Asset Pricing in Equilibrium:
CAPM and APT
1.
In the previous lecture, we considered the definition of
the returns on portfolios of assets, given the expected
returns and variances of assets with normal return
distributions. We were interested in portfolios with the
properties that they offered investors the highest returns
for a given level of risk (standard deviation of return).
2.
Here we want to show how some of those results are
used to get a model that defines what returns should be
for individual assets in equilibrium where all assets are
held, i.e., supply equals demand.
3.
We will consider two models of asset-pricing.
-
CAPM - older and not strongly supported by the
data. It is a special case of the APT when the
market portfolio is assumed to be the only factor.
-
APT – newer and more supported by data but the
inability to specify the priced factors has reduced its
use.
1.
Assumptions for CAPM
a.
Investors are risk averse and maximize expected
utility of end-of-period wealth.
b.
Investors are price takers with homogeneous
expectations about joint normal asset returns.
c.
A risk-free asset exists with borrowing and lending.
d.
Asset quantities are fixed, assets are marketable
and divisible.
e.
Asset markets are frictionless with information
available costlessly to all instantaneously.
f.
No taxes, regulations and short-selling restrictions.
3. These assumptions may seem reasonable because they
are frequently used in many other economic models.
In fact, I believe that the assumptions of costless information
and homogeneous expectations are too strong.
The purpose of a market is to provide a means for trade,
however, if everyone sees assets the same way, there is little
reason for individuals to trade assets except for liquidity
reasons.
4. The reasoning underlying the CAPM is somewhat circular.
Since we know everyone sees assets the same way and each
chooses an efficient portfolio, then since the market portfolio
is just a combination of these efficient portfolios, the market
portfolio will also be efficient.
The efficiency of the market portfolio is the primary testable
implication of the CAPM but because the market portfolio
includes all assets and data on all assets is unavailable, the
CAPM is largely untestable.
Derivation of the CAPM
Consider the market portfolio M consisting of all assets. The
weighting of each asset i in the market portfolio is
Wi = (market value of asset i)/(Market value of all assets)
1. Form a portfolio of a% of the risky asset i and (1-a)% of the
market portfolio, its mean and standard deviation are
E(Rp) = aE(Ri) + (1-a)E(Rm)
(Rp) = [a2i2 + (1-a)2m2 + 2a(1-a)im].5
2. Now define the change in each of these with respect to a
change in the weight a,
E(Rp)/a = E(Ri) - E(Rm)
(Rp)/a = .5[a2i2 + (1-a)2m2 + 2a(1-a)im]-.5
x [2ai2 - 2m2 + 2am2 + 2im – 4aim]
3. This is equivalent to saying, “consider the market portfolio
and assume we increase the weight a of asset i in the
market portfolio”. But if we add more asset i, by definition
we are no longer in equilibrium since the market portfolio
is an equilibrium portfolio. Thus, in equilibrium, a = 0.
Assuming a=0, then the previous derivatives are
E(Rp)/a = E(Ri) - E(Rm)
(Rp)/a = .5[m2]-.5[ - 2m2 + 2im ] = [im - m2 ]/ m
4. The risk-return tradeoff evaluated at point M is thus
[E(Rp)/a]/[(Rp)/a] = [E(Ri) - E(Rm)]/{[im - m2 ]/ m}
This defines the equilibrium risk-return trade-off. This is the
marginal return (marginal benefit to investor) offered for
marginal risk (marginal cost to investor) in equilibrium.
5. The slope of the investment opportunity set defined by
asset i and the market portfolio must be equal to the slope of
the capital market line (because the frontier defined by i and
M is a parabola that includes M). Assuming efficiency, the
portfolio tangent to the CML must be the market portfolio M.
Capital Market Line Slope = [E(Rm) - Rf]/ m
Therefore,
[E(Ri) - E(Rm)]/{[im - m2 ]/ m} = [E(Rm) - Rf]/ m
Solve for E(Ri)
E(Ri) = Rf + [E(Rm) - Rf] im/m2
This defines the Security Market Line (CAPM) where the
required return, E(Ri), of any security must fall on the line that
begins at Rf, has a slope of [E(Rm) - Rf] and a domain over
i = im/m2 .
 measures the units of risk for a particular asset.
6. Therefore, an asset’s return equals the risk-free return if
 = 0, i.e., its returns are uncorrelated with that of the
market’s.
E(Ri) = Rf
if i = 0
E(Ri) = Rf + [E(Rm) - Rf]i
if i  0
E(Ri) = risk-free return + (price of unit of risk)(units of risk)
7. The main implication of the CAPM is that all assets should
plot along the Security market line in (E(R), ) space.
Contrast this with what we had from the previous lecture that
all assets plot somewhere within or on a parabola in (E(R), )
, with the edge of the parabola defined by the minimum
variance for a given return.
8. What explains the difference between results for (E(R), )
space and (E(R), ) space ?
Consider what we call the Market Model.
Ri = ai + biRm + ei
It relates the return on the asset to the market return in a
regression where e is assumed to be an independent error
term. (versions of this are often used in empirical studies).
Assuming normal distributions, we can define the return
variance as
i2 = bi2m2 + e2
Here, the first term on the RHS is the market-related variance,
called nondiversifiable risk and the second term in assetspecific variance or diversifiable risk.
One can show that bi = i so that beta defines an asset’s
nondiversifiable portion of return variance, however, the
diversifiable portion differs for different assets, therefore,
even though all assets plot on a line in (E(R), ) space, they
are more spread out in (E(R), ) space.
Because no one will pay higher returns to those bearing more
diversifiable variance, assets that have the same betas will
have the same expected return even though some have larger
variances than others.
9. To get the risk of a portfolio of assets, one can simply take
the weighted average of the betas, where the weights are the
proportions of the total portfolio placed in each asset .
Annual return pairs f or the S&P 500 and Homestake Mining's stock
H
o
m
e
s
t
a
k
e
'
s
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1
0.8
0.6
Slope is 0.54
0.4
0.2
R
0
e
t
-0.2
u
r
n -0.4
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation betw een Homestake and the S&P 500 is 0.18 and its beta is 0.54
S&P Homestake
0.23
0.01
0.06
-0.26
0.32
0.1
0.18
0.09
0.05
0.39
0.17
-0.27
0.31
0.55
-0.03
-0.09
0.3
-0.16
0.08
-0.25
0.1
0.83
0.01
-0.19
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
Annual return pairs f or the S&P 500 and gasoline
0.8
G
a
s
o
l
i
n
e
R
e
t
u
r
n
0.7
0.6
0.5
Slope is -2.11
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
Gasoline's correlation w ith the S&P 500 is -0.47 and its beta is -2.11.
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
Gas
0.08
-0.1
0.09
-0.45
0.19
-0.04
-0.08
0.73
-0.33
-0.07
-0.29
0.2
Annual return pairs f or the S&P 500 and Gold
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
0.4
G
o
l
d
0.2
R
e
t
u
r
n
Slope is zero
S&P
0.23
0.06
0.32
0.18
0.05
0.17
0.31
-0.03
0.3
0.08
0.1
0.01
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
S&P 500 Return
The correlation betw een gold and the S&P 500 and its beta is approximately zero.
Gol
-0.1
-0.1
0
0.25
0.2
-0.1
0
-0.0
-0.0
-0.0
0.12
0.02
High Beta
Stock
Return
Market
Low Beta
During this time period the market rises, falls, and then rises again. A high (low)
beta stock varies more (less) than the market.
Positiv e Beta
Stock
Return
Negativ e Beta
Positive and negative beta stock returns
move opposite one another.
Applications and Extensions
1.
Conditional CAPM - The derivation of the CAPM above is
in a one-period context. In an intertemporal (multiperiod)
context, things can change. In particular, unless we
assume that a company’s risk does not change over
time, then Beta will change and this adds additional
uncertainty to the model.
2.
The CAPM Model can be restated in intertemporal form
Et-1 Rit = Rft + Bit-1 Et-1 [Rmt - Rft ]
Where Bit-1 = Covt-1(Rit , Rmt )/Var(Rmt)
To get unconditional expectations we us the Law of Iterated
Expectations to get
Et-1 E(Rit) = Rft + E(Bit-1 Et-1 [Rmt - Rft ])
Which equals
E(Rit) = Rft + E(Bit-1 Et-1 [Rmt - Rft ])
From the definition of covariance we can substitute for the
expectation of a product of random variables to get
E(Rit) = Rft + E(Bit-1)Et-1 [Rmt - Rft ] + Cov(Bit-1, Et-1 [Rmt - Rft ])
Now, this ICAPM looks similar to the CAPM except for an
additional term that measures the degree to which a
company’s beta moves with the market risk premium.
•
The extra term can be explained as follows. Consider
two assets with the same unconditional betas (E(Bit-1)),
that is, the same average beta over the full range of the
economic cycle. Asset 1, whose beta tends to be larger
than average in good times (when Et-1[Rmt - Rft ] is large)
but smaller in bad times should have a larger expected
return. Note that Et-1[Rmt - Rft ] is always positive so the
asset will earn less in the bad times because its beta is
smaller than average in bad times. This asset is riskier
than asset 2 whose beta stays fixed at the average over
time. That asset earns more in the bad times than the
asset 1.
•
The CAPM is often used to estimate a company’s cost of
equity.
a. First, estimate a company’s beta with the market
model and 3-5 years of return data on the stock and the
market portfolio (usually the S&P 500).
b. Next, assume that the risk-free rate is the long-term
U.S. Treasury bond rate, now at about 6%.
c. Also assume that the expected market return is, say
12%, which is the average return on the S&P 500 over
the last 70 years.
d. Plug these numbers into the CAPM, e.g.
ki = .06 + 1.2[.12 - .06] = .132
for i = 1.2
2. There are many extensions of the CAPM. Most are
associated with relaxing one of the assumptions. For
example, human capital is not tradable (no slavery) so the
model has been derived under the conditions of untradable
assets.
The most important extension involves the case where no
riskless asset exists.
When no riskless asset exists, we can still find a minimum
variance portfolio for which  = 0, and the return on this
portfolio Rz can be substituted for the riskless rate in the
CAPM. All of the properties of the CAPM, like linearity and
beta as the risk measure, still hold.
This form of the of the CAPM is called the two-factor model.
Recall from the previous lecture that any portfolio on the
efficient frontier could be seen as a combination of the
minimum variance portfolio and another portfolio on the
frontier. Here, the other portfolio is the market portfolio.
3. Other types of extensions of the CAPM are typically solved
by adding another portfolio to the model in addition to the
market portfolio. Then we will get, say, three fund separation.
The extra fund is part of the CAPM in order to allow for a new
risk to be hedged. For example, a portfolio that captures the
covariance between the market portfolio and untraded human
capital allows individuals to either reduce their human capital
risk (short the portfolio) or increase it (purchase the portfolio).
Something like APT with a human capital factor.
4. The most damaging assumption is the assumption that
investors have homogeneous expectations. When this
assumption does not hold, the market portfolio will be
inefficient and thus the CAPM is not testable.
5. Early tests of the CAPM supported the model but most
recent tests have rejected it. One study showed that
earlier studies largely relied on a unique sample period
where the model fit well – in other samples it performs
poorly.
6. Roll’s critique of the CAPM suggests that
a.
To test the CAPM we need data on all assets in the
market portfolio and one must show that the market
portfolio is efficient. This is nearly impossible to do.
b.
If we use an ex post efficient portfolio to measure
investment performance, then efficient set mathematics
implies that no security will have abnormal performance
– all will lie on the SML.
c.
If we use an ex post inefficient set to measure
performance, any result is possible so no inference
concerning performance should be made. This means
that if we find that all assets (don’t) lie on the SML, the
CAPM is supported (rejected) assuming we measured
the market portfolio correctly, or we could have gotten
lucky (unlucky) with an incorrect market portfolio.
7. What underlies Roll’s critique is found in the previous class
on selection of the investment opportunity set. NOTE: The
definitions of the scalars have been changed to Ingersol’s
(1987, p. 84) definitions.
7a. When there are many assets instead of just two we get,
p2 = [A2 – 2B + C]/D
Where A = 1’-11 > 0, B = 1’-1E, C = E’ -1E > 0, D=AC – B2 >
0 and -1 is the inverse of the variance-covariance matrix of
returns and E is the vector of expected returns. This is also a
parabola.
7b. The portfolio weights (wi) which are very complex can be
represented more easily for the general case of many assets
in matrix form as follows.
W =  -11/A +  -1E/B
where  = A[C – B]/D and  = B[A – B]/D.
One can show that  +  = 1. Also note that this shows that
the weights are functions of the minimum variance portfolio
[1/A] and the portfolio represented by [E/B]. This will be called
“two fund separation”.
7c. Roll’s point: portfolio [E/B] can be any efficient portfolio
and [1/A] is its associated minimum variance portfolio
8. Why is the only test of the model a test of the efficiency of
the market portfolio?
Because the model is an “equilibrium” model it requires that
supply equals demand for all assets. This implies that the
weight of each asset in the market portfolio is its value as a
percent of total market value.
With homogeneous expectation assumed, then we know all
investors view asset distributions the same and will hold
assets in a way to minimize the variance of their returns. This
requires everyone to have the same portfolio weights for the
risky assets they hold.
Summed across individuals, this happens only when the asset
weights equal their percent of total market asset value.
Equilibrium and homogeneous expectations severely limit the
number of observable implications of the model.Thus, the only
testable proposition is to see whether the market portfolio is
efficient, i.e., ex ante, could investors do better if they held
their risky assets in some other portfolio.
Hansen-Jagannathan (1991 JPE)
lower bound condition on asset
pricing kernel (SDF) and relation to
risk-free and risk premium puzzles.
The price of a simple security at time t (Pt) that pays an
uncertain cash flow at time t+1 (Xt+1) is:
Pt = Et[dt+1Xt+1]
where dt+1 is the stochastic discount factor.(SDF)
U
dt  1  MRSt  1  Ct  1
U
Ct
Note that dt+1 is random because it depends upon the random
amount of consumption in t+1. If consumption is high in t+1,
then dt+1 will be small because the marginal utility of
consumption in t+1 will be small. This makes the asset price
small because consumers value the cash flow Xt+1 less. Any
asset is priced according to the cash flow it provides multiplied
by the same dt+1, hence, the SDF is called a pricing kernel.
Note that the expectation of dt+1 equals the risk free discount
rate (1/(1 + Rf)) because across all states, the average return
equals the risk free rate. That is, form a portfolio by
purchasing one state security for each possible state. The
return for that portfolio equals the risk free rate.
Hansen-Jagannathan (1991 JPE)
lower bound condition on asset
pricing kernel (SDF) and relation to
risk-free and risk premium puzzles.
If we divide through by current time t price to get return
1 = Et[dt+1Rt+1]
This holds for all states of nature so it holds unconditionally
and for any two assets such as the market portfolio (m) and
risk free asset (f), or the difference between them is:
E[d(Rm – Rf)] = 0
or
E(d)E(Rm-Rf) + cov(d, Rm-Rf) = 0
And after substitutions for cov and rearranging
E(Rm-Rf)/m-f = -ρd,mf[d/E(d)]
Because correlation cannot exceed |1| then,
d/E(d) = d(1 + Rf) > | E(Rm-Rf)/m-f |
For the CCAPM the SDF is the MRS of current for future
consumption and its expectation equals 1/(1+Rf) which equals
say, 0.96 (Rf=0.04). The average risk premium equals 0.062
and has a standard deviation of 0.167. This implies that
d > 0.96(0.062/0.167) = 0.355
But measured standard deviation of d is much lower at 0.002.
The stand. dev. Is small because consumption is smooth. This
might hold if Rf was much larger or Rm-Rf was much smaller.
These are called the risk free rate and risk premium puzzles.
APT – Arbitrage Pricing
1. Assumptions
a.
Perfect competition and frictionless markets.
b.
The number of assets n must be much larger than the
number of factors that generate returns.
c.
The unsystematic risk component of asset i is ei. It must
be independent of all factors and all other assets’ ej.
d.
Investors have homogeneous beliefs that asset returns
are governed by a linear k-factor model such as
Ri = E(Ri) + i = E(Ri) + bi1F1 + bi2F2 + …. + bikFk + ei
where Ri = asset i’s random return.
E(Ri) = asset i’s expected return.
bik = the sensitivity of asset i’s return to the kth factor.
Fk = the mean zero kth common factor (non-zero factors
can be restated as deviations from their mean)
i, ei = the mean zero error terms for asset i.
2. The APT relies on the simple concept of arbitrage. The noarbitrage condition states that in equilibrium, portfolios of
assets with
a.
No net investment – i.e., some assets purchased and
some assets sold short.
b.
No net risk – i.e. no probability of loss should earn no
return. Otherwise, infinite riskless returns are available.
Interpreting the APT ReturnGenerating Model
The linear k-factor model is:
Ri = E(Ri) + i = E(Ri) + bi1F1 + bi2F2 + …. + bikFk + ei
It generates the actual return for a particular period, e.g. day.
It says that stock i’s return today will be its expected return as
long as the realized values of all the factors equals their
expected values of zero, and the firm-specific return e is
zero. Of course, the F and e are random, so the actual
return for a particular day will depend on their realized
values as well as a firm’s factor betas b. This holds also
for the CAPM model with the market return Rm as:
Ri = E(Ri) + B[Rm – E(Rm)] + ei
Here the market return is demeaned to give a zero mean
factor.
The APT is a model of expected returns, not actual returns.
Below we will derive a two factor APT from the return
generating model above to get.
E(Ri) = Rf + [E(RM) - Rf ]biM + [E(RO) - Rf ]biO
The factor risk premiums are in square brackets and a
particular firm’s factor loadings (betas) are applied to the
premiums to obtain its expected return.
Note that the Fama-French factor data are zero investment
portfolio returns but they are not mean zero returns. For
the size factor, you go long small and short large to get:
{.5[E(Rs) – Rf] - .5[E(RL) – Rf]} = .5[E(Rs) – E(RL)].
Derivation of APT
1.
Create an arbitrage portfolio. Let wi be the change in the
dollar amount invested in asset i as a percentage of an
individual’s wealth. Then the summation over longs and
shorts balance so
iwi = 0.
2. The returns on this portfolio are then
Portfolio Return = Rp = iwiRi
= iwiE(Ri) + iwibi1F1 + iwibi2F2 + …. + iwibikFk + iwiei
3. Select the the weights wi so as to eliminate both systematic
and unsystematic risk. This can be done by selecting
wi = 1/n with n large
and iwibik = 0 for each factor k.
This means wi is tiny, and that the arbitrage portfolio “betas”
for each factor are zero, so that all risk is eliminated and
Portfolio Return = Rp = iwiRi = iwiE(Ri)
4. If markets are in equilibrium then Rp = iwiRi = iwiE(Ri) =0.
This riskless portfolio, requiring no net investment, must
earn zero return because, otherwise, everyone would
want to own the portfolio and prices would change –
contradicting the assumption of equilibrium.
5. Restated using vectors we have:
w’1 = 0, w’bk = 0 for each factor k, implies w’E = 0
where E is the vector of expected returns.
A theorem in linear algebra states that if the fact that a vector
(here, w) is orthogonal to k+1 vectors (here, 1 and each of k
vectors bk ) implies it is orthogonal to another vector (here, E)
then this other vector can be expressed as a linear
combination of the first k+1 vectors.
This implies that
E(Ri) = 01 + 1bi1 + 2bi2 + …. + kbik
Here, the i represent the price of risk specific to each factor
k.
6. An intuitive restatement: When we selected a portfolio with
w’1 = 0 and w’bk = 0 for each factor k, we know that we have
spanned the expected return space because w’E = 0. w’E = 0
means that there are no net returns to this portfolio, i.e., there
is nothing left in the vector of expected returns to be
explained. Therefore, it must be that E is simply a linear
combination of the vector 1 and the k vectors bk.
Of course, this is not so surprising. We started by assuming
that all investors believed that asset returns are governed by a
linear model with k factors. This just proves that greedy, risk
averse investors will, in fact, price assets to reflect these
factor risks. If they don’t, then arbitrage is possible.
7. A riskless asset has bik = 0 for all k so that
Rf = 0
And therefore
E(Ri) = Rf + 1bi1 + 2bi2 + …. + kbik
8. Suppose we assume that the only risk factor is the general
market risk represented by the market portfolio M and returns
are joint normal, then
E(Ri) = Rf + MbiM
This is just the CAPM where the units of risk are biM = i and
the price per unit of risk is M = [E(RM) - Rf ].
To see how this comes about, note that this must hold for the
market portfolio itself, which has a biM = i =1. Therefore,
E(RM) = Rf + M(1)
Or
M = [E(RM) - Rf ].
Now we are back to the CAPM where for any asset i,
E(Ri) = Rf + [E(RM) - Rf ]biM
9. Now suppose there are 2 factors, the market portfolio and
oil prices. The APT requires that expected returns follow
E(Ri) = Rf + MbiM + ObiO
We can select a portfolio such that it has biO =1 but biM = 0.
This portfolio must also follow the relationship so that
E(RO) = Rf + M(0) + O(1)
Or
O = [E(RO) - Rf ]
The two factor APT for the expected return of any asset i is
E(Ri) = Rf + [E(RM) - Rf ]biM + [E(RO) - Rf ]biO
10. More generally for factors numbered 1 to k, select k
portfolios such that each loads only on one factor, that is,
each portfolio’s return can represent the return on a factor.
E(Ri) = Rf + [E(R1) - Rf ]bi1 + [E(R2) - Rf ]bi2 + … + [E(Rk) - Rf ]bik
The notation often used is to have k = E(Rk) which gives
E(Ri) = Rf + [1 - Rf ]bi1 + [2 - Rf ]bi2 + … + [k - Rf ]bik
Like the CAPM beta, each bik = ik/ k2
In a one factor APT depicted above, the expected return (i) risk (bi) combinations for all of the assets must lie on a line
that passes through rf. Thus all assets must fall on this
Security Market Line or else arbitrage exists. In the case
above, we would sell short a portfolio of assets 1 and 3 which
has a risk level of b2. Then take the short sale proceeds and
purchase asset 2. This portfolio is riskless but should
produce a positive return. If APT is correct, this should not be
possible.
Advantages of the APT
Over the CAPM
1.
No assumptions about return distributions.
2.
No assumptions on utility except greed and risk aversion.
3.
APT applies to any subset of assets – don’t need to
know the return distribution of all assets.
4.
No special role for the market portfolio – easier testing.
5.
Easy to extend to multi-period framework.
6.
Most important: Many factors can impact returns – if the
risk-return world is multi-faceted, using only the market
return will give poorer return predictions – like being in a
plane trying to land and offered only the fact that you are
200 miles from the destination (no latitude, longitude or
altitude).
For example, some Wall Street investment banks used
CAPM type models to determine that some Japanese
stock warrants were under-priced. They bought the
warrants and hedged using Japanese futures.
Unfortunately, more than just a market factor was
important so their hedges worked poorly and they lost.
Problems and Applications
1.
Major empirical problem: the factors are not specified,
making testing more difficult. Some factors people have
used are
a.
Industrial production
b.
Changes in default risk premiums (AAA – Baa)
c.
Yield curve (Long – short treasury yields)
d.
Unanticipated inflation
2. Like the CAPM, the APT can be used to estimate the cost
of equity for a firm given its betas on the factors.
3. The factor betas for a portfolio are just a linear combination
of the individual asset betas with their value weights.