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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 1832926, 18 pages
https://doi.org/10.1155/2018/1832926
Research Article

Multiperiod Telser’s Safety-First Portfolio Selection with Regime Switching

1Research Center for International Trade and Economic, Guangdong University of Foreign Studies, Guangzhou 510006, China
2China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China

Correspondence should be addressed to Huiling Wu; moc.liamtoh@861gnilynnus

Received 24 August 2017; Revised 11 January 2018; Accepted 13 February 2018; Published 11 April 2018

Academic Editor: J. R. Torregrosa

Copyright © 2018 Chuangwei Lin and Huiling Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates a multiperiod Telser’s safety-first portfolio selection model with regime switching where the returns of the assets are assumed to depend on the market states modulated by a discrete-time Markov chain. The investor aims to maximize the expected terminal wealth and does not want the probability of the terminal wealth to fall short of a disaster level to exceed a predetermined number called the risk control level. Referring to Tchebycheff inequality, we modify Telser’s safety-first model to the case that aims to maximize the expected terminal wealth subject to a constraint where the upper bound of the disaster probability is less than the risk control level. By the Lagrange multiplier technique and the embedding method, we study in detail the existence of the optimal strategy and derive the closed-form optimal strategy. Finally, by mathematical and numerical analysis, we analyze the effects of the disaster level, the risk control level, the transition matrix of the Markov chain, the expected excess return, and the variance of the risky return.

1. Introduction

Nowadays, the portfolio selection theory has been one of the main areas of research in the financial field. The earliest approach to portfolio optimization is the mean-variance approach pioneered by Markowitz [1]. In the past decades, Markowitz’s mean-variance approach, where the variance of return is used as a risk measure, has received a lot of attention. In addition to the mean-variance criterion, there is another important school of thought called the safety-first criterion, which can be traced back to the work by Roy [2] based on the recognition that avoiding loss of a significant magnitude is a matter of great concern to most investors. According to Roy’s safety-first rule, the investor aims to minimize the disaster probability of the final return falling below a prespecified critical return. In a follow-up paper, Kataoka [3] prespecifies the probability that the final return is less than a critical return and selects the strategy that maximizes the critical return. The third form of the safety-first criterion proposed by Telser [4] presents another form of the safety-first (TSF for short) criterion, which tries to maximize the expected final return subject to the constraint that the probability of the final return no greater than a disaster level is less than a predetermined acceptable number. The safety-first criterion can actually be regarded as a significant complement to the prevailing mean-variance criterion for portfolio optimization. First, the mean-variance approach views the risk as return variability, but in the real world, investors might perceive risk in different ways. For example, as Hagigi and Kluger [5] note, when the time horizon is long, the investor might not care much about the short-term fluctuation of the return. He might instead aim to maximize the expected return while ensuring that the probability of disaster is less than a given number. Second, the results obtained under the safety-first framework are different from those under a mean-variance criterion. The findings of Shefrin and Statman [6] indicate that, in general, optimal safety-first portfolios are not mean-variance efficient. Moreover, according to the empirical findings in Lopes [7], De Bondt [8], and Neugebauer [9], there actually exists a comparative advantage of the safety-first approach over deviation risk measures, such as the variance, because it seems to better fit with the way investors perceive risk.

Nowadays, people are more concerned about risk and try their best to minimize inevitable losses in the face of the intensified economic turmoil and political unrest over the past years. For example, Haque and Varela [10] apply safety-first portfolio principles to optimize the portfolios of risk-averse US investors considering the harmful influence of the 911 terrorist attacks on US financial markets. Therefore, the safety-first approach now receives as much attention as Markowitz’s theory. In terms of Roy’s safety-first criterion, Norkin and Boyko [11] consider a static portfolio selection model by improving Roy’s safety-first approach to the case with a better estimation of the negative return probabilities. Li et al. [12] extend Roy’s safety-first approach to a multiperiod setting. In view of Tchebycheff inequality, they adopt an approximation approach, which is to replace Roy’s disaster probability by its upper bound, to obtain an analytical solution. Their paper represents the first pioneering work in dynamic safety-first. In some follow-up papers that adopt the solution scheme of Li et al. [12], Chiu and Li [13] study asset-liability management; Yan [14] deals with a continuous-time portfolio selection under the assumption that the evolution of the stock price is a jump-diffusion process. However, the approximation approach actually deviates from the original conceptual framework set by Roy. Therefore, Chiu et al. [15] study the dynamic Roy’s original safety-first formulation and its application in asset and liability management. In addition, Li and Yao [16] investigate a continuous-time Roy’s portfolio selection problem in a Black-Scholes setting and obtain closed-form solutions of the best constant-rebalanced portfolios. Li et al. [17] compare the optimal constant-rebalanced portfolio, dynamic-rebalanced portfolio, and buy-and-hold strategies under Roy’s safety-first principle. In terms of Kataoka’s safety-first (KSF for short) principle, Ding and Zhang [18] study a static KSF investment choice model. They obtain conditions under which the KSF model has a finite optimal strategy without normality assumption and derive the optimal portfolios in two cases where the short-sell is allowed or it is not allowed. Ding and Zhang [19] give a further study on KSF model with regular distribution by providing geometrical properties of the KSF model and establishing a model for risky asset’s pricing. Nico [20] investigates a static Telser’s safety portfolio model with two kinds of targets, the fixed target and the stochastic target, and tries to determine which target choice results in a better investment performance. Arzac and Bawa [21] analyze the existence of the optimal solution for the TSF model and derive the conclusion that when the asset returns are normally or stably Pareto distributed, the CAPM can be derived from the TSF model. Engels [22] gives an intuitive and analytical solution for the TSF model under the assumption that the portfolio returns are, respectively, normally and elliptically distributed. For more details about this topic, interested readers are referred to Pyle and Turnovsky [23], Levy and Sarnat [24], Bigman [25], Milevsky [26], Stutzer [27], and Haley and Whiteman [28].

From the above-mentioned papers, the common points of the existing portfolio optimization under these three safety-first criterions can be summarized as follows. They only consider the risk from the asset prices but do not take into account the risk resulting from the change of the financial market states. To fill the gap, this present paper investigates a multiperiod portfolio selection problem under the TSF criterion with regime switching, in which the asset returns depend on the market state modulated by a Markov chain. To the best of our knowledge, no work in the existing literature has considered this topic. In reality, financial markets usually have a finite number of states, and these states would switch among each other. The empirical analysis indicates that the returns of the assets are actually sensitive to the change of the market states. For example, the findings of Hardy [29] show that the regime-switching log-normal model is better than any other asset pricing model. For this reason, many papers have studied portfolio selection with regime switching. Among others, Zhou and Yin [30], Yin and Zhou [31], Çakmak and Özekici [32], Çelikyurt and Özekici [33], Chen et al. [34], Costa and Araujo [35], Wu and Li [36], Wu et al. [37], and Wu and Chen [38] consider the investment model with regime switching under a mean-variance criterion. Cheung and Yang [39], Zeng et al. [40], and Wu [41] study this topic for investors with a power utility. In contrast, this paper takes the step of investigating portfolio optimization under the TSF framework. Actually, the three basic safety-first models mentioned above have the same constraint condition but different optimization objectives. There are two reasons that the authors choose Telser’s safety-first criterion. First, an overwhelming majority of portfolio selection models under safety-first criterion adopt Roy’s safety-first criterion, while Telser’s safety-first portfolio selection models deserve greater attention. In addition, the authors prefer Telser’s criterion because it can take into account both utility maximization and downside risk control.

The rest of the paper is organized as follows. In Section 2, we introduce the model and separate its solving process into three steps. Prime notations and assumptions are also described in this section. Sections 3 and 4 are devoted to the existence and the explicit expressions of the optimal strategies for the auxiliary problem, the Lagrangian optimal control problem, and the original problem, respectively. Mathematical and numerical analysis of some results is given in Section 5. This paper is concluded in Section 6. Proofs of the lemmas and theorems are given in the appendixes.

2. Problem Formulation and Notations

This paper assumes that an investor accesses the market at time with initial wealth and plans to invest her wealth in the financial market for consecutive periods. Moreover, we assume that the financial market has multiple states , and its dynamics are described by a time-homogeneous Markov chain where represents the market state at time . There are one risk-free asset and one risky asset available in the financial market whose returns depend on the states of the financial market. Denote by and , respectively, the random return of the risky asset and the risk-free return over period (time interval ) given . In this paper, is assumed to be independent of for any given as long as . The Markov chain and the returns are mutually independent in the following sense:for all , , and , where is the probability based on the information up to time and is the Borel -algebra on . Furthermore, we use the following notations in this paper.

(N1) The transition matrix of the Markov chain is denoted by . The matrix is the th power of . In particular, we define as an identity matrix.

(N2) For any matrix and any vector , denote by the th row of and the th component of . Furthermore, let where and be a column vector whose th component is .

(N3) If vectors have the same dimension, then denotes a vector whose th entry is and a vector with .

(N4) . , which is assumed to be nonzero for . , and are -dimension column vectors whose th components are, respectively,, and are column vectors whose th components are, respectively,For the sake of convenience, we set

If we define as the amount invested in the risky asset at time and as the wealth under the strategy at time , then the wealth dynamics are

In this paper, we consider the optimal investment choice with the TSF criterion where the investor does not want the probability of her final wealth falling below a disaster value to exceed the risk control level . Hence the strategy subject to is an admissible action, and then the investor tries to select an admissible action to maximize the expected terminal wealth. Given the initial market state and the initial wealth , we formulate the portfolio selection problem as follows:where stands for the disaster value and is a given real number representing the risk control level. Referring to Tchebycheff inequality, we haveThis means that if the upper bound satisfiesthen . Therefore, we modify the above-mentioned problem as follows:

When we adopt the Tchebycheff inequality to replace the probability by its upper bound , the resulting modified formulation degenerates to a mean-variance formulation, thus losing the spirit of safety-first, as indicated in Chiu et al. [15]. For this reason, we admit that this is a weakness of the current approach. Nonetheless, due to the absence of detailed knowledge or empirical estimates of the cumulative distribution function of the final wealth when the returns of the available assets are assumed to depend on the regimes, we have to fall back on the Tchebycheff inequality to calculate the maximum probability of the final wealth below and then use this approximation approach to derive the optimal strategy. In what follows, we aim to derive the optimal strategy and the optimal value function for the problem and analyze the conditions that and satisfy when the optimal strategy exists. In order to solve , we introduce a Lagrangian multiplier and formulate the Lagrangian optimal control problem as follows:The relationship of and is summarized in Lemma 1.

Lemma 1. Denote by and the optimal strategy and the value function of the problem , respectively. Ifexists, then the optimal value function of is and the optimal strategy is .

Now, we can obtain the optimal solution of by solving . However, the problem is not separable in the sense of dynamic programming due to the term . In view of Zhu et al. [42], we construct an auxiliary problem of as follows:Definethen the relationship between and is summarized in the following lemma.

Lemma 2. For any , ; conversely, if , then a necessary condition for is

The proof of Lemma 2 is similar to that of Zhu et al. [42]; thus, it is omitted here. Lemma 2 implies that . We can obtain the optimal strategy of the problem by first solving the auxiliary problem and then finding a suitable that can make become the optimal strategy of problem . The second part of Lemma 2 gives the necessary condition that should satisfy. In the next section, we shall solve the auxiliary problem , which is separable in the sense of dynamic programming.

3. Solution to Problems and

We first introduce Lemma 3 to solve , and the proof can be found in Wu et al. [37].

Lemma 3. Given any vectors , one has

Now we define the value functions Then, according to Bellman’s principle of optimality, we havefor , with the boundary conditionAccording to the recursive formulas (14)-(15), we have Theorem 4.

Theorem 4. The value function of problem is given byand the corresponding optimal strategy is given byfor and .

Proof. See Appendix A.

In order to derive the solution of problem , we give the explicit expressions for and in the following theorem.

Theorem 5. Under the optimal strategy (17) of the auxiliary problem,

Proof. See Appendix B.

In order to obtain the optimal strategy for the problem , we need to summarize some properties of the coefficient of and . In view of the notations of and , we first obtainTherefore, For convenience, denote Then, by Theorem 5, and can be written as

Lemma 6. , .

Proof. See Appendix C.

Now, we begin to seek the optimal strategy of the problem . To this end, we define a function as follows:By (22), we haveDifferentiating (24) with respect to , we obtainThe optimal strategy of the problem exists if and only if , that is, . Otherwise, the optimal solution of the problem does not exist. In view of Lemma 6, when the probability satisfiesthe optimal solution for the problem exists. When (26) holds, let , and then we derive the optimal solution of atWe have verified that solving equationin Lemma 2 yields the same expression of as (27). Substituting (27) back into (17) gives the optimal policy of problem , which is summarized in the following theorem.

Theorem 7. The optimal strategy of the problem is given byfor and .

In the next section, we shall seek the optimal strategy of the problem referring to the relationship between and .

4. Optimal Solution of Problem

We define a function by substituting (27) into (24) as follows:Substituting and into (30) results in Lemma 8.

Lemma 8. where

Proof. See Appendix D.

According to Lemma 6 and (26), the coefficient of is strictly greater than . Then, the formula (31) implies that the finite minimum value of exists in if and only if . If for some , then is a decreasing function with respect to , and then the minimum value of does not exist. This means that the prespecified critical return has to satisfy specific conditions so that the problem has the optimal solution. Considering that is a quadratic curve with respect to , we definewhich is not less than zero by (26) and Lemma 6. Hence, the roots of exist and are given asWhenby solving the equation , exists and is given asThen, the corresponding optimal value of at isReferring to (27) and (37), we haveIt first follows from Theorem 5 thatConsequently, according to (39), we obtain Now, we have verified thatAs mentioned in Section 2, also satisfies the condition Together with (36), we havewhich we will make further efforts to simplify later.

Lemma 9. Condition (44) can be reduced to .

Proof. See Appendix E.

Under this circumstance, the optimal strategy and optimal value of problem are summarized in the following theorem.

Theorem 10. When and , the optimal strategy of iswhere and satisfy (27) and (37), respectively, and the corresponding optimal value satisfies (38).

Now, we tend to derive the variance of the terminal wealth under in Theorem 10. According to (39) and (C.2), the variance of the terminal wealth under is given asMoreover, according to (40), we have . Thus, the relationship between the expected terminal wealth and the terminal risk is given as

5. Analysis of the Obtained Results

5.1. Effects of and

In view of Theorem 10, the risk control level should be less than a given value; otherwise, the maximum value of the problem can be positive infinite. In addition, to guarantee the existence of the optimal strategy of , the disaster level also needs to be less than . An intuitive understanding of these results is that if is large enough, then the constraint to the strategy might disappear. In other words, the condition might hold naturally when is large enough. Without this constraint, the investor just aims to maximize the expected terminal wealth without any risk control, so the maximum value of might be positive infinite. When holds, the disaster level should satisfy . Otherwise, does not have the optimal solution. This conclusion also makes sense. On the one hand, in reality, in order to reflect the awareness of the risk control, the disaster level should not be a very large number. On the other hand, if the value of is very large, the probability that the wealth is less than is very high so that there might be no strategy satisfying the condition . Because there is also no strategy satisfying the condition

In what follows, we shall analyze the effects of the disaster level and the risk control level on the optimal strategy in Theorem 10, the expected terminal wealth and the variance of the terminal wealth . The obtained results are summarized in Theorems 11-12.

Theorem 11. When , the optimal strategy of the problem , the expected terminal wealth , and the variance of the terminal wealth are decreasing along with the disaster level .

Proof. See Appendix F.

We will explain the results in Theorem 11. When is a very small number and especially when is negative and small enough, the constraint might disappear, causing the investor to invest more wealth in the risky asset in order to obtain more expected terminal wealth. At the same time, a higher expected return is often accompanied by higher risk; thus, we have a larger when is smaller.

Theorem 12. When and , the optimal strategy of the problem , the expected terminal wealth , and the variance of the terminal wealth are increasing along with the risk control level .

Proof. See Appendix G.

When the risk control level is larger, the investor has more tolerance for wealth that is less than the disaster level. In other words, he will tend to invest more wealth in the risky asset. Consequently, the expected terminal wealth is larger, along with a higher terminal risk. Theorems 11 and 12 indicate that and have the opposite influence on the obtained results. People with different attitudes toward the disaster level and the risk control level will have different investment behavior.

5.2. Effects of the Regime Switching

In this subsection, we numerically analyze the effects of the mechanism of regime switching on some obtained results. To this end, we investigate how the transition matrix, , and at each state affect the investment strategy, the disaster level , and the risk control level . Suppose that there are three market states; the initial wealth is 10; the investor adjusts the strategy every three months and there are time periods; that is, . For convenience, we assume that the risk-free return is assumed to be a constant 1.0135 over time and the return of the risky asset depends on the market states only. Therefore, for convenience, denote by the expected excess return at state and the variance of the excess return at state .

5.2.1. Effects of and

In this part, we first study the strategy at the initial time as an example to show the effects of and . Then, their influence on and is also investigated. To this end, let , , , , , , and the transition matrix be . Because state 1 has the highest Sharpe ratio and state 3 has the lowest, we call state 1 the best state, state 2 a normal state, and state 3 the worst state.

We first demonstrate the impact of and . To this end, we assume that , , and increase from 0.3 to 0.33 with the step size 0.01 while other parameters are kept the same value given above. In a similar way, when we increase from 0.4 to 0.43, we do not change the values of other parameters. Thus, the influence of the market states on the strategy is demonstrated in Table 1, which indicates that is increasing along with and is decreasing with respect to . Moreover, Table 1 also shows that (a) is very sensitive to the change in and ; (b) , the optimal investment strategy at state 1, is the most sensitive to the change in and . For example, when is increased from 0.3 to 0.31, that is, the growth rate is , the growth rates of the investment strategy at each state are , , , respectively. When the growth rate of is , the decrement rates of the investment strategy at each state are , , and , respectively. As for the impact of and , here we emphasize that two similar experiments have been conducted and the results also indicate that is the most sensitive to the change in and .

Table 1: Influence of and on the strategy.

Second, we want to know the influence of and on and . Because and , actually we just need to observe how and change along with and . In Table 2, both the rows and the columns indicate that a worse market state with a smaller excess return or a larger variance results in a longer interval, which belongs to. We explain this phenomenon as follows. In a worse market state, the probability that the running wealth is below the disaster level might be increasing, leading to a larger risk control level. Two similar experiments have also been conducted to study the influence of and , , and we obtain similar results. That is, a worse market environment leads to a larger interval . As for , Table 2 shows that has almost no change regardless of the excess return and the variance. When there is only one market state, referring to (34), can be simplified as , which shows that the investor will regard the value less than the risk-free return of the initial wealth from the initial time to the terminal time as a disaster level when the risk of the market-state fluctuation is neglected. In this subsection, substituting and into yields . These findings suggest that the value of in Table 2 is roughly equal to . In other words, the investor almost does not consider the financial risk when she sets the value range of the disaster level. She would like to choose the risk-free return of the initial wealth as the disaster level.

Table 2: Influence of and on and .
5.2.2. Influence of the Transition Matrix

In this part, the influence of the transition matrix is studied. To do this, we change , the th row of , while keeping the values of other parameters, and then we obtain Tables 3 and 4. We find from Table 3 that when the transition probability staying at the best state (state 1) is increased from 0.3 to 0.7, the optimal strategy is increasing accordingly. In particular, strategy at state 1 is increased the most rapidly. However, when the probability of staying at state 2 or state 3 is increased, is decreased accordingly. Table 4 suggests that a high probability of staying at the best market state yields a shorter interval , while the maximum value of the disaster level will also not be affected by the transition matrix.

Table 3: Influence of the transition matrix on the strategy.
Table 4: Influence of the transition matrix on and .

6. Conclusion

This paper investigates a multiperiod Telser’s safety-first portfolio selection problem with regime switching. There are one risk-free asset and one risky asset available in the financial market whose returns depend on the market states. The investor aims to maximize the expected terminal wealth subject to a constraint that the probability of the terminal wealth no greater than a disaster value is less than a predetermined acceptable number. Referring to Tchebycheff inequality, we modify Telser’s safety-first model to the problem that aims to maximize the expected terminal wealth subject to a constraint that the upper bound of the disaster probability is less than the risk control level. We find that when the risk control level and the disaster value satisfy and , the optimal strategy of the problem exists and is obtained by the Lagrange multiplier technique and the embedding method. We investigate the effects of the disaster level , the risk control level , the expected excess return , the variance , and the transition matrix . Mathematics analysis indicates that the optimal strategy , the expected terminal wealth , and the variance of the terminal wealth are decreasing along with the disaster level , while they are increasing with respect to the risk control level . By numerical analysis, we find the following: (a) the excess returns have a positive effect, while the variances have a negative impact on the investment strategy; (b) a smaller expected excess return or a larger variance of the risky return leads to a longer interval that the risk control level lies in. However, the disaster value is almost not affected by the expected excess return or the variance of the risky return and is roughly equal to the risk-free return of the initial wealth from the initial time to the terminal time; (c) when the probability of staying at the best state is increased, the investment amount is increased accordingly. Meanwhile, a higher probability staying at the best market state yields a shorter interval that belongs to, while the disaster value is also insensitive to the change in the transition matrix.

Appendix

A. Proof of Theorem 4

Proof. Referring to (14) and (15), when , we haveObviously, the optimal solution of (A.1) exists and isSubstituting (A.2) into (A.1) yieldsHence, (16) and (17) hold true for . Now, we assume that (16) and (17) are true for ; then, for ,