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

Authors: Chuangwei Lin; Huiling Wu
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## 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.


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## Body

## 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 Section2, 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 time1 with initial wealth x1>0 and plans to invest her wealth in the financial market for T consecutive periods. Moreover, we assume that the financial market has multiple states 1,2,…,L, and its dynamics are described by a time-homogeneous Markov chain Sn,n=1,2,… where Sn represents the market state at time n. 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 Rn(i) and rnf(i), respectively, the random return of the risky asset and the risk-free return over period n (time interval [n,n+1),n=1,2,…,T) given Sn=i. In this paper, Rn(i) is assumed to be independent of Rm(j) for any given i,j∈1,2,…,L as long as n≠m. The Markov chain and the returns are mutually independent in the following sense:(1)PnSn+1=j,RnSn∈B=PnSn+1=jPnRnSn∈Bfor all B∈BRm+1, j∈S, and n=1,2,…,T, where Pn is the probability based on the information up to time n and B(Rm+1) is the Borel σ-algebra on Rm+1. Furthermore, we use the following notations in this paper.(N1) The transition matrix of the Markov chainSn,n=1,2,… is denoted by Q. The matrix Qk is the kth power of Q. In particular, we define Q0 as an identity matrix.(N2) For any matrixAL×L and any vector aL×1, denote by Ai the ith row of A and ai the ith component of a. Furthermore, let Aa=Aai,jL×L where Aai,j=A(i,j)a(j) and A¯ be a column vector whose ith component is A¯i=∑j=1L‍Ai,j.(N3) If vectorsa,b,c have the same dimension, then a·c/b denotes a vector whose ith entry is a(i)c(i)/b(i) and a2 a vector with a2(i)=a(i)2.(N4)Rnei=Rni-rnfi. rne(i)=ERnei, which is assumed to be nonzero for n=1,2,…,T. hn,gn, and qn(n=1,2,…,T) are L-dimension column vectors whose ith components are, respectively,(2)hni=rnei2ERnei2,gni=rnfiVar⁡RneiERnei2,qni=rnfi2Var⁡RneiERnei2.h~n,h¯n, and α¯n(n=2,3,…,T+1) are column vectors whose ith components are, respectively,(3)h~ni=∏m=nTQgm¯i∏m=nTQqmihn-1i,h¯ni=∏m=nTQgm¯i2∏m=nTQqm¯i2hn-1i,α¯ni=∏m=nTQgm¯i2∏m=nTQqm¯ihn-1i.For the sake of convenience, we set(4)∑k=mnAk=0,ifn<mfor anyAk;∏k=mnQgk=I,ifn<mwhereI is an identity matrix;∏k=mnQgk=Qgm×Qgm+1×⋯×Qgn,ifn≥m.If we defineun as the amount invested in the risky asset at time n and Wnu as the wealth under the strategy u at time n, then the wealth dynamics are(5)Wn+1u=rnfSnWnu+RneSnun,n=1,2,…,T.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 valueγ to exceed the risk control level β. Hence the strategy u subject to PWT+1u≤γ∣S1=i1,W1=x1≤β 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 S1=i1 and the initial wealth x1, we formulate the portfolio selection problem as follows:(6)maxu1,u2,…,uTEi1,x1WT+1us.t.Wn+1u=rnfSnWnu+RneSnun,n=1,2,…,T,PWT+1u≤γ∣S1=i1,W1=x1≤β,where γ stands for the disaster value and 0<β<1 is a given real number representing the risk control level. Referring to Tchebycheff inequality, we have(7)PWT+1u≤γ∣S1=i1,W1=x1≤Vari1,x1WT+1uEi1,x1WT+1u-γ2,Ei1,x1WT+1u>γ.This means that if the upper bound satisfies(8)Vari1,x1WT+1uEi1,x1WT+1u-γ2≤β,then PWT+1u≤γ∣S1=i1,W1=x1≤β. Therefore, we modify the above-mentioned problem as follows:Pγ,βmaxu1,u2,…,uTEi1,x1WT+1us.t.Wn+1u=Rn,0SnWnu+RneSnun,n=1,2,…,T,Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2.When we adopt the Tchebycheff inequality to replace the probabilityPWT+1u≤γ∣S1=i1,W1=x1 by its upper bound Vari1,x1WT+1u/Ei1,x1WT+1u-γ2, 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 γ 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 Pγ,β and analyze the conditions that γ and β satisfy when the optimal strategy exists. In order to solve Pγ,β, we introduce a Lagrangian multiplier ω>0 and formulate the Lagrangian optimal control problem as follows:PLω,γ,βmaxu1,u2,…,uTEi1,x1WT+1u-ωVari1,x1WT+1u+ωβEi1,x1WT+1u-γ2s.t.Wn+1u=rnfSnWnu+RneSnun,n=1,2,…,T.The relationship of Pγ,β and PLω,γ,β is summarized in Lemma 1.Lemma 1.
Denote byuPLω,γ,β and H(ω,γ,β) the optimal strategy and the value function of the problem PLω,γ,β, respectively. If(9)ω∗=argminω>0⁡Hω,γ,βexists, then the optimal value function of Pγ,β is Hω∗,γ,β and the optimal strategy is uPL(ω∗,γ,β).Now, we can obtain the optimal solution ofPγ,β by solving PLω,γ,β. However, the problem PLω,γ,β is not separable in the sense of dynamic programming due to the term Vari,x1(WT+1u). In view of Zhu et al. [42], we construct an auxiliary problem Aλ,ω of PLω,γ,β as follows:Aλ,ωmaxu1,u2,…,uTEi1,x1λWT+1u-ωWT+1u2s.t.Wn+1u=rnfSnWnu+RneSnun,n=1,2,…,T.Define(10)ΠAλ,ω=u∣uis an optimal strategy ofAλ,ω,ΠPLω=u∣uis an optimal strategy ofPLω,γ,β,du=1-2ωβγ+2ω1+βEi1,x1WT+1u,then the relationship between ΠAλ,ω and ΠPL(ω) is summarized in the following lemma.Lemma 2.
For anyuPL∈ΠPLω, uPL∈ΠAduPL,ω; conversely, if u~∈ΠAλ,ω, then a necessary condition for u~∈ΠPLω is(11)λ=1-2ωβγ+2ω1+βEi1,x1WT+1u~.The proof of Lemma2 is similar to that of Zhu et al. [42]; thus, it is omitted here. Lemma 2 implies that ΠPLω⊆⋃λΠA(λ,ω). We can obtain the optimal strategy of the problem PLω,γ,β by first solving the auxiliary problem Aλ,ω and then finding a suitable λ∗ that can make u∈ΠAλ∗,ω become the optimal strategy of problem PLω,γ,β. The second part of Lemma 2 gives the necessary condition that λ∗ should satisfy. In the next section, we shall solve the auxiliary problem Aλ,ω, which is separable in the sense of dynamic programming.


## 3. Solution to ProblemsAλ,ω and PLω,γ,β

We first introduce Lemma3 to solve Aλ,ω, and the proof can be found in Wu et al. [37].Lemma 3.
Given anyn-k(k≥1) vectors ak+1=ak+1(j),j∈S,…,an=an(j),j∈S, one has(12)E∏l=k+1nalSl∣Sk=i=Qak+1Qak+2⋯Qan¯i,n=k+1,k+2,….Now we define the value functions(13)fni,xn=maxun,…,uTEλWT+1u-ωWT+1u2∣Sn=i,Wn=xn. Then, according to Bellman’s principle of optimality, we have(14)fni,xn=maxun⁡Efn+1Sn+1,Wn+1un∣Sn=i,Wn=xn=maxun∑j∈SQi,jEfn+1j,rnfixn+Rneiunfor n=1,…,T, with the boundary condition(15)fT+1i,xT+1=λxT+1-ωxT+12.According to the recursive formulas (14)-(15), we have Theorem 4.Theorem 4.
The value function of problemAλ,ω is given by(16)fni,xn=-ωqni∏m=n+1TQqm¯ixn2+λgni∏m=n+1TQgm¯ixn+λ24ωα¯n+1i+λ24ω∑k=n+2T+1Qk-n+2Qα¯k¯i,and the corresponding optimal strategy is given by(17)u~ni,xn=rneiERnei2λ2ω∏m=n+1TQgm¯i∏m=n+1TQqm¯i-rnfixnfor n=1,2,…,T and i=1,2,…,L.Proof.
See AppendixA.In order to derive the solution of problemPLω,γ,β, we give the explicit expressions for Ei1,x1WT+1u~ and Ei1,x1(WT+1u~)2 in the following theorem.Theorem 5.
Under the optimal strategy (17) of the auxiliary problem,(18)Ei1,x1Wnu~=g1i1∏k=2n-1Qgk¯i1x1+λ2ω∑k=2nQk-2h~k·∏l=kn-1Qgl¯i1,Ei1,x1Wnu~2=q1i1∏k=2n-1Qqk¯i1x12+λ24ω2∑k=2nQk-2h¯k·∏l=kn-1Qql¯i1,n=2,3,…,T+1.Proof.
See AppendixB.In order to obtain the optimal strategy for the problemPLω,γ,β, we need to summarize some properties of the coefficient of Ei1,x1(WT+1u~) and Ei1,x1(WT+1u~)2. In view of the notations of h¯n(i) and h~n(i), we first obtain(19)∑k=2T+1Qk-2h¯k·∏l=kTQql¯i1=∑k=2T+1Qk-2i1h¯k·∏l=kTQql¯=∑k=2T+1Qk-2i1∏l=kTQgl¯2∏l=kTQql¯·hk-1,∑k=2T+1Qk-2h~k·∏l=kn-1Qgl¯i1=∑k=2T+1Qk-2i1h~k·∏l=kTQgl¯=∑k=2T+1Qk-2i1∏l=kTQgl¯2∏l=kTQql¯·hk-1.Therefore,(20)∑k=2T+1Qk-2h¯k·∏l=kTQql¯i1=∑k=2T+1Qk-2h~k·∏l=kn-1Qgl¯i1. For convenience, denote(21)ϕi1=g1i1∏k=2TQgk¯i1,φi1=q1i1∏k=2TQqk¯i1,ξi1=∑k=2T+1Qk-2h~k·∏l=kTQgl¯i1. Then, by Theorem 5, Ei1,x1WT+1u~ and Ei1,x1WT+1u~2 can be written as(22)Ei1,x1WT+1u~=ϕi1x1+λ2ωξi1,Ei1,x1WT+1u~2=φi1x12+λ24ω2ξi1.Lemma 6.
0 < ξ ( i 1 ) < 1, (1-ξ(i1))φ(i1)-(ϕ(i1))2≥0.Proof.
See AppendixC.Now, we begin to seek the optimal strategy of the problemPLω,γ,β. To this end, we define a function Uλ as follows:(23)Uλ=Ei1,x1WT+1u~-ωVari1,x1WT+1u~+ωβEi1,x1WT+1u~-γ2=1-2ωβγEi1,x1WT+1u~+ω1+βEi1,x1WT+1u~2-ωEi1,x1WT+1u~2+ωβγ2.By (22), we have(24)Uλ=1-2ωβγϕi1x1+λ2ωξi1+ω1+βϕi1x1+λ2ωξi12-ωφi1x12+λ24ω2ξi1+ωβγ2.Differentiating (24) with respect to λ, we obtain(25)
U
′
λ
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1
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2
ω
β
γ
2
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.The optimal strategy of the problem PLω,γ,β exists if and only if U′′(λ)<0, that is, (1+β)ξ(i1)<1. Otherwise, the optimal solution of the problem PLω,γ,β does not exist. In view of Lemma 6, when the probability β satisfies(26)0<β<1ξi1-1,the optimal solution for the problem PLω,γ,β exists. When (26) holds, let U′(λ)=0, and then we derive the optimal solution of maxλ∈RU(λ) at(27)λ∗=11-1+βξi1+2ω1+βϕi1x1-βγ1-1+βξi1≔c1i1+2ωc2i1.We have verified that solving equation(28)λ=1-2ωβγ+2ω1+βEi1,x1WT+1u~in Lemma 2 yields the same expression of λ∗ as (27). Substituting (27) back into (17) gives the optimal policy of problem PLω,γ,β, which is summarized in the following theorem.Theorem 7.
The optimal strategy of the problemPLω,γ,β is given by(29)unPLi,xn=rneiERnei2λ∗2ω∏m=n+1TQgm¯i∏m=n+1TQqm¯i-rnfixn=rneiERnei212ω11-1+βξi1∏m=n+1TQgm¯i∏m=n+1TQqm¯i+1+βϕi1x1-βγ1-1+βξi1∏m=n+1TQgm¯i∏m=n+1TQqm¯i-rnfixnfor n=1,2,…,T and i=1,2,…,L.In the next section, we shall seek the optimal strategy of the problemPγ,β referring to the relationship between PLω,γ,β and Pγ,β.


## 4. Optimal Solution of ProblemPγ,β

We define a functionΓ(ω) by substituting (27) into (24) as follows:(30)
Γ
ω
=
1
-
2
ω
β
γ
ϕ
i
1
x
1
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c
1
i
1
+
2
ω
c
2
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1
2
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ξ
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1
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ϕ
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ω
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.Substituting c1(i1) and c2(i1) into (30) results in Lemma 8.Lemma 8.
(31) F ω = ϕ i 1 x 1 - β γ ξ i 1 1 - 1 + β ξ i 1 + ξ i 1 1 - 1 + β ξ i 1 1 4 ω + y γ 1 - 1 + β ξ i 1 ω ,where(32)yγ=β1-ξi1γ2-2βϕi1x1γ+1+βϕi12-1-1+βξi1φi1x12.Proof.
See AppendixD.According to Lemma6 and (26), the coefficient of 1/4ω is strictly greater than 0. Then, the formula (31) implies that the finite minimum value of Γω exists in ω>0 if and only if y(γ)>0. If y(γ)≤0 for some γ, then Γω is a decreasing function with respect to ω>0, and then the minimum value of Γ(ω) does not exist. This means that the prespecified critical return γ has to satisfy specific conditions so that the problem Pγ,β has the optimal solution. Considering that y(γ) is a quadratic curve with respect to γ, we define(33)Δ=4β2ϕi12x12-4β1-ξi11+βϕi12-1-1+βξi1φi1x12=4β1-1+βξi1φi1-ϕi12-ξi1φi1x12,which is not less than zero by (26) and Lemma 6. Hence, the roots of yγ=0 exist and are given as(34)y1=βϕi1-1-1+βξi1φi1-ϕi12-ξi1φi1β1-ξi1x1,(35)y2=βϕi1+1-1+βξi1φi1-ϕi12-ξiφi1β1-ξi1x1.When(36)γ∈γ∣γ<y1∪γ∣γ>y2,by solving the equation Γ′(ω)=0, ω∗=argminω>0Γ(ω) exists and is given as(37)ω∗=12ξi1yγ.Then, the corresponding optimal value of Pγ,β at ω∗ is(38)minω>0Γω=ξi1yγ+ϕi1x1-βγξi11-1+βξi1.Referring to (27) and (37), we have(39)λ∗2ω∗=c1i12ω∗+c2i1=c1i1yγξi1+c1i11+βϕi1x1-βγ.It first follows from Theorem 5 that(40)Ei1,x1WT+1u~λ∗ω∗=ϕi1x1+λ∗2ω∗ξi1.Consequently, according to (39), we obtain(41)Ei1,x1WT+1u~λ∗ω∗=ϕi1x1+c1i1yγξi1+c1i11+βϕi1x1-βγξi1=ϕi1x1-ϕi11+βξi1x1+1+βϕi1x1-βγξi1+yγξi11-1+βξi1=yγξi1+ϕi1x1-βγξi11-1+βξi1. Now, we have verified that(42)minω>0Γω=Ei1,x1WT+1u~λ∗ω∗=yγξi1+ϕi1x1-βγξi11-1+βξi1.As mentioned in Section 2, γ also satisfies the condition (43)γ<Ei1,x1WT+1u~λ∗ω∗.Together with (36), we have(44)γ∈γ∣γ<y1∪γ∣γ>y2∩γ∣γ<Ei1,x1WT+1u~λ∗ω∗,which we will make further efforts to simplify later.Lemma 9.
Condition (44) can be reduced to γ∈γ∣γ<y1.Proof.
See AppendixE.Under this circumstance, the optimal strategy and optimal value of problemPγ,β are summarized in the following theorem.Theorem 10.
Whenγ∈γ∣γ<y1 and 0<β<min1,1/ξi1-1, the optimal strategy of Pγ,β is(45)un∗i,xn=rneiERnei2λ∗2ω∗∏m=n+1TQgm¯i∏m=n+1TQqm¯i-rnfixn=rneiERnei2yγξi111-1+βξi1∏m=n+1TQgm¯i∏m=n+1TQqm¯i+1+βϕi1x1-βγ1-1+βξi1∏m=n+1TQgm¯i∏m=n+1TQqm¯i-rnfixn,where λ∗ and ω∗ satisfy (27) and (37), respectively, and the corresponding optimal value satisfies (38).Now, we tend to derive the variance of the terminal wealth underu∗ in Theorem 10. According to (39) and (C.2), the variance of the terminal wealth under u∗ is given as(46)
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2
+
φ
i
1
-
ϕ
i
1
2
-
ξ
i
1
φ
i
1
1
-
ξ
i
1
x
1
2
=
ξ
i
1
1
-
ξ
i
1
c
1
i
1
y
γ
ξ
i
1
+
c
1
i
1
1
+
β
ϕ
i
1
x
1
-
β
γ
-
ϕ
i
1
x
1
1
-
ξ
i
1
2
+
φ
i
1
-
ϕ
i
1
2
-
ξ
i
1
φ
i
1
1
-
ξ
i
1
x
1
2
=
ξ
i
1
1
-
ξ
i
1
c
1
i
1
y
γ
ξ
i
1
+
c
1
i
1
β
ϕ
i
1
x
1
1
-
ξ
i
1
-
γ
2
+
φ
i
1
-
ϕ
i
1
2
-
ξ
i
1
φ
i
1
1
-
ξ
i
1
x
1
2
.Moreover, according to (40), we have λ∗/2ω∗=Ei1,x1WT+1u∗-ϕ(i1)x1/ξ(i1). Thus, the relationship between the expected terminal wealth and the terminal risk is given as(47)Vari1,x1WT+1u∗=ξi11-ξi1Ei1,x1WT+1u∗-ϕi1x1ξi1-ϕi1x11-ξi12+φi1-ϕi12-ξi1φi11-ξi1x12=1-ξi1ξi1Ei1,x1WT+1u∗-ϕi1x11-ξi12+φi1-ϕi12-ξi1φi11-ξi1x12.


## 5. Analysis of the Obtained Results

### 5.1. Effects ofγ and β

In view of Theorem10, the risk control level β should be less than a given value; otherwise, the maximum value of the problem Pγ,β can be positive infinite. In addition, to guarantee the existence of the optimal strategy of Pγ,β, the disaster level γ also needs to be less than y1. An intuitive understanding of these results is that if β is large enough, then the constraint to the strategy might disappear. In other words, the condition Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2 might hold naturally when β 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 Pγ,β might be positive infinite. When 0<β<min1,1/ξ(i1)-1 holds, the disaster level γ should satisfy γ<y1. Otherwise, Pγ,β 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 γ should not be a very large number. On the other hand, if the value of γ is very large, the probability that the wealth is less than γ is very high so that there might be no strategy satisfying the condition PWT+1u≤γ∣S1=i1,W1=x1≤β. Because(48)PWT+1u≤γ∣S1=i1,W1=x1≤Vari1,x1WT+1uEi1,x1WT+1u-γ2, there is also no strategy u satisfying the condition(49)Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2.In what follows, we shall analyze the effects of the disaster levelγ and the risk control level β on the optimal strategy un∗ in Theorem 10, the expected terminal wealth Ei1,x1WT+1u∗ and the variance of the terminal wealth Vari1,x1WT+1u∗. The obtained results are summarized in Theorems 11-12.Theorem 11.
Whenγ<y1, the optimal strategy un∗ of the problem Pγ,β, the expected terminal wealth Ei1,x1WT+1u∗, and the variance of the terminal wealth Vari1,x1WT+1u∗ are decreasing along with the disaster level γ.Proof.
See AppendixF.We will explain the results in Theorem11. When γ is a very small number and especially when γ is negative and small enough, the constraint Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2 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 Vari1,x1WT+1u∗ when γ is smaller.Theorem 12.
Whenγ<y1 and 0<β<min1,1/ξ(i1)-1, the optimal strategy un∗ of the problem Pγ,β, the expected terminal wealth Ei1,x1WT+1u∗, and the variance of the terminal wealth Vari1,x1WT+1u∗ are increasing along with the risk control level β.Proof.
See AppendixG.When the risk control levelβ 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 β and γ 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,re(·), and Var(Re(·)) at each state affect the investment strategy, the disaster level γ, and the risk control level β. Suppose that there are three market states; the initial wealth x1 is 10; the investor adjusts the strategy every three months and there are 12 time periods; that is, T=12. 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 rei the expected excess return at state i and Var(Re(i)) the variance of the excess return at state i.


#### 5.2.1. Effects ofre· and Var(Re(·))

In this part, we first study the strategy at the initial time as an example to show the effects ofre(·) and Var(Re(·)). Then, their influence on β and γ is also investigated. To this end, let re(1)=0.30, re(2)=0.2, re(3)=0.1, Var(Re(1))=0.4, Var(Re(2))=0.4, Var(Re(3))=0.3, and the transition matrix be Q=0.30.40.30.40.30.30.30.30.4. 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 ofre· and VarRe·. To this end, we assume that β=0.2, γ=11, and increase re(1) 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 VarRe1 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 u1∗·,x1 is increasing along with re1 and is decreasing with respect to Var(Re(1)). Moreover, Table 1 also shows that (a) u1∗·,x1 is very sensitive to the change in re(1) and Var(Re1); (b) u1∗(1,x1), the optimal investment strategy at state 1, is the most sensitive to the change in re(1) and VarRne1. For example, when re1 is increased from 0.3 to 0.31, that is, the growth rate is (0.31-0.3)/0.3=3.33%, the growth rates of the investment strategy at each state are (2.1485-1.8123)/1.8123=18.55%, 1.1707-1.0666/1.0666=9.76%, (0.6977-0.6456)/0.6456=8.07%, respectively. When the growth rate of VarRe1 is 0.41-0.4/0.4=2.5%, the decrement rates of the investment strategy at each state are 6.76%, 3.16%, and 2.66%, respectively. As for the impact of re(i) and VarRei,i=2,3, here we emphasize that two similar experiments have been conducted and the results also indicate that u1∗i,x1 is the most sensitive to the change in re(i) and VarRei,i=2,3.Table 1
Influence ofre(1) and Var(Re(1)) on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) r e ( 1 ) = 0.30 1.8123 1.0666 0.6456 r e ( 1 ) = 0.31 2.1485 1.1707 0.6977 r e ( 1 ) = 0.32 2.6113 1.2977 0.7593 r e ( 1 ) = 0.33 3.2883 1.4555 0.8330 V a r ( R e ( 1 ) ) = 0.40 1.8123 1.0666 0.6456 V a r ( R e ( 1 ) ) = 0.41 1.6897 1.0329 0.6284 V a r ( R e ( 1 ) ) = 0.42 1.5824 1.0024 0.6127 V a r ( R e ( 1 ) ) = 0.43 1.4878 0.9747 0.5984Second, we want to know the influence ofre(·) and Var(Re(·)) on γ and β. Because 0<β<1/ξ(i1)-1 and γ<y1(i1), actually we just need to observe how 1/ξ(i1)-1 and y1(i1) change along with re(·) and Var(Re(·)). In Table 2, both the rows and the columns indicate that a worse market state with a smaller excess return re1 or a larger variance VarRe1 results in a longer interval, which β 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 re(i) and Var(Re(i)), i=2,3, and we obtain similar results. That is, a worse market environment leads to a larger interval 0,1/ξ·-1. As for y1·, Table 2 shows that y1 has almost no change regardless of the excess return and the variance. When there is only one market state, referring to (34), y1 can be simplified as y1=ϕ/1-ξx1=x1∏n=1T‍rnf, which shows that the investor will regard the value less than the risk-free return of the initial wealth x1 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 x1=10 and rnf=1.0135 into x1∏n=1T‍rnf yields y1≈11.7459. These findings suggest that the value of y1 in Table 2 is roughly equal to x1∏n=1T‍rnf. 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 ofre(1) and Var(Re(1)) on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) r e ( 1 ) = 0.30 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.31 ( 0.3101,0.3589,0.4016 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.32 ( 0.2891,0.3390,0.3793 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.33 ( 0.2696,0.3202,0.3583 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.40 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.41 ( 0.3414,0.3882,0.4345 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.42 ( 0.3501,0.3962,0.4435 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.43 ( 0.3586,0.4041,0.4523 ) ( 11.7459,11.7459,11.7459 )


#### 5.2.2. Influence of the Transition Matrix

In this part, the influence of the transition matrixQ is studied. To do this, we change Q(i), the ith (i=1,2,3) row of Q, 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 u1∗(·,x1) is increasing accordingly. In particular, strategy u1∗(1,x1) at state 1 is increased the most rapidly. However, when the probability of staying at state 2 or state 3 is increased, u∗(·,x1) is decreased accordingly. Table 4 suggests that a high probability of staying at the best market state yields a shorter interval [0,1/ξ(·)-1], while the maximum value of the disaster level will also not be affected by the transition matrix.Table 3
Influence of the transition matrixQ on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) Q ( 1 ) = [ 0.3,0.4,0.3 ] 1.8123 1.0666 0.6456 Q ( 1 ) = [ 0.5,0.4,0.1 ] 3.2986 1.4842 0.8450 Q ( 1 ) = [ 0.7,0.2,0.1 ] 12.1759 2.2759 1.1666  Q ( 2 ) = [ 0.4,0.3,0.3 ] 1.8123 1.0666 0.6456 Q ( 2 ) = [ 0.3,0.5,0.2 ] 1.7873 1.0521 0.6395 Q ( 2 ) = [ 0.2,0.7,0.1 ] 1.7496 1.0301 0.6302  Q ( 3 ) = [ 0.3,0.3,0.4 ] 1.8123 1.0666 0.6456 Q ( 3 ) = [ 0.3,0.2,0.5 ] 1.6685 0.9969 0.5980 Q ( 3 ) = [ 0.3,0.1,0.6 ] 1.5168 0.9214 0.5471Table 4
Influence of the transition matrixQ on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) Q ( 1 ) = [ 0.3,0.4,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.5,0.4,0.1 ] ( 0.2650,0.3173,0.3554 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.7,0.2,0.1 ] ( 0.2161,0.2707,0.3036 ) ( 11.7459,11.7459,11.7459 )  Q ( 2 ) = [ 0.4,0.3,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.3,0.5,0.2 ] ( 0.3351,0.3835,0.4285 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.2,0.7,0.1 ] ( 0.3388,0.3889,0.4335 ) ( 11.7459,11.7459,11.7459 )  Q ( 3 ) = [ 0.3,0.3,0.4 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.2,0.5 ] ( 0.3475,0.3977,0.4525 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.1,0.6 ] ( 0.3673,0.4212,0.4901 ) ( 11.7459,11.7459,11.7459 )


## 5.1. Effects ofγ and β

In view of Theorem10, the risk control level β should be less than a given value; otherwise, the maximum value of the problem Pγ,β can be positive infinite. In addition, to guarantee the existence of the optimal strategy of Pγ,β, the disaster level γ also needs to be less than y1. An intuitive understanding of these results is that if β is large enough, then the constraint to the strategy might disappear. In other words, the condition Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2 might hold naturally when β 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 Pγ,β might be positive infinite. When 0<β<min1,1/ξ(i1)-1 holds, the disaster level γ should satisfy γ<y1. Otherwise, Pγ,β 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 γ should not be a very large number. On the other hand, if the value of γ is very large, the probability that the wealth is less than γ is very high so that there might be no strategy satisfying the condition PWT+1u≤γ∣S1=i1,W1=x1≤β. Because(48)PWT+1u≤γ∣S1=i1,W1=x1≤Vari1,x1WT+1uEi1,x1WT+1u-γ2, there is also no strategy u satisfying the condition(49)Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2.In what follows, we shall analyze the effects of the disaster levelγ and the risk control level β on the optimal strategy un∗ in Theorem 10, the expected terminal wealth Ei1,x1WT+1u∗ and the variance of the terminal wealth Vari1,x1WT+1u∗. The obtained results are summarized in Theorems 11-12.Theorem 11.
Whenγ<y1, the optimal strategy un∗ of the problem Pγ,β, the expected terminal wealth Ei1,x1WT+1u∗, and the variance of the terminal wealth Vari1,x1WT+1u∗ are decreasing along with the disaster level γ.Proof.
See AppendixF.We will explain the results in Theorem11. When γ is a very small number and especially when γ is negative and small enough, the constraint Vari1,x1WT+1u≤βEi1,x1WT+1u-γ2 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 Vari1,x1WT+1u∗ when γ is smaller.Theorem 12.
Whenγ<y1 and 0<β<min1,1/ξ(i1)-1, the optimal strategy un∗ of the problem Pγ,β, the expected terminal wealth Ei1,x1WT+1u∗, and the variance of the terminal wealth Vari1,x1WT+1u∗ are increasing along with the risk control level β.Proof.
See AppendixG.When the risk control levelβ 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 β and γ 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,re(·), and Var(Re(·)) at each state affect the investment strategy, the disaster level γ, and the risk control level β. Suppose that there are three market states; the initial wealth x1 is 10; the investor adjusts the strategy every three months and there are 12 time periods; that is, T=12. 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 rei the expected excess return at state i and Var(Re(i)) the variance of the excess return at state i.


### 5.2.1. Effects ofre· and Var(Re(·))

In this part, we first study the strategy at the initial time as an example to show the effects ofre(·) and Var(Re(·)). Then, their influence on β and γ is also investigated. To this end, let re(1)=0.30, re(2)=0.2, re(3)=0.1, Var(Re(1))=0.4, Var(Re(2))=0.4, Var(Re(3))=0.3, and the transition matrix be Q=0.30.40.30.40.30.30.30.30.4. 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 ofre· and VarRe·. To this end, we assume that β=0.2, γ=11, and increase re(1) 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 VarRe1 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 u1∗·,x1 is increasing along with re1 and is decreasing with respect to Var(Re(1)). Moreover, Table 1 also shows that (a) u1∗·,x1 is very sensitive to the change in re(1) and Var(Re1); (b) u1∗(1,x1), the optimal investment strategy at state 1, is the most sensitive to the change in re(1) and VarRne1. For example, when re1 is increased from 0.3 to 0.31, that is, the growth rate is (0.31-0.3)/0.3=3.33%, the growth rates of the investment strategy at each state are (2.1485-1.8123)/1.8123=18.55%, 1.1707-1.0666/1.0666=9.76%, (0.6977-0.6456)/0.6456=8.07%, respectively. When the growth rate of VarRe1 is 0.41-0.4/0.4=2.5%, the decrement rates of the investment strategy at each state are 6.76%, 3.16%, and 2.66%, respectively. As for the impact of re(i) and VarRei,i=2,3, here we emphasize that two similar experiments have been conducted and the results also indicate that u1∗i,x1 is the most sensitive to the change in re(i) and VarRei,i=2,3.Table 1
Influence ofre(1) and Var(Re(1)) on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) r e ( 1 ) = 0.30 1.8123 1.0666 0.6456 r e ( 1 ) = 0.31 2.1485 1.1707 0.6977 r e ( 1 ) = 0.32 2.6113 1.2977 0.7593 r e ( 1 ) = 0.33 3.2883 1.4555 0.8330 V a r ( R e ( 1 ) ) = 0.40 1.8123 1.0666 0.6456 V a r ( R e ( 1 ) ) = 0.41 1.6897 1.0329 0.6284 V a r ( R e ( 1 ) ) = 0.42 1.5824 1.0024 0.6127 V a r ( R e ( 1 ) ) = 0.43 1.4878 0.9747 0.5984Second, we want to know the influence ofre(·) and Var(Re(·)) on γ and β. Because 0<β<1/ξ(i1)-1 and γ<y1(i1), actually we just need to observe how 1/ξ(i1)-1 and y1(i1) change along with re(·) and Var(Re(·)). In Table 2, both the rows and the columns indicate that a worse market state with a smaller excess return re1 or a larger variance VarRe1 results in a longer interval, which β 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 re(i) and Var(Re(i)), i=2,3, and we obtain similar results. That is, a worse market environment leads to a larger interval 0,1/ξ·-1. As for y1·, Table 2 shows that y1 has almost no change regardless of the excess return and the variance. When there is only one market state, referring to (34), y1 can be simplified as y1=ϕ/1-ξx1=x1∏n=1T‍rnf, which shows that the investor will regard the value less than the risk-free return of the initial wealth x1 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 x1=10 and rnf=1.0135 into x1∏n=1T‍rnf yields y1≈11.7459. These findings suggest that the value of y1 in Table 2 is roughly equal to x1∏n=1T‍rnf. 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 ofre(1) and Var(Re(1)) on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) r e ( 1 ) = 0.30 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.31 ( 0.3101,0.3589,0.4016 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.32 ( 0.2891,0.3390,0.3793 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.33 ( 0.2696,0.3202,0.3583 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.40 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.41 ( 0.3414,0.3882,0.4345 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.42 ( 0.3501,0.3962,0.4435 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.43 ( 0.3586,0.4041,0.4523 ) ( 11.7459,11.7459,11.7459 )


### 5.2.2. Influence of the Transition Matrix

In this part, the influence of the transition matrixQ is studied. To do this, we change Q(i), the ith (i=1,2,3) row of Q, 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 u1∗(·,x1) is increasing accordingly. In particular, strategy u1∗(1,x1) at state 1 is increased the most rapidly. However, when the probability of staying at state 2 or state 3 is increased, u∗(·,x1) is decreased accordingly. Table 4 suggests that a high probability of staying at the best market state yields a shorter interval [0,1/ξ(·)-1], while the maximum value of the disaster level will also not be affected by the transition matrix.Table 3
Influence of the transition matrixQ on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) Q ( 1 ) = [ 0.3,0.4,0.3 ] 1.8123 1.0666 0.6456 Q ( 1 ) = [ 0.5,0.4,0.1 ] 3.2986 1.4842 0.8450 Q ( 1 ) = [ 0.7,0.2,0.1 ] 12.1759 2.2759 1.1666  Q ( 2 ) = [ 0.4,0.3,0.3 ] 1.8123 1.0666 0.6456 Q ( 2 ) = [ 0.3,0.5,0.2 ] 1.7873 1.0521 0.6395 Q ( 2 ) = [ 0.2,0.7,0.1 ] 1.7496 1.0301 0.6302  Q ( 3 ) = [ 0.3,0.3,0.4 ] 1.8123 1.0666 0.6456 Q ( 3 ) = [ 0.3,0.2,0.5 ] 1.6685 0.9969 0.5980 Q ( 3 ) = [ 0.3,0.1,0.6 ] 1.5168 0.9214 0.5471Table 4
Influence of the transition matrixQ on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) Q ( 1 ) = [ 0.3,0.4,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.5,0.4,0.1 ] ( 0.2650,0.3173,0.3554 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.7,0.2,0.1 ] ( 0.2161,0.2707,0.3036 ) ( 11.7459,11.7459,11.7459 )  Q ( 2 ) = [ 0.4,0.3,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.3,0.5,0.2 ] ( 0.3351,0.3835,0.4285 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.2,0.7,0.1 ] ( 0.3388,0.3889,0.4335 ) ( 11.7459,11.7459,11.7459 )  Q ( 3 ) = [ 0.3,0.3,0.4 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.2,0.5 ] ( 0.3475,0.3977,0.4525 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.1,0.6 ] ( 0.3673,0.4212,0.4901 ) ( 11.7459,11.7459,11.7459 )


## 5.2.1. Effects ofre· and Var(Re(·))

In this part, we first study the strategy at the initial time as an example to show the effects ofre(·) and Var(Re(·)). Then, their influence on β and γ is also investigated. To this end, let re(1)=0.30, re(2)=0.2, re(3)=0.1, Var(Re(1))=0.4, Var(Re(2))=0.4, Var(Re(3))=0.3, and the transition matrix be Q=0.30.40.30.40.30.30.30.30.4. 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 ofre· and VarRe·. To this end, we assume that β=0.2, γ=11, and increase re(1) 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 VarRe1 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 u1∗·,x1 is increasing along with re1 and is decreasing with respect to Var(Re(1)). Moreover, Table 1 also shows that (a) u1∗·,x1 is very sensitive to the change in re(1) and Var(Re1); (b) u1∗(1,x1), the optimal investment strategy at state 1, is the most sensitive to the change in re(1) and VarRne1. For example, when re1 is increased from 0.3 to 0.31, that is, the growth rate is (0.31-0.3)/0.3=3.33%, the growth rates of the investment strategy at each state are (2.1485-1.8123)/1.8123=18.55%, 1.1707-1.0666/1.0666=9.76%, (0.6977-0.6456)/0.6456=8.07%, respectively. When the growth rate of VarRe1 is 0.41-0.4/0.4=2.5%, the decrement rates of the investment strategy at each state are 6.76%, 3.16%, and 2.66%, respectively. As for the impact of re(i) and VarRei,i=2,3, here we emphasize that two similar experiments have been conducted and the results also indicate that u1∗i,x1 is the most sensitive to the change in re(i) and VarRei,i=2,3.Table 1
Influence ofre(1) and Var(Re(1)) on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) r e ( 1 ) = 0.30 1.8123 1.0666 0.6456 r e ( 1 ) = 0.31 2.1485 1.1707 0.6977 r e ( 1 ) = 0.32 2.6113 1.2977 0.7593 r e ( 1 ) = 0.33 3.2883 1.4555 0.8330 V a r ( R e ( 1 ) ) = 0.40 1.8123 1.0666 0.6456 V a r ( R e ( 1 ) ) = 0.41 1.6897 1.0329 0.6284 V a r ( R e ( 1 ) ) = 0.42 1.5824 1.0024 0.6127 V a r ( R e ( 1 ) ) = 0.43 1.4878 0.9747 0.5984Second, we want to know the influence ofre(·) and Var(Re(·)) on γ and β. Because 0<β<1/ξ(i1)-1 and γ<y1(i1), actually we just need to observe how 1/ξ(i1)-1 and y1(i1) change along with re(·) and Var(Re(·)). In Table 2, both the rows and the columns indicate that a worse market state with a smaller excess return re1 or a larger variance VarRe1 results in a longer interval, which β 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 re(i) and Var(Re(i)), i=2,3, and we obtain similar results. That is, a worse market environment leads to a larger interval 0,1/ξ·-1. As for y1·, Table 2 shows that y1 has almost no change regardless of the excess return and the variance. When there is only one market state, referring to (34), y1 can be simplified as y1=ϕ/1-ξx1=x1∏n=1T‍rnf, which shows that the investor will regard the value less than the risk-free return of the initial wealth x1 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 x1=10 and rnf=1.0135 into x1∏n=1T‍rnf yields y1≈11.7459. These findings suggest that the value of y1 in Table 2 is roughly equal to x1∏n=1T‍rnf. 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 ofre(1) and Var(Re(1)) on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) r e ( 1 ) = 0.30 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.31 ( 0.3101,0.3589,0.4016 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.32 ( 0.2891,0.3390,0.3793 ) ( 11.7459,11.7459,11.7459 ) r e ( 1 ) = 0.33 ( 0.2696,0.3202,0.3583 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.40 ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.41 ( 0.3414,0.3882,0.4345 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.42 ( 0.3501,0.3962,0.4435 ) ( 11.7459,11.7459,11.7459 ) V a r ( R e ( 1 ) ) = 0.43 ( 0.3586,0.4041,0.4523 ) ( 11.7459,11.7459,11.7459 )


## 5.2.2. Influence of the Transition Matrix

In this part, the influence of the transition matrixQ is studied. To do this, we change Q(i), the ith (i=1,2,3) row of Q, 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 u1∗(·,x1) is increasing accordingly. In particular, strategy u1∗(1,x1) at state 1 is increased the most rapidly. However, when the probability of staying at state 2 or state 3 is increased, u∗(·,x1) is decreased accordingly. Table 4 suggests that a high probability of staying at the best market state yields a shorter interval [0,1/ξ(·)-1], while the maximum value of the disaster level will also not be affected by the transition matrix.Table 3
Influence of the transition matrixQ on the strategy.
u 1 ∗ ( 1 , x 1 ) u 1 ∗ ( 2 , x 1 ) u1∗(3,x1) Q ( 1 ) = [ 0.3,0.4,0.3 ] 1.8123 1.0666 0.6456 Q ( 1 ) = [ 0.5,0.4,0.1 ] 3.2986 1.4842 0.8450 Q ( 1 ) = [ 0.7,0.2,0.1 ] 12.1759 2.2759 1.1666  Q ( 2 ) = [ 0.4,0.3,0.3 ] 1.8123 1.0666 0.6456 Q ( 2 ) = [ 0.3,0.5,0.2 ] 1.7873 1.0521 0.6395 Q ( 2 ) = [ 0.2,0.7,0.1 ] 1.7496 1.0301 0.6302  Q ( 3 ) = [ 0.3,0.3,0.4 ] 1.8123 1.0666 0.6456 Q ( 3 ) = [ 0.3,0.2,0.5 ] 1.6685 0.9969 0.5980 Q ( 3 ) = [ 0.3,0.1,0.6 ] 1.5168 0.9214 0.5471Table 4
Influence of the transition matrixQ on β and γ.
1 / ξ ( ⋅ ) - 1 y1(⋅) Q ( 1 ) = [ 0.3,0.4,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.5,0.4,0.1 ] ( 0.2650,0.3173,0.3554 ) ( 11.7459,11.7459,11.7459 ) Q ( 1 ) = [ 0.7,0.2,0.1 ] ( 0.2161,0.2707,0.3036 ) ( 11.7459,11.7459,11.7459 )  Q ( 2 ) = [ 0.4,0.3,0.3 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.3,0.5,0.2 ] ( 0.3351,0.3835,0.4285 ) ( 11.7459,11.7459,11.7459 ) Q ( 2 ) = [ 0.2,0.7,0.1 ] ( 0.3388,0.3889,0.4335 ) ( 11.7459,11.7459,11.7459 )  Q ( 3 ) = [ 0.3,0.3,0.4 ] ( 0.3327,0.3801,0.4254 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.2,0.5 ] ( 0.3475,0.3977,0.4525 ) ( 11.7459,11.7459,11.7459 ) Q ( 3 ) = [ 0.3,0.1,0.6 ] ( 0.3673,0.4212,0.4901 ) ( 11.7459,11.7459,11.7459 )


## 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 problemPγ,β 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<β<1/ξ(i1)-1 and γ<y1, the optimal strategy un∗(i,xn) of the problem Pγ,β exists and is obtained by the Lagrange multiplier technique and the embedding method. We investigate the effects of the disaster level γ, the risk control level β, the expected excess return re(i), the variance Var(Re(i)), and the transition matrix Q. Mathematics analysis indicates that the optimal strategy un∗(i,xn), the expected terminal wealth Ei1,x1WT+1u∗, and the variance of the terminal wealth Vari1,x1WT+1u∗ are decreasing along with the disaster level γ, while they are increasing with respect to the risk control level β. By numerical analysis, we find the following: (a) the excess returns re(i),i=1,2,3 have a positive effect, while the variances Var(Re(i),i=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 β lies in. However, the disaster value γ 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 β belongs to, while the disaster value γ is also insensitive to the change in the transition matrix.



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