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Capital Asset Pricing Model
BKM: Chapter 7
Optimal Portfolios

Given current expected returns, standard
deviations, and correlations:



Choose risky portfolio to maximize Sharpe
ratio.
Tailor the risk by investing (long or short) in the
risk-free asset.
This is the right approach whether or not
markets are “efficient”.
Beating the Market

Beating the market means more than
earning a higher average return


You could do this by simply investing in
small-cap stocks.
Beating the market means developing a
portfolio with a higher Sharpe ratio.
Beating the market

To beat the market: you should continually
be searching for stocks such that



E[r]>rf+bE[rM]-rf) tilt towards these
E[r]<rf+b(E[rM]-rf) tilt away from these
By tilting the market portfolio as above, you
increase the Sharpe ratio of the portfolio.
Efficient Markets

If markets are efficient
P
 k e
P



We can’t predict “e” – our best guess is zero
k is then the expected return
What determines k across stocks?
Models of k

Capital Asset Pricing Model



Arbitrage Pricing Model



Requires many assumptions
Model is very specific
Requires light assumptions
Model is more general
Accounting Ratios


Can be difficult to interpret
Relies more on gut instinct
Expected return

E[Get ]
Expected return =
1
Pay

Anything that raises the price, lowers
the expected return

Anything the lowers the price, raises
the expected return
Intuition of the CAPM

Basic Idea: everyone is trying to beat the market

If E[r]>rf+bE[rM]-rf) then everyone should buy these


If E[r]<rf+b(E[rM]-rf) then everyone should tilt away


Increases price, lowers the expected return
Lowers price, lowers the expected return
In equilibrium: E[r]=rf+b(E[rM]-rf) for any asset
Intuition of the CAPM

But what is the market portfolio?

In equilibrium, the “market portfolio”
should be the portfolio with highest
Sharpe ratio.

To answer this question, let’s look at a
sketch of the proof of the CAPM.
CAPM: Assumption 1
Aggregation Assumption

Assumption: all investors have the same forecasts
of variances, covariances, and expected returns.



Very strong assumption
Although not true, there may be considerable consensus
among investors on what these parameters are.
Result (1):

We can think of the aggregate investing community as
one big investor called the “representative investor” (the
R-investor)
CAPM: Assumption 2
Preferences Assumption

The representative investor only cares about
maximizing the Sharpe ratio of his portfolio

Result (2.1):

In equilibrium, the R-investor holds the tangency
portfolio, or the portfolio with maximum Sharpe
ratio.
CAPM: Assumption 2
Preferences Assumption

Result (2.2): For any asset i in equilibrium:
E[ri ]  rf  bi ( E[r ]  rf )
*
p

bi is the slope coefficient in the regression
ri  ai  bi rp*  ei
ri  return for stock i observed at time t
rp*  return for tangen cy portfolio observed at time t
CAPM: Assumption 2
Preferences Assumption

What if the R-investor finds an asset i such
that
*
E[ri ]  rf  bi ( E[rp ]  rf ) ?

He knows he can increase his Sharpe ratio
by tilting towards this asset.
But he can’t tilt



He already owns all shares of all assets!
Supply of shares is fixed.
CAPM: Assumption 2
Preferences Assumption
CAPM: Assumption 2
Preferences Assumption

What if the R-investor finds an asset i such
that
*
E[ri ]  rf  bi ( E[rp ]  rf ) ?

He knows he can increase his Sharpe ratio
by tilting away from this asset.
But he can’t tilt



He owns all shares of all assets.
No one around to borrow from!
CAPM: Assumption 2
Preferences Assumption
CAPM: Assumption 3
Supply=Demand Assumption

The representative investor must hold all shares
of all risky assets (supply=demand)

Result (3)

Representative Investor’s total equity in asset i:


Representative Investor’s total equity in the tangency
portfolio:


pricei*sharesi = market-cap of asset i
total market-cap of all risky assets.
Portfolio weight of asset i in tangency portfolio:

Value weight
CAPM


Given the assumptions, the tangency portfolio held
by the representative investor is the value-weighted
portfolio of all assets.
Notationally, we call the slope coefficient of any
asset on the value-weighted portfolio “beta” (b).
ri  ai  b i rm  ei
ri  return for stock i observed at time t
rm*  return for value - weighted portfolio observed at time t
CAPM

In equilibrium, expected returns, or
discount rates are given by:
E[ri ]  rf  bi ( E[rM ]  rf )
*
CAPM: Intuition 1

Assets with negative betas tend to pay high
realized returns when the market tanks

Assets with positive betas tend to pay low
realized returns when the market tanks

Asset with negative betas are like hedging
instruments (insurance contracts). They should
therefore have high prices and low expected
returns.
Pumpkin Crusher


You need $2 to feed your family tonight
Assets:

Two green pumpkins:


One white pumpkin



Each when opened, has $0.50 inside
When opened, has $1 inside
75 cents
Pumpkin crusher chooses pumpkins to crush at
random.




If a pumpkin is crushed, you get nothing
PC crushes white with 50% probability
PC crushes gold with 50% probability
PC never crushes more than 1 color
Pumpkin Crusher

If PC crushes white, how much do you
have tonight?


$1.75 (not enough to feed family)
If PC crushes gold, how much do you
have tonight?

$2.75 (more than enough to feed family)
Pumpkin Crusher



Assume you can buy gold, white, or
green pumpkins for 45 cents.
What to do?
Suppose you buy white



If gold is crushed: $3.30
If white is crushed: $1.30
Suppose you buy Green


If gold is crushed: 2.80
If white is crushed: 1.80
Pumpkin Crusher

Suppose you buy gold:



If white is crushed: 2.30
If gold is crushed: 2.30
At 45 cents expected returns are



White: 50/45-1=0.111
Gold: 50/45-1=0.111
Green: 50/45-1=0.111
Pumpkin Crusher

You probably would be willing to pay
more for gold than either white or
green



I
Gold acts as a hedging instrument –
“insurance” against bad times
For such assets, we are willing to accept
low or even negative expected returns.
Pumpkin Crusher

You probably would pay more for green
than white.



Green makes you better off in bad times
than white does, although not as much as
gold.
So: P(gold)>P(green)>P(white)
This implies for expected returns:


Gold<green<white
But then Gold has a negative Sharpe ratio,
and we want to buy it!
Stock Return
Stock Return
Intuition
x
x
y
y
Market Portfolio Return
Asset with a positive Beta
Market Portfolio Return
Asset with a negative Beta
“Hedging Instrument”
CAPM Intuition 2
ri  ai  b i rm  ei
Var (ri )  b i2Var (rm )  Var (ei )
(by rule #6)

But we know any fluctuations in ei are “diversified away”

The market does not require compensation to bear this
type of volatility risk.

The only firm-specific characteristic that matters in
determining expected returns is the firm’s beta.
CAPM Intuition 2

Total Risk of a Stock:
Var (ri )  b i2Var (rm )  Var (ei )

Systematic Risk of a Stock:
b i2Var (rm )

Idiosyncratic Risk of a Stock:
Var (ei )

Idiosyncratic risk gets diversified away

Beta indicates the amount of risk the stock is
contributing to the portfolio.
Example
Standard Deviation
A
.9
B
.2
Market
.18

Systematic Risk



Idiosyncratic Risk



A: (.05^2)*.18^2=0.00008
B: (1^2)*.18^2=.0324
A:
B:
0.9^2-.00008=0.81
0.2^2-.0324=0.0076
Percent of risk that is systematic


A: .00008/(.9^2)=.000010
B: .0324/.2^2=0.81
beta
.05
1
Example



“A” has more total risk, but virtually all of
this is diversified away in the market
portfolio.
The representative should not require a
premium to bear this risk
Therefore, the representative investor
should demand a higher expected return
to hold “B” even though its total risk is
lower than that of “A”.
Using the CAPM

How can we evaluate a fund manager?


Is manager’s Sharpe ratio above that of the
value-weighted portfolio?
Many fund managers try to create value
by specializing in a certain style:



Small-cap
Growth
Emerging market equity
Using the CAPM

Fund managers who are style focused
create “separate pieces” that we can
combine to create portfolios with high
Sharpe ratios if they are consistently
finding securities such that


E[r]>rf+bE[rM]-rf) and buying these
E[r]<rf+b(E[rM]-rf) and shorting these
Using the CAPM

If a fund manager is skilled, then for the
portfolio he manages:
E[rp]>rf+bp(E[rM]-rf)


We can increase the Sharpe ratio of our
portfolio by tilting toward the fund manager’s
portfolio.
If the above condition is not satisfied, then
we should not tilt towards the fund manager’s
portfolio – rather we should invest in a
passive value-weighted index.
Alpha

If a fund manager has been adding value in the
past then
rP  rf  b P (rM  rf )

Alpha is defined as
a  rP  rf  b P (rM  rf )

If a>0 then by tilting our portfolio toward the fund
manager’s portfolio, we can increase our Sharpe
ratio, if the past somehow reflects the future.
Example

Suppose a small cap manager has earned
an average return of 18% over the last 10
years.





CAPM beta of portfolio = 1.8
risk-free rate = 0.05
Average return on value-weighted portfolio =
0.13
What is fund manager’s alpha?
How could you create a portfolio using t-bills
and the market portfolio with the same beta?
Example

Fund manager’s Alpha:


Assume





.18-.05-1.8(.13-.05) = -1.4%
w1 is portfolio weight in asset 1
w2 is portfolio weight in asset 2
…
wn is portfolio weight in asset 3
Beta of portfolio is

w1b1+ w2b2+…+wnbn
Example

Beta of value-weighted portfolio=1.0
Beta of risk-free bonds = 0.0

Portfolio with same beta

w(1.0)+(1-w)(0)=1.8
w=1.8
So invest 180% of equity in market portfolio by
shorting the t-bill.

Expected Return of this portfolio:


.05+1.8*(.13-.05)=19.4%
1.4% higher than fund