FM 300 Section A
The Capital Asset Pricing Model:
Overview
Lecture Note 1
Prof. Konstantinos E. Zachariadis
London School of Economics
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Questions
How much should we expect to receive as a return from a
risky asset?
How do we know if a trading strategy is profitable?
How do we estimate the impact of announcing an acquisition
plan?
What benchmark should we use to evaluate the performance
of a mutual fund manager?
How do we discount the cash-flows of a project?
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Outline
Basic statistics rules for portfolio returns
Brief overview of portfolio theory and the CAPM
– How are expected returns determined?
– How are they related to risk?
Risk and return with a risk-free asset
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Basic statistics for asset returns
Suppose you had bought a stock at the end of December 2020
for $42.25 and sold it a year later, at the end of December
2021, for $34.375. During this year, the stock paid a per-share
dividend of $3.1 Your realized return from holding GM:
34.375 + 3.1 - 42.25
= -11.54%
42.25
E(ri) – rf : The expected excess return of asset i is the expected
asset’s return minus the return on the riskless asset
E(rM) – rf : The market risk premium is the expected excess
return of the market portfolio.
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Basic statistics for asset returns
GM example
Date
Price
Dividend
Return
29-Dec-89
42.2500
-
31-Dec-90
34.3750
3.00
-11.54%
31-Dec-91
28.8750
1.60
-11.35%
31-Dec-92
32.2500
1.40
16.54%
31-Dec-93
54.8750
0.80
72.64%
30-Dec-94
42.1250
0.80
-21.78%
29-Dec-95
52.8750
1.10
28.13%
31-Dec-96
55.7500
1.60
8.46%
31-Dec-97
60.7500
5.59
19.00%
31-Dec-98
71.5625
2.00
21.09%
31-Dec-99
72.6875
14.15
21.34%
Average return
14.25%
Variance of return
0.0638
Standard deviation of return
25.25%
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Basic statistics for asset returns
Expected Return (Mean Return)
1
N
N
åR
t =1
t
= RA ® E ( RA )
Law of large numbers
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Expected return on a portfolio
The expected rate of return on a portfolio of stocks is:
( )
E (rP ) = å j =1 x j E rj
j =n
where
å j =1 x j = 1
j =n
Portfolio expected rate of return is a weighted average of the
expected rates of return on the individual stocks.
In the two-asset case:
E (rP ) = x1 E (r1 ) + (1 - x1 ) E (r2 )
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Portfolio returns
We measure the (realized) return rP on a portfolio in period t as:
rPt = å j =1 x j rjt
j=N
where xj = fraction of portfolio’s total value invested in stock j.
– xj > 0 is a long position; xj < 0 is a short position;
Stock market indices:
– Equally weighted: x1=x2=…=xN=1/N
– Value weighted: xj= Proportion of market capitalization
We measure the average return of a portfolio over time as:
1 t =T
r P = åt =1 rPt
T
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A digression: short sales
A short sale consists of selling a stock that we do not own.
Steps:
– Date 0: Borrow the stock from a broker.
– Date 0: Sell the stock in the market.
– Between dates 0 and 1: Compensate the broker for any
dividends the stock pays.
– Date 1: Buy the stock in the market.
– Date 1: Return the stock to the broker.
A short sale is profitable if the stock price goes down.
Portfolio return is still weighted average of returns on the
individual stocks, but weights of shorted stocks are negative.
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Variance
Variance: Average value of squared deviations from the
mean. A measure of volatility.
Var( X ) = E{[ X - E ( X )]2 }
Standard deviation: A measure of volatility that is easier to
interpret because it is in the same units as the portfolio return.
Std ( X ) = Var( X )
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Covariance and correlation
Covariance between two random variables X and Y:
Cov ( X , Y ) = E[{X - E ( X )}{Y - E (Y )}]
The covariance measures how returns on different stocks
move in relation to each other.
Correlation between two random variables X and Y:
Corr ( X , Y ) =
Cov( X , Y )
Std ( X ) Std (Y )
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Rules of means and variances
Let a, b be two constants. Then expected values are:
E (ar1 ) = aE (r1 )
E (r1 + r2 ) = E (r1 ) + E (r2 )
E (ar1 + br2 ) = aE (r1 ) + bE (r2 )
where the third rule is implied by the first two. In our applications
the constants a and b will typically be portfolio weights.
Similarly, for variances:
Var (ar1 ) = a 2Var (r1 )
Cov(ar1 , r2 ) = aCov(r1 , r2 )
Var (r1 + r2 ) = Var (r1 ) + Var (r2 ) + 2Cov(r1 , r2 )
Var (ar1 + br2 ) = a 2Var (r1 ) + b 2Var (r2 ) + 2abCov(r1 , r2 )
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Measuring Portfolio Risk
The risk of a portfolio is measured by its standard
deviation or variance.
The variance for the two stock case is:
var( rp ) = s p2 = x12s 12 + x22s 22 + 2 x1 x2s 12
s i2 = Variance of asset i
s ij = Covariance of returns of assets i and j
or, equivalently,
var( rp ) = s p2 = x12s 12 + x22s 22 + 2 x1 x 2 r12s 1s 2
rij = Coefficient of correlation of the returns of i and j
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Covariance and correlation
GM and MSFT
Date
GM return
MSFT return
31-Dec-90
-11.54%
72.99%
31-Dec-91
-11.35%
121.76%
31-Dec-92
16.54%
15.11%
31-Dec-93
72.64%
-5.56%
30-Dec-94
-21.78%
51.63%
29-Dec-95
28.13%
43.56%
31-Dec-96
8.46%
88.32%
31-Dec-97
19.00%
56.43%
31-Dec-98
21.09%
114.60%
31-Dec-99
21.34%
68.36%
Average return
14.25%
62.72%
Variance of return
6.38%
14.43%
Standard deviation of return
25.25%
37.99%
Covariance of returns
Correlation
-0.0552
-0.5755
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Review of Portfolio Theory
What happens to risk if we combine more and more stocks
into a portfolio?
Combining stocks into portfolios can reduce standard
deviation.
Correlation coefficient is crucial to diversification
– Perfectly correlated;
– Imperfectly correlated;
– Perfectly and negatively correlated.
Efficient portfolios: have the smallest standard deviation given
expected return; with the highest expected return given risk.
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Portfolio standard deviation
Portfolio Risk
Unique
risk
Market risk
0
5
10
15
Number of Securities
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2-stock Portfolios
We can plot frontiers for different correlations.
Diversification depends on correlations.
0.15
Expected Return (R)
0.14
IBM
0.13
0.12
0.11
0.1
0.09
ExxonMobil
0.08
0.07
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Standard Deviation (σ)
ρ=-1
ρ=-0.5
ρ=0
ρ=0.35
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ρ=0.5
ρ=1
17
Efficient Frontier with N stocks
Formally, the problem of finding the efficient frontier of Nstock portfolios is as follows:
σ p = min
w1 ,w 2 ,...,w N
subject to:
N
N
∑i=1 ∑ j=1w i w jσ iσ j ρij ,
w1 +... + w N = 1,
w1r1 +... + w N rN = rP .
Investor achieves minimal risk for given expected return r p
Hence, the optimal portfolio is on the efficient frontier.
This optimization problem can be solved explicitly. However,
this is beyond our scope. You can use Excel solver to solve
for optimal weights and σp given r p .
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Example
Consider 3 stocks with the following characteristics:
Stock
Expected Return
Standard Deviation
IBM
15%
29.7%
ExxonMobil
7.9%
19.2%
Starbucks
12.3%
29.9%
and correlations:
IBM
IBM
ExxonMobil
1
Starbucks
0.35
0.2
1
-0.1
ExxonMobil
Starbucks
1
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Efficient Frontier with 3 stocks
The picture below shows 2-stock and 3-stock frontiers.
Efficient frontier is the upward sloping (dark blue) part.
16%
Expected Return (R)
15%
Efficient Frontier with 3 stocks
14%
IBM
13%
12%
11%
Starbucks
10%
9%
8%
7%
14%
ExxonMobil
16%
18%
20%
22%
24%
26%
28%
30%
Standard Deviation (σ)
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Lending and Borrowing allowed
How do the combinations of risky portfolios and riskless assets
look like? Consider a new portfolio Q with fraction (1-w) in riskless
asset (long or short position in bonds) and w in risky portfolio P:
rQ = (1- w )rf + wrP = rf + w (rP - rf ).
Weight w > 1 means we borrow (or sell short Treasury Bills).
Expected returns and standard deviations of returns rQ:
rQ = rf + w (rP - rf ),
sQ = (1- w ) 2 var(rf ) + w 2sP2 + 2(1- w )w cov(rf , rP )
=0
= w s = ws P .
2
2
P
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=0
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Lending and Borrowing allowed
Substituting out w=σQ/σP we obtain expected returns of Q as
functions of standard deviations:
æ r P - rf ö
rQ = rf + sQ ç
÷.
è sP ø
This is an equation of a straight line with the slope given by
the Sharpe ratio: (rP - rf ) / sP .
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Lending and Borrowing allowed
Portfolios on R-P line are not the best. Portfolios on a steeper
line that passes through R dominate those on R-P line.
Expected Return (%)
The efficient frontier with the riskless asset is the steepest
possible line, R-T, where T is tangency portfolio. Efficient
portfolios have the largest Sharpe ratio.
ing
w
rro
o
B
T
ing
d
n
Le
rf
Q1
R
W<1
P
Q2
W>1
Standard Deviation (σ)
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Lending and Borrowing allowed
Lending/borrowing at a riskless rate rf (or buying/selling
bonds) helps achieve higher returns for a given risk.
Efficient frontier becomes a tangent line.
Expected Return (%)
tangency
portfolio
ng
i
w
rro
o
B
ing
d
n
Le
rf
Standard Deviation
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Efficient Portfolios with Multiple Assets
Return
Low Risk
High Risk
High Return
High Return
Low Risk
High Risk
Low Return
Low Return
Risk
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CAPM
Capital market line (CML) passes through the market portfolio and
the riskless asset. Market and tangency portfolios coincide because
supply=demand in equilibrium.
Therefore, CML coincides with the efficient frontier.
Expected Return (%)
CML
rM
rf
IBM
Coca-Cola
ExxonMobil
Standard Deviation
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The Capital Market Line
The CML gives the trade-off between risk and return for
portfolios consisting of the risk-free asset and the tangency
portfolio M.
The equation of the CML is:
E ( rp ) = r f + s p
E ( rM ) - r f
sM
The expected rate of return on a risky asset can be thought of
as composed of two terms: risk-free rate and risk premium:
– E(r) = rf + Risk x [Market Price of Risk]
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Basic assumptions of the CAPM
The goal is to determine expected returns by combining
optimal portfolio choices of all investors in the economy.
Assumptions:
– Investors have a one-period horizon
– Each investor holds a mean-variance efficient portfolio,
i.e., a portfolio with the highest expected return given its
volatility
– Investors have the same beliefs.
– Lending and borrowing at a single riskless rate.
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CAPM
Capital Asset Pricing Model (CAPM):
cov 𝑟! , 𝑟#
𝐸 𝑟! − 𝑟" =
(𝐸 𝑟# − 𝑟" )
$
𝜎#
stock excess
return
beta, βi
market excess
return
This formula can be rewritten in terms of beta:
𝐸 𝑟! − 𝑟" = 𝛽! (𝐸 𝑟# − 𝑟" )
Expected returns depend on market risk measured by beta and
not by individual risk which can be diversified away.
Market portfolio M is well-diversified, and hence, contains only the
market risk.
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CAPM
Excess return is a linear function of β. This function is called
Security Market Line. All possible portfolios will be somewhere on
this line.
Expected Return (%)
– Note that market portfolio has β = 1.
r = rf + b (rM - rf ),
rM
rf
1.0
Beta (Risk)
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Beta
The appropriate measure of risk for a stock is beta.
Beta measures the stock’s sensitivity to market risk factors.
– The higher the beta, the more sensitive the stock is to
market movements.
The relevant risk for pricing asset i is its contribution to the
variance of the market portfolio:
bi =
s iM Asset i's contribution to Market variance
=
2
sM
Total Market variance
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Summary
Optimal investments depend on trading off risk and return
– Investors with higher risk tolerance invest in riskier assets;
– Only risk that cannot be diversified matters.
If investors can borrow and lend, then everybody holds a
combination of two portfolios: the market portfolio of all risky
assets and the the riskless asset.
In equilibrium, all stocks must lie on the security market line.
– Beta measures the amount of non-diversifiable risk;
– Expected returns reflect only market risk;
– These expected returns can be used as required returns in
project evaluation, performance measurement, etc.
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Appendix: CAPM Derivation (Not Required)
Derived from the fact that market portfolio coincides with tangency
portfolio => is on the steepest line with highest Sharpe ratio.
Denote by wf proportion of wealth in riskless asset, and wi
proportion of wealth in asset i. Then:
w f + w1 + ... + w N = 1.
The return on a portfolio with expected value rP (after substituting
out wf) and volatility σP is given by:
n
n
i =1
i =1
rP = w f rf + å w i ri = rf + å w i (ri - rf ),
N
(1)
N
s = åå w i w j s i s j rij .
2
P
i =1 j =1
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… CAPM: Derivation (Not Required)
Consider now a portfolio P with stock weights equal to the weights
of market portfolio, except for stock k:
w1 = w1M ,..., w k -1 = w kM-1, w k ¹ w kM , w k +1 = w kM+1,..., w N = w NM .
Suppose, we fix all portfolio weights except for weight wk and
consider the Sharpe ratio (rP - rf ) / sP as a function of wk.
We know, that the market portfolio has highest possible Sharpe
ratio. Hence, Sharpe ratio of portfolio P will be maximized when
w k = w kM .
Therefore, the following first order condition should hold:
d [(rP - rf ) / sP ]
= 0.
dw k
w k =w kM
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(2)
34
… CAPM: Derivation (Not Required)
Using chain and product rules we obtain:
d [(rP - rf ) / sP ] 1 d (rP - rf ) rP - rf dsP
=
- 2
.
dw k
sP dw k
sP dw k
From this equation and (2), evaluating the expressions at
w k = w kM we obtain:
d (rP - rf ) rM - rf dsP
=
,
dw k
sM dw k
(3)
where all derivatives are evaluated at w k = w kM .
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… CAPM: Derivation (Not Required)
Taking into account (1), computing the derivatives with respect to
wk and evaluating them at w k = w kM we obtain:
d (rP - rf )
= rk - rf ,
dw k
dsP
1 dsP2 cov(rk , rM )
=
=
.
dw k 2sP dw k
sM
(4)
The last equality holds because:
s =w s +2
2
P
2
k
2
k
N
M
w
w
å k i cov(rk , ri ) + terms without w k .
i =1, i ¹k
Hence, the derivative evaluated at w k = w kM is given by:
dsP2
= 2(w 1M cov(rk , r1 ) + ... + w kM cov(rk , rk ) + ... + w NM cov(rk , rN ) )
dw k
= 2 cov(rk , w1M r1 + ... + w NM rN ) = 2 cov(rk , rM ).
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