-.-^^'
ifEB SISBB
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A Note on the Valuation of Stochastic Cash Flows
Rajnish Mehra
Sloan School of Management
Massachusetts Institute of Technology
and
University of California
Santa Barbara
WP 7/1913-87
Revised July 1986
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A Note on the Valuation of Stochastic Cash Flows
Rajnish Mehra
Sloan School of Management
Massachusetts Institute of Technology
and
University of California
Santa Barbara
WP-:-1913-87
Revised July 1986
I would like to thank John Donaldson for his helpful comments.
Financial support from the National Science Foundation and the Faculty
Research Fund of the Graduate School of Business, Columbia University
is gratefully acknowledged.
FEB
8 1988
Abstract
This paper considers the implications of the equilibrium asset
pricing model developed by Lucas (1978) for the valuation of risky cash
flows.
It
is
shown that even in the context of the simple economy
studied, discounting stochastic cash flows becomes computationally
complex and of limited practical use.
.
I.
Introduction
The paper by Lucas
:
Economy" initiated
a
"Asset Prices in an Exchange
(1978)
paradigmatic change in the Theory of Equilibriam
Pricing of Risky Assets.
It
was followed by several papers including
those by Brock (1979 and 1982), Prescott and Mehra
and Mehra
(1984) who generalize Lucas'
model to
a
(1980) and Donaldson
production setting.
Research in the area of multiperiod valuation of risk assets has predominantly focussed on the conditions necessary to generalize the single
period partial equilibrium model of Sharpe (1964), Lintner (1965) and
Mossen (1966) to
above,
a
multiperiod context.
addition to the papers cited
In
related papers include the work of Bhattacharya (1981), Breeden
(1979), Cox.
(1985"),
Ingersol and Ross
(1985), Fama
(1977), Mehra and Prescott
Merton (1973), Rubinstein (1976) and Stapelton and Subrahmanyam
(1978).
This paper considers the logical implications of the equilibrium
asset pricing model of Lucas
flows.
(1978)
for the validation of risky cash
Since this paper is extensively quoted
m
the Finance and
Economics Literature, it seems appropriate that its implications be
svstematically analysed to further its understanding.
The paper is pedagogical in nature since much of the analysis such
as
tne existence,
uniqueness and optimality of equilibrium has been
developed earlier.
The paper is divided into four sections.
scribes the economy under consideration.
developed in section
comments
3
while in section
Section
2
briefly de-
The valuation formulae are
4 we
provide some concluding
The Economy
II.
We consider
:
pure exchange economy of the
a
As shown in Prescott and Mehra
considered in Lucas (1978).
example 2, such an economy can be cast as
librium.
a
ty7)e
(1980),
recursive competitive equi-
We have deliberately abstracted from the more general struc-
tures considered by Brock fl979 and 1982) and Prescott and Mehra (1980)
which allow for capital accumulation since they are not necessary to
establish the results in this paper.
The reader is referred to
A brief sketch of the economy follows.
Lucas (1978) for details.^
The economy has
single representative ''stand in" household.
a
This
agent orders its preferences over random consumption paths by
X
I
2 ^ ulc
E;
(1)
)]
t=0
where
IS
c
is a
stochastic process representing per capita consumption,
the subjective discount factor,
operator and
u:
R
-
R
is
Production
be the output vector at
P
<
1,
E|-}
is
the expectation
the current period utility function.
consumption good is produced by
in period t.
<
is
time
firnis.
n
entirely "exogenous."
t.
X
=
Let X
assumed to follow
is
The
be the output of
Let x.
p
a
{x.
.x,
firm
.
i
..x
Markov process
defined by its transition function
FIX'IXJ = priX^^^
1
X'lX^
=
Xj.
Each period ownership in these productive units is determined in
petitive stock market.
equity share.
a
com-
Each unit has outstanding one perfectly divisible
A share entitles
its owner as of the beginning of
t
to all
of the unit's output
period
in
t.
Shares are traded ex-dividend at
competitively determined price vector
=
p
t
=
Let Z
share holdings.
consumed
(c
=
...z
,z^
(z,
In this
Ix
denote
economy,
and all
)
.
)
m
a
a
.)•
(Pw,Po^---P
t
nt
1
1
i_
consumer's beginning of period
equilibrium, all output will be
shares held, that is
=
Z
If we
(1,1. ..1).
1
let the price of the consumption good be the numeraire,
of the firms can be thought of as their "cash flows".
then the output
We will use the
latter interpretation in the analysis that follows.
III.
(2)
Valuation:
The price dynamics of firm
u'(c^)p^(X^)
--
P
J-
u'U,,i)[x^_^^^
i
are characterized by
.
P:'Vl^J
m
This IS the celebrated stochastic Euler equation
sufficient to value risky cash flows from firm
i.
''^
'Vl'^t^
U97Sj and
Lucas
is
The same equation is
obtained in models incorporating capital accumulation developed by Brock
(1979 and 1982), Donaldson and Mehra
and others.
(19S4j, Prescott and Mehra
(1980)
These models incorporate other equations that must be
consistent with (2j; nevertheless, equation
(2
>
must necessarily be
satisfied in each case.
In the analysis
Since there
is
follow we will value the cash flow
froni
firm
i.
little room for ambiguity, we will suppress the subscript
so that at any time
(3)
to
t
equation (2) may be written as
u'(c^) p(X^) -
p
J
u'(c^^^)lx^^^
The object is to solve for p(^X
future cash flows
{x,X-, ...)
)
which
+
P'\.i^l
i_s
of this firm
.
^^
'\+l'-\)
the present value of all
i
To this end leL us recursively define the sequence of conditional
distribution functions {F (X^^
t+n
^2^\+2'^t^
|X
according^ to
))
t
F(X^^JZJ dF (ZIX^)
= /
F3(X^^3|X^) = / F2(X^^3|ZJ dF (Z|X^)
with
F„(X^,JX^)
=
/
Vl'^^.s'^^
''
^2l-\'^
and hence
''Jh^n^h^'- {'\-l^\.n^^'
Let n^
.
t,j
be
''
the price of a riskless bond at time
^'^V
t
paving
one unit
^
^
-
*^
of the consumption good at time t+j
(4)
t ,J
t
.
Then
"^
u
(c^J
and bv definition
(5)
^'J
where
r^
.
t,j
is
the return on
To explicitly solve
a
(1
i
for p(X
+
r,
iJ
period pure discount bond.
)
in equation
(3)
we use the technique
of recursive substitution.
(6)
Defi ne g(X^,
--
f.
J-
u'lc^^jlx^^^ dF (X^.^IX^)
and
f(Xj_)
(7)
=
p(X^)u'(c^)
then equation (3J can be written as
f(XJ
(8)
Lucas
(.1978)
=
g(X^) +
3
/ f(X^^^)
dF (X^^jlX^)
shows that equation (8) has
a
unique solution f(*)
This implies that
f(X^)
(using equation ^)
(9)
To find fl*),
observe that the operator
T:
C
- C
(where
C
is
the
space of bounded continuous functions] defined by
(Tf)(x^) = g(x^j
(10)
is
contraction.
a
+
p
;
Hence for any
^^\^i' ^^
f-
e
C.
'^t+i'-'^t^
lim T'f- =
f.
n-3;
(f^
=
T"f.^
=
T(T"'^f^l
we can generate
a
n
=
1,2,3.
sequence of functions
f^CX^J = g(X^)
Letting
*"
jf
f,,
be the null
(i
.f..,...;
using (10)
function
f^lX^) = g(X^) + p ; fj(X^^j) dF (X^^j|Xj_)
and
=8(X^)
P/8(X^,^)
Mu'(c^^^)x^^^
=
^
r
Proceeding recursively
since f(X
)
t
dF (X^^^IX^)
=
lim
J u'(c^,,)x^^, dF,
f
n
(x^.oi^t)
can be expressed
as
^
(X,) we have
f
n
t
f(X ,)
)
-1
J
=
and using equation (9) we get
(12)
dF (X^^^IX^)
f,(X^) = p J u'(c^^^)x^^^ dF (X^^jlX^)
or
(11)
-
P^E^lu'lc
)x
-^
l
]
This can be written after some simplification (see the appendix) as
*
J
where
6
t ,j
=
-
[1
t"
^-J
^
(I
l
E
-
[x
)>
r
Cov^(-"'(c^,J,x^^^
6^^^
^
-
and 6
=
e
-EJu'(c^^^)J
E^
Ix^,^
Equation (13) exhausts the implications for valuing stochastic cash
m
flows
Lucasian economies
--
economies characterized by "representative
agents" who behave optimally in light of their objectives.
All prices
and price distributions are endogenous and are determined through market
Expectations are formed rationally; i.e., the prices and price
clearing.
distributions on which the economic agents base their decisions coincide
with those implied by their behavior.
The economy is thus "informa-
tionally efficient."
Superficially equation (13) resembles the familiar discounting
formulations of "text book finance" with
adjustment.
complex.
being the certainty equivalent
implications and applications are, however, far more
Its
The adiustment
term 6,,
J
in the
t+j
from period to period.
It
6
It
is
numerator will normallv vary'
precisely this variability in
difficult to generalize the Sharpe (1964), Lintner (1965
(1966)
asset pricing model to
generalization
m
a
multiperiod context.
effect constraint
6
to be constant,
achieved by imposing restrictive conditions.
Co\-
(-u'(c
•),
t-^j
x
.)'
t+j
utility of consumption
is
= 0,
'
is
G
j
that makes
and Mossin
The attempts at
which can only be
In the case where
I.e., where the covariance of marginal
<=
'
not correlated with the return from
a
project
—
J
--
perhaps the project is an insignificant component of
consumption basket -- then
6
=
.
1
t+j
the individuals were risk neutral.
V
a
diversified
This would also be true if
i.
Equation (13) could then be written
as
P(X
(14)
)
=
z
^^
^
{
y
(1+r
j=l
'-
.
1
which is consistent with our intuition for risk neutral valuation.
.Although equation (13) makes explicit the informational requirements
for valuing stochastic cash flows in this simple economy, we can gain
additional intuitive insights by rewriting it as
(15)
P(X^)
=
X
1
j=l
E
(X
-
)
\ Gov
{^^^
(-u'(c
^
J
{r.
.
'-'
.}
>
X
^
)
jJ
(1+r
^
where we have set t=0
),
J
:
>
J
the sequence of returns on pure discount
is
J
bonds of various maturities and A. =
J
=
—ry—ry
f
Eq[u'(c^)]
is a
constant for each
i.
If we take a Taylor's expansion of the utility function u(") around
E(c.) - c. and retain only the second order terms,
(15)
can be rewritten
as
oc
(16)
P(X.)
=
1
E
(X
{^—i
)
-
Cov (c
^
J
5-J
X
)
L]
The approximation is exact if u(*) is quadratic or if the analysis
is
conducted in continuous time (Ross (1977)).
0.
is
the reciprocal of
the coefficient of relative risk aversion and is
Equation (16)
is
constant for each
a
j.
"similar" to the various multiperiod valuation
relations in the existing finance literature.
this note to critique these studies,
the purpose of
is not
It
rather it illustrates
a
logical
and cogent methodology for the valuation of stochastic cash flows in
general equilibrium setting.
a
Clearly the information required to con-
duct such an exercise in its general form is considerable.
By imposing
additional structure (restrictions) these requirements can be reduced.
For example,
has been postulated that the coefficient of relative
it
risk aversion is
a
constant;
if this
and (16) is further simplified.
is
then
the case,
($).
= $ for all
j
(1970) have argued that
.Arrow and Kurz
for evaluating public projects risk netural valuation is appropriate.
In this case,
equation (14) derived earlier would be the appropriate
expression for valuation, significantly reducing the informational
requirements.
Clearly, other restrictions may be placed on (16) de-
pending on the application to make it more "operational".
4
.
Concluding Comments
This note has examined the implications of Lucas'
model for valuing stochastic cash flows.
The assumption of consumer
homogeneity is admittedly an unrealistic one.
Mehra (1980) point out,
asset pricing
However, as Prescott and
complete market setting the equilibrium
in a
process for economic aggregates and prices for
heterogeneous consumer
a
economy will be observa tiona ly equivalent to that for some homogeneous
1
consumer economy.
This provides our rationale for introducing
a
repre-
sentative "stand-in" indivi.lual.
It should be
apparent that cash flows from
a
finitely lived firm
(project) can be valued within the framework developed above by simply
defining them to zero after the project terminates.
10
Finally, as observed by Constantinides (1980),
it
is
evident that
even in the context of the simple economy studied here, discounting
stochastic cash flows is, in general, computationally complex and of
limited practical use.
11
Footnotes
For a succinct review and critique of the earlier work see
1.
Constantinides (1980).
For a comprehensive survey see LeRoy (1982).
Conditions necessary for the existence and uniqueness of
rium are also discussed in that paper.
2.
3.
Lucas (1978), pp.
a
equilib-
1434-1435.
See Arrow (1971).
He further argues that the coefficient of relative
risk aversion should be approximately one.
4.
5.
For
a
formal proof see Constantinides (1982).
12
Appendix
Rewriting equation (12) we have
P(X^)
»
=
/
(
BM
Z
._^
^
E
t
u'(c^^.)\
I
u
\
rM-T
U^)
/
)E
/
(x
)
t' t+j'
1
u'(c^^,)
+
Substituting from equation (4) we get
P(\)
=
^^^
"t.j
^t^Vj)(^
where
'
J
^.j c-t^-'^'^t.j^'Vj ))
-
E^iu'.c^^^jJ E^ [x^^^J
or using equation (5)
p(X
)
1
J
=
-
\h^\^^^^^
r
=
\^^
Cov^(-u'(c^^^).x^^^^))
J
l
^'''t,P
or
""
j
where
e,.. =
[1
-
=l
6^^^
I
(1
+
r,
jJ
'
CovJ-u'(c^^J,x^^^)
This IS precisely equation (13) in the text.
13
References
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^_^
and M. Kurz "Public Investment, The Rate of Return and
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.
Breeden, D.T.:
".-Xn Intertemporal Asset Pricing Model with Stochastic
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,
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,
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,
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,
,
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,
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,
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,
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,
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34
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The
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,
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,
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.A
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,
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,
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