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Journal of Financial Economics 2 (1975) 187-203. 0 North-Holland
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OPTIXlAL CONSUMPTION,
PORTFOLIO
AND LIFE INSURANCE
RULES FOR AN UNCERTAIN
LIVED INDIVIDUAL
IN
A CONTINUOUS
TIME MODEL
Scott F. RICHARD*
Carnegie-Mellon
University. Pittsburgh. PO. 15213, U.S. A.
Received September 1974, revised version received January 1975
A continuous time model for optimal consumption, portfolio and life insurance ales,
for an
investor with an arbitrary but known distribution of lifetime, is derived as a generalization of
the model by Met-ton (1971). The classic Tobin-Markowitz
separation theorem obtains with
the mutual funds being identical to those obtained under the assumption of certain lifetime.
The investor is found to have a ‘human capital’ component of wealth, which is independent of
his preferences and risky market opportunities and represents the certainty equivalent of his
future net (wage) earnings. Explicit solutions, which are linear in wealth, are found for the
investor with constant relative risk aversion.
1. Introduction and conclusions
In this paper the work of Merton (1971) is generalized to include optimal
consumption,
investment and life insurance decisions for an investor with an
arbitrary,
but known, distribution
of lifetime. Throughout
this paper all markets
are assumed to be perfect and frictionless with securities traded continuously.
The price of all securities traded in these markets follows a stationary geometric
Brownian motion so that prices at any future time have a log-normal distribution. Under these assumptions
Merton has shown that portfolio separation will
obtain for a decision-maker
with known lifetime Twishing to maximize
Ejo’ WC, 0 df,
where U is the instantaneous
utility function, C is consumption,
and E is the
expectation operator. By portfolio separation we mean a ‘mutual fund’ theorem
can be proven, which states that each investor would be indifferent between
choosing his portfolio by selecting from all available securities or choosing his
*I would like to acknowledge the suggestions of George Constantinides to generalize an
earlier version of Theorem 3 and the helpful comments of Robert Kaplan and an anonymous
referee. Allerrors are, ofcourse. my own.
188
S.F. Richard,
Consamption.
portfolio
and life insurance rules
portfolio by only buying shares in two mutual funds.’ Furthermore,
if a riskless
security* exists then one of the mutual funds may be chosen to be the riskless
security and the other a weighted combination
of all the risky securities with the
risky fund’s price having a log-normal distribution.
It will be shown below that the separation theorem also obtains if the investor
behaves to
Max. E[ji
U(C, 1) df +B(Z, T)],
where now T is the uncertain time of death, Z is the legacy he leaves at death, B
is his utility for bequest, and E is an expectation over time of death, as well as
future distribution
of prices. Furthermore,
the composition
of the risky mutual
fund will be exactly the same as in Merton’s case of known lifetime, i.e., all
securities in the risky fund will be represented in the same proportions they would
be if the investor had a certain lifetime. However, the uncertain lived individual
will not, in general, invest the same proportion of his wealth in the risky fund as
would the certain lived individual. Thus, within the context of the Merton model,
we find that uncertain
life time and life insurance in no way affect the investment opportunity
set not the efficient portfolios. This is in part due to the
individual’s expending (receiving) funds in order to buy (sell) life insurance.
The life insurance offered is of an instantaneous
term’ variety where payment
at a premium rate P(r) dollars per unit time buys insurance of P(r)/p(r) dollars,
where F(I) is specified in the insurance contract. The only general result concerning the bequest is that if an individual’s imputed utility for wealth is strictly
concave (i.e., he is risk-averse),
then his legacy will be a strictly increasing
function of wealth. In general this does not mean he will purchase insurance,
but will either buy or sell it depending on his feelings for risk and his relative
liking for consumption
or legacy.
An interesting result obtained is a new ‘separation’ theorem which proves that
for every investor there is an amount of ‘human capital’, independent
of risky
market opportunities
and preferences, which is a certainty equivalent
for the
investor’s future (wage) earnings stream which is assumed to be sure if the
investor is alive. This ‘human capital’ term is calculated by discounting
the
future earnings stream until the maximum time of death at a discount rate equal
to the sum of the risk-free rate plus the insurance rate applicable at that time.
This rate is a generalization
of a result by Yaari (1965) who shows that it is the
correct discount rate to use for cash flows that will be received only if an
individual does not die when life insurance is priced to be a fair game.
‘Of course the mutual funds will not be the same for any two investors unless they agree on
the distribution of the securities.
‘The riskless security has a sure return and will not default. Furthermore.
all returns are
stated in terms ofnct price levelchanges.
‘There is no loss of generality in using term insurance since all available life insurance is a
linear combination
of one period (year) term insurance and a savings plan of some sort.
S.F. Richard,
Conswnption.
portfolio
and life inrurmce
rules
189
Lastly the assumption
is made that the individual’s
utility for both consumption and bequest have constant relative risk aversion. In this case explicit
solutions, which are linear in wealth, are found for optimal consumption,
life
insurance and portfolio rules.
Hakansson (1969) has also examined the problem analyzed herein with constant relative risk aversion utility, but in a discrete time environment.
Unfortunately,
he was unable to solve explicitly for the optimal level of life insurance but he was able to find conditions under which zero insurance is optimal.
Lastly, Hakansson
finds that the optimal mix of risky securities will depend
upon the investor’s utility for consumption.
We find the leoel of investment in
risky assets will depend upon the utility for consump.tion,
but not’the mix of
risky assets.
In an extension of Hakansson’s discrete time formulation,
Fischer (1973) also
examines optimal lifetime insurance purchases. Fischer obtains a human capital
theorem similar to the one proved in this paper, but only for utility functions
with constant relative risk aversion.
2. The economic environment and asset dynamics
As Merton does, we assume that there are n securities and that the prices per
share, {s,(t)}, are generated by a non-stationary
geometric Brownian motion
process, i.e.,
dS,(f)lSi(f)
= a,(f) df+ o,(f) dqi(r),
i=
l,...,n,
(1)
where dq,(r) is the stochastic diflcrcntia14 of qi(r) which is a standard normal
random variable, a,(t) is the constant5 instantaneous
expected percentage change
*The variate dy is often rcfcrrcd to as Gaussian White Noise. The ‘diffcrcntiation’
of a normal
random variable has rules dificrcnt from those found in the calculus. The correct ditTcrcntiation
of functions ofq is done according to a thcorcm known as Ito’s Lemma. For a relatively easy to
follow (but rigorous) explanation
of stochastic integrals and stochastic calculus see Kushner
(1967, cspccially ch. 10.5-10.7).
‘By constant we mean non-stochastic
functions of time and, in general. all parameters may
be known functions off. However. the gain from this generalization
is illusory since no revision
of the functions is allowed and all results are identical for the case where the parameters are
true constants, except that the distributions
would be linear and non-stationary
for true constants. As an example we use Ito’s Lemma and (I) tocalculate
d
Integrating
In S,(r) = [dS&)/S&)l- f[dS,(r)/S,(r)l’
= [a,(r)- 1a,‘(r)l dr+a,(O d&J.
we find that
In(S,(r)/S,(O))
= fi [a,(s)-
io,Ys)l
ds+lb
U,(S) dq,(s).
The tirst term in the right-hand side of this equation is simply a function of r. while the second
term is a non-stationary
normal random variable. Thus S,(r) has a non-stationary
log-normal
distribution.
lfa, and 0, arc trueconstants,
then
In (S,(r)/S,(O)
where q,(r) is normal
1 = (ai - taf)r + a,q,(f).
with mean zero and variance r.
190
S.F. Richard,
Consumption.
portfolio
and life insurance rules
in price per unit time and a:(r) is the constant instantaneous
variance per unit
time. Actually (1) is shorthand symbolism for the stochastic integral equation
Si(l) = Si(O)+ IOSi(s)ai(s) CIS
+ I:, Si(s)ai(s) @i(s),
i=
(1’)
l,...,n,
and (1) should always be interpreted
as meaning (1’). The investor’s wealth
time I (assuming he has not died previously) is given by the stochastic integral
at
w, -f. c(s) ds - I0 P(s) dr + I0 Y(s) ds
W(t) =
’
+f
[f
I=1
w(s)
dSi(s) v
S,(s)
1
wi(s)
0
where W(0) = W, is his wealth at time 0; Y(s) is his non-stochastic
rate of
earnings (presumably wages) at times; w,(s) is the fraction of his wealth invested
in security i at time s so that
Thus, w,(s) W(s)/S,(s) is the number of shares of security i that the investor holds
at times. Substituting
(1) into (2) for dS,/S, and writing the result as a stochastic
dill’erential equation,we
have6
d W(r) = -C(t)
+ ,i,
s
dr-P(r)
df+ Y(r) dt
wWW)[a~(~)
dr + c,(r) ddO1.
(3)
If one of the n assets is risklcss, say the nth one, so that 0, = 0 and a, = r,
the instantaneous
riskless rate of return, then (3) is rewritten as
dW=
-Cdl-Pdt+
Ydr
+ t w,W[(a,-r)dr+a,dq,]+rWdr.
1-I
wherem
=n-landw,,...,
w,=
w, are unconstrained
I-2
(3’)
since we have substituted
w,.
l- 1
“Merfon derives the equafion for wealth dynamics by carefully taking limits of a discrete
time process and using Ito’s Lemma. Our equation is ofcourse identical to his except for the life
insurance
premium P(r)dr.
S.F. Richard,
Consumption,
porrfoiio
and life insurance
rules
191
Before proceeding with the stochastic dynamic programming
formulation
we
will first specify the probability law governing the investor’s lifetime. We assume
that he will die at a time TE [0, T] and that T is specified by a probability
density function n, such that
x(f) 2 0
for all f E [0, T],
n(r) > 0
for all t E (0, T),
(4)
s; n(r) dt = 1,
and n(t) is continuous
cumulative probability
in the neighborhood
distribution,
i.e.,
G(t) = jJ n(r) dr,
of r’
Let G(t) be the right-hand
0stsT.
(5)
Denote by n(T; t) the conditional
probability
density
conditional upon the investor being alive at time t, so that
W; t) = n(T)IG(t),
In particular
Ojt$TST,
t#T
for death
at time
T
(6)
we will define A(t) by
A(t) = n(t: I).
ost<T.
(7)
When n(‘P) > 0, it is easily seen that for t -+ 7, A(t) -+ co. If n(T) = 0, then, as
Yaari (1965) does for each t E [0, 7’). we define T, by
?t(T,) = sup n(r) > n(T) = 0.
rq 1.T1
Now G(t) s (T-
t)n(r,), so that
40 2 W/~V- t)dr,)l.
(8)
Now since n(t) is continuous,
it must bc monotone decreasing in the neighborhood of ?‘, so that n(t)/n(r,) = 1 and hence A(t) diverges.
Now that the investor’s probability
distribution
for death has been given, we
can specify the life insurance rates. We assume that paying premiums at the
rate P(t) buys life insurance in the amount P(t)/p(t) for the next (very) short
period. Thus the investor’s total legacy should he die at time t with wealth W is
at) = W(t)+ vYNP(t)l.
(9)
‘We will not be overly concerned with the regularity conditions on non-stochastic exogenous
functions of time; in general we will assume them to bc continuous in the neighborhood
of T,
unless otherwise noted.
192
S.F. Richard,
Consumprion.
portfolio
and life insurance rules
We assume that
P(r) = 4r)+q(l),
where q(r) is any non-negative
of 7, i.e.,
(10)
integrable function
H(f) = {Fq(r) dr < co,
continuous
in the neighborhood
r E [O, n
(11)
and
lim {q(r)/A(f)} = 0.
r-7
Hence
lim {&)/A(r)}
r-F
= 1.
The investor
wishes to choose optimal
controls (i.e., consumption,
life
insurance and portfolio functions) to maximize his expected lifetime utility for
consumption
and legacy. That is, to
(h$x;
.
E,Jj; U(C(s), 4 ds+ W(T), 7-11,
(12)
,w
where E; is the conditional
including T conditional
upon
in C and E is assumed strictly
the constraints
that W(0) =
asset exists (3’).
expectation
operaton over all random variables
W(0) = W e ; U is assumed to be strictly concave
concave in Z. The maximization
(I 2) is subject to
IV, and the budget constraint
(3) or, if a riskless
3. The stochastic dynamic programme formulation
WC will USCstochastic
Define
J( W, r) I
dynamic
programming
to derive the optimal
Max. E, It rr(T; f>[j: U(C, s) dsi- B(Z(T),
(C.P,Wl
controls.
T)] dT,
(I 3)
where E, is the conditional
expectation
operator
over all random variables
except T, conditional
upon W(f) = W. Now if the integral (13) is absolutely
convergent then we may integrate by parts to find that
J(W, t) =
Max.
E,
(C,P,wi
I’ [n(T)B(Z(T),
T)+CGV’(C.
G(r)
VI dT.
t,4I
S.F. Richard,
Consumption.
porrfilio
and life insurance rules
193
Define
r&c, P, w; W, 1) = L(r)B(Z(r), r)+ UC,
+J,+
wiziw+
i
t
i=l
Y-C-P
i
J,
1
1
[ i=
+$
t)-U)J
UijWiWjW2J,,,
(15)
j=l
wTi(f) = wi, C(t) = C, W(r) = W, Y(r) = Y, P(r) = P and Z(r) = Z;
given
and where J, = dJ/dt, J, = dJ/dW, Jww = d2J/dW2, and o,~ = aiajpij, where
pij is the instantaneous
correlation
coefficient between dq, and dq,; i.e.,
E(dqidqj) = oijdt.* The optimal controls
C*, P* and w* are given by the
following theorem.
Theorem I. 9 If the Si(t) are generated by a geometric Brownian motion, i.e.,
they are log-normally distributed, U is strictly concave in C, and B is strictly
concaw in Z, then there exists a set of optimal controls C*, P* and w* satisfying
(3), CT_, wi = I and J( W, T) = B(Z(T), ip) and these controls are such that
0 = O(C’. p*, w*; W, I) L O(C, P, n’; W, f),
Thcorcm
I E [O, ip].
I implies that
0 =
,yljtx;
* .w
{&XC, P, w; IV/,I)}.
(16)
WC form the Lagrangian
L=0+Y
and differcntiatc
[ 1=1
1
1-i:
w,,
to find the first-order
0 = L,(Cl,
conditions
p*, w’) = Q(c*,
O = L,&(C*, P+,
w*)
=
r)-(J,,
-‘f’+J,?,W+I,,,i,
(17)
Okj’“fW’*
(18)
k = l,...,n,
“We note that R = [n,,] is n x n if there is no risk-free asset and is m x m if there is a risk-free
asset and assunw that it is non-singular in either case.
PThis theorem is almost identical to Merton’s Theorem I. For a heuristic proof based upon
taking limits of a discrete time process. see Merton (1969) or Drefus (1965). For a more formal
proof, see Kushner (1971,ch.
I 1.3).For a rigorous proof, set Kushner (1967. ch. IV. Theorem
7).
194
S.F.
Richard,
0 = Lr(C+,P+,
0 = L,(c*,
Consumption.
portfolio
1
w’) = - B,
c1
p*, w*) = I-
and life insurance rules
-Jw,
2
(19)
WT.
(20)
i=l
The second-order conditions are
&c
= ucc < 0,
L wrw1 =
L PP
=
~,/W2Jww,
~~l~~V$z
<
0.
all other second partials being zero. Thus a sufficient condition for a maximum
is
J ww < 0 which we will henceforth assume. Total differentiation of (17) with
respect to W yields X*/aH’
> 0 and of (19) yields JZ*/a W > 0, where Z* =
W+ P*/p is the optimal of legacy.
We now observe that OUI eq (18), which determines the optimal
portfolio
of
securities, is identical to Merton’s (20) [except for the last term in (20) which is
zero under the assumption of log-normality of prices]. Thus without further
derivation
Markowitz
we may conclude that Merton’;
generalization
of the Tobinseparation theorem is valid under the assumption of uncertain
lifetime.
Gi~vn n a.s.sct.s wifh prices Si which arc log-normally
Theorcrrl 2 (Morton).
rlis~ribu~d, rhcn (I) /here c~.vi.s,sa unique (up 10 a non-.tingular transformation)
pair of mutual finds which arc tincar combinarion of ~hc.ve assets such rhat,
indepcntlcn~ of prcft*rcncrs, wrald clisrriburion, probability clisrribulion of li/e~irnc
or ii/e insurance opporlunilirs irulic~id~ralswill be ind#iircn~ between choosing from
a lintpar conlbinarion of ~hc*sseMO funds or a lincar conrbinarion of the original
assels. (2) If S, is Ihc price pvr share of eilhpr fund Ihen S, is log-normally distribured.
As Merton shows the proportion lo of values each mutual fund would invest
in each risky security depends only on the physical distribution of returns, i.e.,
(ai, ali}, and hence these proportions for the uncertain lived investor arc the
same as they are for his certain lived counterpart. Moreover, if one of the assets
is risk-free then the following corollary is valid.
‘%ee Mcrton (1971, p. 384) for the proportions
of each asset held in each fund. Our notation
is the same as Merton’s
for ease of comparison.
Also Merton shows (p. 399). for the case of
exponential
distribution
of lifetime, that the only effec! is to add a time discount factor IO the
utilily function.
S.F. Richard, Conswnption. portfolio and life inrwance rules
195
Corollary (Merlon). If one of the assets is risk-free (say the nth) then one
mutual fund can be chosen to contain only the risk-free security and the other to
contain only the risky assets in the proportions
Sk =
,zlvkj(aj-f)/,Fl
,f, vij(aj-fh
=
k = 1, - - - , m,
(21)
s
where
br,l = b*,l-l.
We will henceforth assume that there is a riskless asset so that we can work
without loss of generality with only two assets, one risk-free and the other a
mutual fund of risky assets with a log-normally distributed price S(f) defined by
dS/S = adr+adq,
(22)
where
k=I
(23)
dq
=
,$
@k”k
dqkb-‘).
I
Hence we may define w+(f) E w;(l)/S, to be the optimal fraction of wealth to
be invested in the risky fund at time f. Now from (IS) with k = n we find that
Y = J,rW. Substituting for w; and Y in (IS), then multiplying (18) by bk and
summing from k = 1, . . . , m, we get
0 =
J&a-r)+JWWw*u2W2,
w # 0.
(18’)
Let R( W, I) G -J,,/J,
be the investors (imputed) risk aversion for wealth
and assume that R > 0. We find that
a-f
w+w = -
1
-_
(24)
u2 R
Differentiating (24) with respect to Wyields
a(w* W)
a-r
3-v-=---* u2
Rw
R2
196
S.F. Richard, Consumption, portfolio and Iij insurance rules
Thus the optimal dollar amount invested in the risky fund will be an increasing
function of wealth if and only if the imputed risk aversion in wealth is decreasing. However, even if Rw < 0 it is not necessarily true that the optimal fraction
of wealth will increase since
a-r
dW’
dCY = 02
R+ WRw
(RW)2
’
positive or negative if Rw
increasing risk aversion, then
fraction of wealth invested in the risky fund decrease
Finally we substitute w: = w* 6, into (16) and
find that
which
R,
may be either
> 0, i.e., positive
+J,[U++
Y-c*-P*]--
1 J$
2&P,
c 0. ‘On the other
hand if
both the dollar amount and
with increasing wealth.
then (18’) into the result to
a-f
’
.
u
(
>
(16’)
Thus (16’) (17). (IS’), (19) and (20). i.e., w.’ = 1 -w*, are a complete set of
partial differential
equations
for C*( W, I), P*( W, I), w*( W, r), wi( W, I) and
J( W, t), the five required functions. WC note that (17) and (19) respectively, are
the normal marginality
requircmcnts
that the marginal utility of current consumption equals the marginal utility of wealth (future consumption)
which in
turn equals the expected marginal
utility of legacy. Furthermore,
we may
regard C+ and P+ as, respectively,
the levels of consumption
and premium
payment net of any minimum
levels. This can be seen by letting /(I) be the
minimum (subsistence) level of consumption
at time I and letting n(r) be the
minimum level of legacy at time f. [Note that n(r) may be negative if nonpurchased
insurance,
such as social security and employee benefit policies,
exceed minimum requirements.]
Then defining
C’(r) = C’(r)-/(r),
P’(f) = P’(f)--/&)“(1),
Z’(r) = W(r)+Y’(t)
E
P’(f)
= Z’(r)-n(r),
P(f)
Y(f)-F(I)-p(f)n(f),
U’(C’(f),
I) = u(c*(f),
B’(Z’(t),
I) 5 B(Z*(r),
I),
I).
S.F. Richard,
Consumption,
portfolio
197
and life insurance rules
we see that the form of (16)-o-(O) remains unchanged with the primed variables
and functions substituted for the unprimed ones. Thus without loss of generality
we can assume that there are no minimum levels of consumption
or legacy [or
more precisely that any such minimums are deducted from the future earnings
stream Y(t) leaving a net future earnings stream].
We have shown that minimum levels of consumption
and legacy are unessential
to our problem by incorporating
them in the future earnings stream. We may
next reasonably question whether or not the future earnings stream is essential
to the problem. The following theorem shows it is not.”
Theorem 3. If security prices have a log-normal distribution and there exists
a riskless security and lyat any time t = ‘5,any investor relinquishes his right to his
future earnings Y(t) for t > T and receives instead a certain addition to wealth
b(r), where
b(t) = 17 Y(s)exp
= F
[-r(s-t)-jI:p(~)dr]ds
Y(s) %)/G(r)
exp [-r(s-t)-{(H(t)-H(s)}]
which is independent of his preferences and the
depends on/y upon the risk-free rate of return and
investor’s optimal consumption rate, legacy, risky
utility will remain unchanged. Furthermore, if we
positions at t = T to be his ‘adjusted wealth’,
ds,
(25)
return on risky securities and
the l1Q-einsurance rates, than the
investment and expected future
d&e the investor’s new wealth
X(r) = W(r)+ b(r),
then following optimal policies at any t > T, the investor’s wealth will be
X(t) = W(r)+b(t).
ProoJ
(26)
Define
b(t) = P:(t) -AM).
(27)
and
1(X, I) = I(W+b(t),
I) = J(W, I),
(28)
so that
Jw = I,,
J,+,w = I,, ,
J, = f, +I,b’(t),
(29)
where
b’(t) = - Y(t)+ [r+p(t)]b(t).
(30)
’ 'Mcrton (1971) obtains the same result for non-stochastic lifetime and wage income
(p. 394) and a similar result for a stochastic wage income and non-stochastic lifetime (p. 397).
198
S.F. Richard.
Consumption,
portfolio
and life insurance rules
Now defining ifx = W* W and C = C* we substitute
(lS’)and(19)tofind
(25x30)
into (16’). (17).
(16”)
(17”)
(18”)
(19”)
subject
to
1(X(T), T) = B(Z(T),
T).
Now an investor with total wealth X and a risky investment
riskless investment 3,X, such that
of RX must have a
2” = l-8.
(20’)
We recognize (16”)-(20”) as the equations satisfied by the optimal policies of an
investor with wealth X(t) and no future earnings. Thus we conclude that C =
c+,xx=
w*w,Z = X+(p/l)
= W+(P*/p)
= Z* are the optimal policies
for the investor with no future earnings and further that f(X, r) = J( W, I) for
any I = T, such that X(7) = W(T) + b(r).
To complete the proof we must show that if the investor without a future
earnings stream chooses C = C*, 2! = Z+ and xX = w+ W for all I > r, then
X(r) = W(r)+b(r). The asset dynamics equations are
dW=
-C*df-P*dr+
Ydl+w*Wadr+(l-w+)Wrdf
+ w+ Wu dq,
(31)
and
dX = -C
dr-P
dr+_QXa dl+(l
-_Q)Xr dr+2Xa
dq.
(32)
Under our assumptions,
dX = dW-
Y(f)dr+(r+/@dr
= d W+b’(r)
dr.
(33)
S.F. Richard.
Consumption,
Using Ito’s Lemma to integrate
portfolio
and life insurance ruler
199
(33). we find that, for t > T,
X(r) = X(r) + W(r) + W(T) - b(t) - b(T)
= W(f) + b(f).
Q.E.D.
The function b(t) represents the investors net ‘human capital’ at time 1. It is the
expected present value of future (wage) earnings found by discounting
back to
time t. The instantaneous
rate of discount is the sum of the risk-free rate and the
cost rate of insurance, i.e., to discount from r+ Ar to r the discount factor is
{I-[r+p(r+Ar)]Ar}
Thus to discount
zz exp {-[r+cl(r)lAr).
from r + T to r the discount
,fi, {I -[r+p(r+iAr)]Ar}
factor is
-+ exp r-j:”
[r+dT)]dr),
as
Ar -+Q,
while
nAr = T.
Hence the more costly insurance is, i.e., the greater is 9. the less that future
earnings are worth. It is interesting to note that the investor’s human capital does
not directly depend on the force of mortality, A(r) [except that p(r) 2 J(r)].
In a sense Theorem 3 is a separation theorem in that the certainty cquivalcnt
of future net earnings b(r) is indcpcndent
of risky market opportunities
and
prcfcrenccs.
Hence X(r) = W(r)+b(r) represents
the ‘objectively’
obtained
certainty equivalent of any investors present wealth and future earnings, i.e.,
independent
of preferences any investors would trade his future net wages for
b(r). Furthermore,
we may interpret
as the premium
necessary to cover b(r) with insurance
and
B(r) = P(r)-p(r)b(r)
as the excess premium paid to insure beyond (or below) b(r). It is commonly
thought that insuring against the risk of losing b(r) requires that p = 0, but in
general this will not be the case.
200
S. F. Richard.
Consumption,
portfolio
and life insurance rules
4. Solutions for utility function with coastaot relative risk aversion
Let
y-Cl,
U(C, r) = VrO)lYJC’,
be the investor’s
that
h>O,
utility for consumption.”
c>o.
By substituting
(34)
(34) into (17). we find
c* = (h/&)“(‘-“.
Similarly,
(35)
let ”
NZ, I) = bW~lZy,
be the investor’s
z*
1,
Y <
utility for bequest. Substituting
w+p_’=
=
P
m > 0,
Z > 0,
(36)
(36) into (19) yields
1/(1-Y)
Am
(zv)
-
(37)
We now use (34)-(37) in (I 6’) and S c I - y to find that
0 =
(38)
+J,[(r+pOW+
VI.
where
k(l) = ({~(r)/~(l)}y’d{~(r)m(l)jlld+h(l)”d].
subject
(39)
to
J(W, T) = B(Z. T) = {iii/y}ZY.
where the overbar denotes evaluation
at 7’. e.g., Z = Z(T). A solution
to (38) is
J( W 11= MO/Y I( W+ W))Yv
where
b(f) is given
u(r) =
(40)
by (25). and
’
IS
,
4s)
“When
y = 0, U(C. I) = h(r) In C.
“It is mathematically
necessary that the exponent
legacy in order to get solutions in closed form.
[&)-H(s)]
1I*,
ds
y be the same for both consumption
(41)
and
S.F. Richard,
Consumption,
portfolio
and life insurance rules
201
where
v = (cr-r)2/2CW.
From (18’). (25), (35). (37). (40) and (41) the optimal
insurance and portfolio rules can be explicitly written as
consumption,
life
and
By using L’Hopital’s rule in (41) we find that 5 = fi. Thus from (43) we find
that z* = W(T), i.e., the investor will purchase no insurance at $(because
the
cost rises without limit). Furthermore,
J(
w, T) = {n/y}Z*Y =
satisfying the boundary conditions.
The stochastic process gcncrating
(44) in (3’)
qz*, T).
wealth can be derived by substituting
(42)-
(45)
+[W(r+p)+
Y]dr,
where
X(f) = W(r)+b(r).
Now by Ito’s Lemma
dX = d W+b’(r)
= d WThus X(r) is the solution
dr
(46)
Y dl+(r+p)b
dr.
to
dX
_ = (a-r)Z+r+P
X
C Sa2
-- X(r)
u(l)“*
1
dr+ac
dq.
(47)
202
S.F. Richurd.
We may use Ito’s
Lemma
Consumption,
porrfolio
again to integrate
and life insurance rules
(47), finding
that
I
+yyody ,
and
(48)
hence the ‘adjusted wealth’ X(r) is log-normally
distributed.
Thus,
W(t) = X(r)-b(r)
is, as Merton
random
found,
variable.
and W(f)
a ‘displaced’
By Ito’s
is continuous
only
find with
exp [(
“+a)
T+i
distributed
probability
H(O)+yJadq].
if iC = 0. i.e., it is optimal
has no bcqucst
probability
log-normally
to (47) with
motive
(49)
to bc penniless
at 7? Substituting
at i’if
(48) into
(42),
so that
will
control
It seems reasonable,
to at any time
kY+6(f).‘4
time
I for
while
for
From
if h = 0. Hence.
his wealth
keeping his consumption
want
if the investor
to make it approach
rate non-negative
and WC shall do so, to assume
(43) we have then
enough
that
the investor
he will
has no legacy motive
zero while
at the same time
and finite.
have a legacy greater
W large enough
N’small
WC
(50)
c = 0 if and only
T, he
and
one that
C’( K I)
at
one
one, so that
[--$]“d
kv = 0 if and only
if the investor
(48) is a solution
with probability
= x(o)
Thus
or ‘three-parameter’
Lemma
than
that the investor
X(r),
i.e., assume
{,,l(r)~(r)/n(r)lr(r)}
will
purchase
certainly
insurance.
i
would
Z*(
not
IV, I) 2
I so that at any
be a seller
of insurance,
It seems quite
reasonable
“II is of course possible that the investor’s preferences for legacy are strong enough so that
he wishes lo insure IO a level beyond rhe present value of his toral wealth (capital plus human
capital). However. this case is not what is commonly thought of as insuring against the ri\k of’
loss of future income.
S. F. Richard,
Consumption,
portfolio
and life insurance rules
and intuitive that, neglecting tax effects, the wealthy will find little need for life
insurance. Moreover, for large enough b(r) the investor will purchase insurance
no matter what his relative likings for consumption
and legacy are. At the same
time we see from(44) that as wealth increases, the/faction
of wealth w* placed in
the risky security shouid decline [provided b(r) > 01, while the dollar amount
w* W increases absolutely. This is again quite reasonable in the following sense.
The individual with small wealth W relative to future prospects b(t), depends
upon his wages both for his consumption
and for his life insurance premium,
which he must pay to protect against the loss of his main asset - his future. The
wealthy individual with W relatively large to 6, depends upon his current wealth
to earn most of his funds for consumption
and to supply the bulk of his estate.
In the same way the ‘poor’ man must protect his assets, the wealthy individual
protects his by, at any given time, placing a greater proportion
of his wealth in
the riskless security, the more wealth he has at that time.
References
Dreyfus, S.L., 1965, Dynamic programming
and the calculus of variations (Academic
Press,
New York).
Fischer, S., 1973, A life cycle model of life insurance purchases, International
Economics
Review 14. no. I. 132-152.
Hakansson.
N.H.,
1969. Optimal investment and consumption
strategies under risk, an uncertain lifetime. and insurance, International
Economic Review IO. no. 3.443466.
Kushncr. H.J.. 1967. Stochastic stability and control (Academic Press, New York).
Kushncr. H.J., 1971, Introduction
to stochastic control (Holt.
Rinehart and Winston,
New
York).
Mcrton.
R.C.,
1969. Lifetime portfolio
sclcction under uncertainty:
The continuous
time
case, Review of Economics and Statistics Lt. 239-246.
Merton. R.C.. 1971, Optimal consumption
and portfolio rules in a continuous-time
model,
Journal of Economic Theory 3, no. 4.373413..
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