Optimal Control of Investment for an Insurer in Two Currency Markets
OOptimal Control of Investment for an
Insurer in Two Currency Markets
Qianqian Zhou, Junyi Guo + School of Mathematical Sciences, Nankai University, Tianjin, China(Email: [email protected], [email protected])
Abstract
In this paper, we study the optimal investment problem of an insurer whosesurplus process follows the diffusion approximation of the classical Cramer-Lundbergmodel. Investment in the foreign market is allowed, and therefore, the foreign exchangerate model is considered and incorporated. It is assumed that the instantaneous meangrowth rate of foreign exchange rate price follows an Ornstein-Uhlenbeck process. Dy-namic programming method is employed to study the problem of maximizing the ex-pected exponential utility of terminal wealth. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal investment strategies and the value functions areobtained. Finally, numerical analysis is presented.
Keywords:
Cramer-Lundberg model; Exponential utility; Hamilton-Jacobi-Bellmanequation; Optimal investment strategy; Foreign exchange rate
In actuarial science and applied probability, risk theory is a traditional and modern field,which uses mathematical models to describe an insurer’s vulnerability to ruin. In orderto decrease (increase) the risk (profit), the insurance companies are allowed to investtheir wealth into risk-free assets and risky assets. And in recent years, there are manyremarkable works of optimal investment problems. Especially, maximizing the expectedexponential utility function and minimizing the probability of ruin have attracted a sub-stantial amount of interest.Browne [5] used Brownian motion with drift to describe the surplus of the insurancecompany and found the optimal investment strategy to maximize the expected exponentialutility of the terminal wealth and minimize the probability of ruin. Later, Yang and Zhang[22] explored the same optimal investment problem for a risk process modeled by a jumpdiffusion process. In Hipp and Pulm [12], the authors considered a risk process modeled asa compound Poisson process and investigated the optimal investment strategy to minimize
This work is supported by the National Natural Science Foundation of China (Grant No. 11931018)and Tianjin Natural Science Foundation. + Corresponding author. a r X i v : . [ q -f i n . P M ] J un he ruin probability of this model. Liu and Yang [15] generalized the model in Hipp andPulm [12] by including a risk-free asset. And then the optimal investment strategy wasinvestigated. Schmidli [19] also studied the compound Poisson risk model and the optimalinvestment strategy of minimizing its ruin probability. In Bai and Guo [4], they consideredthe optimal problem with multiple risky assets and no-shorting constraint. By solving thecorresponding Hamilton-Jacobi-Bellman equations, the optimal strategies for maximizingthe expected exponential utility and minimizing the ruin probability were obtained. Wang[21] considered the optimal investment strategy to maximize the exponential utility of aninsurance company’s reserve. The claim process was supposed to be a pure jump process,which is not necessarily compound Poisson, and the insurer has the option of investing inmultiple risky assets.However, prior studies did not consider the condition that the insurers are allowed toinvest their wealth in more than one currency market. Thus, in this paper we investigatethe case that the insurance company is allowed to invest its wealth into more than onecurrency market, such that it can invest its wealth into domestic risk-free assets andforeign risky assets.The connection between the domestic currency market and the foreign currency marketis the exchange rate. Foreign exchange rate plays an important role as a tool used toconvert foreign market cash flows into domestic currency. And there are many factorsthat influence exchange rate prices, such as inflation, balance of international payment,interest-rate spread, etc.Inflation is the most important fundamental factor affecting the movements of ex-change rate. If the inflation rate of a domestic country is higher than that of a foreigncountry, the competitiveness of the domestic country’s exports is weakened, while thisincreases the competitiveness of foreign goods in domestic country’s market. It wouldcause the domestic country’s trade balance of payments deficit and the demand of foreignexchange is larger than the supply. Then it leads to the increase of foreign exchange rateprice. Conversely, the price of foreign exchange rate declines.The balance of payments is the direct factor that affects the exchange rate. Forexample, when a country has a large balance of payments surplus, i.e., the country’simports are less than its exports, its currency demand will increase, which will lead toan increase in foreign exchange flowing into the country. In this way, in the foreignexchange market, the supply of foreign exchange is greater than the demand then theforeign exchange rate price goes down. But if a country has a large balance of paymentsdeficit, i.e., the country’s imports are more than its exports, the supply of foreign exchangeis greater than the demand which leads to the increase of foreign exchange rate price.Under certain conditions, interest rates have a great short-term impact on exchangerate price. This effect is caused by the difference of interest rates between differentcountries. In general, if interest- rate spread is increasing then the demand for domesticcurrency increases. This leads to the increase of the domestic currency price. And thusthe foreign exchange rate price declines. Conversely, if the interest- rate spread goes down,the price of foreign exchange rate is increasing.As we all known, the classical geometric Brownian motion is the most commonly usedmodel to describe the dynamics of exchange rate price. In Musiela and Rutkowski [16],the explicit valuation formulas for various kinds of currency and foreign equity optionswere established in which the foreign exchange rate was modeled by means of geometric2rownian motion. Veraart [20] considered an investor in the foreign exchange market whotrades in domestic currency market and foreign currency market with the exchange ratemodeled as a geometric Brownian motion. After that, a more general model of exchangerate was used. In Eisenberg [6], the author considered an insurance company seeking tomaximize the expected discounted dividends whereas the dividends are declared or paidin a foreign currency. It was assumed that the insurance company generates its income ina foreign currency but pays dividends in its home currency. In that paper, the exchangerate was modeled by a geometric L´ e vy process. However, taking into account of a varietyof factors which affect the exchange rate, the classical geometric Brownian motion can notbetter reflect the real dynamics of exchange rate. Thus other models have been created todescribe exchange rate. One of the most popular model is the one in which the interest-rate spread is incorporated into the geometric Brownian motion. For example, in Guo,et al. [11], the authors investigated the optimal strategy of an insurer who invests inboth domestic and foreign markets. They assumed that the domestic and foreign nominalinterest rates are both described by extended Cox-Ingersoll-Ross (CIR) model. And theexchange rate price is modeled by geometric Brownian motion with domestic and foreigninterest rates. For more details of the model of exchange rate price described by geometricBrownian motion with interest rates one can refer [1–3].Although interest-rate spread has a certain impact on exchange rate price and themodels with interest-rate spread are excellent, from the perspective of the basic factorsdetermining the trend of exchange rate fluctuation that the effect of interest- rate spreadis limited. Moreover, from the above descriptions we obtain that the supply and demandof domestic and foreign currencies is the most paramount and direct factor on the changeof exchange rate price. Thus in our paper, the foreign exchange market’s supply anddemand is incorporated into the model of exchange rate price.In Liang, et al. [14] and Rishel [17], they used the model of geometric Brownian motion,where the mean growth rate is given by Ornstein-Uhlenbeck process, to describe thedynamics of risky asset price which can have features of bull and bear markets. Inspiredby the models of risky assets in Liang, et al. [14] and Rishel [17] and the discussions ofthe effect factors of foreign exchange rate, we consider that the foreign exchange rate alsohas the bull and bear markets. That is when the demand of foreign currency is largerthan the supply, the foreign exchange rate price increases, we call it the ”bull foreignexchange rate”. In addition, when the demand of foreign currency is less than the supply,the foreign exchange rate price goes down, we call it the ”bear foreign exchange rate”.The commonly-used model for the exchange rate price is the geometric Brownian motionin which the expected instantaneous rate and the volatility of the exchange rate price areboth constants. This seems to rule out bull and bear markets.In this paper, the price of foreign exchange rate Q t is described by the followingdifferential equation: dQ t = Q t (cid:8) a ( t ) dt + σ Q dW t (cid:9) , Q = q, (1.1)where a ( t ) = u Q + m ( t ) and m ( t ) is given by the Ornstein-Uhlenbeck equation dm ( t ) = αm ( t ) dt + βdW t , m (0) = m . Here u Q , σ Q , q, α, β are known constants and they are all positive except α and β. In(1.1), u Q is the target mean growth rate for the exchange rate. If m ( t ) > , then a ( t ) is3ubstantially larger than u Q , this could be considered as a bull market of exchange rate.Conversely, if m ( t ) < a ( t ) is substantially less than u Q , this could be considered asa bear market of exchange rate. Especially, if a ( t ) < m ( t ) is to let the random mean growth rate of foreign exchange rateprice be close to the target mean growth rate. Once the mean growth rate of foreignexchange rate price is larger than the target mean growth rate for a long time, then m ( t ) < . Otherwise, m ( t ) > . The insurance company is allowed to invest its wealth into domestic and foreign cur-rency markets with the exchange rate price Q t described by (1.1). Our target is tomaximize the expected exponential utility of terminal wealth over all admissible strate-gies.The rest of the paper is organized as follows. The model is described in Section2. Our main results are given in Section 3. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal value functions and optimal strategies are explicitlyderived. In particular, we find that if the insurance company only invests in foreign riskyassets and the price of exchange rate is modeled by geometric Brownian motion then theoptimal investment strategy is a constant, regardless of the level of wealth the companyhas. In the last section, numerical examples and analysis are presented. And we find that,in some cases, investing into two currency markets can produce a higher value functionthan investing into only one currency market. We start from the classical Cramer-Lundberg model in which the surplus of the insurancecompany is modeled as X t = X + pt − N ( t ) (cid:88) i =1 Z i , with X = x, where p is the premium rate, N ( t ) is the Poisson process stating the number of claims,and Z i is a sequence of independent random variables which are identically distributedrepresenting the size of claims. Without the loss of generality we assume that the intensityof the process N ( t ) is 1, then the dynamics of X t can be approximated by dX t = udt + σdW t , X = x, where u = p − E [ Z ] > , σ = E [ Z ] , and W t is a standard Brownian motion. For moredetails of the diffusion approximation of the surplus process, we can refer to [7, 9, 13, 18].The insurance company invests its wealth into domestic risk-free asset and foreignrisky asset. The price of domestic risk-free asset is given by dB dt = B dt r d dt, B d (0) = B d . The foreign risky asset price S ft is modeled by means of geometric Brownian motion suchthat dS ft = S ft ( u f dt + σ f dW t ) , S f (0) = S f . (2.2)4e adopt here the convention that the price S ft is denominated by foreign currencyand the exchange rate is denominated in units of domestic currency per unit of foreigncurrency. This means that Q t represents the domestic price at time t of one unit of theforeign currency. Thus let g t := g ( S ft , Q t ) = Q t S ft , then g t is the price of foreign riskyasset denominated by domestic currency. By Itˆ o (cid:48) s formula and the formulas of (1.1) and(2.2) we find that g t satisfies the following stochastic differential equation dg t = g t (cid:8) ( u f + a ( t )) dt + σ f dW t + σ Q dW t (cid:9) . The total amount of money invested in foreign risky asset at time t is denoted by π t and the rest of the surplus is invested into domestic risk-free asset. Under the strategy π t , the surplus of the insurance company is as follows: dX πt = (cid:8) π t A + u + r d X t + π t m ( t ) (cid:9) dt + σdW t + π t σ f dW t + π t σ Q dW t , (2.3)where A = u f + u Q − r d and the initial surplus is X π = x . Here it is allowed that π t < π t > X πt which means that the company is allowed to short sell the foreignrisky asset and borrow money for investment in foreign risky asset. And we assume that W t , W t , W t , W t are independent standard Brownian motions on the same probabilityspace (Ω , F , P ) . We are going to maximize the expected exponential utility of terminal wealth over alladmissible strategies π t . A strategy π t is said to be admissible, if π · is F t -adapted, where F t is the filtration generated by X πt , and for any T > , E [ (cid:82) T π ( t ) dt ] < ∞ . The set ofall admissible strategies is denoted by Π . Suppose now that the insurance company is interested in maximizing the utility of itsterminal wealth at time T. Denote the utility function as u ( x ) with u (cid:48) ( x ) > u (cid:48)(cid:48) ( x ) < . For a strategy π, the utility attained by the insurer from state x, m at time t is definedas V π ( t, x, m ) = E [ u ( X πT ) | ( X πt , m ( t )) = ( x, m )] . Our objective is to find the optimal value function V ( t, x, m ) = sup π ∈ Π V π ( t, x, m ) (3.4)and the optimal investment strategy π ∗ such that V π ∗ ( t, x, m ) = V ( t, x, m ) . Assume now that the investor has an exponential utility function u ( x ) = λ − γθ e − θx , (3.5)where γ > θ > . The utility function (3.5) plays a remarkable part in insurancemathematics and actuarial practice, since it is the only utility function under which theprinciple of ”zero utility” gives a fair premium that is independent of the level of reserveof an insurance company (see Gerber [10]). 5pplying the dynamic programming approach described in [8], from standard argu-ments, we see that if the optimal value function V ( t, x, m ) and its partial derivatives V t , V x , V xx , V m , V mm are continuous on [0 , T ] × R × R , then V ( t, x, m ) satisfies the fol-lowing Hamilton-Jacobi-Bellman (HJB) equation V t + sup π (cid:8) [ πA + πm + xr d + u ] V x + 12 [ σ + π ( σ f + σ Q )] V xx (cid:9) + αmV m + 12 β V mm = 0 , (3.6)with boundary condition V ( T, x, m ) = u ( x ) . In order to solve the HJB equation (3.6), we first find the value π ( x, m ) which maxi-mizes the function (cid:0) πA + πm + xr d + u (cid:1) V x + 12 [ σ + π ( σ f + σ Q )] V xx (3.7)Differentiating with respect to π in (3.7) the optimizer π ∗ = − A + mσ f + σ Q V x V xx (3.8)is obtained.Assume that HJB equation (3.6) has a classical solution V such that V x > V xx < . Inspired by the form of the solution in [5], we try to find the solution of (3.6) asthe form V ( t, x, m ) = λ − γθ exp (cid:8) − θxe r d ( T − t ) + h ( t, m ) (cid:9) , (3.9)where h ( t, m ) is a suitable function such that (3.9) is a solution of (3.6). And the boundarycondition V ( T, x, m ) = u ( x ) implies that h ( T, m ) = 0 . From (3.9) we can calculate that V t = (cid:2) V ( t, x, m ) − λ (cid:3)(cid:8) θxr d e r d ( T − t ) + h t (cid:9) V x = − (cid:2) V ( t, x, m ) − λ (cid:3) θe r d ( T − t ) , V xx = (cid:2) V ( t, x, m ) − λ (cid:3) θ e r d ( T − t ) V m = (cid:2) V ( t, x, m ) − λ (cid:3) h m , V mm = (cid:2) V ( t, x, m ) − λ (cid:3) ( h m + h mm ) , where V t , V x , V xx , V m , V mm are the partial derivatives of V ( t, x, m ) and h t , h m , h mm are thepartial derivatives of h ( t, m ) . Substituting V t , V x , V xx , V m , V mm back into (3.6) yields h t + sup π (cid:110) − π ( A + m ) θe r d ( T − t ) − uθe r d ( T − t ) + 12 π θ ( σ f + σ Q ) e r d ( T − t ) (cid:111) + 12 θ σ e r d ( T − t ) + αmh m + 12 β ( h m + h mm ) = 0 . (3.10)And from (3.8) π ∗ = A + mθ ( σ f + σ Q ) e − r d ( T − t ) . (3.11)6ut π ∗ into (3.10) and calculate then h t − uθe r d ( T − t ) + 12 θ σ e r d ( T − t ) −
12 ( A + m ) σ f + σ Q + αmh m + 12 β ( h m + h mm ) = 0 . (3.12)It can be shown that (3.9) is a solution to (3.10) if h ( t, m ) is a solution to (3.12). Theorem 3.1.
With the terminal condition h ( T, m ) = 0 , the partial differential equation(3.12) has the solution of the form h ( t, m ) = K ( t ) m + L ( t ) m + J ( t ) , (3.13) where K ( t ) is a solution to K (cid:48) ( t ) + 2 β K ( t ) + 2 αK ( t ) − σ f + σ Q ) = 0 , K ( T ) = 0; (3.14) L(t) is a solution to L (cid:48) ( t ) + ( α + 2 β K ( t )) L ( t ) − A σ f + σ Q = 0 , L ( T ) = 0; (3.15) and J(t) is a solution to J (cid:48) ( t ) − uθe r d ( T − t ) + 12 θ σ e r d ( T − t ) − A σ f + σ Q ) + 12 β L + β K = 0 , J ( T ) = 0 . (3.16) Proof.
Substituting (3.13) into (3.12) and combining like terms with respect to thepowers of m, we have that m (cid:8) K (cid:48) ( t ) + 2 β K ( t ) + 2 αK ( t ) − σ f + σ Q ) (cid:9) + m (cid:8) L (cid:48) ( t ) + αL ( t ) + 2 β K ( t ) L ( t ) − A σ f + σ Q (cid:9) + (cid:8) J (cid:48) ( t ) − uθe r d ( T − t ) + 12 θ σ e r d ( T − t ) − A σ f + σ Q ) + 12 β L ( t ) + β K ( t ) (cid:9) = 0 . (3.17)Then it is obvious that (3.13) is a solution to (3.12) if K ( t ) , L ( t ) , J ( t ) are solutions to thedifferential equations (3.14), (3.15) and (3.16), respectively.Then we are going to solve the differential equations (3.14), (3.15) and (3.16), respec-tively.Let B := 2 β , C := 2 α, D := − σ f + σ Q ) , then the Riccati equation (3.14) becomes K (cid:48) ( t ) + BK ( t ) + CK ( t ) + D = 0 , K ( T ) = 0 . (3.18)7f B (cid:54) = 0 , i.e. β (cid:54) = 0 , integrating dK ( t ) BK ( t ) + CK ( t ) + D = − dt on both sides with respect to t we obtain that (cid:90) dK ( t ) BK ( t ) + CK ( t ) + D = − t + E, (3.19)where E is a constant. Since ∆ = C − BD = 4 α + β σ f + σ Q > , the quadratic equation BK ( t ) + CK ( t ) + D = 0 has two different real roots given by K , K = − C ± √ C − BD B . (3.20)Substituting (3.20) into (3.19) and considering the boundary condition K ( T ) = 0 thenwe obtain K ( t ) = K − K e B ( K − K )( t − T ) − ( K /K ) e B ( K − K )( t − T ) . (3.21)If B = 0 , i.e. β = 0 , then K ( t ) = 14 α ( σ f + σ Q ) − α ( σ f + σ Q ) e α ( T − t ) . (3.22)With the value of K ( t ) defined in (3.21) or (3.22), the linear ordinary equation (3.15)has the solution of the form L ( t ) = e (cid:82) Tt ( α +2 β K ( s )) ds (cid:2) (cid:90) Tt − A σ f + σ Q e (cid:82) Tt − ( α +2 β K ( y )) dy ds (cid:3) . (3.23)And the solution of (3.16) is given by J ( t ) = uθr d (1 − e r d ( T − t ) ) − θ σ r d (1 − e r d ( T − t ) ) − A σ f + σ Q ) ( T − t )+ (cid:90) Tt ( 12 β L ( s ) + β K ( s )) ds. (3.24)From [8] the following verification theorem exists. Theorem 3.2.
Let W ∈ C , ([0 , T ] × R ) be a classical solution to the HJB equation (3.6)with the boundary condition W ( T, x, m ) = u ( x ) , then the value function V given by (3.4)coincides with W such that W ( t, x, m ) = V ( t, x, m ) . In addition, let π ∗ be the optimizer of (3.6), that is for any ( t, x, m ) ∈ [0 , T ] × R V t + (cid:2) π ∗ ( A + m ) + xr d + u (cid:3) V x + 12 (cid:2) σ + π ∗ ( σ f + σ Q ) (cid:3) V xx + αmV m + 12 β V mm = 0 . Then π ∗ ( t, X ∗ t , m ( t )) is the optimal strategy with V π ∗ ( t, x, m ) = V ( t, x, m ) , where X ∗ t is the surplus process under the optimal strategy π ∗ . Theorem 3.3.
With the utility function (3.5), the optimal strategy for the optimizationproblem (3.4) subject to (2.3) is π ∗ t = A + m ( t ) θ ( σ f + σ Q ) e − r d ( T − t ) , ∀ t ∈ [0 , T ] . And the value function is given by the form V ( t, x, m ) = λ − γθ exp (cid:8) − θxe r d ( T − t ) + h ( t, m ) (cid:9) with h ( t, m ) = K ( t ) m + L ( t ) m + J ( t ) , where K ( t ) , L ( t ) and J ( t ) are given by (3.21)-(3.24). If m ( t ) = 0 in (1.1) then the exchange rate price Q ( t ) is degenerated into the processwhich is modeled by means of geometric Brownian motion dQ ( t ) = Q ( t )( u Q dt + σ Q dW t ) . In this case, under the control of π, X πt satisfies the following stochastic equation dX πt = (cid:8) π t A + u + r d X t (cid:9) dt + σdW t + π t σ f dW t + π t σ Q dW t . (3.25)Then the HJB equation in (3.6) becomes to be V t + sup π (cid:110)(cid:2) πA + xr d + u (cid:3) V x + 12 (cid:2) σ + π ( σ f + σ Q ) (cid:3) V xx (cid:111) = 0 . (3.26)By solving the above HJB equation (3.26) the following corollary is obtained. Corollary 3.4.
With the X πt in (3.25) the optimal investment strategy is given by π ∗ t = A θ ( σ f + σ Q ) e − r d ( T − t ) , ∀ t ∈ [0 , T ] . Furthermore, the value function has the form V ( t, x ) = λ − γθ exp (cid:8) − θxe r d ( T − t ) + f ( T − t ) (cid:9) where f ( T − t ) = θur d (1 − e r d ( T − t ) ) − θ σ r d (1 − e r d ( T − t ) ) − A σ f + σ Q ) ( T − t ) . Let S dt be the price of domestic risky asset described by the following stochastic dif-ferential equation dS dt = S dt ( u d dt + σ d dW t ) , (3.27)where u d and σ d are positive constants and W t is the standard Brownian motion whichis independent of W t . Assume the insure invests his wealth only in domestic currencymarket, i.e., domestic risk-free asset and domestic risky asset, then the surplus under thecontrol π is that dX πt = (cid:8) π ( u d − r d ) + u + r d X t (cid:9) dt + σdW t + πσ d dW t . (3.28)9 orollary 3.5. In the domestic currency market, the optimal strategy for the optimizationproblem (3.4) is π ∗ t = u d − r d θσ d e − r d ( T − t ) , ∀ t ∈ [0 , T ] , and the corresponding value function has the form V ( t, x ) = λ − γθ exp (cid:8) − θxe r d ( T − t ) + g ( T − t ) (cid:9) where g ( T − t ) = θur d (1 − e r d ( T − t ) ) − θ σ r d (1 − e r d ( T − t ) ) − ( u d − r d ) σ d ( T − t ) . Remark 3.6.
By comparing the optimal investment strategies and value functions inCorollary 3.4 and Corollary 3.5, it is not difficult to see that(i) Suppose the insurer invests the same amount of his wealth into domestic riskyassets and foreign risky assets. We can see thatIf u f + u Q ≥ u d , then the value function with exchange rate is always larger than thevalue function without exchange rate. Conversely, if u f + u Q < u d it is better for theinsurer to invest in domestic risky assets.(ii) Assume that the insurer wants to get the same value functions in the two kinds ofcurrency markets.If u f + u Q ≥ u d , in order to get the same value functions in the two cases, the amountof wealth invested in foreign risky assets are less than that invested in domestic riskyassets. In addition, if u f + u Q < u d , in order to get the same values of the value functionsthe insurer should invest more in foreign risky assets.(iii) When u f + u Q ≥ u d if the insurer invests more in foreign risky assets the valuefunction is higher than the value function in domestic risky assets. Remark 3.7.
From Corollaries 3.4 and 3.5, it is not difficult to see that if no domesticrisk-free assets are traded and only risky asset is considered even in domestic or foreignmarket, the optimal strategies are always constants, regardless of the level of wealth theinsurer has.
In order to demonstrate our results, numerical examples are presented for the optimalinvestment strategies and value functions in two kinds of currency markets. Our objectiveis to study the effect of exchange rate on the insurer’s decision and the value function.The particular numbers of basic parameters are given in the following tables.
T r d λ θ γ u σ u f σ f u Q σ Q x u d σ d √ . √ . √ . a) (b) The numbers in Table 1 satisfy that σ d ( u f + u Q − r d ) = 0 .
08 = ( σ f + σ Q )( u d − r d )and u f + u Q = 0 . > u d = 0 . . The graphs of the optimal investment strategies and value functions corresponding to thedata in Table 1 are shown in (a) and (b). They show that if the insure invests the sameamount of its wealth into foreign and domestic risky assets, then the former produces alarger value function. Thus it is better for the insurer to invest in foreign risky assets.And the results in graphs (a) and (b) also coincide with the conclusions in (i) of Remark3.6.
T r d λ θ γ u σ u f σ f u Q σ Q x u d σ d √ .
12 2 0.3 0.2Table 2 (c) (d)
By employing the numbers in Table 2, the graphs of optimal investment strategiesand value functions are given in (c) and (d). From graphs (c) and (d) it is not difficult11o see that if the insurer wants to get the same value functions in the currency marketswith and without exchange rates, he should invest more in domestic risky assets than inforeign risky assets. And the numbers in Table 2 satisfy that σ d ( u f + u Q − r d ) = ( σ f + σ Q )( u d − r d )and u f + u Q > u d . Thus graphs (c) and (d) reflect the results in (ii) of Remark 3.6.The graphs (e) and (f) are obtained from the numbers listed in Table 3. We firststudy the effect of the exchange rate on optimal investment strategies. When the insurerhas exponential preferences, the realization of their optimal investment strategies areillustrated in Figure (e). It can be seen that, when there are two currency markets theinsurer invests a larger proportion of her wealth in the risky asset.Secondly, we explore the effect of the exchange rate on the value function. From figure(f), we can easily find that, it is much better for the insurer to invest her surplus in foreignrisky asset to decrease the risk. The value function with exchange rate is always largerthan the value function without exchange rate, except the terminal value. It indicatesthat it is better to incorporate the exchange rate in the model.
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