Keywords

1 Introduction

In financial asset management, portfolio technique is important for hedging the risk and it is used to make asset management stable. As a classical portfolio theory, Markowitz’s mean-variance model is studied by many researchers and fruitful results have been achieved, and the variance-minimizing is also important to minimize the risk in portfolio [5, 7,8,9]. Recently, value-at-risk (VaR) is used widely in finance to estimate the risk of worst-scenarios. VaR is a risk-sensitive criterion based on percentiles, and it is one of the standard criteria in asset management [3, 6, 12,13,14]. VaR is a kind of risk values of the asset prices at a specified risk-level probability and it is for selecting portfolios to get rid of bad scenarios in investment. An extension of VaR is average value-at-risk (AVaR), and it is known that AVaR is a coherent risk measure but VaR is not coherent [1]. Markowitz’s mean-variance criterion and variance-minimizing criterion are represented by quadratic programming, and AVaR criterion is analyzed based on the results regarding Markowitz’s criterion. In this paper dynamic AVaR portfolio selection problem is proposed in order to maximize both of AVaR and the expected rates of return, and owing to AVaR we can maximize the expected rate of return after due consideration of the worst-scenarios. This paper derives analytical solutions for the AVaR portfolio problem at each step, which is strongly related to the bankruptcy and the falling in the asset prices [4]. Introducing AVaR based on conditional probability, we discuss a dynamic optimization problem and we derive the optimality equation and the optimal solution for the model by dynamic programming.

2 A Dynamic Portfolio Model

First we explain a portfolio model with n stocks, where n is a positive integer. Let \(\{ 0,1,2,\ldots ,T \}\) be the time space with an expiration date T, and \({\mathbb {R}}\) denotes the set of all real numbers. Let \((\varOmega , \mathcal{M}, P)\) be a probability space, where \(\mathcal{M}\) is a \(\sigma \)-field and P is a non-atomic probability on a sample space \(\varOmega \). For an asset \(i=1,2,\ldots ,n\), a stock price process \(\{ S^{i}_{t} \}_{t=0}^{T}\) is given by rates of return \(R_{t}^{i}\) [10, 11]:

$$\begin{aligned} S_{t}^{i} = S_{t-1}^{i} (1+R_{t}^{i}) \end{aligned}$$
(1)

for \(t=1,2,\ldots ,T\), where \(\{ R^{i}_{t} \}_{t=1}^{T}\) is an integrable sequence of independent real-valued random variables. Hence \((w^{1}_{t}, w^{2}_{t}, \ldots , w^{n}_{t})\) is called a portfolio weight vector if it satisfies \(\sum _{i=1}^{n} w^{i} = 1\), and further the portfolio is said to allow for short selling if \(w^{i}_{t} \ge 0\) for all \(i=1,2,\ldots ,n\). Then the rate of return with a portfolio \((w^{1}_{t}, w^{2}_{t}, \ldots , w^{n}_{t})\) is given by

$$\begin{aligned} R_{t} = w^{1}_{t} R^{1}_{t} + w^{2}_{t} R^{2}_{t} + \cdots + w^{n}_{t} R^{n}_{t}. \end{aligned}$$
(2)

Therefore, the reward at time \(t(=1,2,\ldots ,T)\) follows

$$\begin{aligned} S_{t} = S_{t-1}\sum _{i=1}^{n} w^{i}_{t} (1+R_{t}^{i}) = S_{t-1} (1+R_{t}). \end{aligned}$$
(3)

This paper deals with a dynamic portfolio model for stock price processes \(\{ S^{i}_{t} \}_{t=0}^{T}\). The falling of asset prices is one of the most important risks in stock markets. The theoretical bankruptcy at time t occurs on scenarios \(\omega \) satisfying \(S_{t}(\omega ) \le 0\), i.e. it follows \(1 + R_{t}(\omega ) \le 0\) from (3). Similarly, for a constant \(\delta \in [0, 1]\), a set of sample paths \( \{ \omega \in \varOmega \mid 1 + R_{t}(\omega ) \le 1 - \delta \} = \{ \omega \in \varOmega \mid R_{t}(\omega ) \le - \delta \} \) is the event of scenarios where the asset price \(S_{t}\) will fall from the current price \(S_{t-1}\) to a lower level than \(100 (1-\delta ) \, \%\) of the current price \(S_{t-1}\), i.e. the rate of falling is \(100 \, \delta \, \%\). The parameter \(\delta \) is called the rate of falling. Then the probability of falling is also given by

$$\begin{aligned} p_{\delta } = P( R_{t} \le - \delta ). \end{aligned}$$
(4)

For example, \(p_{\delta }\) denotes the probability of the falling below par value if ‘\(\delta = 0\)’ and it indicates the probability of the bankruptcy if ‘\(\delta = 1\)’. In this paper, we discuss dynamic portfolio optimization regarding the rate of falling \(\delta \).

For a positive probability p, VaR regarding the rate of return \(R_t\) is given by a real number v satisfying

$$\begin{aligned} P( R_t \le v ) = p \end{aligned}$$
(5)

since P is non-atomic. VaR v is the upper bound of the rate of return \(R_t\) at the worst scenarios under a given risk probability p, and then VaR v in (5) is denoted by \(\text{ VaR }_{p}(R_{t})\). From (4) and (5), for a risk probability \(p = p_{\delta }\), the rate of falling is

$$\begin{aligned} \delta = - \text{ VaR }_{p}(R_{t}). \end{aligned}$$
(6)

In next section we discuss the minimization of the rate of falling (6). In this paper, we deal with a case where VaR v in (5) has the following representation.

$$\begin{aligned} \text{(VaR } \ v) = \text{(the } \text{ mean) } - \text{(a } \text{ positive } \text{ constant } \ \kappa _p ) \times \text{(the } \text{ standard } \text{ deviation) }, \end{aligned}$$
(7)

where the positive constant \(\kappa _p\) is given corresponding to the probability p. Equation (7) holds if the distribution of the rate of return \(R_t\) is Gaussian [2, 6].

3 A Dynamic AVaR Portfolio Model

First we introduce mathematical notations of average value-at-risk [12, 13]. Let \(\mathcal{X}\) be the set of all integrable \(\mathcal{M}\)-adapted real-valued random variables X on \(\varOmega \) with a continuous distribution function \(x \mapsto F_{X}(x) = P(X < x)\) for which there exists a non-empty open interval I such that \(F_{X}(\cdot ) : I \rightarrow (0,1)\) is strictly increasing and onto. Then there exists a strictly increasing and continuous inverse function \(F_{X}^{-1} : (0,1) \rightarrow I\). We note that \(F_{X}(\cdot ) : I \rightarrow (0,1)\) and \(F_{X}^{-1} : (0,1) \rightarrow I\) are one-to-one and onto. The value-at-risk (VaR) at a probability \(p (\in (0,1))\) is given by the percentile of the distribution function \(F_{X}\), i.e. \(\text{ VaR }_{p}(X) = \sup \{x \in I \mid F_{X}(x) \le p \}\). Then we have \(F_{X}(\text{ VaR }_{p}(X)) = p\) and \(\text{ VaR }_{p}(X) = F_{X}^{-1} (p)\). The average value-at-risk (AVaR) at a probability p is given by

$$\begin{aligned} \text{ AVaR }_{p}(X) = \frac{1}{p} \int _{0}^{p} \text{ VaR }_{q}(X) \, d q \end{aligned}$$
(8)

for \(p \in (0,1)\). It is known that AVaR is a coherent risk measure [1] but VaR is not coherent. From (3), AVaR for the reward \( S_{t} \) at time t is given by \( \text{ AVaR }_{p} (S_{t} ) = \text{ AVaR }_{p} \left( S_{t-1} (1+R_{t}) \right) . \) To discuss the dynamics, we introduce VaR based on conditional expectations. Let \(\mathcal{G}\) be a sub-\(\sigma \)-field of \(\mathcal{M}\). Define a map \(x \mapsto F_{X}(x \mid \mathcal{G}) = P(X< x \mid \mathcal{G}) = E(1_{\{ X < x \}}\mid \mathcal{G})\). We introduce VaR of \(X (\in \mathcal{X})\) under a condition \(\mathcal{G}\) at a probability p by \( \text{ VaR }_{p}(X\mid \mathcal{G}) = \sup \{x \in I \mid F_{X}(x \mid \mathcal{G}) \le p \}. \) Then AVaR under a condition \(\mathcal{G}\) is given by \( \text{ AVaR }_{p}(X\mid \mathcal{G}) = \frac{1}{p} \int _{0}^{p} \text{ VaR }_{q}(X\mid \mathcal{G}) \, dq \) for \(p \in (0,1)\), and we note \(\text{ AVaR }_{p}(X\mid \mathcal{G})\) is a \(\mathcal{G}\)-measurable random variable and \(F_{X}(\text{ AVaR }_{p}(X \mid \mathcal{G}) \mid \mathcal{G}) \le p\) for \(p \in (0,1)\). From (3), portfolio weights \((w^{1}_{t}, w^{2}_{t}, \ldots , w^{n}_{t})\) are decided sequentially and predictably. We note that \(S_{t}\) is estimated under information \(\mathcal{M}_{t-1}\) up to time \(t-1\). Then VaR of \(S_{t}\) under information \(\mathcal{M}_{t-1}\) at a probability p is \( \text{ AVaR }_{p} (S_{t} \mid \mathcal{M}_{t-1} ) = \text{ AVaR }_{p} \left( S_{t-1} (1+R_{t}) \mid \mathcal{M}_{t-1} \right) . \) This term means the risk of worst scenarios which occur on the transition from time \(t-1\) to time t. Therefore, taking the sum of these risks which occur at each time, we demonstrate the following dynamic portfolio problem regarding the total AVaR under information \(\{ \mathcal{M}_{t-1} \}_{t=1}^{T}\). Let a discount rate \(\beta \) be a positive constant and take \(S_0 =1\) for simplicity. Define the set of portfolios by \( \mathcal{W} = \{ (w^{1}, w^{2}, \ldots , w^{n}) \in {\mathbb {R}}^n \mid \sum _{i=1}^{n} w^{i} = 1 \ \text{ and } \ w^{i} \ge 0 \, (i=1,2,\ldots ,n) \}. \)  

Dynamic problem for AVaR: :

Maximize the total AVaR

$$\begin{aligned} E\left( \sum _{t=1}^{T} \beta ^{t-1} \text{ AVaR }_{p} (S_{t} \mid \mathcal{M}_{t-1} ) \right) = \sum _{t=1}^{T} \beta ^{t-1} \prod _{s=1}^{t-1} (1+E(R_{s})) \cdot ( 1+\text{ AVaR }_{p}(R_{t}) ) \end{aligned}$$
(9)

with portfolio weights \((w^{1}_{t}, w^{2}_{t}, \ldots , w^{n}_{t}) \in \mathcal{W}\) \((t=1,2,\ldots ,T)\).

 

By dynamic programming, we obtain the following equations.

Theorem 1

The optimal AVaR in Dynamic problem for AVaR is given by \(v_{1}\) which is defined inductively by the sequence \(\{ v_{t} \}\) of sub-total AVaR after time \(t-1\) satisfying the following backward optimality equations:

$$\begin{aligned} v_{t-1} = \max _{(w^{1}, \ldots , w^{n}) \in \mathcal{W}} \left\{ 1+\text{ AVaR }_{p} \left( \sum _{i=1}^{n} w^{i} R^{i}_{t-1} \right) + \beta \left( 1 + \sum _{i=1}^{n} w^{i} E(R^{i}_{t-1}) \right) v_{t} \right\} \end{aligned}$$
(10)

for \(t=2,3,\ldots ,T\), and

$$\begin{aligned} v_{T} = \max _{(w^{1}, \ldots , w^{n}) \in \mathcal{W}} \left\{ 1+\text{ AVaR }_{p} \left( \sum _{i=1}^{n} w^{i} R^{i}_{T} \right) \right\} . \end{aligned}$$
(11)

4 An Optimal Portfolio for AVaR

First we estimate the rate of return with a portfolio [12, 13]. Let the mean, the variance and the covariance of the rate of return \(R^{i}_{t}\), which is given in (1), respectively by \( \mu ^{i}_{t} = E( R^{i}_{t} ), \) \( (\sigma ^{i}_{t})^2 = E(( R^{i}_{t} - \mu ^{i}_{t} )^2) \) and \( \sigma ^{ij}_{t} = E( (R^{i}_{t} - \mu ^{i}_{t} ) (R^{j}_{t} - \mu ^{j}_{t} ) ) \) for \(i,j=1,2,\ldots ,n\). Hence we assume that the determinant of the variance-covariance matrix \(\varSigma _{t} = [\sigma ^{ij}_{t}]\) is not zero and there exists its inverse matrix \(\varSigma _{t}^{-1}\). This assumption is natural and it can be realized easily by taking care of the combinations of assets. For a portfolio \((w^{1}, w^{2}, \ldots , w^{n}) \in \mathcal{W}\), we calculate the expectation and the variance regarding \(R_{t} = \sum _{i=1}^{n} w^{i} R^{i}_{t}\). The expectation \(\mu _{t}\) of the rate of return \(R_{t}\) with the portfolio is \( \mu _{t} = E(R_{t}) = \sum _{i=1}^{n} w^{i} E(R_{t}^{i}) = \sum _{i=1}^{n} w^{i} \mu ^{i}_{t}. \) On the other hand, the variance \((\sigma _{t})^2 \) of the rate of return \(R_{t}\) with the portfolio is \( (\sigma _{t})^2 = E(( R_{t} - \mu _{t} )^2) = \sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t} \) for \(i = 1,2,\ldots ,n\). Therefore, for a positive probability p, AVaR of the rate of return \(R_{t}\) is evaluated as \( \text{ AVaR }_{p}(R_{t}) = \sum _{i=1}^{n} w^{i} \mu ^{i}_{t} - \kappa \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t}} \) with a positive constant \(\kappa = \frac{1}{p} \int _0^p \kappa _q \, dq\) with \(\kappa _p\) in (7). Let \({\mathbf {1}}\) be a unit vector and let \( \mu _{t} = [ \mu _{t}^{i} ], \varSigma _{t} = [ \sigma _{t}^{ij} ], \) \( A_{t} = {\mathbf {1}}^{{\!\scriptscriptstyle \mathrm T}} \varSigma _{t}^{-1} {\mathbf {1}}, B_{t} = {\mathbf {1}}^{{\!\scriptscriptstyle \mathrm T}} \varSigma _{t}^{-1} \mu _{t}, C_{t} = \mu _{t}^{{\!\scriptscriptstyle \mathrm T}} \varSigma _{t}^{-1} \mu _{t} \) and \( \varDelta _{t} = A_{t} C_{t} - B_{t}^2, \) where \({\!\scriptscriptstyle \mathrm T}\) denotes the transpose of a vector. Now step by step we discuss a portfolio problem to minimize the rate of falling. First, we deal with a variance-minimizing problem. For a given constant \(\gamma \), which is the total expected rate of return to be guaranteed for portfolios, we consider the following quadratic programming with respect to portfolios.  

Variance-minimizing (P1): :

Minimize the variance \( \sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t} \) with respect to portfolios \((w^{1}, w^{2}, \ldots , w^{n})\) satisfying \(\sum _{i=1}^{n} w^{i} = 1\) under a condition \( \sum _{i=1}^{n} w^{i} \mu _t^{i} = \gamma . \)

 

Lemma 1

[12, 13]. The solution of the variance-minimizing (P1) is given by \( w = \xi \varSigma ^{-1} {\mathbf {1}}+ \eta \varSigma ^{-1} \mu \) and then the corresponding variance is \( \rho = \frac{A \gamma ^2 - 2 B \gamma +C}{\varDelta }. \)

Solution w in Lemma 1 is called a minimal risk portfolio, and a set \( \mathcal{E} = \{ (\rho , \mu ) \mid \rho = \frac{A (\mu )^2 - 2 B \mu +C}{\varDelta } \ \text{ and } \ \mu \ge \frac{B}{A} \} \) is also called the efficient frontier [7, 8], where \(\rho \) is a variance and \(\mu \) is an expected rate of return for acceptable minimal risk portfolios. Next we consider a risk-sensitive model in order to deal with a portfolio problem to minimize the rate of falling in the third step. For a constant \(\gamma \), we discuss the following risk-sensitive portfolio problem.  

Risk-sensitive problem (P2): :

Maximize risk-sensitive expected rate of return

$$\begin{aligned} \sum _{i=1}^{n} w^{i} \mu _t^{i} - \kappa \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t}} \end{aligned}$$
(12)

with respect to portfolios \((w^{1}, w^{2}, \ldots , w^{n})\) (\(\sum _{i=1}^{n} w^{i} = 1\)) under the condition \( \sum _{i=1}^{n} w^{i} \mu _t^{i} = \gamma . \)

 

Lemma 2

Let A and \(\varDelta \) be positive. If \(\kappa \) satisfies \(\kappa ^2 > \varDelta /A\), then a function \(v : {\mathbb {R}}\rightarrow {\mathbb {R}}\) given by \( v(\gamma ) = \gamma - \kappa \sqrt{\frac{A \gamma ^2 - 2 B \gamma +C}{\varDelta }} \) (\(\gamma \in {\mathbb {R}}\)) is concave and it has the maximum \( v(\gamma ^*) = \frac{B - \sqrt{A \kappa ^2 - \varDelta }}{A} \) at \( \gamma ^* = \frac{B}{A} + \frac{\varDelta }{A \sqrt{A \kappa ^2 - \varDelta }}. \)

Now we discuss the following portfolio problem to minimize the rate of falling (6) without allowance for short selling. The rate of falling (6) is given by the following (13).  

Portfolio minimizing the risk of falling (P3): :

Minimize the risk of falling

$$\begin{aligned} \delta = - \text{ AVaR }_{p}(R_{t}) = - \sum _{i=1}^{n} w^{i} \mu _t^{i} + \kappa \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t}} \end{aligned}$$
(13)

with portfolio weights \((w^{1}, w^{2}, \ldots , w^{n}) \in \mathcal{W}\).

 

Hence since we have \( \inf _{w} (13) = \inf _{\gamma } ( \inf _{w : \sum _{i=1}^{n} w^{i} \mu _t^{i} = \gamma } (13) ) = - \sup _{\gamma } \ (12), \) by Lemma 2 we arrive at the following analytical solutions for (P3).

Lemma 3

Let A and \(\varDelta \) be positive, and let \(\kappa \) satisfy \(\kappa ^2 > C\). Then an optimal solution for (P3) is given by \( w^* = \xi \varSigma ^{-1} {\mathbf {1}}+ \eta \varSigma ^{-1} \mu , \) and then the corresponding rate of falling is \( \delta (\gamma ^*) = - \frac{B - \sqrt{A \kappa ^2 - \varDelta }}{A} \) at the expected rate of return \( \gamma ^* = \frac{B}{A} + \frac{\varDelta }{A \sqrt{A \kappa ^2 - \varDelta }}, \) where \( \xi = \frac{C - B \gamma ^*}{\varDelta } \) and \( \eta = \frac{A \gamma ^* - B}{\varDelta }. \)

Condition (V). \(v_{t} > 0\) for all \(t=1,2,\ldots ,T\).

6pt Condition (V) is a natural one because if Condition (V) is not satisfied at some time t, it has already gone bankrupt. Under Condition (V), from Theorem 1 and Lemma 3, we obtain the following results. We can calculate numerical optimal solutions by the optimality equation in Theorem 2 and numerical optimal solutions at each time given in Lemma 3.

Theorem 2

Suppose Condition (V) is satisfied. The optimal AVaR \(v_{1}\) in Theorem 1 is given by the sequence \(\{ v_{t} \}\) of sub-total AVaR after time \(t-1\) satisfying the following backward optimality equations:

$$\begin{aligned} v_{t-1} = \max _{(w^{1}, \ldots , w^{n}) \in \mathcal{W}} (1 + \beta v_{t}) \left( 1 + \sum _{i=1}^{n} w^{i} \mu ^{i}_{t-1} - \frac{\kappa }{1 + \beta v_{t} } \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{t-1}} \right) \end{aligned}$$
(14)

for \(t=2,3,\dots ,T\), and

$$\begin{aligned} v_{T} = \max _{(w^{1}, \ldots , w^{n}) \in \mathcal{W}} \left( 1 + \sum _{i=1}^{n} w^{i} \mu ^{i}_{T} - \kappa \sqrt{\sum _{i=1}^{n} \sum _{j=1}^{n} w^{i} w^{j} \sigma ^{ij}_{T}} \right) . \end{aligned}$$
(15)