1. finance
2. personal finance
3. financial planning
4. asset and portfolio management

Equilibrium Asset Returns
George Pennacchi
University of Illinois
George Pennacchi
Equilibrium Asset Returns
University of Illinois
1/ 38
Introduction
We analyze the Intertemporal Capital Asset Pricing Model
(ICAPM) of Robert Merton (1973).
The standard single-period CAPM holds when investment
opportunities are constant, but with changing investment
opportunities, a multi-beta ICAPM results.
Breeden (1979) shows that this multi-beta ICAPM can be
written as a single “consumption beta” CAPM (CCAPM).
We also analyze the general equilibrium production economy
model of (Cox, Ingersoll and Ross,1985) that is useful for
pricing contingent claims.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
2/ 38
ICAPM Model Assumptions
Individuals can trade in a risk-free asset paying rate of return
of r (t) and in n risky assets whose rates of return are
dSi (t) = Si (t) =
where i = 1; :::; n, and (
i
i
(x; t) dt +
dzi )(
j
dzj ) =
i
(x; t) dzi
ij
dt.
(1)
The k state variables follow the process:
dxi = ai (x; t) dt + bi (x; t) d
i
(2)
where i = 1; :::; k, and (bi d i )(bj d j ) = bij dt and
( i dzi )(bj d j ) = ij dt.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
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Constant Investment Opportunities
When r and the i ’s, i ’s, and ij ’s are constants, we showed
previously that it is optimal for all individuals to choose the
risky assets in the relative proportions
k
n
X
kj ( j
j =1
n
n X
X
=
ij ( j
r)
(3)
r)
i =1 j =1
This “single” risky asset portfolio’s mean and variance is
n
X
i =1
George Pennacchi
Equilibrium Asset Returns
i
i
2
n X
n
X
i j
ij :
(4)
i =1 j =1
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CAPM Equilibrium
Similar to our derivation of the single-period CAPM, we argue
that in equilibrium this common risky-asset portfolio must be
the market portfolio; that is, = m and 2 = 2m .
Moreover, the continuous-time market portfolio is exactly the
same as that implied by the single-period CAPM.
Thus, asset returns in this continuous-time environment
satisfy the same relationship as the single-period CAPM:
i
where
i
=
George Pennacchi
Equilibrium Asset Returns
r=
i
(
m
r) ;
i = 1; : : :, n
(5)
2
im = m .
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5/ 38
CAPM Implications
Thus, the constant investment opportunity set assumption
replicates the standard, single-period CAPM.
Yet, rather than asset returns being normally distributed as in
the single-period CAPM, the ICAPM has asset returns that
are lognormally distributed.
While the standard CAPM results continue to hold for this
more realistic intertemporal environment, the assumptions of
a constant risk-free rate and unchanging asset return means
and variances are untenable.
Clearly, interest rates vary over time, as do the volatilities of
assets such as common stocks.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
6/ 38
Stochastic Investment Opportunities
For a single state variable, x, the system of n equations that
an individual’s portfolio weights satisfy are:
0 =
A(
i
r) +
n
X
ij ! j
W
H i ; i = 1; : : : ; n (6)
j =1
where A = JW =JWW = UC = [UCC (@C =@W )] and
H = JWx =JWW = (@C =@x) = (@C =@W ).
Rewrite (6) in matrix form, and use the superscript p to
denote the values for the p th person (individual):
Ap (
r e) = ! p W p
Hp
(7)
where = ( 1 ; :::; n )0 , e is an n-dimensional vector of ones,
! p = (! p1 ; :::; ! pn )0 and = ( 1 ; :::; n )0 .
George Pennacchi
Equilibrium Asset Returns
University of Illinois
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Aggregate Asset Demands
P
Sum (7) across all persons and divide by
re = a
P
P
p=
p WP
p
p
p! W =
p
p
h
P
Ap :
(8)
P
p
p
Ap , h
pH =
p A , and
is the average investment in each
asset. These must be the market weights.
The i th row (i th asset excess return) of equation (8) is
wherePa
p
Wp
i
Pre-multiply (8) by
0
m
r =a
im
h
(9)
i
to obtain
r =a
2
m
h
(10)
mx
where mx = 0 is the covariance between the market
portfolio and the state variable, x.
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Equilibrium Asset Returns
University of Illinois
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Portfolio that Best Hedges the State Variable
1
De…ne
1 , which are the weights of the portfolio that
e0
best hedge changes in the state variable, x.
The expected excess return on this portfolio is found by
pre-multiplying (8) by 0 :
r =a
m
h
(11)
x
where m and x are the hedge portfolio’s covariances with
the market portfolio and the state variable, respectively.
Solving the two linear equations (10) and (11) for a and h,
and substituting them back into equation (9) gives:
i
r=
x
im
2
x
m
i
mx
m
m
(
m
r )+
i
2
m
2
m
x
im mx
mx
r
m
(12)
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Equilibrium Asset Returns
University of Illinois
9/ 38
ICAPM
It can be shown that (12) is equivalent to
i
r
2
im
2 2
m
m
(
m
i
=
m
i
(
2
m
r) +
m
r) +
i
2
m
2 2
m
im m
2
m
r
(13)
r
i
where i is the covariance between the return on asset i and
that of the hedge portfolio.
i = 0 i¤ i = 0. If x is uncorrelated with the market so that
m = 0, equation (13) simpli…es to
i
r=
im
2
m
(
m
r) +
i
2
r
(14)
Equation (13) generalizes to multiple state variables with an
additional “beta” for each state (c.f., APT).
George Pennacchi
Equilibrium Asset Returns
University of Illinois
10/ 38
Extension to State-Dependent Utility
If an individual’s utility is a¤ected by the state of economy, so
that U (Ct ; xt ; t), the form of the …rst order conditions for
consumption (Ct ) and the portfolio weights (! i ) remain
unchanged and equation (13) continues to hold.
The only change is the interpretation of H, the hedging
coe¢ cient. Taking the total derivative of envelope condition
JW = U C :
@C
JWx = UCC
+ UCx
(15)
@x
so that
@C =@x
UCx
H=
(16)
@C
@C =@W
UCC @W
implying that portfolio holdings minimize the variance of
marginal utility.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
11/ 38
Breeden’s Consumption CAPM
Substitute in for Ap and H p in equation (7) for the case of k
state variables and rearrange to obtain:
!pW p =
UCp
p
p
UCC CW
1
(
1
r e)
p
p
p
Cpx =CW
(17)
0
p
@C
, and is the
where CW
= @C p =@W p , Cpx = @C
@x1 ::: @xk
n k asset return - state variable covariance matrix whose
p
i,j th element is ij . Pre-multiplying (17) by CW
:
UCp
p (
UCC
r e) =
p
Wp CW
+ Cpx
(18)
where Wp is n 1 vector of covariances between asset
returns and investor p’s wealth.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
12/ 38
Covariance between Asset Returns and Consumption
Using Itô’s lemma, individual p’s optimal consumption,
C p (W p ; x; t), follows a process whose stochastic terms for
dC p are
p
CW
! p1 W p
1 dz1
+ ::: + ! pn W p
n dzn
+(b1 d
1
b2 d
p
2 :::bk d k ) Cx
(19)
The covariance of asset returns with changes in individual p’s
consumption are given by calculating the covariance between
each asset (having stochastic term i dzi ) with the terms given
in (19). Denoting Cp as this n 1 vector of covariances:
Cp
George Pennacchi
Equilibrium Asset Returns
=
p
Wp CW +
Cpx
(20)
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Consumption Covariance
The right-hand sides of (20) and (18) are equal, implying:
Cp
=
UCp
p (
UCC
r e)
(21)
De…ne C as aggregate consumption per unit time and de…ne
T as an aggregate rate of risk tolerance where
X Up
C
T
(22)
p
U
CC
p
Then aggregate (21) over all individuals to obtain
re = T
1
C
(23)
where C is the n 1 vector of covariances between asset
returns and changes in aggregate consumption.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
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Excess Expected Returns on Risky Assets
Multiply and divide the right-hand side of (23) by current
aggregate consumption to obtain:
1
r e = (T=C )
ln C
(24)
where ln C is the n 1 vector of covariances between asset
returns and changes in log consumption growth.
Let a portfolio, m, have weights ! m that pre-multiply (24):
m
r = (T=C )
1
m;ln C
(25)
where portfolio m’s expected return and covariance with
consumption growth is m and m;ln C . Portfolio m may or
may not be the market portfolio.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
15/ 38
Consumption CAPM
Use (25) to substitute for (T=C )
re = (
= (
1
in (24):
ln C = m;ln C ) ( m
C = mC ) ( m
r)
r)
(26)
where C and mC are the “consumption betas” of all asset
returns and portfolio m’s return.
The consumption beta for any asset is de…ned as:
iC
= cov (dSi =Si ; d ln C ) =var (d ln C )
(27)
Equation (26) says that the ratio of expected excess returns
on any two assets or portfolios of assets is equal to the ratio
of their betas measured relative to aggregate consumption.
Hence, the risk of a security’s return can be summarized by a
single consumption beta.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
16/ 38
A Cox, Ingersoll, and Ross Production Economy
The ICAPM and CCAPM are not general equilibrium models
since they take the form of equilibrium asset price processes as
given.
However, their asset price process can be justi…ed based on
the Cox, Ingersoll, and Ross (1985) continuous-time,
production economy model.
The model assumes that there is a single good that can be
either consumed or invested. This “capital-consumption”
good can be invested in any of n di¤erent risky technologies
that produce an instantaneous change in the amount of this
good.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
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Model Assumptions
If an amount i is physically invested in technology i, then the
proportional change in the good produced is
d i (t)
=
i (t)
i
(x; t) dt +
i
(x; t) dzi , i = 1; :::; n
(28)
where ( i dzi )( j dzj ) = ij dt. The rate of change in the
invested good produced has expected value and standard
deviation of i and i .
Note that each technology displays “constant returns to
scale” and i and i can vary with time and with a k 1
vector of state variables, x(t).
Thus, the economy’s technologies for transforming
consumption into more consumption re‡ect changing
(physical) investment opportunities.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
18/ 38
Model Assumptions cont’d
The i th state variable is assumed to follow the process
dxi = ai (x; t) dt + bi (x; t) d
i
(29)
where i = 1; :::; k, and (bi d i )(bj d j ) = bij dt and
( i dzi )(bj d j ) = ij dt.
If each technology is owned by an individual …rm, …nanced
entirely by shareholders’equity, then the rate of return on
shareholders’equity of …rm i, dSi (t) =Si (t), equals the
proportional change in the value of the …rm’s physical assets
(capital), d i (t) = i (t).
Here, dSi (t) =Si (t) = d i (t) = i (t) equals the instantaneous
dividend yield where dividends come in the form of a physical
capital-consumption good.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
19/ 38
Model Assumptions cont’d
CIR’s speci…cation allows one to solve for the equilibrium
prices of securities other than those represented by the n risky
technologies.
This is done by imagining there to be other securities that
have zero net supplies.
Assuming all individuals are identical in preferences and
wealth, this amounts to the riskless investment having a zero
supply in the economy, so that r is really a “shadow” riskless
rate.
Yet, this rate would be consistent, in equilibrium, with the
speci…cation of the economy’s other technologies.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
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Solving for the Equilibrium Riskless Rate
An individual’s consumption and portfolio choice problem
allocates savings to …rms investing in the n technologies.
An equilibrium is de…ned as a set of interest rate,
consumption, and portfolio weight processes
fr ; C ; ! 1 ; :::; ! n g such that the individual’
s …rst order
P
conditions hold and markets clear: ni=1 ! i = 1 and ! i
0.
Since the capital-consumption good is physically invested in
the technologies, the constraint against short-selling, ! i 0,
applies.
Because, in equilibrium, the representative individuals do not
borrow or lend, the situation is exactly as if a riskless asset did
not exist.
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Equilibrium Asset Returns
University of Illinois
21/ 38
Optimization without a Riskless Asset
Consider the individual’s problem as before except that the
process for wealth excludes a risk-free asset:
Z T
max
Et
U (Cs ; s) ds + B(WT ; T )
(30)
C s ;f ! i ;s g;8s ;i
t
subject to
dW =
n
X
!i W
i
dt
Ct dt +
i =1
P
and ni=1 ! i = 1 and ! i
consumption is
0 =
George Pennacchi
Equilibrium Asset Returns
n
X
!i W
i
dzi
(31)
i =1
0. The …rst order condition for
@U (C ; t)
@C
@J (W ; x; t)
@W
(32)
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Optimal Portfolio Weights with Short Sale Constraints
Let be the Lagrange multiplier for
…rst order conditions for ! i is
i
0 =
@J
@W
iW
+
n
@2J P
@W 2 j =1
ij ! j
Pn
i =1
W2 +
i = 1; : : : ; n
i !i
! i = 1. Then the
k
@2J P
@xi @W j =1
ij W
(33)
Kuhn-Tucker conditions (33) imply that if i < 0, then
! i = 0, so that i th technology is not employed.
Assuming (28) and (29) are such that all technologies are
employed so i = 0 8i, then the solution is
!i =
JW
JWW W
George Pennacchi
Equilibrium Asset Returns
n
P
j =1
ij j
k P
n
P
JWxm
m=1 j =1 JWW W
ij jm +
0
n
P
JWW W 2 j =1
(34)
ij
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Equilibrium Portfolio Allocation
Using matrix notation, (34) becomes
! =
A
W
1
A
JW W 2
1
e+
k
X
Hj
W
1
j
(35)
j =1
where A = JW =JWW , Hj = JWxj =JWW , and
0
j = ( 1j ; :::; nj ) .
The weights are a linear combination of k + 2 portfolios.
The …rst two portfolios are mean-variance e¢ cient portfolios:
1
is the portfolio on the e¢ cient frontier that is tangent
to a line drawn from the origin (a zero interest rate) while
1 e is the global minimum variance portfolio.
1
The last k portfolios,
j , j = 1; :::; k, hedge against
changes in the technological investment opportunities.
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Equilibrium Asset Returns
University of Illinois
24/ 38
Equilibrium Consumption and Portfolio Weights
The proportions of these k + 2 portfolios chosen depend on
the individual’s utility.
An exact solution is found in the usual manner of substituting
(35) and C = G (JW ) into the Bellman equation.
For speci…c functional forms, a value for the indirect utility
function, JP
(W ; x; t) can be derived. This, along with the
restriction ni=1 ! i = 1, determines the speci…c optimal
consumption and portfolio weights.
Since in the CIR economy the riskless asset is in zero net
supply, portfolio weights in (35) must be those chosen by the
representative individual even if o¤ered the opportunity to
borrow or lend at rate r .
George Pennacchi
Equilibrium Asset Returns
University of Illinois
25/ 38
Portfolio Allocation with a Riskless Asset
Recall that portfolio weights for the case of including a
risk-free asset are
A
! =
W
1
k
X
Hj
r e) +
W
(
1
j;
i = 1; : : : ; n
(36)
j =1
(35) and (36) are identical when r = = (JW W ). Hence,
r
=
(37)
WJW
= !
0
W
!
A
0
! +
k
X
Hj
j =1
A
!
0
j
(37) is the same as (10) extended to k state variables.
Hence, the ICAPM and CCAPM hold for the CIR economy.
George Pennacchi
Equilibrium Asset Returns
University of Illinois
26/ 38
Valuing Contingent Claims in Zero Net Supply
The CIR model also can be used to …nd the equilibrium
“shadow prices” of other contingent claims.
Suppose the payo¤ of a zero-net-supply contingent claim
depends on wealth, time, and the state variables, so that
P (W ; t; fxi g). Itô’s lemma implies
P
P
dP = uPdt + PW W ni=1 ! i i dzi + ki=1 Pxi bi d i (38)
where
uP = PW W !
+
Pk
George Pennacchi
Equilibrium Asset Returns
i =1
0
C +
PWxi W !
0
i
Pk
i =1
+
Pxi ai + Pt +
PWW W 2
!
2
1 Pk Pk
Px x bij
2 i =1 j =1 i j
0
!
(39)
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Expected Return under ICAPM
The ICAPM relation (9) extended to k states is
u=r+
W
Cov (dP=P; dW =W )
A
Pk
i =1
Hi
Cov (dP=P; dxi )
A
(40)
Noting that the representative agent’s wealth is the market
portfolio:
Pk Hi
1
Cov (dP; dW )
Cov (dP; dxi )
i =1
A
A
P
1
= rP +
PW W 2 ! 0 ! + ki=1 Pxi W ! 0 i
A
Pk Hi
P
PW W ! 0 i + kj=1 Pxj bij
(41)
i =1
A
uP = rP +
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Equilibrium Asset Returns
University of Illinois
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Equilibrium Partial Di¤erential Equation
Equating (39) and (41) and recalling the value of the
equilibrium risk-free rate in (37), we obtain a partial
di¤erential equation for the contingent claim’s value:
0 =
1 Pk Pk
PWW W 2 0
! ! +
Px x bij +
2
2 i =1 j =1 i j
Pk
0
C ) + Pt
i + PW (rW
i =1 PWxi W !
Pk
Pk Hj bij
W 0
! i + j =1
rP (42)
i =1 Pxi ai
A
A
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Equilibrium Asset Returns
University of Illinois
29/ 38
An Example with Log Utility
Let U(Cs ; s) = e
s
ln (Cs ) and B (WT ; T ) = e
T
ln (WT ).
Then we previously showed that J (W ; x; t) =
d (t) e t ln (Wt ) + F (x; t) where
d (t) = 1 1 (1
) e (T t) , so that Ct = Wt =d (t).
Since JWxi = 0, Hi = 0, and A = W , the portfolio weights in
(35) are
1
! =
( r e)
(43)
where r = = (JW W ).
George Pennacchi
Equilibrium Asset Returns
University of Illinois
30/ 38
Market Portfolio Weights
Market clearing requires e0 ! = 1, and solving for r (t):
r=
1
e0
e0
1
1e
(44)
Substituting (44) into (43),the equilibrium portfolio weights
are:
e0 1
1
1
! =
e
(45)
0
1
e
e
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Equilibrium Asset Returns
University of Illinois
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Square Root Process State Variable
Assume that a single state variable, x (t) ; a¤ects all
production processes:
p
d i = i = bi x dt + bi xdzi , i = 1; :::; n
(46)
where bi and bi are assumed to be constants and the state
variable follows the square root process
p
(47)
dx = (a0 + a1 x) dt + b0 xd
where dzi d =
i dt.
If a0 > 0 and a1 < 0, x is a nonnegative, mean-reverting
random variable.
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Equilibrium Asset Returns
University of Illinois
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Equilibrium Interest Rate Process
Write the technologies’n 1 vector of expected rates of
return as = b x and their n n matrix of rate of return
covariances as = b x.
Then from (44), the equilibrium interest rate is
r=
e0 b
1b
e0 b
1e
1
x= x
(48)
where
e0 b 1 b 1 =e0 b 1 e is a constant.
Thus risk-free rate follows the “square root” process:
p
dr = dx = (r r ) dt +
rd
(49)
p
where
a1 > 0, r
a0 =a1 > 0, and
b0
.
2 , if r (t) > 0 it remains positive at all future
When 2 r
dates, as would a nominal interest rate.
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Equilibrium Asset Returns
University of Illinois
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Valuing Default-Free Discount Bonds
Consider the price of a default-free bond that matures at
T t.
PW , PWW , and PWx in (42) are zero, and since r = x, the
bond’s price can be written as P (r ; t; T ). The PDE (42)
becomes
2r
2
Prr + [ (r
r)
r ] Pr
rP + Pt = 0
(50)
b0 b. !
b equals the right-hand side of equation
where
!
(45) but with replaced by b and replaced by b , while b
is an n 1 vector whose i th element is bi i .
r = ! 0 is the covariance of interest rate changes with
(market) wealth, or the interest rate’s “beta”.
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Equilibrium Asset Returns
University of Illinois
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Solution for Bond Prices
Solving PDE (50) subject to P (r ; T ; T ) = 1 leads to
where
=T
A( )
"
P (r ; t; T ) = A ( ) e
q
t,
( + )2 + 2
( +
2 e( + +
+ ) (e
B ( )r
(51)
2,
)2
1) + 2
#2
r=
2
(52)
2 e
1
(53)
( + + ) (e
1) + 2
Note that the bond price is derived from an equilibrium model
of preferences and technologies rather than the absence of
arbitrage (c.f., Vasicek).
B( )
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Equilibrium Asset Returns
University of Illinois
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Equilibrium Bond Price Process
Note that Itô’s lemma implies the bond price follows
1
dP = Pr dr + Prr 2 rdt + Pt dt
2
1
=
Prr 2 r + Pr [ (r r )] + Pt
2
From (50) 12 Prr
2r
+ Pr [ (r
dP=P = r
1+
= r (1
Equilibrium Asset Returns
dt + Pr
p
rd
r )] + Pt = rP + rPr , so that
Pr
Pr p
dt +
rd
P
P
p
B ( )) dt B ( )
rd
where from (51) we substituted Pr =P =
George Pennacchi
(54)
(55)
B ( ).
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Equilibrium Market Price of Interest Rate Risk
Hence,
p
(r ; ) r
rB ( )
= p
=
rB ( )
p (r ; )
p
r
(56)
Thus we see that the market price of interest rate risk is
proportional to the square root of the interest rate.
When < 0, which occurs when the interest rate is
negatively correlated with the return on the market portfolio
(and bond prices are positively correlated with the market
portfolio), bonds will carry a positive risk premium.
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Equilibrium Asset Returns
University of Illinois
37/ 38
Summary
The Merton ICAPM shows that when investment
opportunities are constant, the expected returns on assets
satisfy the single-period CAPM
In general, an asset’s risk premia include the asset’s
covariances with asset portfolios that hedge against changes
in investment opportunities.
The multi-beta ICAPM can be simpli…ed to a single
consumption beta CCAPM.
The Cox, Ingersoll, and Ross model shows the ICAPM results
are consistent with a general equilibrium production economy.
The model also is used to derive the equilibrium interest rate
and the shadow prices of securities in zero net supply.
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Equilibrium Asset Returns
University of Illinois
38/ 38