Capital Market Equilibrium and
the Capital Asset Pricing Model
Econ 422
Investment, Capital & Finance
Spring 2010
June 1, 2010
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
1
The Risk of Individual Assets
•
•
•
Investors require compensation for bearing
risk.
We have seen that the standard deviation of
the rate of return is an appropriate measure
of risk for one’s portfolio.
Standard deviation is not the best measure of
risk for individual assets when investors hold
diversified portfolios.
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
2
1
The Risk of Individual Assets (continued)
•
For people holding a diversified portfolio it is
the contribution of the individual asset to the
portfolio’s standard deviation that matters.
•
[If your portfolio involved only one asset, e.g.
young Bill Gates,
Gates the portfolio standard
deviation would be the standard deviation of
the single asset.]
© 2006 R.W.Parks/E. Zivot
ECON 422:CAPM
3
The contribution of an individual
asset to the portfolio’s standard
deviation: Beta
•
•
Beta measures the sensitivity of an asset’s
asset s
rate of return to variation in the market
portfolio’s return.
Beta for asset i can be computed as
βi =
ECON 422:CAPM
cov(ri , rm ) σ im
= i2
V (rm )
σm
© 2006 R.W.Parks/E. Zivot
4
2
Beta as a Measure of Portfolio Risk:
Class Example
•
•
Suppose you hold an equally weighted
portfolio with 99 assets
What happens to the portfolio variance if a
new asset, say IBM, is added to the portfolio?
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
5
Measuring Betas
The Market Model
z
z
Beta can be interpreted as the slope coefficient in a
regression of the return on the ith security, ri, on the
return for the market portfolio, rm.
The interpretation rests on the market model:
rit = α i + β i rmt + ε it
z
rm represents “market risk” and εi represents “firm
specific” risk independent of the market. That is,
cov(rmt, εit) = 0.
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
6
3
Beta as a Measure of Risk for
Individual Assets
Asset i’s
is
Return ri
The slope is given by
β=cov(ri,rm)/V(rm)
+10%
The intercept is
given by α
10%
-10%
+10%
Market
return
rm
-10%
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
7
The Capital Asset Pricing Model
(CAPM)
z
CAPM describes the relationship between an
asset’s beta risk and its expected return as
follows:
E ( ri ) = r f + β i E ( rm ) − rf
= riskfree rate + beta x market risk premium.
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
8
4
The Security Market Line (SML)
Describes the CAPM Relationship
E(ri)
Security Market
Li (SML)
Line
E(rm)
Slope is the market risk
premium = E(rm)-rf
rf
1.0
β
E ( ri ) = rf + β i E ( rm ) − rf
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
9
The Capital Asset Pricing Model
Example
z
z
z
Annual T-bill
T bill yield = 1%
Beta for Microsoft = 0.93 (from Yahoo! See link for
key statistics)
Historical market risk premium =7.1%
Then
E[rmsft ] = rf + β msft ( E[ Rmt ] − rf )
= 1% + 0.93*(7.1%) = 7.603%
Note: the return predicted from the CAPM is
sometimes called the “risk-adjusted” return
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
10
5
CAPM and Efficient Portfolios
Example: Starbucks Stock
E[Rsbux]= 0
0.10,
10 SD(Rsbux) = 0
0.20
20
E[Rm] = 0.15, SD(Rm) =0.10, rf = 0.03
z
z
Find the beta for SBUX
Find the efficient portfolio of T-bills and the
market with the same expected return as
SBUX
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
11
The Market Model and the Measures
of Risk
z
z
The total risk of an asset, held alone, i.e. not as part
of a diversified portfolio
portfolio, would be measured by its
variance, V(ri).
According to the market model
rit = α i + β irmt + ε it
and
V(rit ) = β 2i V (rmt ) + V (ε it )
z
z
Total risk = systematic market risk + unique risk
The unique risk can be eliminated through
diversification.
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
12
6
Portfolio Beta
The beta of a portfolio is a weighted average of
the individual asset betas
β p = x1 β 1 + x 2 β 2 + " + x n β n
x i = portfolio share for asset i
β i = beta for asset i
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
13
Example
z
z
z
z
Microsoft has a beta of 1
1.2
2
Starbucks has a beta of 0.8
Portfolio weights are xmsft = 0.25, xsbux =0.75
Portfolio beta = 0.25(1.2) + 0.75(0.8) = 0.9
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
14
7
Applying the CAPM to Valuation
Recall the one-period holding period rate of return:
r=
P1 − P0 + D1
P0
At time t = 0, P0 is known, but P1 and D1 are not known; they are random variables.
Take expectations:
E(P1) − P0 + E(D1)
E(r) =
P0
Solve for P0 :
P0 =
E(P1) + E(D1)
E(P1) + E(D1)
=
1+ E(r)
1+ rf + β[E(rM ) − rf ]
using the CAPM relation: E(r) = rf + β[E(rM ) − rf ]
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
15
Applying the CAPM to Valuation
(continued)
P0 =
z
E ( P1 ) + E ( D1 )
1 + rf + β [ E (rM ) − rf ]
To value a future risky cash flow, discount the
expected value of the cash flow to present value
using the risk
risk-adjusted
adjusted expected return based on the
CAPM.
ECON 422:CAPM
© 2006 R.W.Parks/E. Zivot
16
8
Example: Stock valuation using CAPM
z
z
E[D1] = 5, g = 0.10, rf = 0.03
β = 1.5,
1 5 E[
E[rm] – rf = 0.075
0 075
E[r] = rf + β(E[rM ] − rf ) = 0.03+1.5(0.075) = 0.1425
Constant growth: P0 =
ECON 422:CAPM
E[D1]
5
5
=
=
=117.64
(E[r] − g) 00.1425
1425−00.1
1 00.0425
0425
© 2006 R.W.Parks/E. Zivot
17
9