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Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model
Journal of Financial Economics 2 (1975) 187-203. 0 North-Holland
Publishing Company
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.
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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.
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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.
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S.F. Richard,
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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$TST,
t#T
for death
at time
T
(6)
we will define A(t) by A(t) = n(t: I).
ost
(7)
When n(‘P) > 0, it is easily seen that for t -+ 7, A(t) -+ co. If n(T) = 0, then, as Yaari (1965) does for each t E [0, 7’). we define T, by ?t(T,) = sup n(r) > n(T) = 0. rq 1.T1 Now G(t) s (T-
t)n(r,), so that
40 2 W/~V- t)dr,)l.
(8)
Now since n(t) is continuous, it must bc monotone decreasing in the neighborhood of ?‘, so that n(t)/n(r,) = 1 and hence A(t) diverges. Now that the investor’s probability distribution for death has been given, we can specify the life insurance rates. We assume that paying premiums at the rate P(t) buys life insurance in the amount P(t)/p(t) for the next (very) short period. Thus the investor’s total legacy should he die at time t with wealth W is
at) = W(t)+ vYNP(t)l.
(9)
‘We will not be overly concerned with the regularity conditions on non-stochastic exogenous functions of time; in general we will assume them to bc continuous in the neighborhood of T, unless otherwise noted.
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S.F. Richard,
Consumprion.
portfolio
and life insurance rules
We assume that
P(r) = 4r)+q(l), where q(r) is any non-negative of 7, i.e.,
(10) integrable function
H(f) = {Fq(r) dr < co,
continuous
in the neighborhood
r E [O, n
(11)
and lim {q(r)/A(f)} = 0. r-7 Hence lim {&)/A(r)} r-F
= 1.
The investor wishes to choose optimal controls (i.e., consumption, life insurance and portfolio functions) to maximize his expected lifetime utility for consumption and legacy. That is, to
(h$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
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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
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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,
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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,
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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.. Yaari, M.E.. 1965, Uncertain lifetime. hfe insurance, and the theory of the consumer. Review of Economic Studies 32.137-150.
Publishing Company
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$TST,
t#T
for death
at time
T
(6)
we will define A(t) by A(t) = n(t: I).
ost
(7)
When n(‘P) > 0, it is easily seen that for t -+ 7, A(t) -+ co. If n(T) = 0, then, as Yaari (1965) does for each t E [0, 7’). we define T, by ?t(T,) = sup n(r) > n(T) = 0. rq 1.T1 Now G(t) s (T-
t)n(r,), so that
40 2 W/~V- t)dr,)l.
(8)
Now since n(t) is continuous, it must bc monotone decreasing in the neighborhood of ?‘, so that n(t)/n(r,) = 1 and hence A(t) diverges. Now that the investor’s probability distribution for death has been given, we can specify the life insurance rates. We assume that paying premiums at the rate P(t) buys life insurance in the amount P(t)/p(t) for the next (very) short period. Thus the investor’s total legacy should he die at time t with wealth W is
at) = W(t)+ vYNP(t)l.
(9)
‘We will not be overly concerned with the regularity conditions on non-stochastic exogenous functions of time; in general we will assume them to bc continuous in the neighborhood of T, unless otherwise noted.
192
S.F. Richard,
Consumprion.
portfolio
and life insurance rules
We assume that
P(r) = 4r)+q(l), where q(r) is any non-negative of 7, i.e.,
(10) integrable function
H(f) = {Fq(r) dr < co,
continuous
in the neighborhood
r E [O, n
(11)
and lim {q(r)/A(f)} = 0. r-7 Hence lim {&)/A(r)} r-F
= 1.
The investor wishes to choose optimal controls (i.e., consumption, life insurance and portfolio functions) to maximize his expected lifetime utility for consumption and legacy. That is, to
(h$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.. Yaari, M.E.. 1965, Uncertain lifetime. hfe insurance, and the theory of the consumer. Review of Economic Studies 32.137-150.