Insurance Mathematics and Economics 39 (2006) 4768

www.elsevier.com/locate/ime
Optimal insurance in a continuous-time model
Kristen S. Moore

, Virginia R. Young
1
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States
Received September 2005; received in revised form January 2006; accepted 26 January 2006
Abstract
We seek the optimal dynamic consumption, investment, and insurance strategies for an individual who seeks to maximize her
expected discounted utility of consumption and bequest over a xed or random horizon, such as her random future lifetime. Thus,
we incorporate an insurable loss and random horizon into the classical consumption and investment framework of Merton. We
determine that if the premium is proportional to the expected payout, then the optimal per-claim insurance is deductible insurance;
thus, we extend this result for static models to our dynamic setting. We compute the value function and optimal controls for many
examples and contrast their qualitative properties, including the impact of the investors horizon (or mortality) on the optimal
controls and the interaction between the demand for insurance and the risky asset. We employ the Markov Chain approximation
method of Kushner for those examples for which closed form solutions are not available.
c 2006 Elsevier B.V. All rights reserved.
JEL classication: D91; C61
Keywords: Optimal insurance; Expected utility; Stochastic control; HamiltonJacobiBellman equations; Markov chain approximation method
1. Introduction
We consider a risk-averse investor who can invest in and trade dynamically between a riskless and risky
asset and who faces a random, insurable loss that is independent of the risky asset. We seek the optimal asset
allocation, consumption, and insurance strategies to maximize her expected utility of terminal wealth (or bequest)
and consumption over the xed horizon T, or the random horizon , where is the time of death. The problem
of optimal consumption and saving under uncertainty was studied by Merton (1969, 1971), Samuelson (1969), and
others; we incorporate a random horizon and a random, insurable loss, which we model as a compound Poisson
process, into the optimal investment and consumption problem.
The investor faces competing objectives. Insurance is costly and therefore reduces the utility that could be
derived from saving or consuming; however, absorbing a large loss without insurance also diminishes utility. Current
consumption reduces future utility from bequest. Investing in the risky asset may yield high returns, but it may also
yield losses. Moreover, in the case of a random horizon, there is an additional source of uncertainty.

Corresponding author. Tel.: +1 734 615 6864.

E-mail addresses: ksmoore@umich.edu (K.S. Moore), vryoung@umich.edu (V.R. Young).
1
Tel.: +1 734 764 7227.
0167-6687/$ - see front matter c 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.insmatheco.2006.01.009
48 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Under fairly general conditions, the optimal insurance is deductible insurance (Arrow, 1963). Mossin (1968)
showed that it is never optimal to purchase full insurance when the premium includes a positive loading above the
actuarially fair price. However, most of the existing models for deductible insurance are static ones. In this paper, we
examine a dynamic model for insurance and determine that if the premium rate is proportional to the expected payout,
then optimal per-claim insurance is deductible insurance. We restrict the form of indemnity to be per-claim because
we model the insurance risk as a compound Poisson process. The optimality of deductible insurance for a compound
Poisson random variable is known in the static case (van Heerwaarden, 1991, Theorem 9.4.1; Schlesinger and Gollier,
1995), so it is no surprise that it holds in the dynamic case.
By considering the dynamic setting, random horizon, and the effect of insurance on investment and consumption,
this work complements and extends the existing literature on optimal insurance. Briys (1986) assumes that the loss is
proportional to wealth and looks for the optimal coinsurance. It turns out that if the loss is proportional to wealth and
if the utility is isoelastic, then the optimal insurance is a coinsurance, if we allow the deductible to be a function of
wealth (and time); see Example 3.5. Gollier (1994) assumes that the loss is not a function of wealth only of time
and looks for the optimal coinsurance. We show that optimal insurance in this case is deductible insurance; the two
cases coincide in special cases. Plus, we extend Briys (1986) and Gollier (1994) by allowing the horizon to be random,
as in the random time of death, and we consider the effect of insurance on investment and consumption decisions.
Other researchers have used stochastic control techniques to solve problems of interest in actuarial science and
insurance economics. Asmussen and Taksar (1997) examine the problem of maximizing the discounted value of
dividend payout for insurance companies. Hjgaard and Taksar (1997) maximize the discounted value of wealth
by optimizing over proportional reinsurance policies. Both papers use diffusion models, which are appropriate for
modeling the cash ows of insurance companies but not for individualsthe perspective we take in this paper.
Young and Zariphopoulou (2002, 2004) use stochastic control techniques, similar to the ones in this paper, to nd
the reservation prices for a buyer and seller of insurance products.
In Section 2, we state the specics of our problem. In Section 3, we determine optimal per-claim insurance for
the case in which the horizon of the insurance buyer is xed and nite. In Section 4, we examine the case in which
the horizon of the insurance buyer is determined by the random time of death. In Section 5, we present numerical
results for which the techniques of Sections 3 and 4 do not yield closed form solutions for the value function and
optimal controls. Specically, we consider the isoelastic case in which the loss amount is a xed number. Gollier
(1994) examines this case but does not fully consider the occasion when self-insurance is optimal. In each example,
we examine the impact of the horizon on the optimal controls and examine the interaction between investment,
consumption, and the insurable loss. We summarize the main results of our work below.
1. We show that in our dynamic setting, for per-claim indemnity coverage, optimal insurance is deductible
insurance. This is consistent with the static results of Mossin (1968), van Heerwaarden (1991, Theorem 9.4.1)
and Schlesinger and Gollier (1995).
2. We prove some general qualitative properties of the optimal deductible; namely, that, under certain conditions, it
increases with wealth and the price of insurance.
3. By contrasting Examples 3.4 and 4.1, we nd that introducing a random horizon eliminates the dependence of the
optimal strategies on the investors horizon in the case of exponential utility.
4. We nd that, when we impose a lump-sum (versus continuous) insurance premium, the investors mortality (or
expected horizon) affects the optimal deductible. Moreover, the parameters of the stock price process affect the
optimal deductible; as the stock becomes more risky, the demand for insurance increases. (See Example 4.2.)
5. We demonstrate that, under power utility, the parameters of the loss process affect the optimal consumption
strategy and that this dependence may be counterintuitive, depending on the relative risk aversion. However, under
logarithmic utility, the optimal consumption strategy is independent of the parameters of the loss process. See
Examples 3.5, 4.3, 5.1a, 5.1b and 5.2.
6. Finally, we observe that when we add an exogenous wage, the parameters of the loss process and the investors
mortality (or expected horizon) affect the optimal strategies; see Examples 5.1b and 5.2.
These results are detailed in Section 3 through 5 and summarized in the conclusion in Section 6.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 49
2. The problem
We assume that an individual can invest in a riskless asset (bond) whose price at time t , X
t
, follows the process
dX
t
= r X
t
dt , for some xed r 0. Also, this individual can invest in a risky asset (stock) whose price at time t , S
t
,
follows the process
_
dS
s
= S
s
(ds +dB
s
), s t,
S
t
= S,
in which > r and > 0 are constants, and B is a standard Brownian motion with respect to a ltration {F
t
} of the
probability space (, F, Pr). Note that S follows a geometric Brownian motion.
Let W
t
be the wealth at time t of the decision maker, and let
t
be the amount that the decision maker invests in
the risky asset at time t . The decision maker earns an exogeneously given wage rate of (t ) and consumes at a rate
of c
t
at time t . Finally, the decision maker is subject to an insurable risk modeled as a compound Poisson process, in
which N is a Poisson process with deterministic parameter (t ), and the random loss amount is L(W
t
, t ). Assume
that N is independent of B, the Brownian motion of the process for the risky asset. Also, the (random) loss amount
L is independent of N. Note that we allow the loss to depend on the wealth at time t . Then, wealth without insurance
follows the process
_
dW
s
= [r W
s
+( r)
s
+(s) c
s
] ds +
s
dB
s
L(W
s
, s)dN
s
, s t,
W
t
= w.
(2.1)
Next, consider per-claim insurance of I
t
. That is, if the individual suffers a loss of L = l at time t , the indemnity
pays I
t
(l). Such indemnity arrangements are common in private passenger automobile insurance and homeowners
insurance, among others. Assume that the premium is payable continuously, so we subtract the premium rate from the
rst term on the right-hand side of expression (2.1). Assume that the premium rate P(t ) is proportional to the expected
payout; specically, P(t ) = [1 +(t )](t )E[I
t
(L)], in which (t ) 0. Then, wealth with insurance follows
2
_
_
_
dW
s
= [r W
s
+( r)
s
+(t ) c
s
(1 +(s))(s)E[I
s
(L)]]ds +
s
dB
s
(L(W
s
, s)) I
s
(L(W
s
, s))dN
s
, s t,
W
t
= w.
(2.2)
3. Fixed time horizon T
We assume that the decision maker seeks to maximize (over allowable {c
t
,
t
, I
t
}) his or her expected utility of
discounted consumption during a xed time interval [t, T] and nal wealth at time T. Allowable {c
t
,
t
, I
t
} are those
that are measurable with respect to the information available at time t , namely F
t
, and that result in (2.2) having a
unique solution.
Thus, the associated value function of this problem is given by
V(w, t ) = sup
{c
t
,
t
,I
t
}
E
__
T
t
e
(st )
u
1
(c
s
) ds +e
(Tt )
u
2
(W
T
)|W
t
= w
_
, (3.1)
in which u
1
and u
2
are twice-differentiable, increasing, concave utility functions, and > 0 is the rate at which
consumption and terminal wealth are discounted. The parameter is a personal discount rate; it is not a market
parameter, but rather part of the utility functional. The function u
1
measures the utility of consumption, while u
2
measures the utility of terminal wealth. Note also that we assume that the utility for consumption and the utility for
terminal wealth are additively separable. This is not necessarily true for all decision makers under expected utility, but
we assume it here.
2
We use subscript t or s to represent (possibly) random processes, while we use parentheses t or s

to denote deterministic functions of time.

50 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
V solves the HamiltonJacobiBellman (HJB) equation:
_

_
V
t
+max

_
( r)V
w
+
1
2

2
V
ww
_
+max
c
[u
1
(c) cV
w
] +(rw +(t ))V
w
+ max
I
[(t ){EV(w (L I (L)), t ) V(w, t )} (1 +(t ))(t )E[I (L)]V
w
]
= V
V(w, T) = u
2
(w).
(3.2)
V
t
, V
w
, and V
ww
denote partial derivatives of V with respect to the given variable. For example, V
ww
is the second
partial derivative of V with respect to w. See Merton (1992, Section 5.8) and Bj ork (1998) for the derivation of related
HJB equations. There also exist verication theorems that tell us if the value function V is smooth and if

V is a smooth
solution of the associated HJB equation, then under certain regularity conditions,

V = V. In those cases for which
there is no smooth solution, we rely on the theory of viscosity solutions; see Young and Zariphopoulou (2002) for
further details and references.
The value function V is increasing and concave with respect to wealth w because the utility functions u
1
and u
2
are increasing and concave and because the differential equation for wealth is linear with respect to the controls. Thus,
the optimal investment strategy {

t
} is given by

t
=
( r)

2
V
w
(W

t
, t )
V
ww
(W

t
, t )
, (3.3)
in which W

t
is the optimally controlled wealth. Also, the optimal consumption {c

t
} solves the equation
u

1
(c

t
) = V
w
(W

t
, t ). (3.4)
Concerning the optimal indemnity process {I

t
}, we have the following proposition via standard arguments from
insurance economics, as in Arrow (1963).
Proposition 3.1. The optimal indemnity process {I

t
} is either no insurance or per-claim deductible insurance, in
which the deductible may vary with respect to time. Specically, at a given time, the optimal deductible d

t
(if it exists)
solves
(1 +(t ))V
w
(W

t
, t ) = V
w
(W

t
d
t
, t ). (3.5)
No insurance is optimal at time t if and only if
(1 +(t ))V
w
(W

t
, t ) V
w
(W

t
ess supL(W

t
, t ), t ).
We remark that we do not necessarily assume that L is a continuous random variable in this proposition. Also, if
(t ) = 0, then d

t
= 0; that is, full insurance is optimal if the premium rate is actuarially fair.
The result in Proposition 3.1 is well known in static models; namely, if insurance is per-claim insurance and
if the premium is proportional to the expected payout, then optimal insurance is deductible insurance. Indeed, van
Heerwaarden (1991, Theorem 9.4.1) proves it in her dissertation, and Schlesinger and Gollier (1995) prove it in the
economics literature. Our contribution is to show that the same result holds in a dynamic setting, a not-so-surprising
result. Note that the condition in (3.5) can be interpreted economically. Indeed, the left-hand-side is the marginal cost
of decreasing d
t
(or increasing insurance coverage), while the right-hand-side is the marginal benet of increasing
insurance coverage. Thus, optimality occurs when the marginal cost equals the marginal benet, an intuitively pleasing
result that parallels the static case.
A well-known property of the demand for insurance in the static case is that insurance can be a Giffen good if
the relative risk aversion is greater than one. If the wealth effect is greater than the substitution effect, insurance is a
Giffen good (Briys et al., 1989). However, insurance is not a Giffen good in the continuous-time model, as stated in
the following corollary to Proposition 3.1. Note that this is identical to Proposition 1 of Gollier (1994), in which he
assumes insurance is proportional coverage.
Corollary 3.2. An increase in the instantaneous price of insurance reduces the instantaneous demand for insurance,
if d

t
exists.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 51
Proof. Suppose (t ) increases at time t for an innitesimal length of time, then this change has no effect on the
characteristics of the value function V. Because V
ww
< 0, d

t
also increases at time t for an innitesimal length of
time.
Note that there is only a substitution effect in the instantaneous case. If there were a permanent increase in the
loading, then demand for insurance might increase or decrease due to the wealth effect. We examine this problem in
specic examples.
In the next corollary of Proposition 3.1, we show that if the value function V exhibits decreasing absolute risk
aversion with respect to wealth, then the demand for insurance decreases with increasing wealth. Here, we follow
Pratt (1964) and dene the absolute risk aversion of V with respect to w by
V
ww
(w,t )
V
w
(w,t )
. Corollary 3.3 directly parallels
the corresponding result in the static case.
Corollary 3.3. If V exhibits decreasing absolute risk aversion with respect to wealth, then the demand for insurance
decreases with increasing wealth, if d

t
exists.
Proof. Differentiate Eq. (3.5) with respect to w to get that
d

t
w
(1 +(t ))V
ww
(w, t ) V
ww
(w d

t
, t )
= (1 +(t ))V
ww
(w, t ) V
ww
(G[(1 +(t ))V
w
(w, t ), t ], t ),
in which G is the inverse of V
w
with respect to wealth. Let w = G(y, t ); then,
d

t
w
(1 +(t ))V
ww
(G(y, t ), t ) V
ww
(G[(1 +(t ))V
w
(G(y, t ), t ), t ], t )
(1 +(t ))
V
ww
(G(y, t ), t )
V
w
(G(y, t ), t )

V
ww
(G[(1 +(t ))y, t ], t )
V
w
(G(y, t ), t )

V
ww
(G(y, t ), t )
V
w
(G(y, t ), t )

V
ww
(G[(1 +(t ))y, t ], t )
(1 +(t ))y

V
ww
(G[(1 +(t ))y, t ])
V
ww
(G[(1 +(t ))y, t ], t )

V
ww
(G(y, t ), t )
V
w
(G(y, t ), t )
0
because G decreases with increasing wealth and V exhibits decreasing absolute risk aversion with respect to
wealth.
Next, we consider several examples.
Example 3.4 (Exponential Utility). Suppose u
1
(c) 0 and u
2
(w) =
1

e
w
, for some > 0. Also, assume that
the random loss L is independent of wealth, although we allow Ls probability distribution to vary deterministically
with respect to time. From u
1
(c) 0, it follows that the optimal consumption is identically 0. A straightforward, but
tedious, calculation shows that
V(w, t ) =
1

exp
_
we
r(Tt )

( r)
2
2
2
(T t )
_
(t ),
in which solves
_
_
_

(t ) +(t )[(t ){M

L(t )d
t
(e
r(Tt )
) 1}
+(t )(1 +(t ))E(L(t ) d
t
)
+
e
r(Tt )
(t )e
r(Tt )
] = 0,
(T) = 1,
where M
L(t )d
t
is the moment generating function of L(t ) d
t
= min(L(t ), d
t
). It follows that the optimal deductible
is given by
d

(t ) = min
_
1

e
r(Tt )
ln(1 +(t )), ess supL(t )
_
,
52 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
and the optimal allocation to the risky asset is

=
( r)

e
r(Tt )
.
These results are consistent with those of Merton (1969; 1992, Section 4.9). Observe that d

(t ) and

(t ) are not
stochastic; in particular, they are independent of wealth. Also, the optimal investment strategy is independent of
wealth and, thus, independent of whether the individual buys insurance. Such independence from wealth is generally
observed in calculations with exponential utility because the absolute risk aversion (Pratt, 1964) is constant (equal to
). Moreover, note that d

(t ) is independent of the parameters of the stock process, another artifact of exponential

utility.
Note that as the risk aversion of the decision maker increases, as measured by , the allocation to the risky
asset decreases. Moreover, the deductible decreases, i.e., the demand for insurance increases. Also, as the load (t )
increases, the demand for insurance decreases because it becomes more expensive. Thus, Corollary 3.2 holds for
permanent increases in (t ) because the wealth effect is zero.
Note also that as the horizon T increases, the deductible and the allocation to the risky asset decrease; thus, an
investor with a longer horizon behaves more conservatively. This contradicts the conventional wisdom that investors
with longer horizons can be more aggressive in assuming risk (see Siegel (1994), for example). Samuelson (1963,
1989a,b) challenged this folk wisdom by pointing out that, although the standard deviation of the annual average
return decreases with time, the standard deviation of the total return increases.
Example 3.5 (Power and Logarithmic Utility). Suppose u
1
(c) =
c

and u
2
(w) = b
w

, for some < 1, = 0, and

for some b 0. The parameter b represents the weight that the individual gives to terminal wealth versus consumption
and can be viewed as a measure of her propensity toward saving. Also, suppose the wage rate is zero, and assume that
the loss is proportional to wealth, namely, L(W
t
, t ) = (t )W
t
, for some deterministic function (t ) between 0 and 1.
Then,
V(w, t ) =
w

(t ),
in which is given by
(t ) =
_
b
1
1
exp
_

_
T
t
H(s)
1
ds
_
+
_
T
t
exp
_

_
s
t
H(u)
1
du
_
ds
_
1
,
with H given by
H(t ) = +(t ) +(1 +(t ))(t ) max
_
0, (1 +(t ))
1
1
(1 (t ))
_
(t ) max
_
(1 +(t ))
1
1
, 1 (t )
_

,
and = r +
(r)
2
2
2
(1 )
. Thus, the optimal deductible is
d

t
= min
_
1 (1 +(t ))

1
1
, (t )
_
W

t
,
the optimal consumption is
c

t
= V
1
1
w
= (t )

1
1
W

t
,
and the optimal investment in the risky asset is

t
=
r

2
(1 )
W

t
,
all multiples of wealth. This linearity in wealth occurs because the power utility function exhibits constant relative
risk aversion (Pratt, 1964), that is,
wu

(w)
u

(w)
= 1 . Note that as the relative risk aversion increases, the proportion
of wealth invested in the risky asset decreases and the demand for insurance increases (d

t
decreases).
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 53
We see that , and hence V and c

are affected explicitly by the horizon T and that

and d

are affected by
T only through c

s impact on wealth. Note also that , and hence c

and V, depend on the frequency and severity

parameters and of the loss process. Moreover, we see that the optimal deductible d

is independent of the stock

price parameters and the optimal allocation

to the risky asset is independent of the insurable loss.

Next, consider the effect of insurance on the optimal consumption. If we solve the problemin (3.2) with I
t
restricted
to be identically zero, then we nd that the value function

V is similar to the value function V above with the function
H replaced by

H(t ) = +(t ) (t )(1 (t ))

.
When < 0, we can use the fact that g(x) = x

is decreasing and convex to show that H(t )

H(t ), for all t T,
from which it follows that the relative consumption with insurance is greater than the relative consumption without
insurance. Here, we measure relative consumption with respect to the optimally controlled wealth. Similarly, when
(0, 1), we can use the fact that g(x) = x

is increasing and concave to show that H(t )

H(t ), for all t T, from
which it follows that the relative consumption with insurance is less than the relative consumption without insurance.
Thus, when (0, 1), an investor with no insurance consumes more.
To explain this counterintuitive behavior, we note that when < 0, the utility approaches as w 0
+
;
however, when (0, 1), the utility approaches 0 as w 0
+
. In other words, when < 0, there is an innite
penalty for losing everything. When (0, 1), one can increase utility in the face of a larger potential loss by
consuming more; the penalty for losing everything is nite.
Young and Zariphopoulou (2004) nd similar counterintuitive results when 0 < < 1. They speculate that such
results for 0 < < 1 are counterintuitive because a variety of studies have estimated the value of the relative risk
aversion, 1 , to lie between 1 and 2 (or equivalently, 1 < < 0). See, for example, the well-cited article by
Friend and Blume (1975) that provides an empirical justication for constant relative risk aversion (CRRA), as well
as the more recent Mitchell et al. (1999) paper in which the CRRA value is taken between 1 and 2. In the context of
estimating the present value of a variable annuity for Social Security, Feldstein and Ranguelova (2001) argue that the
value of CRRA is less than 3 and probably even less than 2. Indeed, this is standard in macroeconomics and nance;
see Cochrane (2001) and Campell and Viceira (2002).
Suppose we let 0, so that u
1
(c) = ln c and u
2
(w) = b ln w. Then,
V(w, t ) = (t ) ln w +(t ),
in which
(t ) =
_
b
1

_
e
(Tt )
+
1

.
The optimal controls are consistent with our results for = 0. However, we remark that the optimal consumption c

is independent of the frequency and severity parameters and of the loss process; this was not the case when = 0.
As in the case of power utility ( = 0), we see that , and hence V and c

, are affected explicitly by the horizon T and

that

and d

are affected by T only through c

s impact on wealth. More specically, if b > 1, is a decreasing

function of T and, therefore, an individual with a longer horizon consumes a larger proportion of wealth. Similarly, if
b < 1, an individual with a longer horizon consumes a smaller proportion of wealth. The intuition behind this result
is not clear. On the one hand, b measures the investors propensity toward saving. On the other hand, measures her
preference for immediate, versus deferred, consumption.
4. Random time of death
Now, suppose that the time horizon is not xed, but rather it is determined by the random time of death of the
decision maker. We still assume that the decision maker seeks to maximize (over allowable {c
t
,
t
, I
t
}) her expected
utility of discounted consumption and nal wealth. That is, the associated value function of this problem is given by
V(w, t ) = sup
{c
t
,
t
,I
t
}
E
__

t
e
(st )
u
1
(c
s
) ds +e
(t )
u
2
(W

)|W
t
= w
_
. (4.1)
54 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
The value function V solves the HJB equation:
_

_
V
t
+max

_
( r)V
w
+
1
2

2
V
ww
_
+max
c
[u
1
(c) cV
w
] +(rw +(t ))V
w
+ max
I
[(t ){EV(w (L I (L)), t ) V(w, t )} (1 +(t ))(t )E[I (L)]V
w
]
+
x
(t )[u
2
(w) V] = V,
lim
s
E
_
e

_
s
t
(+
x
(u))du
V(W

s
, s)|W

t
= w
_
= 0,
(4.2)
in which
x
(t ) is the hazard rate, or force of mortality, for a person (x) at time t , or at age x + t . See Merton (1992,
Section 4.6) for a derivation of the boundary condition. As before, V is increasing and concave with respect to wealth
w. Thus, the optimal controls are as in Section 3; in particular, Proposition 3.1 and Corollaries 3.2 and 3.3 hold in this
case.
Example 4.1 (Exponential Utility). In this example, we reconsider Example 3.4, but with a random horizon and with
the riskless rate of return r = 0. Suppose u
1
(c) 0 and u
2
(w) =
1

e
w
, for some > 0, and suppose the random
loss L is independent of wealth. As in Example 3.4, we can show that
V(w, t ) = f (t )e
w
,
in which f (t ) solves
f

+ G(t ) f =

x
(t )

,
and
G(t ) = (t )
_
e
d

1 +(1 +) (L(t ) d

)
_

1
2

2
+(t ) +
x
(t ) +.
We assume that the parameter values are such that f (t ) < 0. Moreover, we have that the optimal consumption,
deductible, and allocation to the risky asset are given by
c

= 0,
d

(t ) = min
_
1

ln (1 +(t )) , ess supL(t )

_
,
and

,
respectively. Thus, as in Example 3.4, the optimal investment strategy is independent of the parameters of the insurable
loss and the optimal deductible is independent of the stock price process.
Recall that in Example 3.4, the optimal deductible and allocation to the risky asset were decreasing functions of
the horizon T. In contrast, in this example, these strategies are independent of the force of mortality
x
, and thus of
the investors horizon. However, the value function V is driven by the horizon through the dependence of f and G on

x
; however, the impact is ambiguous. For example, in the simpler case in which L(t ) = 1 and the model parameters
, ,
x
, and are constant, we have that
V(w, t ) = V(w) =
1

B
e
w
,
where
B = + + +
1
2

2
+ +(1 +) [ln(1 +) (1 +)] ;
thus, V(w) increases with if and only if
[ (1 +) ] < (1 +) [1 ln (1 +)]
_
+ +
1
2

2
_
.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 55
Example 4.2 (Exponential Utility with Lump-sum Premium). Suppose we have the conditions given in Example 4.1.
Also, suppose we have a stationary model; in particular, (t ) , (t ) , (t ) ,
x
(t ) , and the loss
random variable L is independent of wealth and time. Suppose also that the premium is payable as a lump sum at
time t equal to the expected present value of the payout times the loading (1 +). Specically, the premium payable
at time t is given by
1

(1 +)E(L d)
+
.
Because the model is stationary, the premium payable at time t is independent of t . For a given level of deductible d,
the value function V(w) solves
_

_
V

2
2
2
(V

)
2
V

+{EV(w L d) V(w)} + [u
2
(w) V] = V,
lim
s
E
_
e
(+)(st )
V(W

s
)|W

t
= w
_
= 0.
It follows that V is given by
V(w) =
1

e
w

+ + + [M
Ld
() 1]
,
in which =

2
2
2
. We assume that + + +[M
Ld
() 1] > 0. Therefore, as in Example 4.1, the optimal
allocation to the risky asset is

2
.
To nd the optimal deductible d

, we maximize
V
_
w
1

(1 +)E(L d)
+
_
with respect to d. It follows that the optimal d

solves
(1 +) ( + + + [M
Ld
() 1]) = e
d
.
This is our rst example in which the optimal deductible depends on the parameters of the stock price process and
on the Poisson frequency parameter . Moreover, d

depends on the investors horizon through its dependence on the

force of mortality .
It is not difcult to determine the following properties of the optimal deductible:
d

> 0,
d

> 0,
d

(1 +) e
d

,
d

> 0,
d

< 0,
d

> 0, and
d

(1 +) (1 +)
_
d

0
(1 +x) e
x
S
L
(x) dx d

e
d

.
Thus, as the insurance gets relatively more expensive ( increases), the demand for insurance decreases (d

increases).
As the stock process gets more risky ( decreases), the demand for insurance increases. As the force of mortality
increases, (i.e., as the horizon of the investor decreases), the demand for insurance is ambiguous. As the value of future
wealth decreases ( increases), the demand for insurance decreases. As claims become more frequent ( increases),
the demand for insurance increases (d

decreases). As the wage rate increases, the demand for insurance decreases;
this is a type of wealth effect that is a bit of a surprise with exponential utility. As the buyer becomes more risk averse
( increases), the demand for insurance is ambiguous. On the one hand, the demand could increase because the buyer
is more risk averse; on the other hand, if the wage rate is great enough, future wages can act as a type of insurance.
Finally, we can show that if the loss L increases in the ordering of rst stochastic dominance (Wang and Young, 1998),
then the demand for insurance increases.
Example 4.3 (Power and Logarithmic Utility). We revisit Example 3.5, but now incorporate a random horizon.
Suppose u
1
(c) =
c

and u
2
(w) = b
w

, for some < 1, = 0 and for some b 0, and suppose the random
loss L is proportional to wealth, namely, L(W
t
, t ) = (t )W
t
, for some deterministic function (t ) between 0 and 1.
56 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Also, assume that the wage rate is identically 0. Then, V is given by
V(w, t ) =
w

(t ),
in which solves the non-linear, non-homogeneous differential equation
0 =

H(t ) +(1 )

1
+b
x
(t ),
with

H(t ) = H(t )+
x
(t ), in which H is given in Example 3.5. We assume that the parameters are such that (t ) > 0.
Thus, the optimal deductible is
d

t
= min
_
1 (1 +(t ))
1
1
, (t )
_
W

t
,
the optimal consumption is
c

t
= V
1
1
w
= (t )
1
1
W

t
,
and the optimal investment in the risky asset is

t
=
r

2
(1 )
W

t
,
all multiples of wealth, as in Example 3.5. Recall that in Example 3.5, , and hence V and c

, were affected explicitly

by the xed horizon T. A similar phenomenon occurs here; these quantities are impacted by the force of mortality

x
. Moreover,

and d

are affected by mortality only through c

s impact on wealth. Finally, we note that the

optimal deductible d

is independent of the stock price parameters and the optimal allocation

to the risky asset is

independent of the insurable loss.
Suppose now that 0, so that u
1
(c) = ln c and u
2
(w) = b ln w. Then, the optimal deductible and allocation to
the risky asset are consistent with our results for = 0. The optimal consumption is
c

t
= V
1
w
= (t )
1
W

t
.
Here is given by
(t ) = a

x+t
+b

A

x+t
,
in which a

x+t
equals the actuarial present value of a life annuity issued to a person aged x + t that pays 1 per year
continuously with the rate of discount equal to (Bowers et al., 1997). Similarly,

A

x+t
equals the net single premium
of a whole life insurance policy issued to a person aged x + t that pays 1 at the moment of death with the rate of
discount equal to . The results in Example 3.5 concerning how the optimal controls are affected by the parameters
of the loss process hold in this case as well. In particular, in the special case of constant hazard rate , we have that

1
(t ) =
+
1+b
. Thus, if b < 1,
1
increases with ; therefore, investors with a longer horizon (smaller ) consume
a smaller proportion of wealth. This is consistent with the result of Example 3.5.
5. Numerical results
In this section, we present numerical results for examples for which the techniques of Sections 3 and 4 do not yield
closed form solutions for the value function and optimal controls. We restrict our attention to the stationary problem
in which the model parameters , , , and are constant. In this case, the HJB equation is an ordinary differential
equation. We employ the Markov Chain Approximation Method (MCAM) of Kushner (2000) to numerically
approximate the value function and controls. We refer the reader to Sections 2.1 and 5.6 of Kushner and Dupuis
(2001) and Fitzpatrick and Fleming (1991) for details on the implementation of the MCAM and to Chapter 13 of
Kushner and Dupuis (2001) for a discussion of convergence of the method.
We describe the intuition behind and some of the details of the MCAM in the context of a specic example, for
which a closed form solution is available, in Appendix A. We demonstrate the convergence of the method for this
same problem in Example 5.1a. We then apply the method to more complicated examples for which closed form
solutions are not available.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 57
Table 1
Parameters for the base scenario in Examples 5.1a and 5.1b
Premium load 0.25 Utility parameter 0.50
Loss frequency 0.10 Force of mortality 0.05
Risk free rate r 0.03 Subjective discount 0.05
Mean stock return 0.07 Propensity to save b 1.00
Stock volatility 0.30 Loss proportion 0.50
Example 5.1a (Convergence of the MCAM and Impact of Varying the Model Parameters). In this example, we
consider an individual who stands to lose a constant proportion (0, 1] of her wealth and who seeks to maximize her
expected utility of consumption and terminal wealth over her lifetime. We assume that her utility of consumption and
wealth are given by u
1
(c) =
c

and u
2
(w) = b
w

, respectively. Thus, we revisit the stationary case of Example 4.3.

In the rst part of the example, we set the wage rate = 0. In this case, one can verify that the function
V(w) = a
w

(5.1)
solves the HJB equation where a solves
a
_
r +
( r)
2
2
2

1
+Q
_
+(1 )a

1
+b = 0 (5.2)
and
Q = max{1 , (1 +)
1
1
}

1 (1 +) ( [1 (1 +)
1
1
])
+
. (5.3)
Moreover, the optimal controls are given by

t
=
r

2
(1 )
W

t
, (5.4)
c

t
= a
1
1
W

t
, (5.5)
and
d

t
= min{, 1 (1 +)
1
1
}W

t
. (5.6)
In the experiments that follow, we demonstrate the convergence of the MCAM to the solution given above and
we examine the impact of varying the model parameters. In Example 5.1b, we include a nonzero exogenous wage .
In this case a closed form solution is not available; we use the MCAM to compute the value function and optimal
controls and we contrast the qualitative properties of the resulting solutions with the = 0 case.
We x as our base scenario the choice of parameters given in Table 1. We choose the parameter values, and in
particular, to ensure convergence of the numerical method; see Fitzpatrick and Fleming (1991). We comment on more
realistic values of in Example 3.5. Observe that by (5.6),
d

t
= min{, 1 (1 +)
1
1
}W

t
= min{0.50, 0.36}W

t
= 0.36W

t
;
thus, the optimal strategy to insure against a loss of 50% of ones wealth is 36% coinsurance.
Convergence of the MCAM
Fig. 1 shows the approximate and actual value function and optimal controls for w [0, 300]. We note that the error
at the right hand boundary is caused by the fact that the transition probability there prohibits wealth from exceeding
300. In effect, we are truncating the domain on which the HJB equation is dened and imposing an articial boundary
condition at w = 300; see Appendix A for details. However, the convergence is excellent away from the boundary;
for that reason, in the experiments that follow, we approximate the value function and optimal controls on [0, 300],
but restrict our attention to the interval [0, 100]. We used a mesh size of dw = 0.10.
58 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Fig. 1. Convergence of the MCAM to the actual solution for Example 5.1a.
Impact of varying the model parameters
In various numerical experiments, we examined the impact of varying individual model parameters , , , , , b,
and from the base scenario described above. In each case, we veried the convergence of the approximate value
function and optimal controls to the actual solution as in Fig. 1. For conciseness of exposition, we do not present all
the details but rather summarize our results in Table 2. However, we do provide the details for two experiments that
yielded counterintuitive results.
Impact of changing the loss proportion
From (5.6), we see that if > 0.36, partial insurance at 36% is optimal. However, if 0.36, no insurance is
optimal; i.e., d

t
= W

t
. The rst graph in Fig. 2 conrms this. It is surprising to note in the third graph that as
the potential loss increases, the individual should consume more. A similar counterintuitive result was described and
explained in Example 3.5 when (0, 1). For a similar choice of parameters with = 1, we nd that c

decreases
as increases, which is more consistent with our expectation of investor behavior.
Impact of changing the loss frequency
In (5.6) we see that the optimal deductible is independent of the Poisson parameter ; the rst graph in Fig. 3
conrms this. The third graph shows that if the loss frequency increases, the individual should consume more. This
counterintuitive result is similar to the one described in the previous paragraph and in Example 3.5.
The impact of varying the other model parameters is summarized in Table 2.
Example 5.1b (Impact of Adding an Exogenous Wage). In the previous example, the value function and optimal
controls were given in closed form by (5.1)(5.6), although one must determine the coefcient a of the value function
by numerically solving the nonlinear equation (5.2). The purpose of that example was twofold: to demonstrate that the
MCAM converges to the actual solution, and to examine the impact of varying the model parameters from the base
scenario.
In this example, we modify the previous example slightly by adding a constant, non-zero exogenous wage to
the wealth evolution. In this case, the HJB equation is not solvable in closed form. This is because the loss is not
proportional to aggregate wealth, including the net present value of future wages. We use the MCAM to approximate
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 59
Table 2
Impact of adding exogenous wage
Change Wage = 0 Wage = 1
Increase premium load from 25% to
35%.
Buy partial insurance.

is
unchanged. Change in c

is nearly
imperceptible.
Buy no insurance.

is unchanged. Change in c

is nearly
imperceptible.
Decrease loss proportion from 50%
to 40%.
Buy partial insurance.

is
unchanged. c

decreases. See Fig. 2.

Buy no insurance.

is unchanged. c

decreases.
Increase loss frequency from 0.1 to
1.0
d

and

are unchanged. c

increases. See Fig. 3.

Deductible and investment in risky asset are higher than when
= 0, but when increases,
lower deductible
lower investment in risky asset
higher consumption are optimal.
Increase force of mortality from
0.05 to 0.15
d

and

are unchanged. c

increases.
Deductible and investment in risky asset are higher than when
= 0, but when increases,
lower deductible
lower investment in risky asset
higher consumption are optimal.
Increase the propensity to save b d

and

are unchanged. c

decreases.
Similar results.
Increase the relative risk aversion
1 by decreasing
Lower deductible Similar results.
lower investment in risky asset
lower consumption are optimal. See
Fig. 7.
Fig. 2. Impact of changing the loss proportion.
60 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Fig. 3. Impact of changing the loss frequency.
the optimal controls and we contrast the optimal strategies with those from the previous example. Thus, we examine
the qualitative impact of an exogenous wage on the optimal controls.
Impact of changing the wage
In Fig. 4, we see that as we increase the wage from 0 to 1 to 20, the investor should purchase less insurance,
invest more in the risky asset, and consume more. This is consistent with our intuition. Cocco et al. (2005) considered a
realistically-calibrated model for optimal investment during a xed time period in the presence of wage income under
CRRA utility. They found that as the wage income decreases, the amount invested in the riskless asset increases,
which is consistent with our results.
Impact of varying the model parameters
We repeated all of the experiments of Example 5.1a to examine the impact of changing the model parameters in
the presence of an exogenous wage. We examined the base scenario described in Table 1 with = 1 and changed the
model parameters , , , , , b, and as in Example 5.1a. For conciseness of exposition, we will not include the
graphs for the experiments. In Table 2 we summarize their results and highlight the ways in which the presence of the
exogenous wage affects the investors optimal strategies.
It is not surprising that, in the presence of a wage, the investor should purchase less insurance (i.e., opt for higher
deductibles). More specically, we found that when we increased the premium load to = 35%, with = 1, the
individual should not purchase any insurance. (When = 0 and = 35%, the optimal strategy was partial insurance;
see Fig. 2.) Similarly, when we decreased the loss proportion to = 40%, with = 1, the individual should not
purchase any insurance. (When = 0 and = 40%, the optimal strategy was partial insurance.)
We found that when = 1, if the loss frequency increases from = 0.1 to = 1.0, the individual should opt for
a lower deductible and invest less in the risky asset. Thus, in the presence of a nonzero wage, the optimal allocation
to the risky asset is affected by the parameters of the insurable loss; this is the rst example in which this happens.
(Recall that when = 0, the optimal insurance and investment strategies were not affected by increasing ; see Fig. 3
and Eqs. (5.4) and (5.6).)
When = 1 and we increased the force of mortality from = 0.05 to = 0.15, we found that higher deductibles
and greater investment in the risky asset are optimal when = 1 than when = 0. However, when = 1, the
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 61
Fig. 4. Impact of adding a wage.
investors insurance and investment strategies are affected by changes in ; they were unaffected when = 0. Thus,
mortality (or, alternatively, the investors horizon) impacts the insurance and investment decisions in the presence
of an exogenous wage. More specically, an investor with a longer horizon (or smaller ) should choose a higher
deductible (i.e., purchase less insurance), invest more in the risky asset, and consume less. Thus, the investor assumes
more risk, but decreases consumption. This is consistent with Golliers (2002) observation that younger investors can
be optimally time diversied by splitting risks on wealth into risks on consumption.
Example 5.2. We assume that the individual faces a possible loss of L = 1 and that she seeks to maximize
her expected utility of terminal wealth and consumption over her lifetime. We assume that the model parameters
(t ), (t ), (t ), and
x
(t ) are constant. Moreover, we assume power utility of consumption and terminal wealth as
in Examples 3.5 and 4.3; i.e., u
1
(c) =
c

and u
2
(w) = b
w

for some < 1, = 0 and b 0. Unlike these earlier

examples, we set the wage rate to be nonzero. Thus, the wealth process is given by
_
dW
t
= [r W
t
+( r)
t
+ c
t
(1 +)(1 d
t
)] dt +
t
dB
t
d
t
dN
t
W
0
= w,
and the value function V is given by
V(w) = sup
{c
t
,
t
,d
t
}
E
__

0
e
s
u
1
(c
s
)ds +e

u
2
(W

)|W
0
= w
_
.
It follows that V solves the HJB equation
_

_
max

_
( r)V

+
1
2

2
V

_
+max
c
[u
1
(c) cV

] +(rw +)V

+ max
0d1
[V(w d) (1 +)(1 d)V

] + u
2
(w) = ( + +)V,
lim
t
E
_
e
(+)t
V(W

t
)|W

0
= w
_
= 0.
62 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Fig. 5. Impact of changing the wage.
In this case, a closed form solution is not available. We set b = 1, r = = = 0.05, = 0.01, = 0.1,
= 0.07, = 0.3, = 0.5, and = 1 and use the MCAM to approximate the optimal strategies and to examine the
impact on the optimal strategies of varying the model parameters. Because of the boundary error that we discussed in
Example 5.1a, we restrict our attention to the domain [20, 80].
Impact of changing the wage
We contrast the optimal strategies for = 1 and = 20. When the wage = 20, a possible loss of L = 1 has a
lower impact on the insureds nancial health; thus, we expect the optimal deductible to be higher when = 20 than
when = 1. As we see in Fig. 5, this is indeed the case; when = 20, zero insurance is optimal. This contradicts
Golliers (1994, pp 85ff) results because we consider conditions for which self-insurance is the optimal strategy.
However, it is consistent with Golliers (2003) observation that the optimal deductible is more sensitive to the wealth
level in the dynamic framework because the individual is able to time-diversify her risk. He argues that this type of
phenomenon could be driven by liquidity constraints at lower wealth levels, which inhibit investors ability to allocate
risk on wealth over future periods.
Note also that the optimal consumption and investment in the risky asset are higher for = 20 than for = 1;
this is consistent with our nancial intuition.
Impact of changing the cost of insurance
We see in Fig. 6 that when we increase the premium load from = 0.01 to = 0.25, the demand for insurance
decreases. When insurance is expensive ( = 0.25), zero insurance is optimal for w [20, 80], but when insurance
is inexpensive ( = 0.01), d

[0.55, 0.95] for the lower wealth levels. This contradicts Golliers (1994, pp 85ff)
result of constant deductible because he did not consider the fact that the deductible will be higher at lower wealth
levels because negative wealth is precluded under power utility. Again, however, it is consistent with Golliers (2003)
observation that wealthier individuals should self-insure. Note that the optimal consumption and investment strategies
are not affected by this change in .
Impact of changing the frequency of loss
We contrast the optimal strategies as we change the loss frequency from = 0.1 to = 1.0. Fig. 7 shows
that when the loss frequency is higher, at the lower wealth levels, the insured should opt for a lower deductible.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 63
Fig. 6. Impact of changing the cost of insurance.
Fig. 7. Impact of changing the loss frequency.
64 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
Fig. 8. Impact of changing the propensity to save.
However, at the higher wealth levels, zero insurance is optimal for both choices of . We see also that the optimal
investment in the risky asset and the optimal consumption are lower when the loss frequency is higher. This interaction
between the insurable loss and the risky asset is consistent with our nancial intuition and the result of Example 5.1b.
Note that increasing the loss frequency had the opposite effect on the consumption strategy when L = w; see Fig. 3
and Table 2 from Examples 5.1a and 5.1b.
Impact of changing the emphasis on consumption versus bequest
In Fig. 8, we see that if we increase the individuals propensity to save b from 1 to 4 (so that the individual values
bequest over consumption), the optimal consumption level decreases. Similarly, if we decrease b from 1 to 0.25, the
optimal consumption level increases. This is consistent with our intuition and with the result from Example 5.1a; see
Table 2. Recall that in Examples 5.1a and 5.1b, the optimal insurance and investment strategies were unaffected by the
value of b; see (5.4) and (5.6), and Table 2. However, in this case, if the individual values bequest over consumption,
she should choose a lower deductible and invest less in the risky asset. Similarly, if she values consumption over
bequest, she should choose a higher deductible and invest more in the risky asset.
Impact of changing the relative risk aversion
We examine the impact of changing the utility parameter from 0.5 to 0.1; these values of correspond to relative
risk aversion of 1 = 0.5 and 1 = 0.9, respectively. Fig. 9 shows that when the relative risk aversion is higher
( = 0.1), the individual should choose a lower deductible, invest less in the risky asset, and consume less. This is
consistent with our intuition and the result from Examples 5.1a and 5.1b; see Table 2.
Impact of changing the force of mortality and subjective discount rate
Fig. 10 shows that as we increase the force of mortality from 0.05 to 0.15, at lower wealth levels, the optimal
deductible decreases; thus, an investor with a shorter horizon purchases more insurance. At higher wealth levels, self-
insurance is optimal. In the third graph in Fig. 10, we see that an investor with a shorter horizon consumes more. These
results are consistent with the results of Example 5.1b and Gollier (2002). However, the impact of the horizon on the
optimal investment strategy is more complicated. At lower wealth levels, an investor with shorter horizon assumes
less nancial risk; the reverse is true at higher wealth levels. The impact of varying the subjective discount rate is
qualitatively similar.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 65
Fig. 9. Impact of changing the relative risk aversion.
Fig. 10. Impact of changing the force of mortality and subjective discount rate.
66 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
6. Conclusions
We showed that in our dynamic setting, for per-claim indemnity coverage, optimal insurance is deductible
insurance. This is consistent with the static results of Mossin (1968), van Heerwaarden (1991, Theorem 9.4.1), and
Schlesinger and Gollier (1995). We considered a xed horizon and a horizon that terminates at the time of death of
the insured.
We analyzed many examples, both analytically and numerically. In each example, we examined the impact of the
investors horizon (or mortality) on the optimal strategies and the interaction between investment, consumption, and
the insurable loss. Some of our results are summarized in the table in Appendix B.
In many examples, the optimal investment and insurance strategies are independent of each other and the demand
for the risky asset is independent of the parameters of the loss process. However, in the presence of an exogenous
wage, the frequency of the loss affects the individuals demand for the risky asset. Moreover, when premiums are paid
as a lump-sum, under exponential utility, the parameters of the stock price affect the demand for insurance.
In most of the examples, the optimal consumption strategy depends on the investors horizon. However, in many of
the examples, the optimal insurance and investment strategies are independent of the horizon except via the effect on
the optimally-controlled wealth. By contrast, in the presence of an exogenous wage, the impact of the horizon on the
optimal controls is pronounced; an individual with a longer horizon assumes more risk by purchasing less insurance
and investing more in the risky asset, but consumes less. This strategy is consistent with the time diversication
argument of Gollier (2002).
Appendix A. Markov chain approximation method (MCAM)
We describe the intuition behind the MCAM in the context of Example 5.1a. First, discretize the state (wealth)
space into a nite set of values {
0
,
1
,
2
, . . . ,
N
} and assume that in a small time interval t , wealth changes from

i
to
j
with transition probability p
i j
. We employ the Dynamic Programming Principle to write the equation for the
approximate value function recursively as
V(
i
) sup
{c
i
,
i
,d
i
}
{t u
2
(
i
) +t u
1
(c
i
)
+e
(+)t
( t V(
i
d) +(1 t )E [V()|W
t
=
i
])},
in which E[V()|W
t
=
i
] =

N
j =0
p
i j
V(
j
). We refer the reader to sections 2.1.1, 2.1.2, and 5.6.2 of Kushner and
Dupuis (2001) for the derivation, but the intuition is as follows. Over the time interval [0, t ], the approximate utility
from consumption is t u
1
(c
i
). If death occurs in the small time interval [0, t ], which happens with probability
1 e
t
t , the individual has approximate utility of terminal wealth u
2
(
i
). If death does not occur in [0, t ],
which happens with probability e
t
, we examine the outcome on the interval [t, ] and discount it back to time
zero (hence the factor of e
t
). If a Poisson loss occurred in [0, t ], which happens with probability t , the
individual pays the deductible d and continues maximizing her expected utility. If a Poisson loss did not occur on
[0, t ], which happens with probability 1 t , again, the individual continues maximizing her expected utility.
We wish to compute the approximate value function V
i
= V(
i
) and to recover the optimal controls c

i
= c

(
i
),

i
=

(
i
), and d

i
= d

(
i
) for i = 0, 1, . . . , N. We sketch the algorithm below.
1. Fix the controls c
i
,
i
and d
i
.
2. Compute the value function for these xed controls; i.e., compute
J
i
:=
_
t u
2
(
i
) +t u
1
(c
i
) +e
(+)t
_
t J(
i
d
i
) +(1 t )

j
p
i j
J
j
__
. (A.1)
The transition probabilities p
i j
depend on the model parameters , r, , , , and , the mesh size dw, and the
strategies c
i
,
i
and d
i
; we briey describe the choice of p
i j
below. We remark that this computation amounts
simply to solving the (N +1) (N +1) linear system above for the unknown vector J.
3. Given J, compute new policies
i
, d
i
that maximize J; i.e., compute the argument of the maximum of J. The
expression that results for the updated
i
is simply a discretized version of the rst order necessary condition

=
(r)

2
V

. Note that since the wealth space is discretized with mesh dw, each d
i
must be an integral
multiple of dw.
K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768 67
4. Repeat, starting at step 2 with the updated controls, until the change in the vector J is sufciently small. (We used
a tolerance of 10
6
.)
We remark that if we choose the transition probabilities p
i j
so that the approximating chain is locally consistent
with the continuous wealth process (i.e., so that the rst two moments are close; see Section 4.1 of Kushner and
Dupuis (2001) for details), then the value function and optimal controls for the discretized problem converge to the
value function and optimal controls for the continuous problem. Moreover, under our choice of transition probabilities,
solving the linear system (A.1) is equivalent to solving the HJB equation under xed controls, i.e., to solving
J =
_
{( r) + +rw c (1 +) (w d)
+
} J

+
1
2

2
J

+ {J(w d) J(w)}
_
+ [u
2
(w) J] +u
1
(c)
via a modied nite difference scheme.
Finally, we remark that it is tempting to apply a straightforward nite difference method directly to the HJB
equation, but as Kushner (1995) points out, for many problems in stochastic optimal control, the associated
HJB equations have only a formal meaning and standard methods of numerical analysis are not usable to prove
convergence; moreover, the classical methods might not converge. However, as Kushner and Dupuis (2001, Chapter
5) and Hanson (1996, Section IVB) point out, when a carefully-chosen, renormalized nite difference is used, the
coefcients of the resulting discrete equation serve as the transition probabilities that satisfy the local consistency
conditions in the discretized dynamic programming equation (A.1). Indeed, this is the case for our example.
Appendix B. Summary of examples
Example u
1
(c) u
2
(w) Other Loss process affects Horizon (or mortality) affects

3.4 0 Exp. r = 0,
L(w, t ) = L(t ),
= 0
No N/A Yes. Decreases
with T.
Yes. Decreases
with T.
N/A
3.5 Power
and
log
Power
and
log
r = 0,
L(w, t ) = (t )w,
= 0
No Power: Yes,
Log: No
Implicitly Implicitly Yes
4.1 0 Exp. r = 0,
L(w, t ) = L(t ),
= 0
No N/A No No N/A
4.2 0 Exp. r = 0, L(w, t ) = L,
= 0. Lump sum
prem.
No N/A Yes. Impact is
ambiguous.
No N/A
4.3 Power
and
log
Power
and
log
r = 0,
L(w, t ) = (t )w,
= 0
No Power: Yes,
Log: No
Implicitly Implicitly Yes
5.1a Power Power r = 0,
L(w, t ) = (t )w,
= 0. All parameters
constant
No Yes. c

increases with
and
Implicitly Implicitly Yes. c

increases
with
5.1b Power Power Same as 5.1a, but
= 0
Yes.

decreases with

Yes. c

increases with
and
Yes. d

decreases with
.
Yes.

decreases with
.
Yes. c

increases
with .
5.2 Power Power r = 0, L(w, t ) = 1,
= 0
Yes.

decreases with
.
Yes. c

decreases with
.
Yes. High
wealth: no ins.
Low wealth: d

decr. with .
Yes. High
wealth:

incr.
with . Low
wealth:

decr. with .
Yes. c

increases
with .
68 K.S. Moore, V.R. Young / Insurance Mathematics and Economics 39 (2006) 4768
References
Arrow, K.J., 1963. Uncertainty and the welfare economics of medical care. American Economic Review 53, 941973.
Asmussen, S., Taksar, M., 1997. Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics 20, 115.
Bj ork, T., 1998. Arbitrage Theory in Continuous Time. Oxford University Press, New York.
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt, C.J., 1997. Actuarial Mathematics, second ed.. Society of Actuaries, Schaumburg,
Illinois.
Briys, E., 1986. Insurance and consumption: The continuous-time case. Journal of Risk and Insurance 53, 718723.
Briys, E., Dionne, G., Eeckhoudt, L., 1989. More on insurance as a Giffen good. Journal of Risk and Uncertainty 2, 415420.
Campell, J.Y., Viceira, L.M., 2002. Strategic Asset Allocation. Oxford University Press, Oxford.
Cochrane, J., 2001. Asset Pricing. Princeton University Press, Princeton, New Jersey.
Cocco, J., Maenhout, P., Gomes, F., 2005. Consumption and portfolio choice over the life-cycle. Review of Financial Studies 18 (2), 491533.
Feldstein, M., Ranguelova, E., 2001. Individual risk in an investment-based social security system. American Economic Review 91 (4), 11161125.
Fitzpatrick, B., Fleming, W.H., 1991. Numerical methods for an optimal investment-consumption model. Mathematics of Operations Research 16,
832841.
Friend, I., Blume, M.E., 1975. The demand for risky assets. American Economic Review 65, 900922.
Gollier, C., 1994. Insurance and precautionary capital accumulation in a continuous-time model. Journal of Risk and Insurance 61, 7895.
Gollier, C., 2002. Time diversication, liquidity constraints, and decreasing aversion to risk on wealth. Journal of Monetary Economics 49,
14391459.
Gollier, C., 2003. To insure or not to insure? An insurance puzzle. The Geneva Papers on Risk and Insurance Theory 28, 524.
Hanson, F.B., 1996. Techniques in computational stochastic dynamic programming. In: Leondes, C.T. (Ed.), Stochastic Digital Control System
Techniques. In: Series Control and Dynamical Systems: Advances in Theory and Applications, vol. 76. Academic Press, New York,
pp. 103162.
van Heerwaarden, A., 1991. Ordering of Risks. Thesis, Tinbergen Institute, Amsterdam.
Hjgaard, B., Taksar, M., 1997. Optimal proportional reinsurance policies for diffusion models. Scandinavian Actuarial Journal 1997, 166180.
Kushner, H.J., 1995. Numerical methods for stochastic control problems in nance. In: Mathematics of Derivative Securities. In: Publications of
the Newton Institute, vol. 15. Cambridge University Press, Cambridge, Massachusetts, pp. 504527.
Kushner, H.J., 2000. Jump-diffusions with controlled jumps: Existence and numerical methods. Journal of Mathematical Analysis and Applications
249, 179198.
Kushner, H.J., Dupuis, P., 2001. Numerical Methods for Stochastic Control Problems in Continuous Time, second ed.. Springer-Verlag, New York.
Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: the continuous time case. Review of Economics and Statistics 51, 247257.
Merton, R.C., 1971. Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373413.
Merton, R.C., 1992. Continuous-Time Finance, revised edition. Blackwell Publishers, Cambridge, Massachusetts.
Mitchell, O.S., Poterba, J.M., Warshawsky, M.J., Brown, J.R., 1999. New evidence on the moneys worth of individual annuities. American
Economic Review 89 (5), 12991318.
Mossin, J., 1968. Aspects of rational insurance purchasing. Journal of Political Economy 79, 553568.
Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32, 122136.
Samuelson, P.A., 1963. Risk and uncertainty: the fallacy of the Law of Large Numbers. Scientia 98, 108113.
Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic stochastic program. Review of Economics and Statistics 51, 239246.
Samuelson, P.A., 1989a. The judgement of economic science on rationale portfolio management: indexing, timing, and long-horizon effects. Journal
of Portfolio Management 16, 312.
Samuelson, P.A., 1989b. A case at last for age-phased reduction in equity. Proceedings of the National Academy of Sciences 86, 90489051.
Schlesinger, H., Gollier, C., 1995. Second-best insurance contract design in an incomplete market. Scandinavian Journal of Economics 97 (1),
123135.
Siegel, J., 1994. Stocks for the Long Run. Irwin, Burr Ridge, IL.
Wang, S.S., Young, V.R., 1998. Ordering risks: Expected utility theory versus Yaaris dual theory of risk. Insurance: Mathematics and Economics
22, 145161.
Young, V.R., Zariphopoulou, T., 2002. Pricing dynamic insurance risks using the principle of equivalent utility. Scandinavian Actuarial Journal
2002 (4), 246279.
Young, V.R., Zariphopoulou, T., 2004. Pricing insurance via stochastic control: optimal consumption and terminal wealth. Finance 25, 141155.