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
References [1] Berrier F., Rogers L.C. and M. Tehranchi: A characterization of forward utility functions, preprint, (2009). [2] Black, F.: Individual investment and consumption under uncertainty, Portfolio Insurance: A Guide to Dynamic Hedging, D. L. Luskin (Ed.), New York, John Wiley and Sons, 207–225 (1988); …rst version (1968). [3] Goldstein, D. G., Johnson, E. J., and Sharpe, W. F.: A new way to measure consumer risk preferences, working paper (2008). [4] Goldstein, D. G., Johnson, E. J. and Sharpe, W. F.: Choosing outcomes versus choosing products: consumer-focused retirement investment advice, Journal of Consumer Research, 35, 440-456 (2008). [5] Merton, R. : Lifetime portfolio selection under uncertainty: the continuoustime case, The Review of Economics and Statistics, 51, 247-257 (1969). [6] Musiela, M. and T. Zariphopoulou: Portfolio choice under space-time monotone performance criteria, SIAM Journal on Financial Mathematics, in press (2010). [7] Musiela, M. and T. Zariphopoulou: Optimal asset allocation under forward exponential criteria, Markov Processes and Related Topics: A Festschrift for Thomas. G. Kurtz, IMS Collections, Institute of Mathematical Statistics, 4, 285-300 (2008). [8] Musiela, M. and T. Zariphopoulou: Portfolio choice under dynamic investment performance criteria, Quantitative Finance, 9(2), 161-170 (2009). [9] Musiela, M. and T. Zariphopoulou: Stochastic partial di¤erential equations in portfolio choice, Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, C. Chiarella and A. Novikov (Eds.), in print. [10] Musiela, M. and T. Zariphopoulou: Investments and forward utilities, preprint (2006). [11] Sharpe W.F., Goldstein D.G. and P.W. Blythe: The distribution builder: A tool for inferring investor’s preferences, preprint (2000). [12] Sharpe W.F.: Expected utility asset allocation, Financial Analysts Journal, 63(5), 18-30 (2007). [13] Vasquez, J.-L.: The porous medium equation, Oxford University Press (2006). [14] Vasquez, J.-L.: Smoothing and decay estimates for nonlinear parabolic equations of porous medium type, preprint (2005). [15] Widder, D.V.: The heat equation, Academic Press (1975). 21
[16] Zariphopoulou, T. and T. Zhou: Investment performance measurement under asymptotically linear local risk tolerance, Mathematical Modeling and Numerical Methods in Finance, Handbook of Numerical Analysis, A. Bensoussan and Q. Zhang (Eds.), North Holland, 227-254 (2009). [17] Zauderer, E. : Partial di¤ erential equations of Applied Mathematics, John Wiley & Sons ((1983). [18] Zitkovic, G.: A dual characterization of self-generation and exponential forward performances, Annals of Applied Probability, 19(6), 2176-2210 (2009).
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