to market movements
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 31
Aside: Linear Regression
linear regression is a widely used tool in statistical
analysis
consider a simple case with a dependent variable y and
a single independent variable x:
y
x
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 32
Cont’d
the simple linear regression model is
yt = α + β xt + εt
basic idea is to minimize the variance of the error term
the “best fit” line is given by
α̂ = ȳ − β̂ x̄
β̂ =
cov(x, y)
var(x)
simple e.g.: let x be monthly returns on TSE 300 and y
be monthly returns on RBC (last half of 2002):
y 2.53 4.77 -5.89 4.00 7.61 -1.20
x -7.45 0.22 -6.29 1.21 5.28 0.91
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 33
Cont’d
we can calculate:
x̄ = −1.02
cov(x, y) = 15.05
ȳ = 1.97
var(x) = 23.81
α̂ = 2.61
β̂ = 0.63
y
2.53
4.77
-5.89
4.00
7.61
-1.20
x
-7.45
0.22
-6.29
1.21
5.28
0.91
ŷ
-2.09
2.75
-1.36
3.38
5.95
3.19
ε
4.62
2.02
-4.53
0.62
1.66
-4.39
since var(y) = 23.18 = β̂ 2 var(x) + var(ε ) = 9.51 + 13.67, the
regression “explains” 9.51/23.18 = 41% of the variation in y
the implication here is that of RBC’s total risk, 41% is market
risk and 59% is unique risk
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 34
Cont’d
RRBC
8
6
4
2
-8
-6
-4
-2
2
4
6
8
RM
-2
-4
-6
-8
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 35
Back to CAPM
there is positive linear relationship between risk and
expected return
E(R j ) = R f + β j × E(RM ) − R f
basic idea: investors expect a reward for waiting (R f )
and worrying (risk premium)
CAPM conclusions:
expected return on a security depends on security’s
risk relative to the risk of the market portfolio, i.e.
undiversifiable risk
β is the only reason that expected returns differ
between securities
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 36
Cont’d
CAPM applies to portfolios as well as to individual
securities
for portfolios:
E(RP ) = R f + βP × E(RM ) − R f
βP =
N
∑ Xj × βj
j=1
for well-diversified portfolios:
2
var(RP ) = βP2 × σM
s.d.(RP ) = σP = βP × σM
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 37
Cont’d
the security market line (SML):
E(R j )
E(RM )
Rf
0
1
βj
there is a positive linear relationship between E(R j ) and β j :
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 38
Cont’d
in equilibrium, all securities must line on the SML, e.g.:
comparison of capital market line and security market
line:
CML
- traces efficient set of portfolios
formed from risky assets and
riskless asset
SML
- describes the return-β relationship
- relates return to total risk
- relates expected return to
systematic/undiversifiable risk
- holds only for efficient portfolios
- holds for all individual
securities and all possible portfolios
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 39
Cont’d
overall summary:
the stock market is risky and investors want a reward for risk
a measure of risk for a single security is σ or σ 2
in a portfolio, do not look at the risk of a security in isolation
risk consists of diversifiable and undiversifiable risk
only undiversifiable/systematic risk is rewarded
a security’s contribution to the total risk of a portfolio is
measured by β , which represents sensitivity of the security
to market changes (i.e. the systematic risk)
CAPM/SML - positive linear relationship between expected
returns and systematic risk
in equilibrium, all stocks must lie on the SML
CAPM is the best known model of risk and return
AFM 271 - Return and Risk: The Capital Asset Pricing Model
Slide 40