Unformatted text preview: 7. Capital Asset Pricing MODEL
Reading: Luenberger Chapter 7 10/19/2011 1 Sharpe (1964), Lintner (1965) Nobel Prize: Sharpe (1990) Based on Mean Variance Portfolio Optimization
‒ Rules for how to invest (normative economics) Main Result Under Equilibrium
‒ How assets are priced (positive economics)
‒ Asset expected return is related to a measure of risk called Beta Requires Strong Assumptions
‒ Unrealistic but necessary to obtain a very simple model
2 7.1 Market Equilibrium
Assumptions All investors use meanvariance optimal portfolios with
‒ Same single period investment horizon
‒ Same values of returns means and covariances Unlimited buy or selling of a common riskfree asset Security markets are perfect
‒ Assets are infinitely divisible
‒ zero taxes
‒ No transaction costs
3 Under the previous assumptions:
The risky asset holdings of all investors: Portfolio T µP . T The capital market line rf σP
The only difference between investors is their mix between
risky assets and the riskfree asset, and this is determined
by their differing risk aversions.
4 Under Equilibrium T is the Market Portfolio
Claim: If all market participants hold the tangency
portfolio T (or any fixed portfolio) of risky assets then
under equilibrium they all hold the market portfolio of
risky assets.
It seems obvious: If every investor holds the same fund in the
proportional weights determined by T (or any other optimization
scheme as long as it is acknowledged by all to be optimal), then
since the totality of all investments in these assets must be the
market capitalization, the market portfolio must have the same
weights as T.
What is more subtle is the claim that such weights are capitalization
weights: the larger the market cap a stock has, the higher the weight
given to that stock in T.
5 N = number of securities in market
V j = market capitalization of asset j, j = 1, L , N
V = ∑ j =1V j
N total market capitalization Market weights based on capitalization: wM , j = Vj
V Investor k s weights: wk , j
Want to show: wM , j = wk , j under the assumption wk , j ≡ w j , j = 1, L , N common investor portfolio weights
6 Wk = total investment of k th investor , k = 1, L , K
V j = ∑ k =1 wk , jWk total of all investments in j th asset
K = w j ⋅ ∑ k =1Wk
K = w j ⋅W , W = total investment of all investors Under equilibrium
and so W =V V j w j ! " k =1W k
=
=
=wj .
V
W
K wM , j !
7 Capital Market Line
The risky asset holdings of all investors: Portfolio T µP . T The capital market line rf σP
On the capital market line, assets obey: µ M − rf
µ − rf =
σ
σM 8 Capital Market Line
Example 7.2 (An oil venture):
The price of a share of this oil venture is $875. It is
expected to yield $1000 after 1 year, with standard
deviation of 40%. The riskfree rate is 10%. The
expected market return is 17% with standard
deviation of 12%.
Expected return of this venture is: µ = (1000 − 875) / 875 = 14%
If it were on the capital market line it should have
returned:
0.17 − 0.10 µ = 0.10 + Not efficient. 0.12 0.40 = 33% 9 7.2 The CAPM Formula: Relation of
expected return to risk for individual stocks ri
rM return for the i th asset, µi = E ( ri )
return for the market, µ M = E ( rM ) µi − rf = β i ⋅ ( µM − rf )
where cov(ri , rM )
βi =
var(rM ) In terms of excess returns: µi ,e = β i ⋅ µM ,e
10 CAPM in vector notation: µ M ,e
µ e = cov(r, rM ) ⋅ 2 = β ⋅ µM ,e
σM
where β= cov(r, rM ) σ 2
M = Ωw M
2
σM 11 Capital Asset Pricing Model: The expected excess return of an asset is
proportional to the excess return of the
market portfolio, and the proportionality
constant is beta. The expected excess return of an asset is
proportional to its covariance with the market. 12 Proof of the CAPM
A proof is provided in Section 7.3 of Luenberger.
Consider a portfolio consisting of a portion α invested in
asset i, and (1α) invested in the market portfolio M.
The expected return is µ = !µi + (1 ! ! )µ M
The standard deviation is
2
! = [" 2! i2 + 2" (1 ! " )! i, M + (1 ! " )2 ! M ]1/2 13 Proof of the CAPM
2
2
dµ
d" !" i + (1 ! 2! )" i, M + (! ! 1)" M
= µi ! µ M
=
d!
d!
"M
d µ d µ / d!
(µi ! µ M )" M
=
=
2
d" d" / d! !" i2 + (1 ! 2! )" i, M + (! ! 1)" M dµ
d" (µi ! µ M )" M µ M ! rf
=
=
2
" i, M ! " M
"M
! =0 since ! = 0 corresponds to the slope of the capital market line.
Solving for µi : µ M ! rf
µi = rf + (
)" i, M = rf + #i (µ M ! rf )
2
"M
14 Alternative Proof of the CAPM
An alternative proof follows from the derivation of the
efficient frontier with a cash mix in Lecture 6. Recall that
the Lagrangian for that derivation is: 1
L(w ) = w′Σw − γ ⎡(1 − w′1) rf + w ′µ − µ P ⎤
⎣
⎦
2
and setting the derivative equal to zero gives Σw − γ ⎡µ − 1rf ⎤ = Σw − γ µ e = 0
⎣
⎦
w = γΣ −1µe
15 This gave the following optimal weight vector on the linear
efficient frontier w = γ ⋅ Σ −1µ e
At the tangency portfolio w ' 1 → 1 or 1 ' w → 1 γ = (1′Σ µ e )
−1 −1 Under the assumptions for the CAPM the tangency
portfolio is the market portfolio so we have the market
weights vector Σ −1µ e
wT =
= wM
−1
1′Σ µ e 16 Thus µ e = (1′Σ −1µ e ) ⋅ Σw M
Noting that ( Σw M )i = ∑ j =1 cov(ri , rj ) ⋅ wM , j
N = cov(ri , ∑ j =1 wM , j rj )
N = cov(ri , rM )
′Σ −1µ e ) ⋅ cov(ri , rM )
µe,i = (1
Calculate (1′Σ −1µ e ) next.
17 The market portfolio has mean excess return and variance: µ M ,e
2
σM µ′ Σ −1µ e
= E (rM − rf ) = E (re′w M ) = µ′ w M = e −1
e
1′Σ µ e
µ′ Σ −1ΣΣ −1µ e
µ′ Σ −1µ e
= var(re′w M ) = e −1 2 = e −1 2
(1′Σ µ e )
(1′Σ µ e ) This give µ M ,e
1′Σ µ e = 2
σM
−1 µ M ,e
and so µe ,i = cov( ri , rM ) ⋅ 2
σM
= cov(ri , rM ) σ 2
M ⋅ µ M ,e = β i ⋅ µ M ,e (CAPM)
18 Two Ways to View CAPM
View 1: E (ri ) − rf as a (straight line) function of E (rM ) − rf with slope βi . This is the usual simple linear regression model view View 2: E (ri ) − rf as a (straight line) function of βi with slope The resulting straightline plot is called the security E (rM ) − rf market line. 19 . Market risk premium (assume positive) µi = rf + β i ⋅ ( µM − rf ) βi = cov( ri , rM ) σ 2
M Risk premium Example: β i = 1.2, µM = 6%, rf = 3% (annual)
µi = 3% + 1.2 ⋅ ( 6% − 3% ) = 6.6% β i = 1 ⇒ µi = µM
β i < 1 ⇒ µi < µM
β i > 1 ⇒ µi > µM β i = ρi , M σi
σM This is good but it comes at the price of increased risk!
20 Portfolio Beta
n E (rP ) = ∑ wi ⋅ E (ri ) Portfolio expected returns: i =1 Use the CAPM for each asset
to get ( E(ri ) − rf = βi ⋅ E(rM ) − rf E(rP ) − rf = β P ( E(rM ) − rf ) where ) n β P = ∑ wi ⋅ βi
i =1 Note linearity: The beta of a sum of stocks is the same as
the weighted sum of the betas of individual stocks.
n β P = ∑ wi ⋅
i =1 cov(ri , rM ) σ 2
M = ⎛n
⎞
cov ⎜ ∑ wi ⋅ ri , rM ⎟
⎝ i =1
⎠ σ 2
M = cov(rP , rM )
2
σM 21 CAPM as a Pricing Model Suppose an asset is purchased at price P and
later (one year later) sold at price Q. P is
known and Q is random. (Annual) Return: r = Q − P
P Expected return: µ = E[r ] = E[Q] − P = rf + β ( µ M − rf )
P Price according to CAPM (efficient):
E[Q]
P=
1 + rf + β ( µ M − rf )
22 Same example for the oil venture The oil venture has an expected payoff of
$1000 a share but with a high standard
deviation of 40% because of the uncertainty
associated with whether or not there is oil at
the site, and with the future price of oil. The
beta is relatively low at 60% because the
uncertainty related to exploration is not
correlated with the market. CAPM price:
$1000
P=
= $876
1.1 + 0.6(0.17 − 0.10)
23 Certainty equivalent pricing formula: Price according to CAPM (efficient):
E[Q]
P=
1 + rf + β ( µ M − rf ) It can be rearranged into the form (LU p189):
cov(Q, rM )(rM − rf )
1
P=
{E[Q] −
}.
2
1 + rf
σM
{E[Q] − cov(Q, rM )(rM − rf ) σ 2
M } is the certainty equivalent of Q 24 7.3 General Portfolio Betas and Risk
For any portfolio B, not necessarily the market portfolio, define the beta
of the ith stock with respect to the portfolio B as βi =
In vector notation: cov(ri , rB ) β= σ 2
B , i = 1, 2,L , n cov(r, rB )
2
σB MAIN APPLICATION: Active Portfolio Management where
B will be a benchmark portfolio. 25 Covariance Reflects Asset Risk Contribution
Just as in the case of the market portfolio we have for any
portfolio B , with cov(r, rB ) = Σw
2
σ B = w′Σw = ∑ wi cov(ri , rB ) = w′ cov(r, rB )
i =σ 2
B ∑ wi ⋅
i !i cov(ri , rB )
2
σB 2
= σ B ∑ wi ⋅ βi
i measures the contribution of asset i to the overall
2
σB
portfolio s variance Also: ∑ wi ⋅ βi = w′β = 1 if beta is defined with respect to B.
i 26 Systematic vs nonsystematic risk
Let the random rate of return of asset i be ( ε is
whatever it needs to be to make the formula
true):
ri = rf + !i (rM ! rf ) + "i Expected return: µi = rf + !i (µ M ! rf ) + E["i ]
E[!i ] = 0
CAPM says:
(It assumes only that the market M is efficient)
27 Systematic vs nonsystematic risk
Variance:
2
! i2 = var("i (rM ! rf ) + #i ) = "i2! M + var(#i ). if !i is uncorrelated with the market.
Risk= systematic risk + nonsystematic risk
2
Systematic risk= !i2" M
The smallest risk is the asset on the capital market
line with variance=systematic risk
Nonsystematic risk can be reduced by diversification
28 References
Book Treatments of CAPM
Connor, G., Goldberg, L. and Korajczyk, R. (2010). Portfolio Risk Management,
Oxford University Press.
Grinold, R. C. and Kahn, R. N. (2000), Active Portfolio Management, Chapter 2.
Luenberger, D. G. (1987). Investment Science, Oxford University Press.
Pennachi, G. (2008). The Theory of Asset Pricing, Pearson AddisonWesley.
CAPM Papers
W. F. Sharpe (1964). Capital Asset Prices: A Theory of Market Equilibrium under
Conditions of Risk , Jour. of finance, pp. 425442.
L. Lintner (1965). Security Prices, risk and Maximal gains from Diversification , Jour.
of Finance, pp. 587615.
E. F. Fama (1968). Risk, Return and Equilibrium: Some Clarifying Comments , Jour.
of Finance, pp. 2940.
29 ...
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 Winter '11
 KK.T
 Capital Asset Pricing Model, rf, Capital Market Line, Market Portfolio

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