Optimal Portfolio With Benchmark for Fund Managers

Transcription

1 Optimal Portfolio With Benchmark for Fund Managers Francesco Menoncin a, a Dipartimento di Economia Politica e Metodi Quantitativi, Università degli Studi di Pavia, Italy Abstract This paper analyses the portfolio problem of a fund manager maximizing the expected utility of fund s terminal performance with respect to a given benchmark. The benchmark is supposed to follow a stochastic process. Furthermore, we suppose that also contributions to and withdrawals form the fund follow a stochastic process. Contrary to the usual approach, in this work, we do not specify any functional form for the involved stochastic variables. If the market is complete, then we are able to find an exact solution. If the market is not complete, we propose an approximated solution. JEL classification: G11, C61. Key words: asset allocation, background risk, Feynman-Kac theorem, stochastic investment opportunities Dipartimento di Economia Politica e Metodi Quantitativi, Via S. Felice, Pavia, Italy. Tel: ; fax: ; francesco.menoncin@unipv.it 1

2 1 Introduction This work analyses the issue of optimal portfolio policy in a multi-period model and, in particular, the problem of a fund manager who must maximize the expected value of a suitable function of fund s performance. More precisely, he is supposed to maximize the distance between the fund s growth wealth and a benchmark which is not one of his control variables. Thus, the fund manager can only change the asset allocation, in order to make this difference optimal. In the financial market, the fund manager must cope with a set of stochastic investment opportunities. For a review of all variables which can affect the asset prices readers are referred to Campbell (2000) who offers a survey of the most important contributions in this field. We underline that, with respect to the usual portfolio-consumption approach first developed by Merton (1969, 1971), in our setting we do not consider the consumption problem. In fact, while the withdrawals from the investment return for a private investor can be used for consumption and give him some utility, the withdrawals from a fund cannot be chosen by the fund manager and so, for him, they are just state variables and not control variables. In the literature about the optimal portfolio for pension funds (see for instance, Blake, 1998, Blake, Cairns, and Dowd, 1998, and Boulier, Huang, and Taillard, 2001), a lot of attempts have been made for introducing in the framework a stochastic process for subscribers wages. Nevertheless, exact solutions to this kind of problem have been found only in the case of deterministic wages. For an analysis of the stochastic wage case the reader is referred to Blake, Cairns, and Dowd (1998). The framework we develop here is able to deal with a stochastic process for contributions and withdrawals. In particular, the presence of the benchmark can solve some of the problems which are generally faced in this type of models allowing us to solve some technicalities which make the problem very difficult to solve when the usual framework is considered. Our model can be applied also to insurance companies (see Young and Zariphopoulou, 2000) and, in general, to all the institutional investors. Furthermore, it is able to deal with the inflation problem (as shown in Menoncin, 2001). In this paper we follow the traditional route to use the stochastic dynamic programming technique (Merton, 1969, 1971) leading to the Hamilton-Jacobi- Bellman (HJB) equation. 1 For the method called martingale approach the reader is referred to Cox and Huang (1989, 1991), and Lioui and Poncet (2000). We just underline that in this work we are able to reach the same qualitative results as Lioui and Poncet even if they do not consider any withdrawal risk. Even if the martingale method can lead to a simpler stochastic partial differential equation than the Bellman approach, we prefer to use the last one because it is able to offer a solution approach even when the considered market is not complete. In our framework, we find that the optimal portfolio is formed by three components: (i) a preference free component minimizing the fund s wealth diffusion 1 Øksendal (2000) and Björk (1998) offer a complete derivation of the HJB equation. 2

3 variance and immunizing the fund s portfolio against the withdrawal risk, (ii) a speculative part proportional to both the Sharpe ratio of investor s portfolio and the inverse of the Arrow-Pratt risk aversion index, and (iii) a component depending on the derivatives of the value function (indirect utility function) with respect to all the state and withdrawal variables. The last component is the only one depending on the fund s time horizon. In order to find the explicit solution for the value function, it is necessary to solve the HJB equation. Unfortunately, solving this highly non-linear PDE is the most difficult task of the stochastic optimal control approach. In fact, some algebraic solutions can only be obtained in very special cases. In particular, we refer to the works of Kim and Omberg (1996), Wachter (1998), Boulier, Huang, and Taillard (2001), and Deelstra, Grasselli, and Koehl (2001). In the present work we show that if the financial market is complete, then we can find an exact solution to our optimal portfolio problem. This result can be obtained thanks to the insertion of the benchmark for the fund manager. In fact, after introducing this benchmark, the riskless asset is no more riskless. In particular, in this framework, an asset could be riskless if it were perfectly correlated with the benchmark. Instead, because the riskless asset has not this characteristic, then it becomes a risky assetfromthepointofviewofthefund manager. Furthermore, we outline that, contrary to the exact solutions above mentioned, we do not specify any particular functional form for the behaviour of the stochastic variables involved in the problem and we reach, in this way, a very general solution indeed. We underline that the exact solutions presented in Kim and Omberg (1996), Boulier, Huang, and Taillard (2001), and Deelstra, Grasselli, and Koehl (2001), consider only one state variable and do not take into account any withdrawal risk. If the completeness hypothesis does not hold, then we propose a general approximated solution to the HJB equation. In particular, our work concentrates on the hypothesis that the value function has a suitable form under which the Feynman-Kac theorem can be applied to the HJB equation. Thus, even if in an incomplete market our solution is exact only under particular conditions that must hold on the value function, we find that it stays valid as an approximated solution under conditions which are not very restrictive. Furthermore, we will underline that our approximated solution overestimates the risk linked to the state variables and so, it gives higher values for the optimal portfolio third component than the exact one. Through this work we consider agents trading continuously in a frictionless, arbitrage-free market until time H, which is the horizon of the economy. Furthermore, we analyse both a complete and an incomplete market structure. The paper is organized as follows. Section 2 details the general economic framework and exposes the stochastic differential equations describing the behaviour of asset prices, state variables, contributions and withdrawals. In Section 3 the optimal portfolio composition is computed. This section presents our main results: (i) an exact solution of the problem if the financial market is complete, and (ii) an approximated solution when this hypothesis does not hold. Section 4 concludes. 3

4 2 The economy In this paper we consider an economy where the behaviour of asset prices is described by stochastic differential equations. Furthermore, the volatility and drift coefficients of these equations depend on a set of state variables which are represented by diffusion processes. For both contributions and the withdrawals we consider a diffusion process. In this section, we also present the behaviour of the stochastic benchmark, and we derive the stochastic differential equation describing the behaviour of the fund s wealth and performance. 2.1 The market structure We suppose that asset prices are affected by a set of stochastic state variables. For a review of all variables which can affect the asset prices readers are referred to Campbell (2000) who offers a survey of the most important contributions in this field. In this paper we suppose that these risk sources follow the stochastic differential equation: dx = f (X, t) dt + g (X, t) 0 dw, (1) s 1 s 1 s k k 1 X (t 0 ) = X 0, where s is the number of state variables and dw is the differential of a k dimensional Wiener process defined on a probability space (Θ, F, P) and whose components are independent. 2 Given these variables we can write the process describing the behaviour of asset prices like the following stochastic differential equation: ds n 1 = µ (t, X, S) n 1 S (t 0 ) = S 0, dt + Σ (t, X, S) 0 dw, (2) k 1 wherethesetofrisksourcesisthesamewehaveusedforthestatevariables. This hypothesis is not restrictive because thanks to the elements of the matrices g and Σ we can model a lot of different situations. On the financial market there exists a riskless asset whose price (G) follows the differential equation: n k dg = r (X, t) Gdt, G (t 0 ) = G 0, where r (X, t) is the nominal risk-free interest rate which is supposed to depend on the state variables X. 2 This condition can be imposed without loss of generality because a set of independent Wiener processes can always be transformed into a set of correlated Wiener processes thanks to the Cholesky decomposition. 4

5 2.2 The withdrawal risk During the fund management, the subscribers can contribute to the fund or withdraw from it. We model the contribution-withdrawal process as a multidimensional diffusion process. Actually, we would need just a scalar dimensional process but we want to make our model as general as possible. In fact, thanks to a market survey, the fund manager could create some different sets of contribution and withdrawal processes for taking into account the different reasons according to which the subscribers decide to put the money into the fund or to take it back. Thus, we suppose that the contribution-withdrawal process follows the stochastic differential equation given by: dl = µ L (X, L, S, t) dt + Λ (X, L, S, t) 0 dw l 1 l 1 l k L (t 0 ) = L 0, k 1 (3) where dw is the same set of risk sources we have for both asset prices and state variables. This hypothesis is not restrictive because we can model a lot of different cases thanks to the elements of the matrices Λ and Σ. We underline that the process L, in this framework, represents the so-called background risk. That is to say, a risk outside the financial market. Thus, in this framework, the notion of completeness must be clarified. In particular, we want to widen the completeness notion to include also the background risk. Since we consider the same risk set (dw ) for both the financial market and the background risk, in order to check the completeness of the market we should disentangle the two sets of risk sources. In particular, Equations (2) and (3) should be written in the following way: ½ ds = µdt + Σ 0 S dw S, dl = µ L dt + Λ 0 S dw S + Λ 0 L dw (4) L, provided that the background risk variables have more risk sources than the asset prices. In this case, if matrix Σ S is invertible, then we have a complete market, but we also have a set of risks (dw L ) which cannot be hedged through a suitable combination of assets. Blake, Cairns, and Dowd (1998) consider a market having the same structure as in System (4). Thus, they define the risk contained in vector dw L as a non-hedgeable risk. When we want to write System (4) with the same risk sources for both processes: dws dw =, dw L as in Equations (2) and (3), then we must create two new matrices: Σ 0 = Σ 0 S 0, Λ 0 = Λ 0 S Λ 0 L, where 0 is a suitable matrix of zeros. We underline that even if Σ S is invertible, the new matrix Σ cannot be. This means that even if the financial market is 5

6 complete, this property does not hold if we include in the market notion also the background risk. Accordingly, in our analysis, we will use the usual definition of completeness ( Σ 1 ) and we will consider a wider concept of market for including also the background risk. Thus, what Blake, Cairns, and Dowd (1998) define as a non-hedgeable risk in a complete market, for us is just an incomplete market, where the non-hedgeable component cannot be distinguished. 2.3 The benchmark With respect to the classical literature about the optimal portfolio rules, in this work we do not study the simple maximization of expected utility of fund s terminal wealth but we consider the maximization of the difference between the fund s gross performance and the growth ratio of a given benchmark. In particular, in the literature, we can find a lot of models considering the optimal portfolio and consumption problem (Merton, 1969, 1971, Kim and Omberg, 1996, Wachter, 1998, Kogan and Uppal, 1999, Chako and Viceira, 1999) or, more often, only the optimal asset allocation problem (Blake, Cairns, and Dowd, 1998, Cox and Huang, 1989, 1991, Lioui and Poncet, 2000, Boulier, Huang, and Taillard, 2001, Deelstra, Grasselli and Koehl, 2001). Nevertheless, at least to our knowledge, nobody has presented a framework where the maximization is computed with respect to the expected utility of a terminal performance given by the difference between the fund s gross performance and the return which could have been obtained by investing the fund s wealth in a defined benchmark. We underline that both Boulier, Huang, and Taillard (2001) and Deelstra, Grasselli and Koehl (2001) consider the problem of a pension fund where there exists a minimum guarantee for the subscribers pensions. Nevertheless, this guarantee is represented by a fixed amount of money, while in our framework the benchmark is stochastic. For setting our framework we must specify the stochastic process describing the behaviour of the benchmark. For the sake of generality, we do not consider a scalar process but a multi-dimensional process, because we let the benchmark be a weighted mean of some market indexes. Thus, all the indexes appearing in the benchmark definition are supposed to follow the stochastic differential equation: " # db b 1 = I B b b µ B (X, B, S, t) b 1 dt + Π (X, B, S, t) 0 dw b k k 1, (5) where I B is a diagonal matrix containing the elements of vector B. Thevector B can also contain the consumption price index (as in Menoncin, 2001) and, in this case, we would be able to consider also the problem of hedging the inflation risk. 6

7 2.4 The fund s wealth Afterwhatwehavepresentedintheprevioussubsections,themarketstructure can be represented in the following way: dx = f (t, X) dt + g (t, X) 0 dw, s 1 s 1 s k k 1 ds = µ (t, X, S) dt + Σ (t, X, S) 0 dw, n 1 n 1 n k k 1 dg = Gr (t, X) dt, dl = µ L (t, L, S, X) dt + Λ (t, L, S, X) 0 (6) dw, l 1 " l 1 l k k 1 # db b 1 = I B b b µ B (X, B, S, t) b 1 dt + Π (X, B, S, t) 0 dw If we indicate with θ (t) R n 1 and θ G (t) R the number of risky assets held and the quantity of riskless asset held respectively, then the fund s budget constraint can be represented as: R = θ (t) 0 S + θ G (t) G + u (t) 0 L, (7) where u (t) R l 1 is the vector containing the parameters indicating in which proportion the elements of vector L affect the wealth level. For instance, we could suppose that the fund manager considers two different processes for the contributions and for the withdrawals. Then, in this case, u (t) is a two dimensional vector whose elements are 1 and 1 because the contributions increase the wealth R while the withdrawals decrease it. After differentiating the budget constraint (7) we obtain: dr = θ (t) 0 ds + θ G (t) dg + u (t) 0 dl. (8) The most common models about the optimal asset allocation maximize the expected utility of the fund s terminal wealth R. Nevertheless, in this framework we want to maximize a performance measure which does not correspond to the fund s wealth. In the next subsection we present this new performance measure. b k k The fund s performance Even if the amount of money which can be invested by the fund manager is given by the constraint (7), we suppose the fund manager wants to maximize the expected utility of a suitable terminal performance measure. For computing this measure we consider the following argument. The manager starts his activity from a period called t 0. When the time reaches t, the performance measure he has reached is given by the amount of fund s wealth generated by his investments diminished by the increase in wealth he could have obtained by investing all the fund s wealth in the benchmark. Algebraically, if we call Φ the performance measure, then these considerations can be written in the following way: Φ (t) =R (t 0 )+ Z t t 0 dr (s) 7 Z t t 0 θ B (s) 0 db (s), (9)

8 where θ B R b 1 is a vector containing the amount of benchmark which could have been bought with the fund s wealth. We underline that at time t 0 all the market structure is known. We just outline that the initial value of the fund s performance measure coincides with the initial fund s wealth (Φ (t 0 )=R (t 0 )). Equation (9) can also be written in the following way: dφ (t) =dr (t) θ B (t) 0 db (t). Now, we have to take into account that the amount of benchmark which can be bought with the fund s wealth is given by the ratio between the wealth level (R) and the benchmark value. This benchmark value is given by a weighted mean of the market indexes taken into account. In particular, if we call v (t) R b 1 the vector containing the weights, we have: θ B (t) =R (t) I 1 B v (t), and, after substituting into the previous equation we obtain: dφ (t) =dr (t) R (t) v (t) 0 I 1 B db (t). It is sufficient to substitute for the values of R and dr found in Equations (7) and (8) respectively, for obtaining the behaviour of fund s performance Φ: dφ = θ 0 (µ Sv 0 µ B )+θ G (Gr Gv 0 µ B )+u 0 (µ L Lv 0 µ B ) dt + + θ 0 (Σ 0 Sv 0 Π 0 ) θ G Gv 0 Π 0 + u 0 (Λ 0 Lv 0 Π 0 ) dw + u 0 η 0 dp, wherewehaveeliminatedallthefunctionaldependences. We underline that in the models where no benchmark is considered, the control variable θ G does not appear in the diffusion term of the wealth differential equation because the riskless asset does not contain any diffusion term. Instead, in this case, also the riskless asset is affected by the benchmark diffusion term because it cannot hedge the fund s wealth against the risk linked to the stochastic behaviour of the benchmark. If we put into the vector B also the consumption price process, then one of our benchmark component is the inflation rate. Since the riskless asset is nominally riskless, then in real term it becomes a risky asset and its diffusion term coincides with the diffusion term of the inflation process. For the sake of simplicity, we prefer to consider the following change of variables: w (n+1) 1 M (n+1) 1 Γ 0 (n+1) k = = = θ θ G, µ Sv 0 µ B Gr Gv 0 µ B Σ0 Sv 0 Π 0 Gv 0 Π 0,, 8

9 and, thus, we can write the differential equation of the performance as: dφ =[w 0 M + u 0 (µ L Lv 0 µ B )] dt +[w 0 Γ 0 + u 0 (Λ 0 Lv 0 Π 0 )] dw + u 0 η 0 dp. (10) With respect to the usual approach, in this differential equation we lack the term containing the level of wealth. 3 This characteristic comes from the following consideration: the fund manager is not interested in the level of the absolute wealth R, and furthermore, he cannot invest the performance measure Φ because this is just a fictitious index (only the absolute wealth can be actually invested). We recall that in this framework the benchmark can be interpreted as the opportunity cost of investing in financial assets different from the benchmark itself. 3 The optimal portfolio Under the market structure (6) and the evolution of the fund s performance measure given in Equation (10), the optimization problem for a manager maximizing the expected utility of the terminal performance, can be written as follows: max E t w 0 [K (Φ (H))] z µ d = z Φ w 0 M + u 0 (µ L Lv 0 dt + µ B ) z (t 0 )=z 0, Φ (t 0 )=Φ 0 = R 0, t 0 t H, where: Ω 0 w 0 Γ 0 + u 0 (Λ 0 Lv 0 Π 0 ) dw, (11) z (s+n+1+l+b) 1 µ z (s+n+1+l+b) 1 Ω k (s+n+1+l+b) X 0 S 0 G L 0 B 0 0, f 0 µ 0 Gr µ 0 L µ0 B I B g Σ 0 Λ ΠI B, 0, and K (Φ) is an increasing and concave function, while H is the fund s time horizon. The vector z contains all the state variables but the performance Φ, and 0 is a vector of suitable dimension containing only zeros. 3 Without any withdrawal or benchmark risk and under market structure (6), the wealth differential equation should be written as follows: dr = Rr + θ 0 (µ rs) dt + θ 0 Σ 0 dw. 9

10 From problem (11) we have the following Hamiltonian: H = µ 0 zj z + J Φ (w 0 M + u 0 (µ L Lv 0 µ B )) tr (Ω0 ΩJ zz )+ (12) +(w 0 Γ 0 + u 0 (Λ 0 Lv 0 Π 0 )) ΩJ zφ J ΦΦ (w 0 Γ 0 Γw+ +u 0 (Λ 0 Lv 0 Π 0 )(Λ ΠvL 0 ) u +2w 0 Γ 0 (Λ ΠvL 0 ) u), where J (Φ,z,t) is the value function solving the Hamilton-Jacobi-Bellman partial differential equation (see Section 3.1), verifying: J (Φ,z,t)=supE t [K (Φ (H))], w and the subscripts on J indicate the partial derivatives. The system of the first order conditions on H is: 4 H w = J ΦM + Γ 0 ΩJ zφ + J ΦΦ (Γ 0 Γw + Γ 0 (Λ ΠvL 0 ) u) =0, from which we obtain the optimal portfolio composition: w = (Γ 0 Γ) 1 Γ 0 (Λ ΠvL 0 ) u J Φ (Γ 0 Γ) 1 M 1 (Γ 0 Γ) 1 Γ 0 ΩJ zφ. (13) {z } J ΦΦ J {z } ΦΦ {z } w (1) For having a unique solution to the optimal portfolio problem we have to guarantee that the matrix Γ 0 Γ R (n+1) (n+1) is invertible. This condition is satisfied if Γ 0 R (n+1) k has rank equal to n +1 and n +1 k. Actually, due to the benchmark risk, the riskless asset becomes a risky asset, acquiring the diffusion term of the benchmark variable. Thus, the completeness must be defined on n +1assets and no more on n assets. During the following work we will define a complete market as the market where the matrix Γ is invertible (that is to say it is a square (n +1) (n +1)matrix and its rank is maximum). Thus, we can state the following result: w (2) w (3) Proposition 1 Under market structure (6), the portfolio composition maximizing the fund s terminal performance (thus solving problem (11)) is formed by three components: (i) a preference free part (w(1) ) depending only on the diffusion terms of assets and withdrawal variables, (ii) a part (w(2) )proportional to both the portfolio Sharpe ratio and the inverse of Arrow-Pratt risk aversion index on the fund performance, and (iii) a part (w(3) ) depending on the state variable parameters. 4 The second order conditions hold if the Hessian matrix of H: H w 0 w = J RRΓ 0 Γ, is negative definite. Because Γ 0 Γ is a quadratic form it is always positive definite and so the second order conditions are satisfied if and only if J RR < 0, that is if the value function is concave in R. The reader is referred to Stockey and Lucas (1989) for the assumptions that must hold on the function K (R) for having a strictly concave value function. 10

11 The preference free portfolio component has an important characteristic: it minimizes the performance variance. In fact, from Equation (10) we can see that the variance of the fund s performance is given by: w 0 Γ 0 Γw + u 0 (Λ 0 Lv 0 Π 0 )(Λ ΠvL 0 ) u +2w 0 Γ 0 (Λ ΠvL 0 ) u, from which we immediately see that: 5 Proposition 2 The preference-free component (w(1) ) of optimal portfolio (solving problem (11)) minimizes the performance diffusion variance. For the second part, we just outline that w(2) increases if the risk premium increases and decreases if the risk aversion or the asset variance increase. From this point of view, we can argue that this component of the optimal portfolio has just a speculative role. The third part w(3) is the only optimal portfolio component explicitly depending on the diffusion terms of the state variables. Thus, while w(1) covers the fund from the withdrawal risks which can be seen as risks outside the financial market, w(3) covers the fund also from the risk inside the financial market. We will investigate the precise role of this component after computing the functional form of the value function. 3.1 The value function For studying the exact role of the portfolio components we have called w(2) and w(3) (see Equation (13)), we need to compute the value function J (Φ,z,t). By substituting the optimal value of w into the Hamiltonian (12) we have: H = µ 0 zj z J Φ u 0 (Λ 0 Lv 0 Π 0 ) Γ (Γ 0 Γ) 1 M + J Φ u 0 (µ L Lv 0 µ B )+ J 2 Φ 1 M 0 (Γ 0 Γ) 1 M J Φ M 0 (Γ 0 Γ) 1 Γ 0 ΩJ zφ J ΦΦ J ΦΦ 2 tr (Ω0 ΩJ zz )+ +u 0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ ΩJ zφ J 2 J zφω 0 0 Γ (Γ 0 Γ) 1 Γ 0 ΩJ zφ + ΦΦ J ΦΦu 0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ (Λ ΠvL 0 ) u. 5 We underline that the second derivative of this term with respect to w is: 2Γ 0 Γ, which is always positive definite because Γ 0 Γ is a quadratic form. 11

12 From this equation we can formulate the PDE whose solution is the value function. This PDE is called Hamilton-Jacobi-Bellman equation (hereafter HJB) and it can be written as follows: ½ J t + H =0, (14) J (H, Φ,z)=K (Φ (H)), One of the most common way to solve this kind of PDE is to try a separability condition. In the literature (since Merton, 1969, 1971), a separability by product is generally found. Here, we suppose that the value function J (z,φ,t) is separable by product in wealth and in the other state variables according to the following form: J (z, Φ,t)=U (Φ) e h(z,t). After substituting this functional form into the HJB equation (14) and dividing by J we obtain: 0 = h t + µ 0 zh z U Φ U u0 (Λ 0 Lv 0 Π 0 ) Γ (Γ 0 Γ) 1 M + U Φ U u0 (µ L Lv 0 µ B )+(15) 1 UΦ 2 2 U ΦΦ U M 0 (Γ 0 Γ) 1 M U Φ 2 U ΦΦ U M 0 (Γ 0 Γ) 1 Γ 0 Ωh z tr (Ω0 Ω (h zz + h z h 0 z)) + U Φ U u0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ Ωh z + 1 UΦ 2 2 U ΦΦ U h0 zω 0 Γ (Γ 0 Γ) 1 Γ 0 Ωh z U ΦΦ 2 U u0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ (Λ ΠvL 0 ) u. The model is consistent if and only if all the ratios containing functions of Φ are constant with respect to Φ. The only function satisfying all these conditions is the exponential function of the form U (Φ) =αe γ+βφ.accordingly, the boundary condition can be written in the following way: U (Φ) e h(z,h) = K (Φ (H)). Nevertheless, because of the separability condition, we can also write: U (Φ) =K (Φ), and the boundary condition becomes: h (z, H) =0. Thus, we can write: Proposition 3 Under market structure (6) the value function is separable by product in the performance and in the other state variables if and only if the fund manager maximizes an exponential utility function of the form K (Φ) =αe γ+βφ. We just underline that we want the value function to be increasing and concave in wealth. This means that the parameters α and β must be such that: α, β < 0, while γ R. If we substitute for the function U (Φ) =αe γ+βφ into Equation (15) we obtain: ( ³ 0 h t + a (z,t) 0 h z + b (z,t)+ 1 2 tr (Ω0 Ωh zz )+ 1 2 h0 zω 0 I Γ (Γ 0 Γ) 1 Γ Ωh z =0, h (z,h) =0, (16) 12

13 where: a (z,t) 0 µ 0 z M 0 (Γ 0 Γ) 1 Γ 0 Ω + βu 0 (Λ 0 Lv 0 Π 0 ) ³I Γ (Γ 0 Γ) 1 Γ 0 Ω, b (z,t) βu 0 (µ L Lv 0 µ B ) βu 0 (Λ 0 Lv 0 Π 0 ) Γ (Γ 0 Γ) 1 M 1 2 M 0 (Γ 0 Γ) 1 M β2 u 0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ (Λ ΠvL 0 ) u. By using the exponential utility function and the separability condition, we can write the optimal portfolio composition in the following way: w = (Γ 0 Γ) 1 Γ 0 (Λ ΠvL 0 ) u 1 β (Γ0 Γ) 1 M 1 β (Γ0 Γ) 1 Γ 0 Ωh z, and, for finding a close form solution we have to solve the HJB equation in order to find the form of function h (z,t). In the following subsections we study how to solve Equation (16) in both a complete and an incomplete market. 3.2 The case of an incomplete market: an approximated solution Unfortunately, we are not able to apply the Feynman-Kac theorem 6 to Equation (16) because of the term containing h z h 0 z. In order to apply the theorem, we should have only the term h zz.ifweimposeh z to be zero, then also h zz must be zero and a solution to (16) can be found only if the term b (z,t) is independent of z. This is a too particular case for being interesting. We could choose a kind of functional form for h (z,t) such that: 1 ³ 0 2 h0 zω 0 I Γ (Γ 0 Γ) 1 Γ Ωh z = h, and also ³ in this case we 0 could apply the Feynman-Kac theorem. Nevertheless, if Ω 0 I Γ (Γ 0 Γ) 1 Γ Ω does not depend on z, then this function must be a quadratic polynomial of the form: h (z,t) = 1 2 B (t)0 Ω 0 ³ I Γ (Γ 0 Γ) 1 Γ 0 ΩB (t)+b (t) 0 z z0 ³ Ω 0 ³ I Γ (Γ 0 Γ) 1 Γ 0 Ω 1 z, where only B (t) can be arbitrarily chosen. ³ Nevertheless, 0 we do not want to impose such a restrictive condition. If Ω 0 I Γ (Γ 0 Γ) 1 Γ Ω actually depends on z then h (z, t) must be an integral function in z which is very difficult to manage. 6 For a complete exposition of the Feynman-Kac theorem the reader is referred to Duffie (1996), Björk (1998) and Øksendal (2000). 13

14 Instead, we prefer to look for a function satisfying h zz = h z h 0 z. After solving this differential equation, we find that h (z, t) must have the following form: h (z, t) =A (t) ln B (t) 0 z + D (t), (17) where A (t),d(t) R, andb (t) R (s+n+1+l+b) 1 such that B (t) 0 z + D (t) > 0. In this case, in fact, we have: h zz = B (t) B (t) 0 B (t) 0 z + D (t) 2 = h zh 0 z. Thus, if the function h (z,t) has the form (17), then the HJB equation can be simplified as follows: ( h ³ 0 i h t + a (z,t) 0 h z + b (z,t)+ 1 2 tr Ω 0 2I Γ (Γ 0 Γ) 1 Γ Ωh zz =0, h (H, z) =0. For applying the Feynman-Kac theorem, we have to find two real numbers x 1 and x 2 such that: ³ x 1 I x 2 Γ (Γ 0 Γ) 1 Γ 0 2 =2I Γ (Γ 0 Γ) 1 Γ 0, from which, because I Γ (Γ 0 Γ) 1 Γ 0 is a symmetric, idempotent matrix, we easily obtain x 1 = 2 and x 2 = 2 ± 1. Thus, by putting: eω 0 Ω 0 h 2I ³ 2 ± 1 Γ (Γ 0 Γ) 1 Γ 0i 0, we can write the HJB equation in the following way: ( ³ h t + a (z,t) 0 h z + b (z, t)+ 1 2 tr Ω e 0 eωh zz =0, h (H, z) =0. to which we can apply the Feynman-Kac theorem and obtain: h (z,t) = where the variables Z s follow: Z H t E t [b (Z s,s)] ds, dz s = a (Z s,s) ds + Ω e (Z s,s) 0 dw, Z t = z. Thus, our result, can be summarized as follows: 14

15 Proposition 4 Under market structure (6), the portfolio composition maximizing the fund s terminal exponential utility function (K (Φ) =αe γ+βφ )isgiven by: w = (Γ 0 Γ) 1 Γ 0 (Λ ΠvL 0 ) u 1 β (Γ0 Γ) 1 M + 1 Z H β (Γ0 Γ) 1 Γ 0 Ω t z E t [b (Z s,s)] ds, if and only if there exist functions A (t),d(t) R, andb (t) R (s+n+1+l+b) 1 such that: Z H E t [b (Z s,s)] ds = A (t) ln B (t) 0 z + D (t), t where: dz s = a (Z s,s) ds + Ω e (Z s,s) 0 dw, Z t = z, a (z,t) 0 µ 0 z M 0 (Γ 0 Γ) 1 Γ 0 Ω + βu 0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ Ω, b (z,t) βu 0 (µ L Lv 0 µ B ) βu 0 (Λ 0 Lv 0 Π 0 ) Γ (Γ 0 Γ) 1 M 1 2 M 0 (Γ 0 Γ) 1 M β2 u 0 (Λ 0 Lv 0 Π 0 ) ³I 0 Γ (Γ 0 Γ) 1 Γ (Λ ΠvL 0 ) u, eω 0 Ω 0 h 2I ³ 2 ± 1 Γ (Γ 0 Γ) 1 Γ 0i 0. We can see that the only component of optimal portfolio explicitly depending on the investor s horizon H is the third one, which hedges the portfolio against the state variable risks and, in particular, against the withdrawal risk. The closed form solution for the portfolio allocation problem obtained in Proposition 4 is valid if and only if the function h (z, t) has the form (17). If the function h (z, t) cannot be written in this way, then our previous result cannot be applied. Nevertheless, we can state that the result obtained in Proposition 4 is still valid as an approximation of the true result. If we develop in Taylor series the function h (z, t) around a given value of z (let us say z 0 ), then we obtain: h (z, t) = A (t) ln B (t) 0 z 0 + D (t) B (t)0 (z z 0 ) B (t) 0 z 0 + D (t) + (18) + (z z 0) 0 B (t) 0 B (t)(z z 0 ) 2 B (t) 0 z 0 + D (t) 2 + O ³kz 3 z 0 k. Thus, if the function h (z,t) can be expressed in the form (18), that is to say as a polynomial in z, then our result can approximate the exact solution. We underline that Kim and Omberg (1996) find, for an incomplete market, an exact solution where the function h (z, t) is a second degree polynomial in z. 15

16 3.3 The case of a complete market: an exact solution In this subsection we analyse a case in which the result presented in Proposition 4 leads to an exact solution rather than to an approximated solution. In particular, we consider the following hypothesis: Hypothesis 1 The financial market is complete for n +1 risky assets ( Γ 1 ). If Hypothesis 1 holds, then the HJB equation can be written in the following way: ½ ht + a (z, t) 0 h z + b (z, t)+ 1 2 tr (Ω0 Ωh zz )=0, h (z,h) =0, without any approximation, and we can apply the Feynman-Kac theorem obtaining the following result: Proposition 5 Under market structure (6), if Hypothesis 1 holds, then the portfolio composition maximizing the fund s terminal exponential utility function (K (Φ) =αe γ+βφ )isasfollows: w = Γ 1 (Λ ΠvL 0 ) u 1 β (Γ0 Γ) 1 M 1 Z H β Γ 1 Ω t z E t [b (Z s,s)] ds, where: dz s = a (Z s,s) ds + Ω (Z s,s) 0 dw, Z t = z, a (z, t) 0 µ 0 z M 0 Γ 1 Ω, b (z,t) βu 0 (µ L Lv 0 µ B ) βu 0 (Λ 0 Lv 0 Π 0 ) Γ 0 1 M 1 2 M 0 (Γ 0 Γ) 1 M. We can see that, for finding an exact solution, we must consider that state variables following a modified stochastic process with respect to the original one. Nevertheless, since the diffusion term Ω has not changed, then we can use the Girsanov theorem 7 for finding a suitable probability measure in order to switch from one process to the other. This result is important because it shows the possibility to obtain an exact close form solution in the case where all the risks affecting the portfolio composition can be represented by diffusion processes. We underline that Proposition 5 is obtained without specifying any functional form for the coefficients of the stochastic processes implied in the solution. In fact, the exact solution already found in the literature by Kim and Omberg (1996), Wachter (1998), Boulier, Huang, and Taillard (2001), and Deelstra, Grasselli, and Koehl (2001) can be obtained under the hypothesis that there exists only one state variable following a Vasicek (1977) process or a Cox, Ingersoll, and Ross (1985) process. 7 For a complete exposition of the Girsanov theorem the reader is referred to Karatzas and Shreve (1998), and Øksendal (2000). 16

17 3.4 The third component of optimal portfolio As we have already underlined, the third part of optimal portfolio is the only one depending on the fund s horizon. Thus, if we want to consider a fund with an infinite time horizon, then we must verify that the integrals in Propositions 5and4converge. From our previous propositions it can be seen that if all parameters of our problem do not depend on state or withdrawal variables but depend only on time, then the derivative term z E t [b (Z s,s)] vanishes (because Z s b (Z s,s)= 0). Thus, we can state: Proposition 6 If the coefficients of the stochastic processes in the market structure (6) depend only on time, then the third component of the optimal portfolio (w (3) ) vanishes. In the work by Lioui and Poncet (2000) it is shown that, if the market is complete, then the third component of the optimal portfolio is formed only by two parts, even though the number of state variables is arbitrarily large. In particular, the first part is associated with the interest rate risk and the second one with the market price of risk. Even if Lioui and Poncet use the martingale approach, here we underline that we obtain the same qualitative result. Because the authors do not introduce any withdrawal variable, 8 then we have to put in our framework u = 0. Under this hypothesis we can see from Proposition 5 that the function h (z, t) is formed only by one term and, more precisely, we have: b (z, t) 1 2 M 0 (Γ 0 Γ) 1 M, where the interest rate r is already contained into the matrix M (see the first section). Accordingly, in our framework, we are not able to disentangle the two risks linked to the interest rate r and to the market price of risk. Furthermore, if we try to distinguish the terms in r, thenwefind a second degree polynomial in r. This is due to the insertion of the benchmark risk. In fact, here, the riskless asset becomes like the other risky assets, and the risk linked to the interest rate r becomes a component of the market price of risk. Nevertheless, the qualitative result after Lioui and Poncet is preserved. By comparing the approximated and the exact solutions presented in Propositions 4 and 5 respectively, we can see that the approximation h z h 0 z = h zz implies that a positive definite matrix is added to the original term containing 8 We outline that they define an investor who is endowed with a portfolio of discount bonds that he chooses not to trade until his investment horizon (H). This hypothesis allows the authors to have a non-zero first portfolio component w(1). 17

18 h zz. In fact, the HJB equation has been written as follows: h t + a (z, t) 0 h z + b (z,t)+ 1 2 tr [Ω0 Ωh zz ] h0 z ³Ω ³ 0 0 I Γ (Γ 0 Γ) 1 Γ Ω h z =0. ³ 0 Accordingly, after putting h z h 0 z = h zz,thematrixω 0 I Γ (Γ 0 Γ) 1 Γ Ω is added to the matrix Ω 0 Ω. We underline that the coefficient of h zz is the diffusion term of the stochastic process according to which the variables z must behave under the Feynman-Kac theorem. Since the first matrix is positive semi-definite, this means that we are going to add some risks to the original ones and so, in the approximated solution we have presented in Proposition 4, the optimal portfolio third component always overestimates the exact solution. Thus, we can conclude: Corollary 1 The approximated solution for the third optimal portfolio component (w(3) ) presented in Proposition 4 systematically overestimates the exact solution. The future work will concentrate on the measure of this overestimation. We have already seen that the error is negligible if the value function can be written as an exponential of a polynomial in the state variables. Actually, in order to offer a good approximation even in the other cases, we need some more considerations. 4 Conclusion In this paper we have analysed the asset allocation problem for a fund manager maximizing the expected value of fund s terminal exponential utility function. The fund manager faces an economic environment with stochastic investment opportunities and he must cope with a withdrawal risk following a diffusion process. A benchmark is imposed to the fund manager who must maximize a performance measure given by the amount of fund s wealth generated by his investments, diminished by the increase in wealth he could have obtained by investing all the fund s wealth in the benchmark. The optimal portfolio is formed by three components: (i) a preference free part depending only on the diffusion terms of assets and withdrawals, (ii) a part proportional to both the portfolio Sharpe ratio and the inverse of Arrow- Pratt risk aversion index, and (iii) a part depending on all the state variable parameters. We show that the preference-free component minimizes the diffusion variance of fund s wealth. 18

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