Under a comonotonicity assumption between aggregate dividends and the market portfolio, the CCAPM formula becomes more tractable and more easily testable. In this paper, we provide theoretical justifications for such an assumption.CAPM, CCAPM, market beta, equilibrium, financial markets

Clotilde Napp

Elyès Jouini

English

Research Papers in Economics

A class of models satisfying a dynamical version of the CAPM￿
ElyŁs Jouiniy
UniversitØ Paris IX-Dauphine and CREST
Clotilde Napp
UniversitØ Paris IX-Dauphine and CREST
June 26, 2007
Abstract
Under a comonotonicity assumption between aggregate dividends and the market portfolio, the
CCAPM formula becomes more tractable and more easily testable. In this paper, we provide theo-
retical justi￿cations for such an assumption.
1. Introduction
In this paper we provide a dynamical version of the CAPM, i.e., an equilibrium model, analogous to the
CCAPM, replacing in the resulting formula aggregate consumption by the market portfolio. Indeed, we
show in a continuous time framework that the instantaneous rate of return of any security in excess of
the riskless interest rate is a multiple common to all securities of the ￿instantaneous covariance￿of the
￿Keywords : CAPM, CCAPM, market beta, equilibrium, ￿nancial markets. JEL numbers D50, G12.
yCorresponding author: ElyŁs Jouini, CEREMADE, UniversitØ de Paris IX Dauphine, Place du MarØchal de Lattre de
Tassigny, 75116 Paris. e-mail : jouini@ceremade.dauphine.fr
1return with the market portfolio changes. In order to derive such a dynamical version of the CAPM,
we need an important assumption: the stocks prices are nondecreasing functions of the dividends. The
main contribution of this paper is to prove that for almost all the classical utility functions, there exist
equilibria satisfying this monotonicity assumption.
The main advantage of our approach is that in order to estimate the risk premium of any asset we do
not need to observe the aggregate consumption process, which, as underlined by Breeden, Gibbons and
Litzenberger (1989), is a delicate issue. Furthermore, as underlined by Mankiw and Shapiro (1986) ￿a
stock￿ s market beta contains much more information on its return than does its consumption beta￿ .
2. The model
We ￿x a ￿nite-time horizon T > 0 and we set T ￿[0;T]. As usual, we let (￿;F;P) denote a ￿xed
probability space on which is de￿ned a k-dimensional Brownian motion W =
￿
W1
t ;:::;Wk
t
￿￿
t2T and a
P-augmentation of the natural ￿ltration generated by W denoted by (Ft)t2T .
We consider a complete ￿nancial market with (k + 1) securities whose prices are denoted by S0;S1;:::;Sk.
Each security Si is in total net supply equal to ￿i with ￿0 = 0 and we denote by Di the dividends process
associated to asset Si for i = 1;:::;k; and by D =
Pk
i=1 ￿iDi the aggregate dividends process. We assume
that
dS0
t = S0
t rtdt, S0
0 = 1 (2.1)
dSi
t = Si
t
￿￿
￿i
t ￿ ￿
i
t
￿
dt + ￿i
tdWt
￿
for i = 1;:::;k; (2.2)
where the real valued processes r, ￿
i (the dividend yield process); ￿i as well as the (1 ￿ k)￿matrix valued
process ￿i are progressively measurable and uniformly bounded. We assume that for all t in T; (￿t)
￿1 =
2￿
￿i
j (t)
￿￿1
1￿i;j￿k exists and is uniformly bounded. The k -dimensional process ￿ can then be de￿ned by:
￿t ￿ (￿t)
￿1 [(￿t ￿ rt1k)]
where ￿ =
￿
￿1;:::;￿k￿￿
and where 1k denotes the k-dimensional vector whose every component is one.
We let for all t 2 T, Mt ￿ Et (￿￿) ￿ exp
nR t
0 (￿￿s)
￿ dWs ￿ 1=2
R t
0 k ￿s k2 ds
o
.
The market is complete, so we can apply the results of Huang (1987) and our economy can be supported
by a representative agent, endowed with one unit of the market portfolio N ￿
Pk
i=1 ￿iSi and whose
preferences for consumption and terminal wealth are characterized by
U (c;X) = E
"Z T
0
u(t;ct)dt + V (X)
#
where the utility functions u and V satisfy the following assumption.
Assumption (U) The function u : T ￿ R+ ! R is continuous. The function V : R+ ! R as well
as for all t 2 T; u(t;￿) are C3 on (0;1), stricly increasing, strictly concave and satisfy infx uc (t;x) =
infx V 0 (x) = 0 . The function Iu : T ￿ R￿
+ ! R is of class C1;2, where Iu (t;￿) denotes the generalized
inverse of uc (t;￿).
Recall that an equilibrium consists of a collection
￿
S0;:::;Sk;(b ￿;b c)
￿
such that (b ￿;b c) is an optimal
investment-consumption process for the representative agent as de￿ned in Merton (1971) and such that
markets clear, i.e., b ct = Dt and b ￿
i
t = ￿iSi
t for all t 2 T. At the equilibrium the optimal consumption
process of the representative agent must equal the aggregate dividends process, and we deduce from e.g.
Karatzas (1989) that a necessary and su¢ cient condition for an equilibrium to be reached is that there
exists ￿ > 0 such that
3￿tMt = ￿uc (t;Dt); t 2 T (2.3)
￿TMT = ￿V 0 (NT) (2.4)
where ￿t ￿ 1=S0
t and with no additional condition because in this case, the budget constraint, i.e.
E
hR T
0 ￿tMtDtdt + ￿TMTNT
i
= N0, is automatically binding.
This leads to the following classical CCAPM formula, where the market clearing conditions permit to
replace aggregate consumption by aggregate dividends
￿i (t) ￿ rt = ￿t￿i (t)(￿D (t))
￿ for all t 2 T and for all i = 1;:::;k (CCAPM)
where ￿t = ￿ucc (t;Dt)=uc (t;Dt) and dDt = ￿D(t)dt + ￿D(t)dWt for all t 2 T.
The market portfolio value satis￿es a stochastic di⁄erential equation of the form
dNt = Nt ((￿N(t) ￿ ￿N (t))dt + ￿N(t)dWt):
When the process N can be written in the form Nt = fn(t;Dt);t 2 Tg where n : T ￿ R+ ! R+ is of
class C1;2 with nx (t;x) > 0 (Assumption (N)), then It￿￿ s Lemma gives us the following Dynamical
CAPM formula
￿i (t) ￿ rt = ￿ ￿t￿i (t)(￿N (t))
￿ for all t 2 T and for all i = 1;:::;k: (Dynamical CAPM)
This means that it su¢ ces to observe e.g. an index (seen as a proxy of the market portfolio) in order to
estimate the risk premium of any asset.
In the next, we prove the existence of equilibria satisfying Assumption (N) for a large class of utility
4functions.
3. Existence of equilibria satisfying Assumption (N)
More precisely, we shall establish that for a given choice of utility functions and of risk level, there exists
an equilibrium satisfying Assumption (N).
Proposition 1. Let ￿ be given satisfying the regularity assumptions of Section 2. For the following
choices of (u;V ), there exists an equilibrium satisfying Assumption (N) :
1. Logarithmic utility function: u(t;c) = exp(￿￿t)￿log(c + ￿), V (x) = exp(￿￿T)log(x + 1), n(t;x) =
(1=￿)x for some positive constant ￿.
2. Power utility function: u(t;c) = c
￿
￿ for ￿ 2 ]0;1[, V (x) = a
(￿￿1)
￿ x￿, n(t;x) = (1=a)x for some
positive constant a.
3. Exponential utility function: u(t;c) = 1 ￿ exp(￿c), V (x) =
(1+x)
￿
￿ , n(t;x) = expfx=(1 ￿ ￿)g ￿ 1
for some ￿ 2 ]0;1[.
Proof An equilibrium is characterized by (2:3) and (2:4). We proceed as in Karatzas-Lehoczky-Shreve
(1990) and by It￿￿ s Lemma, we obtain that (2:3) and (2:4) as well as Assumption (N) are satis￿ed if there
exists a function d of class C1;2 such that for all t, dx (t;￿) > 0 and
8
> > > > > > <
> > > > > > :
￿N (t)
h
1 +
fx(t;Nt)
f(t;Nt) Nt
i
= ￿
ft(t;Nt)
f(t;Nt) +
fx(t;Nt)
f(t;Nt)
h
d(t;Nt) ￿ k￿N (t)k
2 Nt
i
￿ 1
2
fxx(t;Nt)
f(t;Nt) k￿N (t)k
2 N2
t :
￿i (t) ￿ rt = ￿
fx(t;Nt)
f(t;Nt) Nt
￿
￿i (t)
￿
(￿N (t))
￿
d(T;NT) ￿ Iu [T;V 0 (NT)]
where f (t;x) ￿ uc (t;d(t;x))=uc (0;d(0;N0)).
In our examples, solutions are given by
51. ￿i
t = ￿ + Nt
Nt+1￿i (t)(￿N (t))
￿ ;rt = ￿; d(t;x) = ￿x
2. ￿i
t = rt + (1 ￿ ￿)￿i (t)(￿N (t))
￿ ;rt = 1
2 (￿ ￿ 1)k￿N (t)k
2 +
a(￿￿1)
￿ ;d(t;x) = ax
3.
8
> > <
> > :
￿N (t) = ￿(￿ ￿ 1)
2 log(1+Nt)
1+￿Nt ￿ (￿ ￿ 1) Nt
1+￿Nt k￿N (t)k
2
h
1 + 1
2 (￿ ￿ 2) Nt
(1+Nt)
i
rt = 1
2￿ (￿ ￿ 1)
(Nt)
2
(1+Nt)(1+￿Nt) k￿N (t)k
2 ￿ (￿ ￿ 1)
2 log(1+Nt)
(1+￿Nt) ; d(t;x) = (1 ￿ ￿)log(1 + x)
which completes the proof.
Proposition 2. If ￿ is given satisfying the regularity assumptions of Section 2 and if x
V
00(x)
V 0(x) and x2 V
000(x)
V 0(x)
are bounded, if
V
0(x)
uc(t;x) is bounded in a neighborhood of 0 and if infx x
V
00(x)
V 0(x) > ￿1, then there is an
equilibrium satisfying Assumption (N).
Proof Immediate using the proof of Proposition 1 and taking d(t;x) = Iu [t;V 0 (x)] and
rt =
￿
V
t h
1 +
V 00(Nt)
V 0(Nt) Nt
i ￿
V 0 (Nt)
V (Nt)
Nt k￿N (t)k
2
￿i
t = rt +
V 00 (Nt)
V 0 (Nt)
Nt￿i
t (￿N (t))
￿
where ￿
V
t = ￿
V
00(Nt)
V 0(Nt) Nt k￿N (t)k
2 +
V
00(Nt)
V 0(Nt) Iu [t;V 0 (Nt)] ￿ 1
2
V
000(Nt)
V 0(Nt) k￿N (t)k
2 (Nt)
2.
We have so far obtained existence of an equilibrium satisfying Assumption (N) when ￿ is given.
Another way of modelling the introduction of uncertainty is to replace uncertainty on the securities price
process by uncertainty on the dividends process.
Proposition 3. Let ￿D;u and V be given.
1. If ￿D = ￿ ￿D for ￿ ￿ =
￿
￿ ￿1;:::; ￿ ￿k￿
in Rk, if V (￿) = u(T;￿) and if ~ u : x 7! xuc (t;x) is increasing, then
for all dividend process D = fDt;t 2 Tg of the form dDt = ￿ ￿Dtdt + ￿D(t)dWt, for ￿ ￿ 2 R, there
exist price processes S0;:::;Sk such that the equilibrium conditions (2:3) and (2:4) are satis￿ed as
well as Assumption (N).
62. In the general case, for all utility functions u and V; there exists ￿D such that Nt = n(t;Dt) for
n(t;￿) = IV [uc (t;￿)] and the equilibrium conditions (2:3) and (2:4) are satis￿ed.
Proof We must have n(T;x) = IV [uc (T;x)]. Adopting the same approach as in the proof of
Proposition 1, it is easy to obtain that there is an equilibrium with Nt = n(t;Dt) if n satis￿es the
following partial di⁄erential equation
nt (t;Dt) + nx (t;Dt)￿D (t) +
1
2
nxx (t;Dt)k￿D (t)k
2 + Dt ￿ Atn(t;Dt)
= ￿
ucc (t;Dt)
uc (t;Dt)
nx (t;Dt)k￿D (t)k
2 , (3.1)
for At ￿ ￿ 1
uc(t;Dt)
￿
uct (t;Dt) + ucc (t;Dt)￿D (t) + 1
2uccc (t;Dt) k ￿D (t) k2￿
and with terminal condition
n(T;x) = IV [uc (T;x)].
This is in turn equivalent to the following equation
￿Zt (t;x) = Zxx (t;x) + F (t;x)
with terminal condition Z (0;x) = H (x) ￿ (ucIV ￿ uc)
￿
T;exp 1 p
2 k￿ ￿kx
￿
where Z (t;x) ￿ (nuc)[T ￿ t;exp(￿x + ￿t)],
F (t;x) ￿ exp(￿x + ￿t)uc (T ￿ t;exp(￿x + ￿t)), ￿ = 1 p
2 k￿ ￿k and ￿ = ￿2 ￿ ￿ ￿
The solution of this partial di⁄erential equation (see Cannon [1984]) is given by
Z (t;x) =
Z +1
￿1
K (t;x ￿ y)H (y)dy +
Z t
0
Z +1
￿1
K (t ￿ ￿;x ￿ y)F (￿;y)dyd￿ (3.2)
where
K (t;x) =
1
p
4￿t
exp
￿
￿
x2
4t
￿
.
7By di⁄erentiation, we obtain
Zx (t;x) = K (t;￿) ￿ H0 (x) +
Z t
0
K (t ￿ ￿;￿) ￿ Fx (￿;￿)(x)d￿.
We check then that under our conditions we have H0 (x) > 0 and Fx (t;x) ￿ 0. Hence Zx (t;x) > 0 which
implies that nx (t;￿) > 0. This ends the proof of 1.
If we do not impose a particular choice for ￿D; it su¢ ces to impose n(t;￿) = IV (uc(t;￿)) in the previous
partial di⁄erential equation and to choose ￿D accordingly.
4. Conclusion
We proved that there exist equilibria satisfying our comonotonicity condition for a large class of utility
functions. In such equilibria, the state price density Mt decreases with the market portfolio value and
the consumption beta is replaced by the market beta in the CCAPM formula. These two features are
consistent with empirical evidence as underlined by A￿t-Sahalia and Lo (2000) for the ￿rst one and by
Mankiw and Shapiro (1986) for the second one.
References
[1] A￿t-Sahalia, Y., Lo, A., 2000. Nonparametric risk management and implied risk aversion. Journal of
Econometrics, 94, 9-51.
[2] Breeden, D.T., M.R. Gibbons and R.H. Litzenberger, 1989. Empirical tests of the consumption oriented
CAPM. Journal of Finance , 44 (2), 231-262.
[3] Cannon, 1984. The one-dimensional heat equation, in Encycl. of Math., vol.23, ed. G-C. Rota, Cam-
bridge University Press.
8[4] Huang, C.F., 1987. An Intertemporal General Equilibrium Asset Pricing Model: The Case of Di⁄usion
Information. Econometrica , 55, 117-142.
[5] Karatzas, I., 1989. Optimization problems in the theory of continuous trading, SIAM J. Control
Optimization, 27, 1221-1259.
[6] Karatzas, I., Lekoczy, 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 (1),
80-128.
[7] Mankiw, N.G., Shapiro M.D., 1986. Risk and return : Consumption beta versus market beta. The
Review of Economics and Statistics, 68 (3), 452-459.
[8] Merton, R., 1971, Optimum consumption and portfolio rules in a continuous time model, Journal of
Economic Theory, 3, 373-413.
9

A class of models satisfying a dynamical version of the CAPM

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A class of models satisfying a dynamical version of the CAPM

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