Revista Brasileira de Finanças
ISSN: 1679-0731
[email protected]
Sociedade Brasileira de Finanças
Brasil
do Valle Costa, Oswaldo Luiz; de Barros Nabholz, Rodrigo
A Multi-Period Mean-Variance Portfolio Selection Problem
Revista Brasileira de Finanças, vol. 3, núm. 1, 2005, pp. 101-121
Sociedade Brasileira de Finanças
Rio de Janeiro, Brasil
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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 otimização multiperı́odo, 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 incluı́dos em sua formulação. 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 otimização. A grande vantagem desta
técnica é que ela permite o controle do comportamento intermediário da variância e do
retorno da carteira. Um exemplo ilustrando esta situação é 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 ões 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 ões 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 S̄(t) ∈
Rn+1 . We denote by Ft the σ−field generated by the random vectors {S̄(s); s =
0, . . . , t}. It will be convenient to write


S1 (t)
S0 (t)


S̄(t) =
, S(t) =  ... 
S(t)
Sn (t)
A portfolio H̄(t − 1) will be a vector belonging to Rn+1 such that it is Ft−1 measurable, t = 1, 2, . . . , T . We write


H1 (t − 1)
H0 (t − 1)


..
H̄(t − 1) =
, H(t − 1) = 

.
H(t − 1)
Hn (t − 1)
and we have that Hi (t − 1) represents the amount of asset i in the portfolio at time
t. Notice that H̄(t − 1) is chosen at the beginning of time t, and thus depends of
S̄(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
V (t) = H̄(t − 1)′ S̄(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
V (t) = H̄(t)′ S̄(t).
(2)
Let us define


R1 (t)
Si (t + 1)
R0 (t)


, R̄(t) =
, R(t) =  ...  .
Ri (t) =
R(t)
Si (t)
Rn (t)
(3)
Notice that Ri (t) is Ft+1 -measurable. We have that R̄(t) can be written as
R̄(t) = η̄(t) + Z̄(t),
where Z̄(t) are null mean vectors and η̄(t) ∈ Rn+1 represents the mean value of
R̄(t). We write
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Revista Brasileira de Finanças v 3 n 1 2005


Z1 (t)
Z0 (t)


Z̄(t) =
, Z(t) =  ...  ,
Z(t)
Zn (t)


η1 (t)
η0 (t)


, η(t) =  ... 
η̄(t) =
η(t)
ηn (t)
and make the following hypothesis:
Hypothesis 1 {Z̄(t); t = 0, . . . , T − 1} are independent random vectors.
Since S̄(t) is a function of {Z̄(s); s = 0, . . . , t − 1}, we have from Hypothesis
2 that Z̄(t), and thus R̄(t), is independent from the sigma field Ft .
Hypothesis 2 E(R̄(t)R̄(t)′ ) > 0 for each t = 0, . . . , T − 1.
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 R̄(t).
it follows from Hypothesis 2 that
104
(4)
A Multi-Period Mean-Variance Portfolio Selection Problem
E(R0 (t)2 )
E(R0 (t)P(t)′ )
E(R0 (t)P(t)) E(P(t)P(t)′ )
1 0
1 −e′
′
=
E(R̄(t)R̄(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 )
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Revista Brasileira de Finanças v 3 n 1 2005
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)))
A Multi-Period Mean-Variance Portfolio Selection Problem
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:
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Revista Brasileira de Finanças v 3 n 1 2005
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
A Multi-Period Mean-Variance Portfolio Selection Problem
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
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Revista Brasileira de Finanças v 3 n 1 2005
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.
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A Multi-Period Mean-Variance Portfolio Selection Problem
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
<
α(t) E ρ(t)VU ∗ (t)2 − λ(t)VU ∗ (t)
T
X
t=1
α(t) E ρ(t)VU (t)2 − λ(t)VU (t)
and thus
T
X
α(t)(ρ(t)E(VU ∗ (t)2 − VU (t)2 ) − λ(t)E(VU ∗ (t) − VU (t))) < 0.
(9)
t=1
Recall now that for a function of the form
f (x, y) = ax − ay 2 − by
where a ≥ 0, we have for x̄, ȳ fixed, that for all x, y
x − x̄
f (x, y) ≤ f (x̄, ȳ) + ∇f (x̄, ȳ)
y − ȳ
′
(10)
where
∇f (x̄, ȳ) =
a
.
−(2aȳ + b)
Setting a = ρ(t), b = ℓ(t), x̄ = E(VU (t)2 ), ȳ = E(VU (t)) and x =
E(VU ∗ (t)2 ), y = E(VU ∗ (t)), so that
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Revista Brasileira de Finanças v 3 n 1 2005
f (x, y) = ρ(t)(E(VU ∗ (t)2 ) − E(VU ∗ (t))2 ) − ℓ(t)E(VU ∗ (t))
= ρ(t)V ar(VU ∗ (t)) − ℓ(t)E(VU ∗ (t))
and
∇f (x̄, ȳ) =
ρ(t)
−(2ρ(t)E(VU (t)) + ℓ(t))
=
ρ(t)
−λ(t)
we conclude from (10) that
ρ(t)V ar(VU ∗ (t)) − ℓ(t)E(VU ∗ (t)) ≤ ρ(t)V ar(VU (t)) − ℓ(t)E(VU (t)) (11)
E(VU ∗ (t)2 ) − E(VU (t)2 )
+ρ(t) − λ(t)
.
E(VU ∗ (t)) − E(VU (t))
From (9)
T
X
α(t) ρ(t) − λ(t)E(VU ∗ (t)2 ) − E(VU (t)2 )E(VU ∗ (t)) − E(VU (t))
=
α(t)(ρ(t)E(VU ∗ (t)2 − VU (t)2 ) − λ(t)E(VU ∗ (t) − VU (t))) < 0
t=1
T
X
t=1
and from (11)
T
X
α(t) (ρ(t)V ar(VU ∗ (t)) − ℓ(t)E(VU ∗ (t)))
t=1
≤
T
X
α(t) (ρ(t)V ar(VU (t)) − ℓ(t)E(VU (t))) +
t=1
T
X
α(t)
t=1
<
T
X
ρ(t) −λ(t)
E(VU ∗ (t)2 ) − E(VU (t)2 )
E(VU ∗ (t)) − E(VU (t))
α(t) (ρ(t)V ar(VU (t)) − ℓ(t)E(VU (t)))
t=1
in contradiction with the fact that U ∈ Π(P M V (ℓ, ρ)).
Proposition 4 Suppose that ℓ(t) = ℓ̄ > 0, t = 1, . . . , T and ρ(t) = 0 for t ∈
/ Iσ .
If U ∈ Π(P M V (ℓ, ρ)) with
σ(t)2 = V ar(VU (t)), t ∈ Iσ
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A Multi-Period Mean-Variance Portfolio Selection Problem
then U ∈ Π(P E(σ)).
Proof Suppose by contradiction that U ∈
/ Π(P E(σ)), so that for some U ∗ ∈
Π(P E(σ)),
V ar(VU ∗ (t)) ≤ σ(t)2 = V ar(VU (t)), t ∈ Iσ
and
T
X
α(t)E (VU ∗ (t)) >
t=1
T
X
α(t)E (VU (t)) .
t=1
Thus
T
X
t=1
α(t) ρ(t)V ar(VU ∗ (t)) − ℓ̄E(VU ∗ (t))
T
X
<
t=1
α(t) ρ(t)V ar(VU (t)) − ℓ̄E(VU (t))
in contradiction with the fact that U ∈ Π(P M V (ℓ, ρ)).
Poposition 5 Suppose that ρ(t) = ρ̄ > 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)) <
T
X
α(t)V ar (VU (t)) .
t=1
Thus
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Revista Brasileira de Finanças v 3 n 1 2005
T
X
α(t) (ρ̄V ar(VU ∗ (t)) − ℓ(t)E(VU ∗ (t)))
t=1
<
T
X
α(t) (ρ̄V ar(VU (t)) − ℓ(t)E(VU (t)))
t=1
in contradiction with the fact that U ∈ Π(P M V (ℓ, ρ)).
From the optimal strategy (6) we have that
V (t + 1) = R0 (t)V (t) + P(t)′ U(t)
q(t + 1)
φ(t)−1 χ(t)
= R0 (t)V (t) − P(t)′ φ(t)−1 ϕ(t)V (t) +
2p(t + 1)
′
−1
= R0 (t) − P(t) φ(t) ϕ(t) V (t)
−
q(t + 1)
P(t)′ φ(t)−1 χ(t)
2p(t + 1)
and taking the expected value we obtain that
E (V (t + 1)) = η0 (t) − χ(t)′ φ(t)−1 ϕ(t) E(V (t))
q(t + 1)
χ(t)′ φ(t)−1 χ(t)
2p(t + 1)
q(t + 1)
= A1 (t)E(V (t)) −
B(t).
2p(t + 1)
−
where
p(t) = α(t)ρ(t) + A2 (t)p(t + 1)
q(t) = −α(t)λ(t) + A1 (t)q(t + 1)
(12)
for t = T − 1, . . . , 0, with
p(T ) = α(T )ρ(T )
(13)
q(T ) = −α(T )λ(T ).
From equation (8) we have that to obtain U ∈ Π(P M V (ℓ, ρ)) we must have
U ∈ Π(AU X(λ, ρ)) with λ such that
λ(t) = ℓ(t) + 2ρ(t)E(VU (t)), t = 1, . . . , T.
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(14)
A Multi-Period Mean-Variance Portfolio Selection Problem
Notice from (12), (13) that p(t) doesn’t depend on λ(t) and can be obtained
by recursion from equations (12), (13). In order to obtain q(t) such that (14) is
satisfied, define recursively for t = T − 1, . . . , 0
a(t) = −α(t)ℓ(t) +
2p(t + 1)
A1 (t)a(t + 1)
2p(t + 1) + b(t + 1)B(t)
b(t) = −2α(t)ρ(t) +
2p(t + 1)
A1 (t)2 b(t + 1)
2p(t + 1) + b(t + 1)B(t)
c(t) =
2p(t + 1)
A1 (t)
2p(t + 1) + b(t + 1)B(t)
d(t) = −
a(t + 1)
B(t)
2p(t + 1) + b(t + 1)B(t)
with
a(T ) = −α(T )ℓ(T )
b(T ) = −2α(T )ρ(T ).
Define also for t = 0, . . . , T − 1
v(t + 1) = c(t)v(t) + d(t)
(15)
with (recall that v0 is the initial value of the portfolio)
v(0) = v0
(16)
(a(t + 1) + b(t + 1)v(t + 1))2
B(t)
4p(t + 1)2
(17)
z(0) = v02 .
(18)
and
z(t + 1) = A2 (t)z(t) +
with
Theorem 2 An optimal solution U ∈ Π(P M V (ℓ, ρ)) is obtained from
U(t) = −φ(t)−1 ϕ(t)VU (t) −
q(t + 1)
φ(t)−1 χ(t)
2p(t + 1)
(19)
with p(t), t = T, . . . , 0, given by equations (12), (13), and q(t) given by
q(t) = a(t) + b(t)v(t)
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Revista Brasileira de Finanças v 3 n 1 2005
for t = T, . . . , 0. For this strategy the expected value is given by
E(VU (t)) = v(t)
and the variance is given by
V ar(VU (t)) = z(t) − v(t)2 .
Moreover q(t) satisfies
q(t) = −α(t)λ(t) + A1 (t)q(t + 1)
with q(T ) = −α(T )λ(T ) and
λ(t) = ℓ(t) + 2ρ(t)E(VU (t)).
Proof Let us show by induction on t = 0, . . . , T that
v(t) = E(VU (t)).
For t = 0 it is true from (16). Suppose it holds for t. Then
q(t + 1)
B(t)
2p(t + 1)
q(t + 1)
= A1 (t)v(t) −
B(t).
2p(t + 1)
E (VU (t + 1)) = A1 (t)E(VU (t)) −
(20)
From (15) it follows that
2p(t + 1)A1 (t)v(t) − a(t + 1)B(t)
2p(t + 1) + b(t + 1)B(t)
⇒ v(t + 1) (2p(t + 1) + b(t + 1)B(t))
= 2p(t + 1)A1 (t)v(t) − a(t + 1)B(t)
v(t + 1) =
that is,
(a(t + 1) + b(t + 1)v(t + 1))
B(t)
2p(t + 1)
q(t + 1)
= A1 (t)v(t) −
B(t)
2p(t + 1)
v(t + 1) = A1 (t)v(t) −
(21)
and from (20) and (21) it follows that E (VU (t + 1)) = v(t + 1). Let us show now
that
q(t) = −α(t)λ(t) + A1 (t)q(t + 1), t = T − 1, . . . , 0
q(T ) = −α(T )λ(T )
λ(t) = ℓ(t) + 2ρ(t)E(VU (t)).
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A Multi-Period Mean-Variance Portfolio Selection Problem
We have by definition that
q(t)
=
a(t) + b(t)v(t)
=
−α(t)(ℓ(t) + 2ρ(t)v(t))
2p(t + 1)
A1 (t) (a(t + 1) + b(t + 1)A1 (t)v(t))
2p(t + 1) + b(t + 1)B(t)
+
(22)
and
A1 (t)v(t) =
2p(t + 1) + b(t + 1)B(t)
a(t + 1)B(t)
v(t + 1) +
.
2p(t + 1)
2p(t + 1)
(23)
Replacing (23) into (22) leads to
q(t) = −α(t)λ(t) + A1 (t) (a(t + 1) + b(t + 1)v(t + 1))
= −α(t)λ(t) + A1 (t)q(t + 1)
yielding to the desired result. Finally notice that after some manipulations, we
conclude that
q(t + 1)2
E VU (t + 1)2 = A2 (t)E VU (t)2 +
B(t)
4p(t + 1)2
so that z(t) = E VU (t)2 , completing the proof of the Theorem.
To solve P V (ν) we take ρ(t) = 1, t = 1, . . . , T , and ℓ(t) = 0 for t ∈
/ Iν
and therefore from Proposition 5 we have to find ℓ(t) ≥ 0 such that for t ∈ Iν =
{̺1 , . . . , ̺ιν }, ̺ιν ≤ T
ν(t) = E(VU (t)) = v(t).
We have in this case ιν variables ℓ(t) and ιν equations that need to be solved
simultaneously.
In a similar way, to solve problem P E(σ) we take ℓ(t) = 1, t = 1, . . . , T and
ρ(t) = 0 for t ∈
/ Iσ and therefore from Proposition 4 we have to find ρ(t) ≥ 0
such that for t ∈ Iσ = {ζ1 , . . . , ζισ }, ζισ ≤ T ,
σ(t)2 = V ar(VU (t)) = z(t) − v(t)2 .
Again in this case we have ισ variables ρ(t) and ισ equations that need to be
solved simultaneously.
It is important to point out that there is no guarantee that these problems will
have a solution. What we have is that if a solution exists, then an optimal solution
for the problem P E(σ) or P V (ν) is obtained from (19).
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Revista Brasileira de Finanças v 3 n 1 2005
5.
Example
As mentioned in Section 1, the main advantage of the technique presented in
this paper is that it is possible to control the intermediate behavior of the portfolio’s
return or variance for this multi-period problem. In this section we present an
example of this situation, in which it is desirable to have the intermediate values
of the variance bounded by some reference values.
We borrow the date from an example presented in Li and Ng (2000). Let us
consider the case in which there are 3 risky assets and finite horizon time T = 12.
We assume that


0.0146 0.0187 0.0145
E(Z̄(t)Z̄(t)′ ) = 0.0187 0.0854 0.0104 ,
0.0145 0.0104 0.0289


1.162
η̄(t) = 1.246 , t = 0, . . . , 12.
1.228
We consider two portfolios. Portfolio 1 is obtained from the multi-period optimization problem P E(σ) as presented in Li and Ng (2000), that is, it is desired
to maximize the expected value of the portfolio E(V (12)), with a restriction on
the final variance V ar(V (12)) ≤ 2. Portfolio 2 is obtained from the multi-period
optimization problem P E(σ) as presented in Section 2. For the vector α, we
considered


0.001
0.0019


0.0035


0.0066


0.0123


0.0231
.

α=

0.0433
0.0811


 0.152 


0.2848


0.5337
1.0
The idea behind this choice is to gradually increase the weights on the expected value of the portfolio E(V (t)) in the performance criterion, given more
importance to the latest values. For Iσ , we considered Iσ = {4, 6, 8, 10, 12}, with
σ(4) = 0.1382, σ(6) = 0.2965, σ(8) = 0.4343, σ(10) = 0.8517, σ(12) = 2.0.
The idea behind this choice is to reduce the value of the variance of the portfolio
at the intermediate times Iσ = {4, 6, 8, 10, 12}, at a cost of a lower value for the
expected return E(V (t)). Figure 1 presents the trajectory of the variances of Port118
A Multi-Period Mean-Variance Portfolio Selection Problem
folios 1, 2 (star and solid line, respectively), and the variance of reference (plus
signal). We can see from this figure that Portfolio 2 follows the reference as desired. The mean value of the portfolios are presented in figure 2. As expected,
the price paid for having a lower variance is that the mean value of Portfolio 1
overcomes the mean value of Portfolio 2.
2.5
variance
2
1.5
1
0.5
0
0
2
4
6
time
8
10
12
Figure 1
Variance of the portfolios; Portfolio 1 in ∗, Portfolio 2 in −, and Variance of reference in +
9
8
expected value
7
6
5
4
3
2
1
0
2
4
6
8
10
12
time
Figure 2
Expected value of the portfolios; Portfolio 1 in ∗, Portfolio 2 in −
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Revista Brasileira de Finanças v 3 n 1 2005
6.
Conclusions
In this paper we have considered a multi-period mean variance optimization
problem, with investing horizon T . First we consider a problem (Problem
P M V (ℓ, ρ)) in which the performance criterion can be written as a linear combination of the expected values E(V (t)) and variances V ar(V (T )) for t = 1, . . . , T .
A solution for this problem is derived in Theorem 4, based on backward equations.
With this solution, a numerical procedure for a problem with performance criterion
written as a linear combination of the expected values E(V (t)) and restrictions on
the variances V ar(V (t)) for t ∈ Iσ (Problem P E(σ)), and performance criterion
written as a linear combination of the variances V ar(V (T )) and restrictions on
the expected values E(V (t)) for t ∈ Iν (Problem P V (ν)) are presented. To solve
P V (ν) we have to find ℓ(t) ≥ 0 such that for t ∈ Iν , ν(t) = E(VU (t)) = v(t),
which leads to ιν variables ℓ(t) and ιν equations that need to be solved simultaneously. In a similar way, to solve problem P E(σ) we have to find ρ(t) ≥ 0
such that for t ∈ Iσ , σ(t)2 = V ar(VU (t)), yielding to ισ variables ρ(t) and ισ
equations that need to be solved simultaneously. If a solution exists, then (19) is
an optimal solution for the problem P E(σ) or P V (ν).
The main relevance of the technique presented in this paper is that it is possible
to control the intermediate behavior of the portfolio’s return or variance for this
multi-period problem. An example of this situation is presented in Section 5. In
this example we consider two portfolios. The first one is obtained from the multiperiod problem as in Li and Ng (2000) with horizon time T = 12 for the case in
which there is only a final restriction on the value of the variance (V ar(V (12)) ≤
2). The second portfolio is obtained from Problem P E(σ), with intermediate
restrictions on σ(t) at times 4, 6, 8, 10, 12. The trajectories of the variances of
these two portfolios, as well as the reference variance, are presented in figure 1. It
can be seen that, as desired, the variance of the second portfolio tracked the desired
reference. We believe that this simple example illustrates the possible usefulness
of the technique presented in this paper for real problems.
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