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Multi-period portfolio optimization under probabilistic risk measure
Accepted Manuscript
Multi-period Portfolio Optimization Under Probabilistic Risk Measure Yufei Sun, Grace Aw, Kok Lay Teo, Yanjian Zhu, Xiangyu Wang PII: DOI: Reference:
S1544-6123(16)30042-3 10.1016/j.frl.2016.04.001 FRL 500
To appear in:
Finance Research Letters
Received date: Accepted date:
23 February 2016 3 April 2016
Please cite this article as: Yufei Sun, Grace Aw, Kok Lay Teo, Yanjian Zhu, Xiangyu Wang, Multiperiod Portfolio Optimization Under Probabilistic Risk Measure, Finance Research Letters (2016), doi: 10.1016/j.frl.2016.04.001
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Highlights • Our paper provides a computationally simple analytical solution to the complex portfolio selection problem in a multi-period setting. • Probabilistic risk measure can cater for investors with different degree of risk aversion. • In our settings, the investors do not have to purchase a huge number of stocks to form an optimal portfolio.
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• Our model is superior over its corresponding single period one, as well as over the market index.
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Multi-period Portfolio Optimization Under Probabilistic Risk Measure Yufei Suna , Grace Awb , Kok Lay Teob , Yanjian Zhu∗c , and Xiangyu Wanga,d Australasian Joint Research Centre for BIM, Curtin University, Australia
b c
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a
Department of Mathematics and Statistics, Curtin University, Australia
Academy of Financial Research, College of Economics, Zhejiang University, China Department of Housing and Interior Design, Kyung Hee University, Korea
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d
Abstract
This paper develops a minimax model for a multi-period portfolio selection problem. An analytical solution is obtained and numerical simulations demonstrate the superiority of the multi-
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period model over its corresponding single period one, as well as over the market index.
namic programming
Introduction
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Keywords: Portfolio optimization, probability risk measure, discrete-time optimal control, dy-
The measure of risk is of great importance in portfolio management, especially when the distribution
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of the portfolio returns are nonsymmetric (Leland , 1999; Pedersen , 2001) and investors are averse to downside loss (Ang et al., 2006; Bali et al., 2009; Kahneman and Tversky , 1979). Researchers
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proposed some quantile-based risk measures in the past decades. Among these measures, Value-atRisk (VaR) and Conditional VaR (CVaR) attract much attention in both academy and practice (Duffie and Pan , 1997; Jorion , 2007; Linsmeier and Pearson , 2000; Rockafellar and Uryasev , 2000, 2002). More recently, Sun et al. introduced a probabilistic risk measure, with allowance to cater for investors with different degree of risk aversion (Sun et al., 2015). It is at least equally important to embed these risk measures into portfolio management techniques although it is not an easy task to do so numerically. Basak and Shapiro theoretically compared the ∗ Corresponding
author: Yanjian Zhu, Phone: +86-571-87952202, Email address: [email protected]
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ACCEPTED MANUSCRIPT portfolio optimization strategies of the LEL (Limited-Expected-Losses) and VaR risk managers in a general equilibrium framework (Basak and Shapiro , 2001). Alexander and Baptista compared the VaR and CVaR constraints on portfolio selection (Alexander and Baptista , 2004). Brandtner modeled the mean-spectral risk preferences in a form of spectral utility function (Brandtner , 2013). Some other researchers proposed some novel methods to search for optimal portfolios with risk measures for downside loss-averse preferences (Cui et al., 2013; Jarrow and Zhao , 2006; Roman et al., 2007; Sengupta and Sahoo , 2013; Yao et al., 2013). However, most of these numerical studies implemented Monte Carlo simulation to search for optimal portfolios, and could not find an analytical solution for
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the optimization problem. A computationally simple analytical solution is highly useful, especially for unsophisticated investors. Sun et al. stood out in the literature by providing a model with an analytical solution (Sun et al., 2015).
This paper extends the work of Sun et al. (Sun et al., 2015) to the multi-period setting. In their paper, the authors construct a minimax portfolio selection model by introducing a probabilistic risk
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measure, and a analytical solution is computationally available. Their paper aimed to maximize the expected portfolio return and minimize the maximum individual risk of the assets in the portfolio for a single period. However, portfolios are dynamically managed over multiple periods in practice. Some other papers also argued for the importance of the dynamic relationship between risk and return in a long horizon (Bali et al., 2009; Bickel , 1969; Harrison and Zhang , 1999; Merton , 1973).
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Our paper contributes to the literature in the following way. Firstly, we provide a computationally simple analytical solution to the complex portfolio selection problem in a multi-period setting.
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Secondly, our model is superior to the Sun et al. model which is superior to others in the meanvariance space (Sun et al., 2015). Thirdly, out model inherits some good features of the single-period
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model. For example, we are still able to derive an analytical solution without computation of the covariances. In addition, the investors do not have to purchase a huge number of stocks to form an optimal portfolio.
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The remainder of this paper is organized as follows. Section 2 describes the problem and formulates it as a bi-criteria optimization problem. Section 3 develops the analytical solution to the problem.
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In Section 4, ASX100 data is applied to our model and to the single period model, and the results compared. Section 5 concludes the paper.
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Problem formulation
We consider a multi-period portfolio optimization problem, where an investor is going to invest in N possible risky assets Sj , j = 1, . . . , N with a positive initial wealth of M0 . The investment will be made at the beginning of the first period of a T -period portfolio planning horizon. Then, the wealth 3
ACCEPTED MANUSCRIPT will be reallocated to these N risky assets at the beginning of the following T − 1 consecutive time periods. The investor will claim the final wealth at the end of the T th period. Let xtj be the percentage of wealth at the end of period t − 1 invested in asset Sj at the beginning of period t. Denote xt = [xt1 , . . . , xtN ]> . Here we assume that the whole investment process is a self-financing process. Thus, the investor will not increase the investment nor put aside fund in any period in the portfolio planning horizon. In other words, the total fund in the portfolio at the end of period t − 1 will be allocated to those risky assets at the beginning of period t. Thus, xtj = 1,
t = 1, . . . , T.
j=1
(2.1)
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Moreover, it is assumed that short selling of the risky assets is not allowed at any time. Hence, we have xtj ≥ 0,
t = 1, . . . , T, j = 1, . . . , N.
(2.2)
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Let Rtj denote the rate of return of asset Sj for period t. Define Rt = [Rt1 , . . . , RtN ]> . Here, Rtj is assumed to follow normal distribution with mean rtj and standard deviation σtj . We further assume that vectors Rt , t = 1, . . . , T , are statistically independent, and the mean E(Rt ) = r t = [rt1 , . . . , rtN ]> is calculated by averaging the returns over a fixed window of time τ .
rtj =
t−1 1 X Rji , τ i=t−τ
t = 1, . . . , T, j = 1, . . . , N.
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Let
(2.3)
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We assume that in any time period, there are no two distinct assets in the portfolio that have the same level of expected return as well as standard deviation, i.e., for any 1 ≤ t ≤ T , there exist no i and j such that i 6= j, but rti = rtj , and σti = σtj .
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Let Vt denote the total wealth of the investor at the end of period t. Clearly, we have Vt = Vt−1 ( 1 + R> t xt ),
t = 1, . . . , T,
(2.4)
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with V0 = M0 .
First recall the definition of probabilistic risk measure, which was introduced in (Sun et al., 2015)
for the single period probabilistic risk measure.
wp (x) = min Pr{ |Rj xj − rj xj | ≤ θε }, 1≤j≤N
(2.5)
where θ is a constant to adjust the risk level, and ε denotes the average risk of the entire portfolio,
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ε=
N 1 X σj . N j=1
(2.6)
The whole idea of this risk measure in (2.5) is to locate the single asset with greatest deviation in the portfolio. With this ‘biggest risk’ mitigated, the risk of the whole portfolio can be substantially reduced as well. For multi-period portfolio optimization, the single ‘biggest risk’ should be selected
wp (x) = min
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over all risky assets and over the entire planning horizon. Thus, we define
min Pr{ |Rtj xtj − rtj xtj | ≤ θε },
1≤t≤T 1≤j≤N
(2.7)
> > where x = [x> 1 , . . . , xT ] θ, is the same as defined in (2.5) to be a constant adjusting the risk level,
and ε denotes the average risk of the entire portfolio over T periods, which is calibrated by
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T N 1 XX ε= σtj . T × N t=1 j=1
(2.8)
Assume that the investor is rational and risk-averse, who wants to maximize the terminal wealth as well as minimize the risk in the investment. Thus, the portfolio selection problem can be formulated
max
min
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as a bi-criteria optimization problem stated as follows.
min f (xtj ), E(VT ) ,
1≤t≤T 1≤j≤N
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s.t. Vt = Vt−1 ( 1 + R> t xt ), N X
xtj = 1,
t = 1, . . . , T,
t = 1, . . . , T,
(2.9a) (2.9b) (2.9c)
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j=1
t = 1, . . . , T, j = 1, . . . , N.
(2.9d)
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xtj ≥ 0,
where f (xtj ) = Pr{ |Rtj xtj − rtj xtj | ≤ θε }. To reach the optimality for the above bi-criteria optimization problem, we recall the following
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definition which is based on Theorem 3.1 in (Vincent and Grantham , 1981).
Definition 2.1. A solution x∗ = {x∗1 , . . . , x∗T } satisfying (2.9c) and (2.9d) is said to be a Pareto˜= minimal solution for the bi-criteria optimization problem (2.9) if there does not exist a solution x ˜ N } satisfying (2.9c) and (2.9d) such that {˜ x1 , . . . , x min
min f (x∗tj ) ≤ min
1≤t≤T 1≤j≤N
min f (˜ xtj ), and E(VT∗ |x∗ ) ≤ E(VT∗ |˜ x),
1≤t≤T 1≤j≤N
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ACCEPTED MANUSCRIPT for which at least one of the inequalities holds strictly. We can transform problem (2.9) into an equivalent bi-criteria optimization problem by adding another decision variable y, and T × N constraints.
y, E(VT ) ,
(2.10a)
s.t. Vt = Vt−1 ( 1 + R> t xt ), y ≤ f (xtj ), N X
xtj = 1,
t = 1, . . . , T,
t = 1, . . . , T, j = 1, . . . , N, t = 1, . . . , T,
j=1
xtj ≥ 0,
(2.10b)
t = 1, . . . , T, j = 1, . . . , N.
(2.10c) (2.10d)
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max
(2.10e)
where y ≤ f (xtj ) is the (N × (t − 1) + j)th probabilistic constraint. The optimization process trying to maximize y will eventually push the value of y to be equal to min min f (xtj ). Thus, the
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1≤t≤T 1≤j≤N
optimization problem (2.10) is equivalent to the optimization problem (2.9).
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Analytical solution to the problem
Recall that in Section 2, the return, Rtj , of asset Sj in period t, follows the normal distribution with
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mean rtj and standard deviation σtj . It is clear that Rtj − rtj is also normally distributed with mean 0 and standard deviation σtj .
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Let qtj = Rtj − rtj . Then it follows that
(3.1)
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n θε o f (xtj ) = Pr{ |Rtj xtj − rtj xtj | ≤ θε } = Pr |Rtj − rtj | ≤ xtj Z xθε 2 o n qtj tj 1 √ =2 exp − 2 dqtj . 2σtj 2πσtj 0
By the property of cumulative distribution function (3.1), f (xtj ) is clearly a monotonically decreasing
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function with respect to xtj . As mentioned in the optimization problem (2.10), in the process of maximizing the objective
function, the value of y is to be pushed to reach min min f (xtj ). As a result, if we choose y to be 1≤t≤T 1≤j≤N
an arbitrary but fixed real number between 0 and 1 (because y is a value of probability), using the monotonic property of f (xtj ), we can find an upper bound for xtj . Let this upper bound be denoted as Utj , which is given by Utj = min{ 1, x ˆtj }, where x ˆtj = f −1 (y). 6
(3.2)
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max E(VT ),
(3.3a)
s.t. Vt = Vt−1 ( 1 + R> t xt ), N X
xtj = 1,
t = 1, . . . , T,
(3.3b)
t = 1, . . . , T,
(3.3c)
j=1
0 ≤ xtj ≤ Utj ,
(3.3d)
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t = 1, . . . , T, j = 1, . . . , N.
We use dynamic programming to solve problem (3.3).
For time periods k = 1, . . . , T , we define a series of optimal control problems with the same dynamics, cost function, and constraints but different initial states and initial times. These problems
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are stated as follows.
max E(VT ),
s.t. Vt = Vt−1 ( 1 + R> t xt ), Vk−1 = ξ, xtj = 1,
j=1
t = k, . . . , T,
(3.4b) (3.4c)
t = k, . . . , T,
(3.4d)
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N X
(3.4a)
0 ≤ xtj ≤ Utj ,
(3.4e)
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t = k, . . . , T, j = 1, . . . , N,
where ξ is a variable. At the time k − 1, the number of steps to go is T − k + 1. This is a family of optimal control problems determined by the initial time k − 1 and initial state ξ. We use Problem
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(Pk−1,ξ ) to denote each different problem. The following theorem from Chapter 9 in (Varaiya , 1972)
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is known as the Principle of Optimality. Theorem 3.1. Suppose that {x∗k , . . . , x∗T } is an optimal control for Problem (Pk−1,ξ ), and that 0
∗ {Vk−1 = ξ, Vk∗ , . . . , VT∗ } is the corresponding optimal trajectory. Then, for any k ≤ k ≤ T , {x∗k0 , . . . , x∗T }
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is an optimal control for Problem (Pk0 −1,V ∗0
).
k −1
To continue, we utilize an auxiliary value function denoted by F (k −1, ξ) to be the maximum value
of the objective function for Problem (Pk−1,ξ ). The following value function F (k − 1, ξ) is defined similarly as in Chapter 9 in (Varaiya , 1972), i.e., F (k − 1, ξ) = max{ F (k, ξ( 1 + R> k xk )) : xk ∈ Xk }, 1 ≤ k ≤ T,
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(3.5)
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N X j=1
(3.6)
xkj = 1 and 0 ≤ xkj ≤ Ukj , j = 1, . . . , N }.
Now, we shall solve the discrete-time optimal control problem (3.3) backwards using dynamic programming method. For k = T , we start from the state VT −1 = ξ at time T − 1. Then, from (3.5) and (3.6), the value
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function becomes: F (T − 1, ξ) = max{ F (T, ξ( 1 + R> T xT )) : xT ∈ XT } = max{ E(ξ( 1 + R> T xT )) : xT ∈ XT } = max{ ξ( 1 + r > T xT ) : xT ∈ XT }.
(3.7)
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Consequently, Problem (PT −1,ξ ) becomes a linear programming problem stated as below. max ξ( 1 + r > T xT ), N X
xT j = 1,
j=1
j = 1, . . . , N.
(3.8b) (3.8c)
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0 ≤ xT j ≤ U T j ,
(3.8a)
The optimal solution to this linear programming problem has been obtained in (Sun et al., 2015).
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The result is quoted in the following as a lemma. Lemma 3.1. Let the assets be sort in such an order that rT 1 ≥ rT 2 ≥ . . . ≥ rT N . Then, there exists
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an integer n ≤ N such that
n−1 X
UT j < 1, and
j=1
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j=1
and
x∗T j
n X
UT j ≥ 1,
UT j , j=1,. . . ,n-1, n−1 X 1− UT j , j=n, = j=1 0, j >n,
is an optimal solution to problem (3.8).
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(3.9)
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Note that x∗T in (3.9) is independent from ξ as the solution is a function of UT j , which, from (3.1) and (3.2), depends only on the mean and standard deviation of RT j . Continuing with this process, we obtain the value function at k = T − 1.
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F (T − 2, ξ) = max{ F (T − 1, ξ( 1 + R> T −1 xT −1 )) : xT −1 ∈ XT −1 }
> ∗ = max{ E(ξ( 1 + R> T −1 xT −1 )(1 + r T xT )) : xT −1 ∈ XT −1 } > ∗ = max{ ξ( 1 + r > T −1 xT −1 )(1 + r T xT ) : xT −1 ∈ XT −1 }.
(3.10)
Again, it is easy to show that this maximizing problem is equivalent to the following linear program-
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ming problem. > ∗ max ξ( 1 + r > T −1 xT −1 )(1 + r T xT ), N X j=1
x(T −1)j = 1,
j = 1, . . . , N.
(3.11b) (3.11c)
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0 ≤ x(T −1)j ≤ U(T −1)j ,
(3.11a)
Since x∗T is solved in (3.8) and known, the above problem is in the same form as (3.8). Thus, it can
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be solved in a similar manner as that for (3.9). Details are given below as a lemma. Lemma 3.2. Let the assets be sort in such an order that r(T −1)1 ≥ r(T −1)2 ≥ . . . ≥ r(T −1)N . Then,
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and
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there exists an integer n ≤ N such that n−1 X j=1
x∗(T −1)j
U(T −1)j < 1, and
n X j=1
U(T −1)j ≥ 1,
U(T −1)j , j=1,. . . ,n-1, n−1 X 1− U(T −1)j , j=n, = j=1 0, j >n,
is an optimal solution to problem (3.11).
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(3.12)
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Similarly, the optimal solution x∗T −1 has no dependency on the state ξ. It depends only on fixed values of U(T −1)j , j = 1, . . . , N . From Lemma 3.1 and Lemma 3.2, it is reasonable to postulate that
k = T, . . . , 1.
(3.13)
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∗ > ∗ > ∗ F (k − 1, ξ) = ξ( 1 + r > k−1 xk−1 )( 1 + r k xk ) × · · · × ( 1 + r T xT ),
Moreover, the optimal solution for every period k, k = 1, . . . , T , can be written in a unified form as given below.
Then, there exists an integer n ≤ N such that n−1 X
Ukj < 1, and
j=1
n X j=1
and
Ukj ≥ 1,
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Ukj , j=1,. . . ,n-1, n−1 X Ukj , j=n, 1− = j=1 0, j >n,
(3.14)
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x∗kj
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Theorem 3.2. For any k = 1, . . . , T , let the assets be sort in such an order that rk1 ≥ rk2 ≥ . . . ≥ rkN .
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where Ukj is defined as in (3.1) and (3.2). Thus, with x∗k = [x∗k1 , . . . , x∗kN ]> , {x∗1 , . . . , x∗T } is an
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optimal solution to problem (3.3).
Numerical simulations
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We use daily return data of stocks on ASX100 dated from 01/01/2007 to 30/11/2011. The portfolio takes effect from 01/01/2009 and is open for trading for 3 years. 3 years is chosen since professionally managed portfolios (e.g., Aberdeen Asset Management, JBWere) usually list the average holding period as 3 − 5 years. At the beginning of each month, the funds in the portfolio are allocated based on the updated asset return data. We use 2 years historical data to decide the portfolio allocation each month. For the first portfolio allocation on 01/01/2009, the return data from 01/01/2007 to 31/12/2008 is used to evaluate the expected return and standard deviation of each stock. In the month following, the return data from 01/02/2007 to 31/01/2009 is used to evaluate the updated 10
ACCEPTED MANUSCRIPT Table 1: Multi-period - Expected Portfolio Returns for Selected θ & y HH θ 0.01 0.1 0.2 0.5 1 H HH y 0.95 0.59% 4.32% 6.35% 10.78% 15.38% 0.90 0.71% 4.77% 7.01% 11.80% 16.85% 0.80 0.96% 5.48% 8.09% 13.40% 19.25%
expected return and standard deviation, and this goes on.
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Table 2: Single Period - Expected Portfolio Returns for Selected θ & y HH θ 0.01 0.1 0.2 0.5 1 H HH y 0.95 -2.01% 2.00% 3.31% 3.30% 4.72% 0.90 -1.89% 2.23% 3.66% 3.83% 5.14% 0.80 -1.69% 2.65% 3.53% 4.38% 5.89%
The formulation of the corresponding portfolio optimization problem is as defined in Section 2. Assume the investor has an initial wealth of M0 = 1,000,000 dollars. There are N = 100 stocks to
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choose from for a portfolio of investment holding period of T = 36 months. The average risk of the portfolio, ε, over T periods is calculated as in (2.8). Table 1 shows the portfolio returns for various combination of θ (risk adjusting parameter), and y (lower bound of the probabilistic constraint). By changing the value of θ and/or y, the investor is able to alter the portfolio composition to cater for different risk tolerance levels. The lower the value of θ, the more diversified the portfolio can be, while
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the lower the value of y, the less stringent the probabilistic constraint.
From Table 1, it can be seen that the expected portfolio return increases when θ increases. Sim-
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ilarly, the expected portfolio return increases when y decreases. This makes sense since when θ increases, the portfolio selection consists of a much smaller number of selected ‘better-performing’ stocks. When y is lower, the risk is higher and hence the return is generally expected to be higher.
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The historical price index value of ASX100, composed of 100 large-cap and mid-cap stocks, was 3067.90 and 3329.40 at the end of December 2008 and at the end of December 2011, respec-
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tively. This translates to a return of 8.52% for a portfolio which comprises the entire stock selection of ASX100.
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From Table 1, it can be seen that when θ > 0.5, the expected returns following our portfolio election criteria outperforms the market index return. The multi-period model outperforms the passive single period model with a period of 3 years. Table 2 shows the expected returns using the single period model with a period of 3 years. When θ = 0.1 and y = 0.95, solving the problem with Theorem 3.2 suggests a total wealth in portfolio of 1,043,223.89 dollars at the end of the 3 year investment, which is a return of 4.32%. Comparing this with the result of the single period investment strategy, the multi-period solution outperforms it by more than one fold (the single period portfolio has a total return of about 2%). 11
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Conclusion
Our multi-period portfolio selection method offers various advantages over the single period model while retaining the latter’s advantages over other existing portfolio selection models. Advantages of our model include simplicity in computation, having a small number of ‘optimal’ stocks in the portfolio, allowance to cater for investors with different degrees of risk aversion and the ability to arrive at an elegant analytical solution without requiring the computation of a cumbersome covariance matrix. Moreover, dynamic management of the portfolio results in higher expected returns for the investor
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while retaining the simplicity of portfolio allocation via its explicit solution. Future research can consider the transaction costs involved in the dynamic management of the portfolio.
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Value at Risk (CVaR). Quantitative Finance, vol. 13, pp. 1637-1651. Sun, Y., Aw, G., Teo, K. L., and Zhou, G. (2015). Portfolio optimization using a new probabilistic
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risk measure. Journal of Industrial and Management Optimization, vol. 11, pp. 1275-1283. Varaiya, P. P. (1972). Lecture Notes on Optimization. http://robotics.eecs.berkeley.edu/~varaiya/papers_ps.dir/NOO.pdf.
Vincent, T. L., and Grantham, W. J. (1981). Optimality in Parametric Systems. John Wiley & Sons, New York. Yao, H., Li, Z., and Lai, Y. (2013). Mean-CVaR Portfolio Selection: A Nonparametric Estimation Framework. Computers & Operations Research, vol. 40, pp. 1014-1022. 13
Multi-period Portfolio Optimization Under Probabilistic Risk Measure Yufei Sun, Grace Aw, Kok Lay Teo, Yanjian Zhu, Xiangyu Wang PII: DOI: Reference:
S1544-6123(16)30042-3 10.1016/j.frl.2016.04.001 FRL 500
To appear in:
Finance Research Letters
Received date: Accepted date:
23 February 2016 3 April 2016
Please cite this article as: Yufei Sun, Grace Aw, Kok Lay Teo, Yanjian Zhu, Xiangyu Wang, Multiperiod Portfolio Optimization Under Probabilistic Risk Measure, Finance Research Letters (2016), doi: 10.1016/j.frl.2016.04.001
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Highlights • Our paper provides a computationally simple analytical solution to the complex portfolio selection problem in a multi-period setting. • Probabilistic risk measure can cater for investors with different degree of risk aversion. • In our settings, the investors do not have to purchase a huge number of stocks to form an optimal portfolio.
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• Our model is superior over its corresponding single period one, as well as over the market index.
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Multi-period Portfolio Optimization Under Probabilistic Risk Measure Yufei Suna , Grace Awb , Kok Lay Teob , Yanjian Zhu∗c , and Xiangyu Wanga,d Australasian Joint Research Centre for BIM, Curtin University, Australia
b c
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a
Department of Mathematics and Statistics, Curtin University, Australia
Academy of Financial Research, College of Economics, Zhejiang University, China Department of Housing and Interior Design, Kyung Hee University, Korea
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d
Abstract
This paper develops a minimax model for a multi-period portfolio selection problem. An analytical solution is obtained and numerical simulations demonstrate the superiority of the multi-
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period model over its corresponding single period one, as well as over the market index.
namic programming
Introduction
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1
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Keywords: Portfolio optimization, probability risk measure, discrete-time optimal control, dy-
The measure of risk is of great importance in portfolio management, especially when the distribution
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of the portfolio returns are nonsymmetric (Leland , 1999; Pedersen , 2001) and investors are averse to downside loss (Ang et al., 2006; Bali et al., 2009; Kahneman and Tversky , 1979). Researchers
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proposed some quantile-based risk measures in the past decades. Among these measures, Value-atRisk (VaR) and Conditional VaR (CVaR) attract much attention in both academy and practice (Duffie and Pan , 1997; Jorion , 2007; Linsmeier and Pearson , 2000; Rockafellar and Uryasev , 2000, 2002). More recently, Sun et al. introduced a probabilistic risk measure, with allowance to cater for investors with different degree of risk aversion (Sun et al., 2015). It is at least equally important to embed these risk measures into portfolio management techniques although it is not an easy task to do so numerically. Basak and Shapiro theoretically compared the ∗ Corresponding
author: Yanjian Zhu, Phone: +86-571-87952202, Email address: [email protected]
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ACCEPTED MANUSCRIPT portfolio optimization strategies of the LEL (Limited-Expected-Losses) and VaR risk managers in a general equilibrium framework (Basak and Shapiro , 2001). Alexander and Baptista compared the VaR and CVaR constraints on portfolio selection (Alexander and Baptista , 2004). Brandtner modeled the mean-spectral risk preferences in a form of spectral utility function (Brandtner , 2013). Some other researchers proposed some novel methods to search for optimal portfolios with risk measures for downside loss-averse preferences (Cui et al., 2013; Jarrow and Zhao , 2006; Roman et al., 2007; Sengupta and Sahoo , 2013; Yao et al., 2013). However, most of these numerical studies implemented Monte Carlo simulation to search for optimal portfolios, and could not find an analytical solution for
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the optimization problem. A computationally simple analytical solution is highly useful, especially for unsophisticated investors. Sun et al. stood out in the literature by providing a model with an analytical solution (Sun et al., 2015).
This paper extends the work of Sun et al. (Sun et al., 2015) to the multi-period setting. In their paper, the authors construct a minimax portfolio selection model by introducing a probabilistic risk
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measure, and a analytical solution is computationally available. Their paper aimed to maximize the expected portfolio return and minimize the maximum individual risk of the assets in the portfolio for a single period. However, portfolios are dynamically managed over multiple periods in practice. Some other papers also argued for the importance of the dynamic relationship between risk and return in a long horizon (Bali et al., 2009; Bickel , 1969; Harrison and Zhang , 1999; Merton , 1973).
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Our paper contributes to the literature in the following way. Firstly, we provide a computationally simple analytical solution to the complex portfolio selection problem in a multi-period setting.
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Secondly, our model is superior to the Sun et al. model which is superior to others in the meanvariance space (Sun et al., 2015). Thirdly, out model inherits some good features of the single-period
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model. For example, we are still able to derive an analytical solution without computation of the covariances. In addition, the investors do not have to purchase a huge number of stocks to form an optimal portfolio.
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The remainder of this paper is organized as follows. Section 2 describes the problem and formulates it as a bi-criteria optimization problem. Section 3 develops the analytical solution to the problem.
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In Section 4, ASX100 data is applied to our model and to the single period model, and the results compared. Section 5 concludes the paper.
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Problem formulation
We consider a multi-period portfolio optimization problem, where an investor is going to invest in N possible risky assets Sj , j = 1, . . . , N with a positive initial wealth of M0 . The investment will be made at the beginning of the first period of a T -period portfolio planning horizon. Then, the wealth 3
ACCEPTED MANUSCRIPT will be reallocated to these N risky assets at the beginning of the following T − 1 consecutive time periods. The investor will claim the final wealth at the end of the T th period. Let xtj be the percentage of wealth at the end of period t − 1 invested in asset Sj at the beginning of period t. Denote xt = [xt1 , . . . , xtN ]> . Here we assume that the whole investment process is a self-financing process. Thus, the investor will not increase the investment nor put aside fund in any period in the portfolio planning horizon. In other words, the total fund in the portfolio at the end of period t − 1 will be allocated to those risky assets at the beginning of period t. Thus, xtj = 1,
t = 1, . . . , T.
j=1
(2.1)
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N X
Moreover, it is assumed that short selling of the risky assets is not allowed at any time. Hence, we have xtj ≥ 0,
t = 1, . . . , T, j = 1, . . . , N.
(2.2)
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Let Rtj denote the rate of return of asset Sj for period t. Define Rt = [Rt1 , . . . , RtN ]> . Here, Rtj is assumed to follow normal distribution with mean rtj and standard deviation σtj . We further assume that vectors Rt , t = 1, . . . , T , are statistically independent, and the mean E(Rt ) = r t = [rt1 , . . . , rtN ]> is calculated by averaging the returns over a fixed window of time τ .
rtj =
t−1 1 X Rji , τ i=t−τ
t = 1, . . . , T, j = 1, . . . , N.
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Let
(2.3)
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We assume that in any time period, there are no two distinct assets in the portfolio that have the same level of expected return as well as standard deviation, i.e., for any 1 ≤ t ≤ T , there exist no i and j such that i 6= j, but rti = rtj , and σti = σtj .
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Let Vt denote the total wealth of the investor at the end of period t. Clearly, we have Vt = Vt−1 ( 1 + R> t xt ),
t = 1, . . . , T,
(2.4)
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with V0 = M0 .
First recall the definition of probabilistic risk measure, which was introduced in (Sun et al., 2015)
for the single period probabilistic risk measure.
wp (x) = min Pr{ |Rj xj − rj xj | ≤ θε }, 1≤j≤N
(2.5)
where θ is a constant to adjust the risk level, and ε denotes the average risk of the entire portfolio,
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ε=
N 1 X σj . N j=1
(2.6)
The whole idea of this risk measure in (2.5) is to locate the single asset with greatest deviation in the portfolio. With this ‘biggest risk’ mitigated, the risk of the whole portfolio can be substantially reduced as well. For multi-period portfolio optimization, the single ‘biggest risk’ should be selected
wp (x) = min
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over all risky assets and over the entire planning horizon. Thus, we define
min Pr{ |Rtj xtj − rtj xtj | ≤ θε },
1≤t≤T 1≤j≤N
(2.7)
> > where x = [x> 1 , . . . , xT ] θ, is the same as defined in (2.5) to be a constant adjusting the risk level,
and ε denotes the average risk of the entire portfolio over T periods, which is calibrated by
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T N 1 XX ε= σtj . T × N t=1 j=1
(2.8)
Assume that the investor is rational and risk-averse, who wants to maximize the terminal wealth as well as minimize the risk in the investment. Thus, the portfolio selection problem can be formulated
max
min
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as a bi-criteria optimization problem stated as follows.
min f (xtj ), E(VT ) ,
1≤t≤T 1≤j≤N
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s.t. Vt = Vt−1 ( 1 + R> t xt ), N X
xtj = 1,
t = 1, . . . , T,
t = 1, . . . , T,
(2.9a) (2.9b) (2.9c)
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j=1
t = 1, . . . , T, j = 1, . . . , N.
(2.9d)
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xtj ≥ 0,
where f (xtj ) = Pr{ |Rtj xtj − rtj xtj | ≤ θε }. To reach the optimality for the above bi-criteria optimization problem, we recall the following
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definition which is based on Theorem 3.1 in (Vincent and Grantham , 1981).
Definition 2.1. A solution x∗ = {x∗1 , . . . , x∗T } satisfying (2.9c) and (2.9d) is said to be a Pareto˜= minimal solution for the bi-criteria optimization problem (2.9) if there does not exist a solution x ˜ N } satisfying (2.9c) and (2.9d) such that {˜ x1 , . . . , x min
min f (x∗tj ) ≤ min
1≤t≤T 1≤j≤N
min f (˜ xtj ), and E(VT∗ |x∗ ) ≤ E(VT∗ |˜ x),
1≤t≤T 1≤j≤N
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y, E(VT ) ,
(2.10a)
s.t. Vt = Vt−1 ( 1 + R> t xt ), y ≤ f (xtj ), N X
xtj = 1,
t = 1, . . . , T,
t = 1, . . . , T, j = 1, . . . , N, t = 1, . . . , T,
j=1
xtj ≥ 0,
(2.10b)
t = 1, . . . , T, j = 1, . . . , N.
(2.10c) (2.10d)
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max
(2.10e)
where y ≤ f (xtj ) is the (N × (t − 1) + j)th probabilistic constraint. The optimization process trying to maximize y will eventually push the value of y to be equal to min min f (xtj ). Thus, the
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1≤t≤T 1≤j≤N
optimization problem (2.10) is equivalent to the optimization problem (2.9).
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Analytical solution to the problem
Recall that in Section 2, the return, Rtj , of asset Sj in period t, follows the normal distribution with
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mean rtj and standard deviation σtj . It is clear that Rtj − rtj is also normally distributed with mean 0 and standard deviation σtj .
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Let qtj = Rtj − rtj . Then it follows that
(3.1)
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n θε o f (xtj ) = Pr{ |Rtj xtj − rtj xtj | ≤ θε } = Pr |Rtj − rtj | ≤ xtj Z xθε 2 o n qtj tj 1 √ =2 exp − 2 dqtj . 2σtj 2πσtj 0
By the property of cumulative distribution function (3.1), f (xtj ) is clearly a monotonically decreasing
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function with respect to xtj . As mentioned in the optimization problem (2.10), in the process of maximizing the objective
function, the value of y is to be pushed to reach min min f (xtj ). As a result, if we choose y to be 1≤t≤T 1≤j≤N
an arbitrary but fixed real number between 0 and 1 (because y is a value of probability), using the monotonic property of f (xtj ), we can find an upper bound for xtj . Let this upper bound be denoted as Utj , which is given by Utj = min{ 1, x ˆtj }, where x ˆtj = f −1 (y). 6
(3.2)
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max E(VT ),
(3.3a)
s.t. Vt = Vt−1 ( 1 + R> t xt ), N X
xtj = 1,
t = 1, . . . , T,
(3.3b)
t = 1, . . . , T,
(3.3c)
j=1
0 ≤ xtj ≤ Utj ,
(3.3d)
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t = 1, . . . , T, j = 1, . . . , N.
We use dynamic programming to solve problem (3.3).
For time periods k = 1, . . . , T , we define a series of optimal control problems with the same dynamics, cost function, and constraints but different initial states and initial times. These problems
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are stated as follows.
max E(VT ),
s.t. Vt = Vt−1 ( 1 + R> t xt ), Vk−1 = ξ, xtj = 1,
j=1
t = k, . . . , T,
(3.4b) (3.4c)
t = k, . . . , T,
(3.4d)
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N X
(3.4a)
0 ≤ xtj ≤ Utj ,
(3.4e)
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t = k, . . . , T, j = 1, . . . , N,
where ξ is a variable. At the time k − 1, the number of steps to go is T − k + 1. This is a family of optimal control problems determined by the initial time k − 1 and initial state ξ. We use Problem
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(Pk−1,ξ ) to denote each different problem. The following theorem from Chapter 9 in (Varaiya , 1972)
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is known as the Principle of Optimality. Theorem 3.1. Suppose that {x∗k , . . . , x∗T } is an optimal control for Problem (Pk−1,ξ ), and that 0
∗ {Vk−1 = ξ, Vk∗ , . . . , VT∗ } is the corresponding optimal trajectory. Then, for any k ≤ k ≤ T , {x∗k0 , . . . , x∗T }
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is an optimal control for Problem (Pk0 −1,V ∗0
).
k −1
To continue, we utilize an auxiliary value function denoted by F (k −1, ξ) to be the maximum value
of the objective function for Problem (Pk−1,ξ ). The following value function F (k − 1, ξ) is defined similarly as in Chapter 9 in (Varaiya , 1972), i.e., F (k − 1, ξ) = max{ F (k, ξ( 1 + R> k xk )) : xk ∈ Xk }, 1 ≤ k ≤ T,
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(3.5)
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N X j=1
(3.6)
xkj = 1 and 0 ≤ xkj ≤ Ukj , j = 1, . . . , N }.
Now, we shall solve the discrete-time optimal control problem (3.3) backwards using dynamic programming method. For k = T , we start from the state VT −1 = ξ at time T − 1. Then, from (3.5) and (3.6), the value
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function becomes: F (T − 1, ξ) = max{ F (T, ξ( 1 + R> T xT )) : xT ∈ XT } = max{ E(ξ( 1 + R> T xT )) : xT ∈ XT } = max{ ξ( 1 + r > T xT ) : xT ∈ XT }.
(3.7)
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Consequently, Problem (PT −1,ξ ) becomes a linear programming problem stated as below. max ξ( 1 + r > T xT ), N X
xT j = 1,
j=1
j = 1, . . . , N.
(3.8b) (3.8c)
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0 ≤ xT j ≤ U T j ,
(3.8a)
The optimal solution to this linear programming problem has been obtained in (Sun et al., 2015).
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The result is quoted in the following as a lemma. Lemma 3.1. Let the assets be sort in such an order that rT 1 ≥ rT 2 ≥ . . . ≥ rT N . Then, there exists
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an integer n ≤ N such that
n−1 X
UT j < 1, and
j=1
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j=1
and
x∗T j
n X
UT j ≥ 1,
UT j , j=1,. . . ,n-1, n−1 X 1− UT j , j=n, = j=1 0, j >n,
is an optimal solution to problem (3.8).
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(3.9)
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Note that x∗T in (3.9) is independent from ξ as the solution is a function of UT j , which, from (3.1) and (3.2), depends only on the mean and standard deviation of RT j . Continuing with this process, we obtain the value function at k = T − 1.
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F (T − 2, ξ) = max{ F (T − 1, ξ( 1 + R> T −1 xT −1 )) : xT −1 ∈ XT −1 }
> ∗ = max{ E(ξ( 1 + R> T −1 xT −1 )(1 + r T xT )) : xT −1 ∈ XT −1 } > ∗ = max{ ξ( 1 + r > T −1 xT −1 )(1 + r T xT ) : xT −1 ∈ XT −1 }.
(3.10)
Again, it is easy to show that this maximizing problem is equivalent to the following linear program-
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ming problem. > ∗ max ξ( 1 + r > T −1 xT −1 )(1 + r T xT ), N X j=1
x(T −1)j = 1,
j = 1, . . . , N.
(3.11b) (3.11c)
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0 ≤ x(T −1)j ≤ U(T −1)j ,
(3.11a)
Since x∗T is solved in (3.8) and known, the above problem is in the same form as (3.8). Thus, it can
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be solved in a similar manner as that for (3.9). Details are given below as a lemma. Lemma 3.2. Let the assets be sort in such an order that r(T −1)1 ≥ r(T −1)2 ≥ . . . ≥ r(T −1)N . Then,
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and
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there exists an integer n ≤ N such that n−1 X j=1
x∗(T −1)j
U(T −1)j < 1, and
n X j=1
U(T −1)j ≥ 1,
U(T −1)j , j=1,. . . ,n-1, n−1 X 1− U(T −1)j , j=n, = j=1 0, j >n,
is an optimal solution to problem (3.11).
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(3.12)
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Similarly, the optimal solution x∗T −1 has no dependency on the state ξ. It depends only on fixed values of U(T −1)j , j = 1, . . . , N . From Lemma 3.1 and Lemma 3.2, it is reasonable to postulate that
k = T, . . . , 1.
(3.13)
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∗ > ∗ > ∗ F (k − 1, ξ) = ξ( 1 + r > k−1 xk−1 )( 1 + r k xk ) × · · · × ( 1 + r T xT ),
Moreover, the optimal solution for every period k, k = 1, . . . , T , can be written in a unified form as given below.
Then, there exists an integer n ≤ N such that n−1 X
Ukj < 1, and
j=1
n X j=1
and
Ukj ≥ 1,
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Ukj , j=1,. . . ,n-1, n−1 X Ukj , j=n, 1− = j=1 0, j >n,
(3.14)
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x∗kj
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Theorem 3.2. For any k = 1, . . . , T , let the assets be sort in such an order that rk1 ≥ rk2 ≥ . . . ≥ rkN .
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where Ukj is defined as in (3.1) and (3.2). Thus, with x∗k = [x∗k1 , . . . , x∗kN ]> , {x∗1 , . . . , x∗T } is an
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optimal solution to problem (3.3).
Numerical simulations
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We use daily return data of stocks on ASX100 dated from 01/01/2007 to 30/11/2011. The portfolio takes effect from 01/01/2009 and is open for trading for 3 years. 3 years is chosen since professionally managed portfolios (e.g., Aberdeen Asset Management, JBWere) usually list the average holding period as 3 − 5 years. At the beginning of each month, the funds in the portfolio are allocated based on the updated asset return data. We use 2 years historical data to decide the portfolio allocation each month. For the first portfolio allocation on 01/01/2009, the return data from 01/01/2007 to 31/12/2008 is used to evaluate the expected return and standard deviation of each stock. In the month following, the return data from 01/02/2007 to 31/01/2009 is used to evaluate the updated 10
ACCEPTED MANUSCRIPT Table 1: Multi-period - Expected Portfolio Returns for Selected θ & y HH θ 0.01 0.1 0.2 0.5 1 H HH y 0.95 0.59% 4.32% 6.35% 10.78% 15.38% 0.90 0.71% 4.77% 7.01% 11.80% 16.85% 0.80 0.96% 5.48% 8.09% 13.40% 19.25%
expected return and standard deviation, and this goes on.
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Table 2: Single Period - Expected Portfolio Returns for Selected θ & y HH θ 0.01 0.1 0.2 0.5 1 H HH y 0.95 -2.01% 2.00% 3.31% 3.30% 4.72% 0.90 -1.89% 2.23% 3.66% 3.83% 5.14% 0.80 -1.69% 2.65% 3.53% 4.38% 5.89%
The formulation of the corresponding portfolio optimization problem is as defined in Section 2. Assume the investor has an initial wealth of M0 = 1,000,000 dollars. There are N = 100 stocks to
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choose from for a portfolio of investment holding period of T = 36 months. The average risk of the portfolio, ε, over T periods is calculated as in (2.8). Table 1 shows the portfolio returns for various combination of θ (risk adjusting parameter), and y (lower bound of the probabilistic constraint). By changing the value of θ and/or y, the investor is able to alter the portfolio composition to cater for different risk tolerance levels. The lower the value of θ, the more diversified the portfolio can be, while
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the lower the value of y, the less stringent the probabilistic constraint.
From Table 1, it can be seen that the expected portfolio return increases when θ increases. Sim-
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ilarly, the expected portfolio return increases when y decreases. This makes sense since when θ increases, the portfolio selection consists of a much smaller number of selected ‘better-performing’ stocks. When y is lower, the risk is higher and hence the return is generally expected to be higher.
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The historical price index value of ASX100, composed of 100 large-cap and mid-cap stocks, was 3067.90 and 3329.40 at the end of December 2008 and at the end of December 2011, respec-
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tively. This translates to a return of 8.52% for a portfolio which comprises the entire stock selection of ASX100.
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From Table 1, it can be seen that when θ > 0.5, the expected returns following our portfolio election criteria outperforms the market index return. The multi-period model outperforms the passive single period model with a period of 3 years. Table 2 shows the expected returns using the single period model with a period of 3 years. When θ = 0.1 and y = 0.95, solving the problem with Theorem 3.2 suggests a total wealth in portfolio of 1,043,223.89 dollars at the end of the 3 year investment, which is a return of 4.32%. Comparing this with the result of the single period investment strategy, the multi-period solution outperforms it by more than one fold (the single period portfolio has a total return of about 2%). 11
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5
Conclusion
Our multi-period portfolio selection method offers various advantages over the single period model while retaining the latter’s advantages over other existing portfolio selection models. Advantages of our model include simplicity in computation, having a small number of ‘optimal’ stocks in the portfolio, allowance to cater for investors with different degrees of risk aversion and the ability to arrive at an elegant analytical solution without requiring the computation of a cumbersome covariance matrix. Moreover, dynamic management of the portfolio results in higher expected returns for the investor
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while retaining the simplicity of portfolio allocation via its explicit solution. Future research can consider the transaction costs involved in the dynamic management of the portfolio.
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