Existence and Uniqueness of Equilibria in the
CAPM with a Riskless Asset
By Thorsten Hens
Fakultat fur Wirtschaftswissenschaft
Universitat Bielefeld
PF 10 01 31
33501 Bielefeld
and
Andreas Loffler
1
Fachbereich Wirtschaftswissenschaft
Freie Universitat Berlin
Boltzmannstr. 20
14195 Berlin
Revision: August 1996
Hens acknowledges nancial support from DFG, Sonderforschungsbereich 303.
Loer acknowledges nancial support from DAAD, Bonn. We thank Lutz Hendricks for helpful comments.
1
Abstract
In the standard CAPM with a riskless asset we give a simple proof of existence of equilibria without assuming concavity of the investor's utility functions. Moreover, we give a uniqueness result using assumptions on the risk
aversion of investors.
JEL Class.: G10, C62
keywords: CAPM, uniqueness, existence, risk aversion
1 Introduction
Nowadays the classical two{period Capital Asset Pricing Model is one of the
cornerstones of modern nance. Developed by Sharpe (1964), Lintner
(1965), and Mossin (1966), it is widely used both by practitioners and
theoreticians, since it gives us a managable and attractive way of thinking
about risk and required return on a risky investment. Given this succesful
theory one is forced to ask why the question of uniqueness of equilibria
was not intensively investigated for a long time. It should be clear that
without uniqueness the CAPM loses much of its relevance: if there are
many equilibria, on which can investors base their investment decisions?
And what is the "correct" risk premium for risky assets? In this paper we
intend to solve the uniqueness problem by using assumptions on the risk
aversion of investors.
It was Nielsen (1988), who showed that the static CAPM equilibrium (even
in a very simplied example) need not to be unique. There may be several
equilibria, all with identical expected total returns, covariances, utility functions, and initial distributions of cash ows. In every CAPM equilibrium,
risk is measured in the same way (via the capital market line) but the risk
premium is indeterminate. And even worse, as it was recently shown by
Bottazzi et al. (1995), the situation described by Nielsen is by no means
exceptional. For every risky market portfolio there are CAPM economies
that have arbitrarily many equilibria. This result shows that one cannot expect a general theorem establishing uniqueness for a broad class of CAPM
economies.
Up to now only special cases are known in which the static CAPM possesses
a unique equilibrium: for example, if utility functions are quadratic or if
investors have expected utility functions with constant risk aversion and
returns are jointly normally distributed. Nielsen (1988), considered the
question on uniqueness but he did not give a result that relies on economic
fundamentals, i.e. endowments or utility functions, alone. Dana (1994),
seems to be the most general result on uniqueness. Unfortunately, her criterion uses an assumption on the functional form of the utility function that
is dicult to interpret economically. Moreover, since Dana requires an additively separable and concave mean variance utility function, her criterion
implies that investors have nondecreasing absolute risk aversion { not a very
plausible property.
Our purpose is to give a uniqueness result that relies only on economically interpretable assumptions. We show that under nonincreasing absolute
risk aversion a joint restriction on utility functions and endowments implies
uniqueness in the CAPM. It is a common contention that nonincreasing absolute risk aversion is a plausible property of utility functions of investors,
see for example Arrow (1971). In particular, our condition is satised if
all investors exhibit constant absolute risk aversion.
1
As a suplementary result we can give a proof of Nielsen (1990) existence
theorem that relies only on the mean value theorem. Nielsen used a higher
dimensional xed point theorem to prove existence in the CAPM.
Merton (1973), has developed a continuous time version of the CAPM.
Recently Karatzas et al. (1990), obtained a uniqueness result for this
continuous time model using a well known condition on the relative risk
aversion of the investors. But this condition is not compatible with {
utility functions as was shown by Loffler (1995). Hence, the idea of
Karatzas et al. (1990), cannot be applied to the static CAPM.
Our analysis relies heavily on the existence of a riskless asset. This assumption is far from being trivial. The question whether our results can be
extended to the case when there is no riskless asset will be left for future
research.
The paper is organized as follows. The rst section presents the model and
section 2 gives the existence and the uniqueness result.
2 The model
The description of the model follows Duffie (1988), section I.11. Let
(M; M; ) be a probability space. Consider the space L of all real{valued
measurable
functions on (M; M; ). We endow L with the scalar product
R
xy = M x(m)y (m) d(m) and with the norm kxk2 = xx. The consumption
set will be the subset of L with nite norm, L2() = fx 2 L j kxk2 < 1g:
Let X denote the marketed subspace of the consumption space L2(). Security markets are called complete if X = L2(); otherwise they are incomplete.
Let 1 be a riskless asset, i.e. we have 1(m) = 1 for all m 2 M . We assume
1 2 X.
To simplify the exposition we assume that X is a nite{dimensional vector
space. Hence, X be generated as the span of (yj )j =1;:::;J , a collection of
basic securities in L2 (). Every traded portfolio x 2 X can be written as a
nite{dimensional vector x 2 RJ of shares yj .
Let be the J J {matrix of covariances of the basic securities and E the
J {vector of their expectations. Then the standard deviation of a portfolio
x can be written as
p
(x) = xT x
and expectation
(x) = E T x:
An investor i = 1; : : : ; I is described by his endowments and his utility
function. We assume spanning, i.e. his endowment is traded. Then the
2
endowment vector is also a linear combination of basic assets and we use
the notation ! i 2 RJ .
The market portfolio is given by
!=
X
i
!i
and it should not be riskless (otherwise we face a trivial case where all
portfolios have the same expected return).
For every investor, a function v i(; ) of standard deviation and mean of
total portfolio return is given.1 Assume that v i is dened on [0; 1) R and
continuously dierentiable with negative derivative vi and positive derivative vi . v i is strictly quasi{concave to ensure uniqueness of the optimal
portfolio of investor i, an assumption that goes back to Sharpe (1964). An
investor i is described last by here utility function
ui (x) := v i((x); (x)):
Denition 1. A vector of security prices p 2 RJ and an allocation of opti-
mal portfolios (xiopt)i=1;:::;I is an equilibrium, if the following two conditions
are satised:
(i) xiopt is optimal for investor i given his utility ui(x) and his budget constraints pT x pT ! i ,
(ii) markets clear, i.e. Pi xiopt = !.
3 The main result
We state an existence result that already goes back to Nielsen (1990). Our
proof will be based on the mean value theorem only.
Proposition 1 (Nielsen (1990)). In the CAPM with a riskless asset there
always exists an equilibrium.
Proof: We start with some preliminary calculations. Let a price vector
p 2 RJ be given. Let furthermore
M := + E E T
(1)
be the matrix of the second moments. By assumption M is nonsingular.
Since the riskless asset has an expectation of unity and zero covariance
1
Duffie (1988), used the seemingly weaker concept of variance aversion instead of a
mean variance utility function. But notice that variance aversion implies that the utility
function of the investor is in fact {, see Loffler (1996).
3
with the remaining assets, straightforward calculation shows the following
property of the matrix M :
M ;1 E = (1; 0; : : : ; 0)T ;
(2)
The Tobin separation property implies that the optimal portfolio of investor
i has the following form
9i1; i2 0 xiopt = i1(1; 0; : : : ; 0)T ; i2 M ;1 p
(3)
(cf. Duffie (1988), p.99). Hence, in equilibrium there are numbers 1; 2 0 such that
X
X
! = !i = xiopt = 1(1; 0; : : : ; 0)T ; 2 M ;1 p:
i
i
Since ! is not riskless 2 (and all i2) cannot be zero and we can rewrite the
last equation as
901; 02 p = 01 M (1; 0; : : : ; 0)T ; 02 M !
= 01E ; 02 M !;
(4)
where the last equation follows from (2). If p is an equilibrium price it must
be of the form (4).
The price of the riskless asset must be positive in equilibrium. Hence, using
(2) we have
0 < pT (1; 0; : : : ; 0) = pT M ;1 E
and using homogeneity of the demand functions we can normalize prices in
(4) such that
pT (1; 0; : : : ; 0)T = pT M ;1 E 1 =) 01 = 1 + 02 E T !:
This, (1) and (4) now yields a parametrization of all prices that can be
equilibrium prices (cf. Dana (1994), p.7)
;
p(02) = E ; 02 M ! ; (E T ! )E ; 02 2 (0; 1)
= E ; 02 !; 02 2 (0; 1):
(5)
We use this parametrization to simplify the optimization problem of the
investor. Expectation (xiopt) and standard deviation (xiopt) of the optimal
portfolio (3) of investor i are characterized by the following optimization
problem (cf. Ingersoll (1987), p.88)2
max ;1 v i (; ):
T i
=p ! +(M p)
In our formulation we have already incorporated the Tobin separation property (3)
and the normalization pT (1; 0; : : : ; 0)T = 1.
2
4
Now substituting (5) into the last equation we obtain the parameterized
optimization problem
max
v
=p(02 )T !i +(M ;1 p(02))
i (; ):
(6)
Let us now look at the constraint. Using (5) we get
p(02)T !i = (!i ) ; 02 Cov (!; ! i))
and
;
(M ;1 p(02)) = M ;1 E ; 02(! ; (E T ! )M ;1 E )
= 02 (! )
Hence, we can rewrite the parameterized optimization problem (6) as
max
=(!i );02 Cov (!;!i )+02 (!)
v i(; ):
(7)
A straightforward
now
calculation
shows that the constraints in (7) have the
Cov (! i ;! )
i
common point (!) ; (! ) .
We will subsequently make use of the following lemma.
Lemma 1. Fix 02 2 (0; 1) and consider a point (^; ^) satisfying the constraint in (7). If the slope of the indierence curve through (^; ^) is smaller
(resp. larger) then the slope of the constraint in (7) and if an optimal portfolio exists, then this portfolio satises
(xiopt) > ^
(resp. (xiopt) < ^ )
(xiopt ) > ^
(resp. (xiopt ) < ^):
and
Proof: Let us consider the rst claim. If we had 0 < (xiopt) ^ , the
indierence curves would intersect as shown in the gure 1. In the case of a
corner solution (xiopt) = 0 the indierence curve through = 0 would have
the same or larger slope and the same argument applies. Since the budget
line has positive slope, this implies furthermore (xiopt) > ^. The second
claim follows analogously.
The proof of the lemma can be made rigorous by using the weak axiom of
revealed preference for { preferences, see Bottazzi et al. (1995), q.e.d.
We are now able to give the proof of the rst proposition. We claim that
for a given parameter 02 and the according solutions to the optimization
5
6
.
..
.......
.
.......
.
.......
.
......
.......
.
.
.
.
.
.
...
.
.......
.
...... ...............
.
.......
..
. ............ ...................
..............................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.......................................................................... .
.
.
.
.
.
....................
...
.
....
... ............
....
.
...
.
...
........
.
..
....... ...
......
..
.
.
.
...... ............
.
.
.
.
.
.
.
.
.
..
......................
.
.
.
.
.
.
.
.
.
................... ......
...
.
.
.
.
.
.
.
.
.
......
.....
.................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
........
.
.
.
.
.
.
.
.
.
.
......................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....
......
.
.
.
.
.
.
.
.
.......
.
.
.
.
.
.
.......
.
.
.
.
.
.
.
.
.......
.
.
.
.
.
.
......
.
.
.
.
.
.
.
......
.
.
.
.
.
.
.
.
.
(xiopt)
-
^
Figure 1: Proof of lemma 1.
problem (7) the price vector p(02) is an equilibrium of the CAPM economy
i
X
i
(xiopt(02)) = (!):
(8)
That (8) is necessary can be seen from the Tobin separation property and
the market clearing condition:
X
i
(xiopt(02)) = X
i
!
xiopt(02) = (! ):
To prove suciency recall from (3) using the parametrization (4)
X
i
xiopt =
X
i
i ; 0 i0
1
!
1 2
(1; 0; : : : ; 0)T +
X
i
!
i202 !:
Now, (8) implies that
X
i
i202 = 1:
By the budget identity (Walras' Law) we deduce that
X
i
i1 ; 01i20 = 0:
Both equations show that we have indeed an equilibrium.
To ensure existence it remains to prove that (8) is true for at least one
parameter 02. Consider the optimization constraints for 02 = 0. Since the
6
constraint is a straight line parallel to the {axis (see gure 2) every investor
with a { utility function demands only a riskless portfolio. Hence we have
X
(xiopt(0)) = 0:
i
02 " 1
-
6
(!iA)
?
...
.....
....
.....
.....
.
.
.
.
.....
.....
.....
....
.....
....
.
.
.
.
.
.....
.....
....
.....
.....
.....
.
.
.
.
..
.....
.....
.....
.....
....
.
.....
.
.
.......
.
.
..
.......
.....
.......
.....
.......
..........................................................................................................................................................................................................................................................................................................
.
.
.
.
.
.
.
.
.
.
.
.
.
...
..
..........
.....
..........
.....
q..........
....
.............
.....
............
....
.............
.
....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....
................
....
....................
.... ............................
....................................................
...
.....
.
.
.
..
....
.....
.....
....
A
Cov(!; ! i)= (! ) A
&
02 = 0
-
Figure 2: Proof of existence.
We prove the proposition by showing that for every investor the function
(xi (0 )) is unbounded for suciently large 02. Then evidently the sum
P opt i 2 0
i (xopt(2)) will be unbounded too and by the mean value theorem equation (8) must hold for at least one 02. Hence, there exists an equilibrium.
We show rst that (xiopt(02)) is bounded from below. Choose an arbitrary
i
point A = ( A; A ) (see gure 2) such that A > Cov((!!);!) and A < (! i ).
There is an indierence curve through A that intersects the {axis at ? .
Consider an intersection point (^ ; ^) of the indierence curve through A and
any constraint of (7). If the slope of the indierence curve were larger than
the slope of the constraint, by quasiconcavity the indierence curve could
not pass through A. Hence, the indierence curve has smaller slope than
the constraint and by the lemma we have
? ^ < (xiopt (02))
or the set of numbers (xiopt (02)) is bounded from below.
Next we show that (xiopt (02)) is unbounded from above. This implies that
(xiopt(02)) is unbounded from above, too. Suppose we had
inf := 0lim
inf (xiopt (02)) < 1
!1
2
)
and consider the indierence curve through the point Cov((!!;!
; inf . The
)
values xiopt (02) are optimal, hence the indierence curves are tangent to the
7
i
0
constraints and have a slope
2 i(! ). By dierentiable continuity we would
)
have an innite slope at Cov((!!;!
; inf , a contradiction.
)
In order to formulate our proposition on uniqueness we need to consider the
slope S i (; ) of an indierence curve in the { plane
i )
S i(; ) := ; vvi ((;
:
; )
This slope is a measure of the investors' absolute risk aversion (for details see
Nielsen (1988), or Lajeri & Nielsen (1994)). We say that the investor
exhibits constant (nonincreasing) risk aversion if S i (; ) is a constant (nonincreasing) function of . Furthermore, dene a simple portfolio problem
as a decision problem where an investor maximizes utility over portfolios
formed by one riskless and one risky asset. As Lajeri and Nielsen show
(their proposition 3) the denition of constant and nonincreasing risk aversion is reasonable since the following two conditions are equivalent:
(i) the investor exhibits constant (nonincreasing) risk aversion,
(ii) in every simple portfolio problem the following is true: if income
wi = pT !i is nondecreasing the demand of the risky asset is constant
(nondecreasing).
The case of constant or nonincreasing risk aversion is considered natural
since a behavior according to (ii) is economically plausible. Now we can
state our main proposition.
Proposition 2. Assume that all investors have
functions
utility
such that
Cov (! i ;! )
i
i
v (; ) has constant risk aversion for (; ) < (!) ; (! ) and nonin
i
creasing risk aversion for (; ) Cov((!!);!) ; (! i) . Then the equilibrium
is unique.
Proof: To prove uniqueness, we will show that for every investor the standard deviation of her optimal portfolio (xiopt(p(02)) as a function of 02 is
monotonic increasing. Then (8) implies that the equilibrium is unique.
Let low < high and let xiopt (low ) be optimal.;The indierence curve is tan
gential on the low {budget line at the point (xiopt(low )); (xiopt(low )) .
We have to consider two cases.
Case 1 Let (xopt(02)) < Cov((!!)i;!) . The budget constraint implies (xiopt(low)) <
(!i). Consider the point ((xiopt(low )); ^) that satises the budget
constraint for high . By constant risk aversion the indierence curve
through that point has the same slope as the low {budget line. But
the slope of the high {budget line is larger and the lemma implies now
(xiopt(high )) > (xiopt(low )):
8
Case 2 Let (xopt(0 )) ! ;!)
! .
Cov ( i
( )
The budget constraint implies (xiopt (low )) (!i). Consider again the point ((xiopt(low )); ^) that satises the
budget constraint for high . From the assumption of case 2 it follows
that we must have
2
^ (xiopt (low )):
Nonincreasing risk aversion implies now that the indierence curve
through the point ( (xiopt(low )); ^) has smaller or equal slope than
the low {budget line. Since the slope of the high {budget line is larger
the lemma implies
(xiopt(high )) > (xiopt(low )):
This nishes the proof.
The condition in Dana (1994), can be compared with our result. Dana
assumes that the function v i has the following functional form
i 00
vi (; ) = f i (ui1() + ui2 ()) with ; ((uu1i ))0(()) < 1
1
and v i is concave. The slope of such a function is given by
i 0
S i(; ) = ; ((uui1))0(())
2
and concavity implies that (ui2)0 () is nondecreasing in . Hence, the slope
is nonincreasing in (since (ui1)0 is negative). Thus, Dana's assumption
implies that the investors exhibit nondecreasing risk aversion.
The above proposition implies a joint restriction on endowments and utility
functions of the investors. As a special case we obtain that equilibrium is
unique if all investors have constant risk aversion. Although, this special
case does not seem very surprising, to the best of our knowledge it has not
been noted before.
4 References
Arrow, K.J. (1971). Essays in the Theory of Risk Aversion. Chicago:
Markham.
Bottazzi, J.-M., Hens, T., & Loffler, A. (1995). Market demand
functions in the CAPM. Discussion Paper A{468. Universitat Bonn,
Bonn.
9
Dana, R.A. (1994). Existence, Uniqueness and Determinacy of Equilibrium in CAPM with a riskless asset. GREMAQ, Toulouse.
Duffie, D. (1988). Security Markets. San Diego: Academic Press, Inc.
Ingersoll, J.E. (1987). Theory of Financial Decision Making. Totowa,
New Jersey: Rowmann & Littleeld.
Karatzas, I., Lehoczky, J.P., & Shreve, S.E. (1990). Existence
and Uniqueness of Multi{Agent Equilibrium in a Stochastic, Dynamic
Consumption/Investment Model. Mathematics of Operations Research,
15, 80{128.
Lajeri, F., & Nielsen, L.T. (1994, April). Risk Aversion and Prudence:
The Case of Mean{Variance Preferences. INSEAD.
Lintner, J. (1965). The Valuation of Risk Assets and the Selection of
Risky Investments in Stock Portfolios and Capital Budgets. Review of
Economics and Statistics, 47, 13{37.
Loffler, A. (1995). A Risk Aversion Paradox and Wealth Dependent
Utility. unpublished, Berlin.
Loffler, A. (1996). Variance Aversion Implies {2{Criterion. Journal
of Economic Theory, 69, 532{539.
Merton, R.C. (1973). An Intertemporal Capital Asset Pricing Model.
Econometrica, 41, 867{888.
Mossin, J. (1966). Equilibrium in a Capital Asset Market. Econometrica,
34, 768{783.
Nielsen, L.T. (1988). Uniqueness of Equilibrium in the Classical Capital
Asset Pricing Model. Journal of Financial and Quantitative Analysis,
23, 329{336.
Nielsen, L.T. (1990). Existence of Equilibrium in CAPM. Journal of
Economic Theory, 52, 223{231.
Sharpe, W. (1964). Capital Asset Prices: A Theory of Market Equilibrium
Under Conditions of Risk. The Journal of Finance, 19, 425{442.
10