Download Initial investment choice and optimal future allocations under time

Initial investment choice and optimal future
allocations under time-monotone performance
criteria
M. Musielayand T. Zariphopoulouz
June 30, 2010
Abstract
The paper o¤ers a new perspective on optimal portfolio choice by
investigating how and to what extent knowledge of an investor’s desirable initial investment choice can be used to determine his future optimal
portfolio allocations. Optimality of investment decisions is built on the
so-called forward investment performance criteria and, in particular, on
the time-monotone ones. It is shown that for this class of forward criteria
the desired initial allocations completely characterize the future optimal
investment strategies. The analysis uses the connection between a nonlinear equation, satis…ed by the local risk tolerance, and the backward heat
equation. Complete solutions are provided as well as various examples.
1
Introduction
The powerful and elegant theory of expected utility yields optimal portfolio
choices for risk averse investors in stochastic market environments. In the classical continuous time framework, introduced in [5], the investor’s preferences
are assigned at the end of the trading horizon. The optimal investment strategy
is, then, the one that delivers the maximal expected (indirect) utility of terminal wealth. Recently, the authors proposed an alternative approach to optimal
portfolio choice which is based on the so-called forward investment performance
criterion (see, among others, [8]). In this approach, the investor does not choose
The authors thank the Fields Institute for its hospitality and partial support during the
"Kick-o¤ Workshop" under the auspices of the semester-long program in Financial Mathematics (January-June, 2010). They also thank an anonymous referee whose comments were
very helpful in improving the …rst version of the paper.
y [email protected]; BNP Paribas, London
z [email protected]; Oxford-Man Institute of Quantitative Finance and the Mathematical Institute, University of Oxford and Departments of Mathematics
and IROM, the University of Texas at Austin. Partial support from the Oxford-Man Institute
and NSF grant DMS-RTG-0636586 is acknowledged.
1
her risk preferences at a single point in time, as it is the case in the Merton
model, but has the ‡exibility to revise them dynamically.
From the methodological point of view, one of the fundamental di¤erences
between the two approaches is that the investor speci…es her utility at the beginning of the trading horizon and not at the end. Whether one works in the traditional or the alternative framework, the speci…cation of the terminal, or initial,
risk preferences remains a challenging task. In the context of forward portfolio
choice, this question was recently examined in [6]. Therein, one shows how the
investor’s preferences may be inferred from desirable distributional properties
of future investment targets. An important consequence of this methodology
is that the individual risk preferences are calibrated to the speci…c market environment and the investment opportunities this environment o¤ers. This is
in contrast to the classical framework in which the utility function is assigned
exogenously, in isolation to the market in which the investor operates. Inference
of risk preferences from similar probabilistic criteria has been also proposed and
analyzed in a static model by W. Sharpe and his coauthors in [3], [4], [11] and
[12].
Herein, we provide an additional point of view on portfolio choice and the
speci…cation of the investor’s risk preferences. The analysis is built on an entirely
di¤erent perspective which is, from one hand, quite unorthodox with respect to
the classical academic ideas but, on the other, it appears to be much closer to
the rationale applied in investment practice. Speci…cally, instead of requiring
that the investor knows his risk preferences at the beginning of the investment
horizon, it is assumed that he is able to specify the portfolio allocations that are
desirable to him at initiation. The motivation to consider this direction comes
mainly from the fact that the concept of utility is quite elusive and di¢ cult to
quantify.
The inspiration comes from a paper1 written by F. Black ([2]) on similar
ideas. One reads:
....“Utility” is a foreign concept for most individuals, and the traditional way
of mapping an individual’s utility function is to ask him about his preferences
among a number of gambles.
Instead of stating his preferences among various gambles, the individual
can specify his consumption function and his initial investment function. He
can give the rate at which he would consume his wealth at various times in the
future as a function of the amount of his wealth at those times. And he can tell
how much he would invest currently in the market portfolio as a function of his
current wealth.
It is possible to eliminate the individual’s utility function entirely from these
equations, giving a set of equations involving only his investment function and
his consumption function....
Therein an in…nitesimal analysis is proposed for the speci…cation of the future optimal investment and consumption policies once their initial values are
1 The
authors are thankful to Peter Carr who brought this manuscript to their attention.
2
chosen. The case when these initial choices are proportional to the investor’s
wealth is solved in detail.
In this paper, we build on the same ideas and extend the results in various directions. We allow for both more general initial allocations and asset
prices2 . However, we do not incorporate intermediate consumption. Optimality
is considered with respect to the forward investment performance criteria and,
speci…cally, for time-monotone ones. For such families of dynamic risk preferences, we show how knowledge of the initial desired allocation can be used to
recover all future portfolio processes, the associated wealth process and, …nally,
the investment criterion with respect to which they (portfolio and wealth) are
optimal.
We note that the entire analysis bypasses the need to know the risk preferences. Indeed, we will see that risk preferences are identi…ed only at the
end, when one seeks the criterion with respect to which the already constructed
processes are optimal. The solution approach is based on a deep connection between the so-called local risk tolerance function and harmonic functions as well
as on the speci…cation of a measure whose bilateral Laplace transform yields
the latter. This measure becomes the de…ning element for all quantities of interest. We show how the initial desired allocation reveals it. A by-product of
this analysis is the speci…cation of the set of initial allocations that lead to well
de…ned future investment choices. One can say that this is the set of feasible
initial investment choices.
We conclude mentioning that the choice to use forward performance criteria is a natural consequence of their suitability to address the questions posed
herein which are formulated "forward" in time. The speci…c choice of timemonotone performances was motivated by the facts that, from one hand, similar
monotone criteria were used in [2] and, from the other, this class o¤ers substantial tractability that helps us producing explicit solutions. Our knowledge of
forward performance processes that are non-monotone is still limited. On the
other hand, even the monotone case presents various challenges and o¤ers a
number of interesting, if not intriguing, insights. As a …rst step towards addressing the questions posed by F. Black, the family we use herein seems to
o¤er a rich enough framework to produce meaningful solutions and to provide
strong motivation for further research in this direction.
The paper is organized as follows. In sections 2 and 3 we introduce the model
and pose the portfolio choice problem. In section 4 we provide auxiliary results
on the representation of the local risk tolerance function and its connection with
harmonic solutions. In section 5 we provide the main results and answers to the
questions in consideration. We conclude with various examples in section 6.
2 For
convenience, we assume that there is a single risky asset but the results easily extend
to the multi-asset case.
3
2
The model and the investment performance
criterion
The market environment consists of a riskless bond and a stock. The stock
price, St ; t 0; is modelled as an Itô process following
dSt = St ( t dt +
t dWt ) ;
(1)
with S0 > 0: The process Wt is a standard Brownian motion, de…ned on a …ltered
probability space ( ; F; P). The coe¢ cients t and t ; t 0; are Ft adapted
processes. The riskless asset, the savings account, has the price process Bt ; t 0;
satisfying
dBt = rt Bt dt;
with B0 = 1; and for a nonnegative, Ft adapted interest rate process rt : The
market coe¢ cients t ; t and rt are taken to satisfy the appropriate integrability
conditions.
We de…ne the process t ; t 0; by
t
=
rt
t
;
(2)
t
and we throughout assume that it is bounded by a deterministic constant c > 0;
i.e., for all t 0; j t j c: We will be occasionally referring to this process as
the market price of risk.
Starting at t = 0 with an initial endowment x 2 R, the investor invests, at
any time t > 0; in the two assets. The present value of the amounts invested in
the bond and the stock are denoted, respectively, by 0t and t .
The present value of her current total investment is, then, given by Xt =
0
+
0: Using (1) we have that it satis…es
t; t
t
dXt =
t t
( t dt + dWt ) ;
(3)
with X0 = x: The investment policies t ; t
0; will play the role of control processes and are taken to satisfy the usual assumptions of being self…nancing, Ft progressively measurable and satisfying the integrability condiRt
2
tion E 0 j s s j ds < 1; t > 0: We denote the set of admissible strategies
by A.
Recently, a selection criterion for investment strategies in A was introduced
in [8] (see, also, [10]). It is represented by a process, the so-called forward
investment performance. This criterion is de…ned for all times and complements
the traditional one which is based on the maximal expected utility of terminal
wealth (value function process), de…ned only on [0; T ] ; T < +1:
As the de…nition below states, a strategy is deemed optimal if it generates a wealth process whose average performance is maintained over time. In
other words, the average performance of this strategy at any future date, conditional on today’s information, preserves the performance of this strategy up
4
until today. Any strategy that fails to maintain the average performance over
time is, then, sub-optimal. The intuition behind these requirements comes from
the analogous properties of the value function process, namely, its martingality
along optimal policies and its supermartingality away from the optimum. We
refer the reader to [8] for a detailed discussion on the new approach.
De…nition 1 An Ft adapted process U (x; t) is a forward investment performance if for t 0 and x 2 D, D R:
i) the mapping x ! U (x; t) is strictly concave and increasing,
+
ii) for each 2 A; E (U (Xt ; t)) < 1; and
E (U (Xs ; s) jFt )
iii) there exists
t
U (Xt ; t) ;
s
t;
(4)
2 A; for which
E U Xs ; s jFt
= U Xt ; t ;
s
t:
(5)
Remark 2 Intuitively speaking, U (x; t) represents the stochastic utility that is
generated by the policy of investing all wealth in the riskless asset across time.
Note that even though this is a riskless strategy, it generates nondeterministic
utility, for it is implemented in a stochastic environment. Note, also, that this
interpretation is directly aligned with the fact that the bond has been chosen as
the numeraire.
Characterizing the class of processes which satisfy the above de…nition is a
challenging problem. An important result in this direction, introduced recently
in [9], is a stochastic partial di¤erential equation for the investment performance
process, namely,
2
dU (x; t) =
1 ( t U (x; t) + ax (x; t))
dt + a (x; t) dWt ;
2
Uxx (x; t)
(6)
where a (x; t) is an Ft adapted process and t ; t 0; as in (2).
The volatility process a (x; t) is the novel element of the forward approach
in portfolio choice. This input is up to the investor to choose, in contrast to
the classical formulation in which the volatility coe¢ cient of the value function
process is uniquely determined.
The above SPDE was derived under various assumptions on existence, uniqueness and regularity of the involved processes. Except for the case of zero volatility which was completely solved in [6] (see, also, [1]), results to whether (6)
provides a necessary and/or su¢ cient condition for its solution, U (x; t) ; to satisfy the above de…nition are lacking. These questions are being currently studied
by various authors. While such theoretical results are not yet available, concrete
examples for various cases of non-zero volatility processes can be found in [9].
Herein, we do not examine such issues. Rather, we investigate how information about preferred personal portfolio choice can be used to build a framework
for investment advice across time.
5
3
The portfolio selection problem under timemonotone performance criteria
We consider an individual who starts today with some initial wealth and faces
investment decisions in upcoming times. The investor speci…es his initial desired
investment allocation between the two accounts.
We are interested in the following questions:
How does the investor’s initial investment choice a¤ect, and to what extent
determine his future allocations?
Is there an investment criterion that is consistent with these policies, in
that these portfolios - initial and future - are optimal with respect to it?
Are all initial investment choices admissible in the sense that they lead
to feasible future investment allocations and, in general, to a well-posed
portfolio choice problem?
We will study these questions focusing on the class of investors who have
time-monotone investment criteria, speci…cally, when the process U (x; t) is
time-decreasing. This is the simplest class of forward investment performance
processes which, nevertheless, o¤ers rich intuition and valuable insights. It is
easy to see that this case corresponds to the zero volatility choice, a (x; t) = 0;
t 0; in the performance SPDE (6).
To ease the exposition, we …rst provide some auxiliary results on monotone
investment performance processes. These results can be found in [8] and are
presented herein without a proof. To this end, we recall that such processes
are represented as a compilation of market related input with a deterministic
function of space and time. Speci…cally, for t 0; U (x; t) is given by
U (x; t) = u (x; At ) ;
(7)
where u (x; t) is increasing and strictly concave in x; and satis…es the fully
nonlinear equation
1 u2x
:
(8)
ut =
2 uxx
The process At depends only on market movements up to current time t and is
given by
Z t
2
At =
(9)
s ds
0
with t ; t 0; as in (2).
In [8] it was also shown that the investor’s optimal wealth and optimal portfolio processes, denoted respectively by Xt and t ; t 0; satisfy an autonomous
system (cf. (13)) of stochastic di¤erential equations whose coe¢ cients depend
6
functionally on the spatial derivatives of u. Speci…cally its coe¢ cients can be
expressed in terms of the so-called local risk tolerance function de…ned by
ux (x; t)
:
uxx (x; t)
r (x; t) =
(10)
Another important result therein is that the above function satis…es an autonomous equation (cf. (11)). The autonomous system and equation will play
pivotal role in our analysis as explained in the sequel.
The results below can be found in Propositions 6 and 9 in [8]. Because later
on we will be working with di¤erent cases of the spatial domain, we use, for the
moment, the generic notation D.
Theorem 3 i) Let u (x; t) solve (8) and r (x; t) be as in (10). Then, for (x; t) 2
D [0; +1) ; r (x; t) solves
1
(11)
rt + r2 rxx = 0:
2
ii) De…ne the process
Rt = r (Xt ; At ) ;
(12)
with At ; t
0; as in (9) and consider the system
8
< dXt = r (Xt ; At ) t ( t dt + dWt )
:
(13)
dRt = rx (Xt ; At ) dXt ;
with X0 = x and R0 = r (x; 0) : Assume that Xt and Rt solve (13) and de…ne
0; by
t; t
t
t
=
t
Rt :
(14)
Then t generates Xt and, moreover, is optimal under the criterion U (x; t)
given in (7).
Returning to the problem we want to analyze herein, let us assume that the
investor speci…es her initial desired allocation as functions of her wealth, say
0 (x) and x
0 (x) ; for each wealth level x 2 D. The questions posed at the
beginning of the section can be, then, formulated as follows:
How does the form of 0 (x) a¤ect and, to what extent, determine the
future allocation processes t ; t > 0?
How can one infer the investor’s performance criterion, u (x; At ) ; t
with respect to which the policies t ; t 0; are optimal?
0;
What is the class of admissible initial allocations 0 (x) ; i.e. initial allocations that generate future optimal investment policies t ; t > 0; that
belong to the set of admissible policies A?
7
Theorem 3 gives valuable insights which will be central in our solution approach. Firstly, note that solving the above system essentially bypasses the need
to specify the investment criterion. Indeed, knowing the function r (x; t) and
solving (13) yields directly the optimal processes Xt and t : This is in contrast
to the classical setting in which one starts with the speci…cation of terminal
utility, then …nds the value function process and, in turn, recovers the optimal
policies.
We will utilize the results of the above theorem as follows. Firstly note
that the investor’s initial preferred allocation 0 (x) yields the initial condition
r (x; 0) ; as (12) and (14) indicate. The function r (x; 0) together with equation
(11), speci…es r (x; t) ; t > 0:The latter and its spatial derivatives give the functional coe¢ cients in (13) which, in turn, yield the processes Xt and Rt . The
optimal allocation and the associated investment criterion, t and U (x; t) ; are,
then, recovered from (14) and (7), respectively.
In order to execute these steps, one needs to solve equation (11) and the
system (13), and to construct the function u: We discuss these problems next.
We start with auxiliary results for solutions to (11) and (8).
4
Analysis
We focus on speci…c classes of solutions to equations (11) and (8), namely,
positive solutions of (11) and strictly concave and increasing solutions of (8). We
stress that, for the moment, we do not assume that these solutions are related
to each other as (33) might indicate (as well as the titles of the subsections
below). This connection will be discussed at the end of the section. For now we
treat the two equations separately.
We recall the backward heat equation,
1
(15)
ht + hxx = 0:
2
We consider strictly increasing solutions to the above equation and construct
the associated solutions to (11) and (8) by appropriate nonlinear and integral
transformations (see Propositions 4 and 8). The motivation to involve (15)
comes from the fact that its harmonic solutions have very explicit, and quite
useful for the investment problem at hand, representations. These representation results are also discussed in this section.
As in the previous section, we use D to denote the generic domain of the
function r:
4.1
Local risk tolerance and harmonic functions
We start with a result that relates solutions of (15) and (11).
Proposition 4 Let h : R [0; +1) ! D be a strictly increasing solution to
(15) and de…ne r : D [0; +1) ! R+ by
r (x; t) = hx h(
8
1)
(x; t) ; t :
(16)
Then r solves (11).
Proof. Di¤erentiating (15) yields
ht (x; t) rx (h (x; t) ; t) + rt (h (x; t) ; t) = hxt (x; t)
and
hxx (x; t) rx (h (x; t) ; t) + h2x (x; t) rxx (h (x; t) ; t) = hxxx (x; t) :
Using (15) we deduce
1
1
hxt (x; t) + hxxx (x; t) = rx (h (x; t) ; t) ht (x; t) + hxx (x; t)
2
2
1
= rt (h (x; t) ; t) + h2x (x; t) rxx (h (x; t) ; t) :
2
Using (15) once more we have
1
rt (h (x; t) ; t) + h2x (x; t) rxx (h (x; t) ; t) = 0;
2
and we easily conclude.
Remark 5 In portfolio choice, a quantity that occasionally appears is the reciprocal of the local risk tolerance that is called, in analogy, the local risk aversion,
(x; t) =
Direct di¤ erentiation of
equation
t
1
:
r (x; t)
(x; t) and use of (11) yield that
+
1
(
2
m
)xx = 0
with m =
(x; t) satis…es the
1:
(17)
.
Equation (17) is a porous medium3 equation (PME) with negative exponent
m: Frequently, for m < 1; PME is referred to as a fast di¤usion equation and
various properties of its solution are di¤erent than the ones when m > 1: We
refer the reader to [13] and [14] for an extensive study on this equation.
4.2
Representation of strictly increasing harmonic functions
We continue with representation results for strictly increasing solutions to (15).
They appeared recently in [6] and we refer the reader therein for the related
proofs. Here, we only highlight the main …ndings and representative cases.
3 The porous medium equation is a nonlinear heat equation of the form f =
t
more generally of the form ft = (F (f ))):
9
(f m ) (or
The key idea in constructing monotone solutions to (15) is to use the classical
representation result of Widder for positive harmonic functions. This result,
recalled below (see, also, [15]), states that every positive solution of (15) can be
represented in terms of the bilateral Laplace transform of a …nite positive Borel
measure, denoted hereafter by . We will be using the notation B (R) to denote
the set of Borel measures in R:
Widder’s theorem is not directly applicable in the portfolio problem we study
because the solutions h we consider might not be positive. Indeed, as we will
see in the next section, these solutions will be ultimately used to construct the
investor’s optimal wealth which is not necessarily positive (see, for instance, examples 6.1). However, Widder’s theorem can be applied to the spatial derivative
hx which is a positive harmonic solution, as it follows from direct di¤erentiation
of (15) and the monotonicity of h.
We introduce the following set of measures,
B + (R) =
2 B (R) : 8B 2 B;
(B)
0 and
Z
R
eyx (dy) < 1; x 2 R :
(18)
Theorem 6 (Widder). Let g (x; t) ; (x; t) 2 R [0; +1) ; be a positive solution
of (15). Then, there exists 2 B + (R) such that g is represented as
Z
1 2
(19)
g (x; t) =
eyx 2 y t (dy) :
R
As the results in the sequel show, there is an interesting interplay between
the support and the integrability properties of the measure ; and the range of
the solution h: We will see that, in turn, this interplay a¤ects the range of the
local risk tolerance, the range and the limiting behavior of the di¤erential input
and, …nally, the range of the investor’s optimal wealth process. We stress that
h is de…ned, throughout, for all (x; t) 2 R [0; +1) : It is the range of h that
varies as the underlying measure changes.
To this end, we introduce the following subsets of B + (R) ;
B0+ (R) =
+
B+
(R) =
2 B + (R) and
2
B0+
(f0g) = 0 ;
(R) :
(( 1; 0)) = 0
2 B0+ (R) :
((0; +1)) = 0
and
B + (R) =
:
It is throughout assumed that the trivial case (R) = 0 is excluded.
In what follows, C represents a generic constant whose special choices lead
to simpli…ed expressions which we will use for our examples (cf. (21) and (22)
below).
10
Proposition 7 i) Let
R [0; +1) by
2 B + (R). Then, the function h de…ned for (x; t) 2
h (x; t) =
Z
1 2
2y t
eyx
1
y
R
(dy) + C
(20)
is a strictly increasing solution to (15).
Moreover, if (f0g) > 0; or ( 1; 0) > 0 and (0; +1) > 0 ; or 2
R +1
R0
(dy)
+
B+ (R) and 0+ (dy)
= +1; or
2 B + (R) and
= 1; then
y
y
1
Range (h) = ( 1; +1) ; for t 0:
R +1
+
On the other hand, if 2 B+
(R) with 0+ (dy)
< +1 then Range (h) =
y
R +1 (dy)
R +1 (dy)
C
0: For C = 0+
y ; +1 ; for t
y ; we have
0+
h (x; t) =
Z
1
eyx
1 2
2y t
(dy) ;
(21)
0+
with
(dy) =
Range (h) =
(dy)
y :
1; C
Similarly, if
R0
(dy)
1
y
h (x; t) =
Z
2 B + (R) with
; for t
0
eyx
R0
0: For C =
1 2
2y t
(dy)
>
y
(dy)
y ; we
1
1
R0
(dy) ;
1 then
have
(22)
1
with (dy) = (dy)
y :
ii) Conversely, let h : R [0; +1) ! R be a strictly increasing solution to
(15). Then, there exists 2 B + (R) such that h is given by (20).
Moreover, if Range (h) = ( 1; +1) ; t
0; then it must be either that
R +1
+
(f0g) > 0, or ( 1; 0) > 0 and (0; +1) > 0; or 2 B+
(R) and 0+ (dy)
=
y
R
0
= 1:
+1; or 2 B + (R) and 1 (dy)
y
On the other hand, if for some x0 2 R; Range (h) = (x0 ; +1) (resp.
R +1
+
(R) with 0+ (dy)
Range (h) = ( 1; x0 )), t 0; then it must be that 2 B+
<
y
R
0
+1 (resp. 2 B + (R) with 1 (dy)
> 1).
y
The proof of the above result can be found in Proposition 9 in [6].
4.3
Di¤erential input and harmonic functions
Next, we discuss how strictly concave and increasing solutions of (8) can be
constructed from strictly increasing solutions to (15). We stress that the harmonic function appearing in (23) and (25) is not necessarily the same with the
one in subsection 4.2. We use, with a slight abuse, the same notation only for
simplicity.
In order to facilitate the exposition, we compress some of the di¤erent cases
and assumptions on the measure. We also omit the case Range (h) = ( 1; 0) :
11
Proposition 8 Let h be a strictly increasing harmonic function and its associated measure (cf. (20) and (21)). We have the following cases:
i) Let be such that Range (h) = ( 1; +1) : Then, u : R [0; +1) ! R
de…ned by
Z
Z x
( 1)
1 t h( 1) (x;s)+ s
( 1)
2h
e
e h (z;0) dz; (23)
h
(x;
s)
;
s
ds
+
u (x; t) =
x
2 0
0
with h given in (20), is strictly increasing and concave in its spatial argument
and satis…es equation (8).
ii) Let
be such that Range (h) = (0; +1) and satisfying, in addition,
R +1
+
((0; 1]) = 0 and 1+ y(dy)
[0; +1) ! R is given
1 < +1: Then u : R
+
by (23), for x 2 R ; and satis…es
lim u (x; t) = 0;
for t
x!0
If, on the other hand,
u (x; t) =
1
2
Z
t
e
h(
((0; 1]) > 0 and/or
1)
(x;s)+ 2s
hx h(
1)
R +1
1+
0:
(dy)
y 1
(24)
= +1; we have
(x; s) ; s ds +
0
Z
x
e
h(
1)
(z;0)
dz; (25)
x0
for (arbitrary) x0 > 0; with
lim u (x; t) =
1; for t
x!0
For each t
0:
(26)
0; the Inada conditions
lim ux (x; t) = +1
x! 1
and
lim ux (x; t) = 0
x!+1
and
lim ux (x; t) = +1
x!0
and
lim ux (x; t) = 0
x!+1
(27)
are satis…ed for the cases (i) and (ii), respectively.
In [6] one also shows that the converse of the above statements is also true,
namely, if u is a strictly increasing and concave solution to (8) (and certain
limiting properties) then, there exists an associated solution h to (15) with the
appropriate representation characteristics (see Propositions 10, 14 and 15 in
[6]).
4.4
Local risk tolerance and di¤erential input
So far we have seen that solutions to (11) and (8) are directly related and can be
constructed by harmonic solutions (see (16) and (23)). How are, then, r (x; t)
and u (x; t) related to each other if they "correspond" to the same harmonic
function?
12
Given that the measure is the de…ning element of h (x; t) it is immediate
that r (x; t) and u (x; t) must be associated with the same measure. This, in turn,
implies that the initial conditions r (x; 0) = h h( 1) (x; 0) ; 0 and u (x; 0) =
R x h( 1) (z;0)
e
dz must be compatible. Direct di¤erentiation yields that
0
ux (x; 0)
= r (x; 0) :
uxx (x; 0)
(28)
Further di¤erentiation in (40) and use of (38) yields that the above relation is
also propagated to all positive times, namely, for t > 0;
ux (x; t)
= r (x; t) :
uxx (x; t)
5
(29)
Synthesis
We are now ready to address the questions posed in section 3. We introduce
the process Mt ; t 0;
Z t
Mt =
(30)
s dWs
0
with t as in (2). We also introduce a stricter than (18) assumption on the
measure ; speci…cally,
Z
1 2
(31)
eyx+ 2 y t (dy) < 1; t 0:
R
This condition is not needed for the representation results of the previous section but it is crucial in establishing the appropriate integrability, and in turn
admissibility, of the candidate optimal investment process.
We start with the case when the investor’s initial wealth is unconstrained,
x 2 R: For the reader’s convenience, we rewrite some of the formulae derived
earlier. On the other hand, we do not write explicitly the conditions on the
measure that yield full range of the solution h, for they are already listed in the
…rst part of Proposition 7. We also take C = 0 in (20).
Theorem 9 Let 0 : R ! R+ be the investor’s initial desired allocation in the
risky asset and let the function r0 : R ! R+ be given by
r0 (x) =
0
(x) :
(32)
r0 (h0 (x)) = h00 (x)
(33)
0
0
Let us assume that the autonomous equation
has a solution h0 : R ! R of the form
Z yx
e
1
(dy)
h0 (x) =
y
R
13
(34)
where satis…es (31) and is such that Range (h0 ) = ( 1; +1) :
Let h : R [0; +1) ! R be given by
Z
h (x; t) =
1 2
2y t
eyx
(dy)
(35)
(x; 0) + At + Mt ; At
(36)
R
and de…ne the processes Xt and
X t = h h(
t;
1)
1
y
t
0; by
and
t
=
t
hx h(
1)
t
(Xt ; At ) ; At ;
(37)
with At and Mt ; t 0; as in (9) and (30), respectively. Consider the function
r : R [0; +1) ! R+ de…ned by
r (x; t) = hx h(
1)
(x; t) ; t
(38)
and the process
Rt = r (Xt ; At ) :
Finally, de…ne u : R ! R by
Z
1 t
u (x; t) =
e
2 0
h(
1)
(x;s)+ 2s
r (x; s) ds +
(39)
Z
x
e
h(
1)
(z;0)
dz
(40)
0
where r and h are as in (38) and (35), respectively. Then,
(i) the portfolio t generates the wealth process Xt ;
(ii) the processes Xt and Rt solve the autonomous system (13),
0; is the optimal portfolio process for the forward
iii) the policy t ; t
investment performance criterion
U (x; t) = u (x; At )
(41)
with u (x; t) as in (40).
Proof. i) Let Nt = h(
1)
(x; 0) + At + Mt . Applying Itô’s formula to Xt yields
dXt = hx (Nt ; At ) dNt +
= hx (Nt ; At )
t
2
t
( t dt + dWt ) +
1
ht (Nt ; At ) + hxx (Nt ; At ) dt
2
2
t
1
ht (Nt ; At ) + hxx (Nt ; At ) dt:
2
From Proposition 7 we recall that h solves the heat equation (15). Therefore,
the above simpli…es to
dXt = hx (Nt ; At )
14
t
( t dt + dWt ) :
On the other hand,
hx (Nt ; At ) = hx h(
1)
h h(
1)
(x; 0) + At + Mt ; At ; At ; At
= hx h(
1)
(Xt ; At ) ; At
(42)
and using (3) and (37) we easily conclude.
ii) The above calculations yield that Xt satis…es the …rst equation in (13).
For the second equation we have
dRt = dr (Xt ; At )
=
2
t
1
rt (Xt ; At ) + r2 (Xt ; At ) rxx (Xt ; At ) dt + rx (Xt ; At ) dXt :
2
Proposition 4 and (38) yield that the function r (x; t) solves equation (11). Thus,
the above drift vanishes and the assertion follows.
iii) This part has several assertions to be established. The arguments are
lengthy and the calculations rather tedious. Moreover, most of them follow from
arguments used in the proof of Theorem 4 in [6]. For these reasons, we only
highlight the main steps.
To this end, we …rst need to show that the policy in consideration t ; t 0;
de…ned in (37) and rewritten for convenience,
t
t
=
hx h(
1)
t
(Xt ; At ) ; At
is admissible. Its Ft measurability follows trivially. To show the integrabilRt
2
ity property E 0 j s s j ds < 1; t > 0; we use (20) and that hx (x; t) =
R yx 1 y2 t
2
(dy) ; as it follows by direct di¤erentiation and Tonelli’s theorem.
e
R
Writing
hx h(
=
Z Z
R
(
e(y1 +y2 )(h
1)
1)
2
(Xt ; At ) ; At
(x;0)+At +Mt )
1
2
(y12 +y22 )At (dy ) (dy ) ;
1
2
R
using (37), (31) and the uniform bound on the market risk premium we conclude.
To show that U (x; t) is a forward investment performance process, it su¢ ces
+
to check that u given in (40) solves (8) and that E U (Xt ; t)
< +1: Finally,
one needs to show that the policy is optimal. For this, one needs to prove that
the drift of the process u (Xt ; At ) vanishes.
Next, we state the results for the semi-in…nite case. Due to space limitations,
we compress some arguments.
Theorem 10 Let 0 : R+ ! R+ be his initial desired allocation in the risky
asset. De…ne the function r0 : R+ ! R+ as in (33) and assume that equation
15
(34) has a solution h0 : R ! R+ of the form
h0 (x) =
Z
+1
eyx (dy)
(43)
0+
+
with 2 B+
(R) with
+
R be given by
R +1
(dy)
y
0+
< +1 and satisfying (31). Let h : R [0; +1) !
h (x; t) =
Z
+1
eyx
1 2
2y t
(dy) :
(44)
0+
Then the processes Xt and
X t = h h(
1)
t;
t
0; given by
(x; 0) + At + Mt ; At
and
t
=
t
hx h(
1)
t
(Xt ; At ) ; At
are optimal. Moreover, Xt > 0; t 0; a:e: Let r : R+ [0; +1) ! R+ be given
by r (x; t) = hx h( 1) (x; t) ; t :
R +1
+
[0; +1) ! R by
If ((0; 1]) = 0 and 1+ y(dy)
1 < +1; de…ne u : R
u (x; t) =
while if
1
2
Z
t
e
1
2
1)
(x;s)+ 2s
r (x; s) ds +
0
Z
Z
x
e
h0
(
1)
(z)
dz;
(45)
e
h0
(
1)
(z)
dz;
(46)
0
((0; 1]) > 0 and/or
u (x; t) =
h(
t
e
R +1
1+
h(
1)
(dy)
y 1
= +1; de…ne
(x;s)+ 2s
0
r (x; s) ds +
Z
x
x0
for x0 > 0: Then, Xt and t are optimal with respect to the forward investment
criterion U (x; t) = u (x; At ) with u (x; t) as above.
The above theorems give answers to the questions we put forward in section
2. Indeed, the desired initial allocation 0 (x) yields the initial risk tolerance
r0 (x) : If equation (33) has a solution that can be represented in the appropriate integral form, then the measure is identi…ed. From there, one readily
constructs the harmonic function h (x; t) :
Using h (x; t) ; hx (x; t) ; the market input processes At and Mt ; and formulae
(36) and (37) we …nd the processes Xt and t ; t > 0: From h (x; t) ; hx (x; t) ;
(38) and (40), one, also, …nds the di¤erential input u (x; t) and, in turn, the timemonotone investment performance criterion from (41). The theorems yield that
the processes Xt and t are optimal with respect to the forward investment
process U (x; t) = u (x; At ) :
Conditions (34) and (43) yield the admissibility requirement for the desired
initial allocations.
16
5.1
Inferring risk preferences from desired asset allocations
The analysis herein highlights how risk preferences can be entirely inferred from
the investor’s allocations. This is in contrast to the traditional approach in
portfolio theory in which one …rst speci…es the utility and then determines the
optimal policies. Herein, one takes the reverse route.
Indeed, the investment criterion is found by (41) as long as the market risk
premium is speci…ed (cf. (9)) and the function u (x; t) is known: The latter is
given by (40) (resp. (45) and (46)) which requires knowledge of h (x; t) and
r (x; t) : Both functions are speci…ed once the measure is known, as it is manifested in (35) (resp. (44)) and (38). The measure is found by (33) and (34).
It is worth mentioning an alternative way which provides a direct construction of u (x; t) from r (x; t) which entirely bypasses the use of h (x; t) ; t > 0: This
result carries useful insights, for it essentially shows how risk preferences can be
recovered simply by propagating the initial risk preferences along the inverted
characteristic curves whose slope is half of the local risk tolerance. On the other
hand, the slope of these curves is directly related to the optimal portfolio itself
as it can be seen from (37). In other words, the construction below alludes
to the fact that the investment performance process U (x; t) can be directly
constructed from the optimal portfolio process t by inverting the appropriate
stochastic characteristic curves. This idea was recently explored by El Karoui.
The arguments that follow are non-formal and are presented only to highlight
the approach. The main idea comes from the fact that the fully nonlinear
equation (8) can be equivalently written as the one below if one takes into
account (29). Similarly, the initial condition in (47) comes from (28).
To this end, let us assume that the function r (x; t) is known for t 0 and
consider the equation
Z x
( 1)
1
e h0 (z) dz;
(47)
ut + r (x; t) ux = 0 with u0 (x) =
2
0
with h0 (x) as in (34).
Classical results in the theory of …rst-order linear partial di¤erential equations yield the solution via the so-called method of characteristic curves (see,
for example, Chapter 2 in [17]). Speci…cally, let x 2 D and consider the solution
X x (t) of the equation
dX x (t)
1
= r (X x (t) ; t)
dt
2
with
X x (0) = x:
(48)
Di¤erentiating the function u (X x (t) ; t) with respect to t and using (47) yields
d
u (X x (t) ; t) = 0:
dt
In turn, for all t > 0;
u (X x (t) ; t) = u (X x (0) ; 0) = u0 (x) ;
17
(49)
which essentially tells us that the solution remains constant along each characteristic curve. One, then, sees that u (x; t) can be found from the above equality
and the initial condition. For illustration, we present an example of this construction in the …rst example below.
6
Examples
In this section we provide some representative examples for both the unconstrained and constrained wealth cases. For the sake of simplicity we present the
results in terms of the initial risk tolerance and not the initial allocation in (32).
This is only to eliminate the multiplicative constant 00 :
6.1
Domain(r ) = ( 1; +1)
r0 (x) = 1
Equation (33) has the solution h0 (x) = x: From (34) we easily deduce that
= 0 ; where 0 is a Dirac measure at 0: Then (35) yields h (x; t) = x: In turn,
(36) and (37) give
Xt = x + At + Mt and
t
=
t
:
t
On the other hand, (38) implies r (x; t) = 1. From (40) we deduce
u (x; t) =
1
2
Z
t
e
x+ 2s
ds +
0
Z
x
e
z
dz = 1
e
x+ 2t
:
(50)
0
The associated investment performance process is given by U (x; t) = 1
1
e x+ 2 At (this class of forward processes has been extensively analyzed; see, for
example, [7] and [18]).
For completeness, we also construct u (x; t) using (47) and the method of
characteristics. Using the form of h0 (x) we easily deduce that the initial condition in (47) becomes u0 (x) = 1 e x : Moreover, the characteristic curves (cf.
(48)) are given by
dX x (t)
1
=
dt
2
with X x (0) = x;
which implies that
1
X x (t) = x + t:
2
Therefore, (49) yields
1
u (X x (t) ; t) = u x + t; t
2
and by inverting the arguments, (50) follows.
18
=1
e
x
;
p
r0 (x) = ax2 + b; a; b > 0
The solution of (33) is given by h0 (x) = ab sinh (ax) : Formula (34) corresponds to = 2b ( a + a ) ; a; b > 0; and a are Dirac measures at a: In turn,
q
1 2
1 2
2
h (x; t) = ab e 2 a t sinh (ax) and h( 1) (x; t) = a1 ln ab xe 2 a t + ab2 x2 ea2 t + 1 :
p
This, in turn, yields hx h( 1) (x; t) ; t = a2 x2 + b2 e a2 t : The optimal processes
in (36) and (37) turn out to be
!!!
r
b 1 a2 At
a2 2 a2 t
a
Xt = e 2
x e + 1 + a (At + Mt )
x+
sinh ln
a
b
b2
and
t
=
t
t
If, a = 1, (40) gives
u (x; t) =
1
2
ln x +
while, if a 6= 1;
=
p
a
a2
a
1
e
1
2
a2 (Xt ) + b2 e
x2 + b2 e
et
x x
b2
t
a2 At :
p
x2 + b2 e
t
1
ln b;
2
t
2
u (x; t)
a
2
p
q
t
b2 e
at
p
+ a (1 + a) ax2 + x a2 x2 + b2 e
ax +
p
a2 x2
+
b2 e a2 t
a2 t
1+ 1
p
a
a2
a
1
b1
1
a
:
Further results on this case can be found in [16]. This two-parameter class is
quite rich and leads to the popular utilities - power, logarithmic and exponential
- for limiting values of the parameters a and b (see remark 12 at the end of this
section).
Rx 1 2
r0 (x) = f F ( 1) (x) with F (x) = 0 e 2 z dz; x 2 R and f = F 0
Equation (33) has the solution h0 (x) = F (x) which corresponds to the mea1 2
x
: Using that h( 1) (x; t) =
sure (dy) = p12 e 2 y dy: Then, h (x; t) = F pt+1
p
1
t + 1F ( 1) (x) we obtain hx h( 1) (x; t) ; t = pt+1
f F ( 1) (x) :
The optimal processes are given by
F(
Xt = F
1)
(x) + At + Mt
p
At + 1
and
1
F ( 1) (x) + At + Mt
p
f
:
At + 1
At + 1
We deduce (after tedious calculations that are omitted) that
p
t + 1 + k2
u (x; t) = k1 F F ( 1) (x)
t
with k1 = e
1
2
=p
and k2 = e
1
2
R0
1
1
2
e 2 z dz:
19
6.2
Domain(r (x )) =(0; +1) or [0; +1)
r0 (x) = x; > 1
Equation (33) yields h0 (x) = 1 e x : From (34) we then have
1
x
1
2
2
t
1
( 1)
,h
(x; t) = ln ( x) +
fore h (x; t) = e
x: In turn, (36) and (37) yield
1
2
Xt = x exp
Since
2
At + Mt
1
2
t and hx h
and
t
=
=
: There-
( 1)
(x; t) ; t =
t
Xt :
t
((0; 1]) = 0; u is given by
u (x; t) =
1
2
Z
1
t
xe
ln( x) + 12 s + 2s
ds +
0
Z
1
x
1
( z)
dz =
1
0
x
1
e
1
2
t
;
(cf. part (ii)in Proposition 8).
r0 (x) = x; 2 (0; 1)
This example is the same as the above. The only di¤erence is that now
((0; 1]) 6= 0. Thus, u is given by (46) for for x0 > 0;
u (x; t) =
1
2
Z
1
t
ln( x) + 2 s + 2s
xe
0
ex
x
( z)
1
dz
x0
1
1
=
ds +
Z
1
x
1
e
1
2
t
+
1
1
x0
:
r0 (x) = 1
Equation (33) yields h0 (x) = ex which corresponds to = 1 :Then, h (x; t) =
1
1
( 1)
2 t ; h( 1) (x; t) = ln x +
(x; t) ; t = x: In turn,
2 t; and hx h
Xt = x exp
Since
1
At + Mt
2
and
t
=
t
t
Xt :
((0; 1]) 6= 0; u is given by (25) for x0 > 0,
Z
Z x
1 t
1
x
ln x+ 12 s)+ 2s
(
u (x; t) =
xe
ds +
dz = ln
2 0
z
x
0
x0
t
:
2
Remark 11 The examples in 6.2 are the ones examined in an in…nitesimal
setting in [2].
Remark 12 It is worth noticing that the examplespin 6.2 can be thought as
limiting cases of the parameters apand b in r0 (x) = ax2 + b. Indeed, the …rst
two correspond to b = 0 and = a while the last one to b = 0 and = 1: The
…rst example in 6.1 can be also thought as a limiting case, namely, for a = 0
and b = 1:
20
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22