A Multi-Period Mean-Variance Portfolio Selection Problem

A Multi-Period Mean-Variance Portfolio Selection Problem Oswaldo Luiz do Valle Costa* Rodrigo de Barros Nabholz** Abstract In a recent paper, Li and Ng (2000) considered the multi-period mean variance optimization problem, with investing horizon T , for the case in which only the final variance V ar(V (T )) or expected value of the portfolio E(V (T )) are considered in the optimization problem. In this paper we extend their results to the case in which the intermediate expected values E(V (t)) and variances V ar(V (t)) for t = 1, . . . , T can also be taken into account in the optimization problem. The main advantage of this technique is that it is possible to control the intermediate behavior of the portfolio’s return or variance. An example illustrating this situation is presented. Resumo Em trabalho recente, Li and Ng (2000) consideraram o problema de otimizaça˜ o multiper´ıodo, com horizonte de investimento T , para o caso onde apenas a variância V ar(V (T )) ou o valor esperado final da carteira E(V (T )) eram inclu´ıdos em sua formulaça˜ o. Neste trabalho estenderemos os resultados apresentados por Li and Ng (2000) para o caso onde valores esperados e variâncias intermediárias, E(V (t)) e V ar(V (t)) para t = 1, . . . , T , também podem ser incorporados no problema de otimizaça˜ o. A grande vantagem desta técnica e´ que ela permite o controle do comportamento intermediário da variância e do retorno da carteira. Um exemplo ilustrando esta situaça˜ o e´ apresentado neste artigo. Keywords: portfolio choice; multi-period optimization; mean variance analysis. JEL codes: C61; C63; G11.

1.

Introduction

Mean-Variance portfolio selection is a classical financial problem introduced by Markowitz (1959) in which it is desired to reduce risks by diversifying assets allocation. The main goal is to maximize the expected return for a given level of risk or minimize the expected risk for a given level of expected return. Optimal portfolio selection is the most used and well known tool for economic allocation of capital (see Campbell et al. (1997), Elton and Gruber (1995), Jorion (1992), Steinbach (2001)). More recently it has been extended to include tracking error optimization (see Roll (1992), Rudolf et al. (1999)) and semivariance models Submited in March 2004. Revised in March 2005. The first author received financial support from CNPq (Brazilian National Research Council), grants 472920/03-0 and 304866/03-2, FAPESP (Research Council of the State of São Paulo), grant 03/06736-7, PRONEX, grant 015/98, and IMAGIMB. The authors would like to thank the reviewers and editors for several very helpful comments and suggestions. *Departamento de Engenharia de Telecomunicaç o˜ es e Controle. Escola Politécnica da Universidade de São Paulo. CEP: 05508-900 - São Paulo, SP, Brazil. E-mail: [email protected] **Departamento de Engenharia de Telecomunicaç o˜ es e Controle. Escola Politécnica da Universidade de São Paulo. CEP: 05508-900 - São Paulo, SP, Brazil. E-mail: [email protected] Revista Brasileira de Finanças 2005 Vol. 3, No. 1, pp. 101–121 c

2004 Sociedade Brasileira de Finanças

Revista Brasileira de Finanças v 3 n 1 2005

(see Hanza and Janssen (1998)). Other objective functions can also be considered (Zenios, 1993), and a unifying approach to these methodologies can be found in Duarte Jr. (1999). All previous papers consider single period optimization problems. But typically portfolio strategies are multi-period, since the investor can re-balance his position from time to time. Recently there have been a continuing effort in extending portfolio selection from the single period to the multi-period case. Using a stochastic linear quadratic theory developed in Chen et al. (1998), the continuoustime version of the Markowitz’s problem was studied in Zhou and Li (2000), with closed-form efficient policies derived, along with an explicit expression of the efficient frontier. In Zhou and Yin (2003) and Yin and Zhou (2004) the authors treated the continuou-time and discrete-time versions of the Markowitz’s mean-variance portfolio selection with regime switching, and derived the efficient portfolio and efficient frontier explicitly. In Li and Ng (2000) the authors extended the mean-variance allocation problem for the discrete-time multi-period case, with final time T . As usual in these problems, the authors only considered the variance and expected value of the portfolio at the final time T . This could lead to too aggressive strategies since the intermediate variances and expected values are not taken into account in the performance criterion or constrains. The main goal of this paper is to generalize the results of Li and Ng (2000) for the case in which the intermediate variances and expected values of the portfolio are also considered in the performance criterion or constrains. First we consider a problem in which the performance criterion can be written as a linear combination of the expected values and variances for t = 1, . . . , T . A solution for this problem is derived in Theorem 2, based on backward equations. From this solution, numerical procedures for a multi-period problem with performance criterion written as a linear combination of the mean values of the portfolio and restrictions on the variances, and performance criterion written as a linear combination of the variances of the portfolio and restrictions on the mean value, are presented. The proposed techniques are based on solving a set of equations so that, if a solution exists, then an optimal solution for the problem is derived. The main relevance of the technique presented in this paper is that it is possible to have a better control of the intermediate values of the variance and expected values of the portfolio, avoiding it to reach undesirable high and low values respectively. We consider only the case in which all the assets are risky. The case with one riskless asset can be easily deduced by considering one of the assets with null variance. This paper follows the same approach and notation as in Li and Ng (2000). It is organized in the following way. The notation, basic results, and problem formulation that will be considered throughout the work are presented in Section 2. Section 3 presents the solution of an auxiliary problem. The main problems are analyzed and solved in Section 4. In Section 5 an example comparing the technique introduced here with the one in Li and Ng (2000) is presented. The 102

A Multi-Period Mean-Variance Portfolio Selection Problem

paper in concluded in Section 6 with some final comments. 2.

Preliminaries

On a probabilistic space (Ω, P, F) we shall consider a financial model in ¯ ∈ which there are n + 1 risky assets represented by the random return vector S(t) ¯ Rn+1 . We denote by Ft the σ−field generated by the random vectors {S(s); s= 0, . . . , t}. It will be convenient to write   S1 (t) .  ¯ = S0 (t) , S(t) =  S(t)  ..  S(t) Sn (t)

¯ − 1) will be a vector belonging to Rn+1 such that it is Ft−1 A portfolio H(t measurable, t = 1, 2, . . . , T . We write   H1 (t − 1)  .. ¯ − 1) = H0 (t − 1) , H(t − 1) =  H(t   . H(t − 1) Hn (t − 1)

and we have that Hi (t − 1) represents the amount of asset i in the portfolio at time ¯ − 1) is chosen at the beginning of time t, and thus depends of t. Notice that H(t ¯ S(s) from s = 0 up to the closing values at time s = t − 1. Let V (t) represent the value of the portfolio at the end of time t. It follows that ¯ − 1)′ S(t). ¯ V (t) = H(t

(1)

To have a self-financing portfolio it must keep the same value at the beginning of period t + 1, when the portfolio is re-balanced, and thus ¯ ′ S(t). ¯ V (t) = H(t)

(2)

Let us define   R1 (t) Si (t + 1) ¯ R0 (t)   , R(t) = , R(t) =  ...  . Ri (t) = R(t) Si (t) Rn (t)

(3)

¯ can be written as Notice that Ri (t) is Ft+1 -measurable. We have that R(t) ¯ ¯ R(t) = η¯(t) + Z(t), ¯ are null mean vectors and η¯(t) ∈ Rn+1 represents the mean value of where Z(t) ¯ R(t). We write 103


  Z1 (t) .  ¯ = Z0 (t) , Z(t) =  Z(t)  ..  , Z(t) Zn (t)   η1 (t) η0 (t)   , η(t) =  ...  η¯(t) = η(t) ηn (t)

and make the following hypothesis:

¯ Hypothesis 1 {Z(t); t = 0, . . . , T − 1} are independent random vectors. ¯ is a function of {Z(s); ¯ Since S(t) s = 0, . . . , t − 1}, we have from Hypothesis ¯ ¯ 2 that Z(t), and thus R(t), is independent from the sigma field Ft . ¯ R(t) ¯ ′ ) > 0 for each t = 0, . . . , T − 1. Hypothesis 2 E(R(t) Define now 

 U1 (t)   Ui (t) = Hi (t)Si (t), U(t) =  ...  . Un (t)

It follows that

V (t) = U0 (t) + e′ U(t) where e is the vector formed by one in all components, and V (t + 1) = H0 (t)S0 (t + 1) + H(t)′ S(t + 1) n X = H0 (t)S0 (t)R0 (t) + Hi (t)Si (t)Ri (t) i=1

= R0 (t)U0 (t) +

n X

Ri (t)Ui (t)

i=1 ′

= R0 (t)(V (t) − e U(t)) + R(t)′ U(t) = R0 (t)V (t) + (R(t) − R0 (t)e)′ U(t). Defining ¯ P(t) = R(t) − R0 (t)e = −e I R(t).

it follows from Hypothesis 2 that 104

(4)


E(R0 (t)2 ) E(R0 (t)P(t)′ ) E(R0 (t)P(t)) E(P(t)P(t)′ ) 1 0 1 −e′ ′ ¯ ¯ = E(R(t)R(t) ) >0 −e I 0 I

and thus E(P(t)P(t)′ ) > 0. Define

φ(t) = E(P(t)P(t)′ ) ϕ20 (t) = E(R0 (t)2 ) ϕ(t) = E(R0 (t)P(t)) χ(t) = E (P(t)) = η(t) − η0 (t)e. From the Schur’s complement, ϕ20 (t) − ϕ(t)′ φ(t)−1 ϕ(t) > 0. Define A2 (t) = ϕ20 (t) − ϕ(t)′ φ(t)−1 ϕ(t) > 0 A1 (t) = η0 (t) − (η(t) − η0 (t)e)′ φ(t)−1 ϕ(t) B(t) = (η(t) − η0 (t)e)′ φ(t)−1 (η(t) − η0 (t)e). Consider a set of positive numbers α(t) for t = 1, . . . , T , ν(t) for t ∈ Iν = {̺1 , . . . , ̺ιν }, ̺ιν ≤ T , and σ 2 (t) for t ∈ Iσ = {ζ1 , . . . , ζισ }, ζισ ≤ T . The investor, with an initial wealth v0 , looks for the best investing strategy (U0 (t), U(t)) t = 0, . . . , T − 1 such that: (1) The sum of the expected value of the portfolio E(V (t)), weighted by the value α(t) at time t, is maximized, subject to the constrains that the variance of the portfolio V ar(V (t)) is less than or equal to a pre-specified upper bound σ 2 (t) at each time t ∈ Iσ . (2) The sum of the variance of the portfolio V ar(V (t)), weighted by the value α(t) ≥ 0 at time t, is minimized, subject to the constrains that the expected value of the portfolio E(V (t)) is greater than or equal to a pre-specified lower bound ν(t) at each time t ∈ Iν . There is no loss of generality in assuming that α(T ) > 0, so that the final period is always considered in the optimization problem (if not so, we could redefine T so that it would coincide with the largest weight α(t) strictly greater than zero). For simplicity we shall denote       σ(ζ1 ) ν(̺1 ) α(1)       σ =  ...  , ν =  ...  , α =  ...  . σ(ζισ )

ν(̺ιν )

α(T )

105


We can mathematically formalize the above problems as follows: Problem P E(σ):

max,

T X

α(t)E(V (t))

t=1

subject to ≤ σ 2 (t), t ∈ Iσ = R0 (t − 1)V (t − 1) + P(t − 1)′ U(t − 1)

V ar(V (t)) V (t)

v0 = U0 (0) + e′ U(0) t = 1, . . . , T. Problem P V (ν): min

T X

α(t)V ar(V (t))

t=1

subject to

E(V (t)) ≥ V (t) = v0 = t =

ν(t), t ∈ Iν R0 (t − 1)V (t − 1) + P(t − 1)′ U(t − 1) U0 (0) + e′ U(0) 1, . . . , T.

An alternative problem that will be considered is as follows. For a sequence of positive numbers ℓ(t) and ρ(t), t = 1, . . . , T , set    ρ(1) ℓ(1)     ℓ =  ...  , ρ =  ...  . 

ℓ(T )

Problem P M V (ℓ, ρ): max

T X t=1

subject to 106

ρ(T )

α(t) (ℓ(t)E(V (t)) − ρ(t)V ar(V (t)))


V (t)

R0 (t − 1)V (t − 1) + P(t − 1)′ U(t − 1)

=

U0 (0) + e′ U(0) 1, . . . , T

v0 = t =

where ρ(t) represents the investor’s risk aversion coefficient, that is, the larger ρ(t) is, the bigger the investor’s risk aversion at time t is. The case in which ρ(t) = 0 means that the investor doesn’t care at all with the risk at time t. Similar comments hold for ℓ. To avoid unfeasible problems to P V (ν) and P E(σ), we make the following assumptions: Hypothesis 3 For t ∈ Iν , t Y

η0 (s) ≥

s=1

Hypothesis 4 For t ∈ Iσ , t−1 Y

ϕ20 (s) −

s=0

t−1 Y s=0

ν(t) . v0

!

η02 (s)

≤

σ 2 (t) . v02

From these hypothesis the following propositions are easily shown. For the proof, just make U(t) = 0 for all t, so that 100% of the resources are applied in the risky asset 0, and verify that from hypothesis 3 and 4 the expected value and variance of the resulting portfolio will satisfy the restrictions of Problems P V (ν) and P E(σ). Proposition 1 Under hypothesis 3, Problem P V (ν) is always feasible. Proposition 2 Under hypothesis 4, Problem P E(σ) is always feasible. 3.

Solution of an Auxiliary Problem Define 

   λ(1) ξ(1)     λ =  ...  , ξ =  ...  . λ(T )

ξ(T )

In order to solve the problems P E(σ), P V (ν) and P M V (ℓ, ρ), we shall consider the following auxiliar problem:

107


Problem AU X(λ, ξ): min

T X t=1

subject to V (t)

=

v0 = t =

α(t) E ξ(t)V (t)2 − λ(t)V (t)

R0 (t − 1)V (t − 1) + P(t − 1)′ U(t − 1) U0 (0) + e′ U(0) 1, . . . , T

where ξ(t) ≥ 0 for each t = 1, . . . , T . It is convenient to set ξ(0) = 0 and α(0) = 0. We make the following backward recursive definitions: p(t) = α(t)ξ(t) + A2 (t)p(t + 1) q(t) = −α(t)λ(t) + A1 (t)q(t + 1) w(t) = −

q(t + 1)2 B(t) + w(t + 1) 4p(t + 1)

for t = T − 1, . . . , 0, with p(T ) = α(T )ξ(T ) q(T ) = −α(T )λ(T ) w(T ) = 0. Notice that A2 (t) > 0 and α(t)ξ(t) ≥ 0 for all t = 0, . . . , T . Since by hypothesis α(T )ξ(T ) > 0 we have that p(t) > 0 for all t = 0, . . . , T . Let us apply dynamic programming to solve problem AU X(λ, ξ). Let us define the intermediate problems. Problem AU X(λ, ξ,τ, v τ ): min

T X t=τ

subject to V (t) vτ

= =

t =

α(t) E ξ(t)V (t)2 − λ(t)V (t) |Fτ

R0 (t − 1)V (t − 1) + P(t − 1)′ U(t − 1) U0 (τ ) + e′ U(τ ) τ,...,T

and let J (τ, v τ ) be the value function of this problem.

108


Theorem 1 The value function is given by J (τ, vτ ) = p(τ )vτ2 + q(τ )vτ + w(τ )

(5)

and the optimal strategy for t = τ, . . . , T − 1 is given by U(t) = −φ(t)−1 ϕ(t)V (t) −

q(t + 1) φ(t)−1 χ(t). 2p(t + 1)

(6)

Proof Let us show the result by induction on t. For τ = T we have that J (T, vT ) = α(T )ξ(T )vT2 − α(T )λ(T )vT + 0 = p(T )vT + q(T )vT + w(T ) proving (7) for τ = T . Suppose (7) holds for τ + 1. Then J (τ, vτ ) = α(τ )ξ(τ )vτ2 − α(τ )λ(τ )vτ + min{E (J (τ + 1, V (τ + 1))|Fτ )} uτ

where V (τ + 1) = R0 (τ )vτ + P(τ )′ uτ . We have that E (J (τ + 1, V (τ + 1))|Fτ ) = p(τ + 1)E((R0 (τ )vτ + P(τ )′ uτ )2 |Fτ ) + q(τ + 1)E ((R0 (τ )vτ + P(τ )′ uτ ) |Fτ ) + w(τ + 1). Notice now that

and

2 E (R0 (τ )vτ + P(τ )′ uτ ) |Fτ = ϕ20 (τ )vτ2 + 2ϕ(τ )uτ vτ + u′τ φ(τ )uτ E (R0 (τ )vτ + P(τ )′ uτ |Fτ ) = η0 (τ )vτ + χ(τ )uτ . Thus J (τ, vτ ) = α(τ )ξ(τ ) + p(τ + 1)ϕ20 (τ ) vτ2

+ (q(τ + 1)η0 (τ ) − α(τ )λ(τ )) vτ + w(τ + 1) + min f (uτ ) uτ

109


where the function f (uτ ) is given by f (uτ ) = 2p(τ + 1)ϕ(τ )′ uτ vτ + p(τ + 1)u′τ φ(τ )uτ + q(τ + 1)χ(τ )′ uτ .

(7)

Taking the derivative and making equal to zero we obtain that the optimal strategy u∗τ is given by u∗τ = −φ(τ )−1 ϕ(τ )vτ −

q(τ + 1) φ(τ )−1 χ(τ ) 2p(τ + 1)

which coincides with the optimal strategy given in (6). Replacing this in (7) leads to f (u∗τ ) = (2p(τ + 1)ϕ(τ )′ vτ + q(τ + 1)χ(τ )′ )uτ + p(τ + 1)u′τ φ(τ )uτ q(τ + 1) ′ ′ −1 χ(τ ) = −(2p(τ + 1)ϕ(τ ) vτ + q(τ + 1)χ(τ ) )φ(τ ) ϕ(τ )vτ + 2p(τ + 1) ′ q(τ + 1) q(τ + 1) + p(τ + 1) ϕ(τ )vτ + χ(τ ) φ(τ )−1 ϕ(τ )vτ + χ(τ ) 2p(τ + 1) 2p(τ + 1) = −2p(τ + 1)ϕ(τ )′ φ(τ )−1 ϕ(τ )vτ2 − 2q(τ + 1)ϕ(τ )′ φ(τ )−1 χ(τ )vτ

q(τ + 1)2 χ(τ )′ φ(τ )−1 χ(τ ) + p(τ + 1)ϕ(τ )′ φ(τ )−1 ϕ(τ )vτ2 2p(τ + 1) q(τ + 1)2 χ(τ )′ φ(τ )−1 χ(τ ) + q(τ + 1)ϕ(τ )′ φ(τ )−1 χ(τ )vτ + 4p(τ + 1) −

= −p(τ + 1)ϕ(τ )′ φ(τ )−1 ϕ(τ )vτ2 − q(τ + 1)ϕ(τ )′ φ(τ )−1 χ(τ )vτ −

q(τ + 1)2 χ(τ )′ φ(τ )−1 χ(τ ). 4p(τ + 1)

Thus,

J (τ, vτ )

= (α(τ )ξ(τ ) + p(τ + 1) ϕ20 (τ ) − ϕ(τ )′ φ(τ )−1 ϕ(τ ) vτ2 + q(τ + 1) η0 (τ ) − ϕ(τ )′ φ(τ )−1 χ(τ ) − α(τ )λ(τ ) vτ + w(τ + 1) −

q(τ + 1)2 χ(τ )′ φ(τ )−1 χ(τ ) 4p(τ + 1)

= (α(τ )ξ(τ ) + A2 (τ )p(τ + 1)) vτ2 + (A1 (τ )q(τ + 1) − α(τ )λ(τ )) q(τ + 1)2 vτ + w(τ + 1) − B(τ ) = p(τ )vτ2 + q(τ )vτ + w(τ ) 4p(τ + 1) proving the desired result. 110


4.

Solution to the Problems

We represent the set of optimal solutions for the problems AU X(λ, ξ), P M V (ℓ, ρ), P E(σ) and P V (ν) by Π(AU X(λ, ξ)), Π(P M V (ℓ, ρ)), Π(P E(σ)) and Π(P V (ν)) respectively. We denote by {VU (t)}Tt=0 the value of the portfolio when we use an investing strategy U = {U(0), . . . , U(T − 1)}. Proposition 3 We have that Π(P M V (ℓ, ρ)) ⊂ ∪ Π(AU X(λ, ρ)). λ∈RT

Proof We shall show that if U ∈ Π(P M V (ℓ, ρ)) then U ∈ Π(AU X(λ, ρ)) with λ(t) = ℓ(t) + 2ρ(t)E(VU (t))

(8)

for t = 1, . . . , T . Suppose by contradiction that U ∈ / Π(AU X(λ, ρ)), so that for some U ∗ ∈ Π(AU X(λ, ρ)), T X t=1

t=1

T X

α(t)E (VU (t)) .

t=1

Thus

T X t=1

¯ α(t) ρ(t)V ar(VU ∗ (t)) − ℓE(V U ∗ (t))

T X

0 for t = 1, . . . , T and ℓ(t) = 0 for t ∈ / Iν . If U ∈ Π(P M V (ℓ, ρ)) with ν(t) = E(VU (t)), t ∈ Iν then U ∈ Π(P V (ν)). Proof Suppose by contradiction that U ∈ / Π(P V (ν)), so that for some U ∗ ∈ Π(P V (ν)), E(VU ∗ (t)) ≥ ν(t) = E(VU (t)), t ∈ Iν and T X t=1

α(t)V ar (VU ∗ (t))