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

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, wemodify 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.


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 meanvariance criterion for portfolio optimization.First, the meanvariance 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 assetliability 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 constantrebalanced portfolio, dynamic-rebalanced portfolio, and buyand-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 safetyfirst 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], C ¸akmak and Özekici [32], C ¸elikyurt 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.

Problem Formulation and Notations
This paper assumes that an investor accesses the market at time 1 with initial wealth  1 > 0 and plans to invest her wealth in the financial market for  consecutive periods.Moreover, we assume that the financial market has multiple states {1, 2, . . ., }, and its dynamics are described by a timehomogeneous Markov chain {  ,  = 1, 2, . ..}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 [,  + 1),  = 1, 2, . . ., ) given   = .In this paper,   () is assumed to be independent of   () for any given ,  ∈ {1, 2, . . ., } as long as  ̸ = .The Markov chain and the returns are mutually independent in the following sense: for all  ∈ B(R +1 ),  ∈ , and  = 1, 2, . . ., , where   is the probability based on the information up to time  and B(R +1 ) is the Borel -algebra on R +1 .Furthermore, we use the following notations in this paper.
(N1) The transition matrix of the Markov chain {  ,  = 1, 2, . ..} is denoted by .The matrix   is the th power of .In particular, we define  0 as an identity matrix.
(N2) For any matrix  × and any vector  ×1 , denote by () the th row of  and () the th component of .Furthermore, let   = {  (, )} × where   (, ) = (, )() and  be a column vector whose th component is () = ∑  =1 (, ).(N3) If vectors , ,  have the same dimension, then ⋅/ denotes a vector whose th entry is ()()/() and 2 a vector with which is assumed to be nonzero for  = 1, 2, . . ., . ℎ  ,   , and   ( = 1, 2, . . ., ) are -dimension column vectors whose th components are, respectively, () =    () h , ℎ  , and   ( = 2, 3, . . .,  + 1) are column vectors whose th components are, respectively, For the sake of convenience, we set 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 {  +1 ≤  |  1 =  1 ,  1 =  1 } ≤  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  1 =  1 and the initial wealth  1 , we formulate the portfolio selection problem as follows: max where  stands for the disaster value and 0 <  < 1 is a given real number representing the risk control level.Referring to Tchebycheff inequality, we have This means that if the upper bound satisfies ((, )) When we adopt the Tchebycheff inequality to replace the probability , 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  > 0 and formulate the Lagrangian optimal control problem as follows: max exists, 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 Var , 1 (  +1 ).In view of Zhu et al. [42], we construct an auxiliary problem ((, )) of ((, , )) as follows: then the relationship between Π  (, ) and Π  () is summarized in the following lemma.

Lemma 2. For any
The proof of Lemma 2 is similar to that of Zhu et al. [42]; thus, it is omitted here.Lemma 2 implies that Π  () ⊆ ⋃  Π  (, ).We can obtain the optimal strategy of the problem ((, , )) by first solving the auxiliary problem ((, )) and then finding a suitable  * that can make  ∈ Π  ( * , ) 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.
Proof.See Appendix A.
In order to derive the solution of problem ((, , )), we give the explicit expressions for in the following theorem.
Theorem 5.Under the optimal strategy (17) of the auxiliary problem, Proof.See Appendix B.
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 have Differentiating (24) with respect to , we obtain The optimal strategy of the problem ((, , )) exists if and only if   () < 0, that is, (1 + )( 1 ) < 1.Otherwise, the optimal solution of the problem ((, , )) does not exist.
In view of Lemma 6, when the probability  satisfies the optimal solution for the problem ((, , )) exists.When (26) holds, let   () = 0, and then we derive the optimal solution of max ∈ () at We have verified that solving equation in 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.
In the next section, we shall seek the optimal strategy of the problem ((, )) referring to the relationship between ((, , )) and ((, )).

𝐹 (𝜔) = 𝜙 (𝑖
where Proof.See Appendix D. According to Lemma 6 and ( 26), the coefficient of 1/4 is strictly greater than 0.Then, the formula (31) implies that the finite minimum value of Γ() exists in  > 0 if and only if () > 0. If () ≤ 0 for some , then Γ() is a decreasing function with respect to  > 0, and then the minimum value of Γ() 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 define which is not less than zero by ( 26) and Lemma 6.Hence, the roots of () = 0 exist and are given as When by solving the equation Γ  () = 0,  * = arg min >0 Γ() exists and is given as Then, the corresponding optimal value of ((, )) Referring to ( 27) and (37), we have It first follows from Theorem 5 that Consequently, according to (39), we obtain Now, we have verified that As mentioned in Section 2,  also satisfies the condition Together with (36), we have which we will make further efforts to simplify later.Proof.See Appendix E.
Under this circumstance, the optimal strategy and optimal value of problem ((, )) are summarized in the following theorem.
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 as Moreover, according to (40), we have  * /2 * = (E  1 , 1 (  * +1 ) − ( 1 ) 1 )/( 1 ).Thus, the relationship between the expected terminal wealth and the terminal risk is given as (47)

Effects of 𝛾 and 𝛽.
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  1 .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 Var  1 , 1 (  +1 ) ≤ [E  1 , 1 (  +1 ) − ] 2 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 0 <  < min{1, 1/( 1 ) − 1} holds, the disaster level  should satisfy  <  1 .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 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 E  1 , 1 (  * +1 ) and the variance of the terminal wealth Var  1 , 1 (  * +1 ).The obtained results are summarized in Theorems 11-12.
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 Var  1 , 1 (  +1 ) ≤ [E  1 , 1 (  +1 ) − ] 2 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 Var  1 , 1 (  * +1 ) when  is smaller.

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.

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 Var(  (⋅)) 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  1 is 10; the investor adjusts the strategy every three months and there are 12 time periods; that is,  = 12.For convenience, we assume that the riskfree 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 Var(  ()) the variance of the excess return at state .

Effects of 𝑟 𝑒 (⋅) and Var(𝑅 𝑒 (⋅)).
In this part, we first study the strategy at the initial time as an example to show the effects of   (⋅) and Var(  (⋅)).Then, their influence on  and  is also investigated.To this end, let   (1) = 0.30,   (2) = 0. ).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.
Second, we want to know the influence of   (⋅) and Var(  (⋅)) on  and .Because 0 <  < 1/( 1 ) − 1 and  <  1 ( 1 ), actually we just need to how 1/( 1 ) − 1 and  1 ( 1 ) change along with   (⋅) and Var(  (⋅)).In Table 2, both the rows and the columns indicate that a worse market state with a smaller excess return   (1) or a larger variance Var(  (1)) 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 Var(  ()),  = 2, 3, and we obtain similar results.That is, a worse market environment leads to a larger interval [0, 1/(⋅) − 1].As for  1 (⋅), Table 2 shows that  1 has almost no change regardless of the excess return and the variance.When there is only one market state, referring to (34),  1 can be simplified as  1 = (/(1 − )) 1 =  1 ∏  =1    , which shows that the investor will regard the value less than the risk-free return of the initial wealth  1 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  1 = 10 and    = 1.0135 into  1 ∏  =1    yields  1 ≈ 11.7459.These findings suggest that the value of  1 in Table 2 is roughly equal to  1 ∏  =1    .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.

Influence of the Transition Matrix.
In this part, the influence of the transition matrix  is studied.To do this, we change (), the th ( = 1, 2, 3) 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  * 1 (⋅,  1 ) is increasing accordingly.In particular, strategy  * 1 (1,  1 ) at state 1 is increased the most rapidly.However, when the probability of staying at state 2 or state 3 is increased,  * (⋅,  1 ) is decreased accordingly.Table 4 suggests that a high probability of staying at the best market state yields a shorter interval [0, 1/(⋅) − 1], while the maximum value of the disaster level will also not be affected by the transition matrix.

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 0 <  < 1/( 1 ) − 1 and  <  1 , 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 Var(  ()), and the transition matrix .Mathematics analysis indicates that the optimal strategy  *  (,   ), the expected terminal wealth E  1 , 1 (  * +1 ), and the variance of the terminal wealth Var  1 , 1 (  * +1 ) 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   (),  = 1, 2, 3 have a positive effect, while the variances Var(  (),  = 1, 2, 3 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.

Substituting (A.2) into (A.1) yields
Hence, ( 16) and ( 17) hold true for  = .Now, we assume that ( 16) and ( 17) are true for  + 1; then, for , where It is clear that ∏  =+1    () > 0 according to the definition of   ().Hence, the optimal solution of (A.5) exists and is given as In view of the tower property of the expectation, we first have Further, by Lemma 3, we derive ( 1 ) .

D. Proof of Lemma 8
Proof.According to (30), we first have Moreover, according to (D.4) the coefficient of  can be simplified as (D.6)

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

Table 4 :
Influence of the transition matrix  on  and .