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Optimal Investment with Multiple Risky Assets for an Insurer in an Incomplete Market
This paper studies the optimal investment problem for an insurer in an incomplete market. The insurer's risk process is modeled by a Lévy process and the insurer is supposed to have the option of investing in multiple risky assets whose price processes are described by the standard Black-Scholes model. The insurer aims to maximize the expected utility of terminal wealth. After the market is completed, we obtain the optimal strategies for quadratic utility and constant absolute risk aversion (CARA) utility explicitly via the martingale approach. Finally, computational results are presented for given raw market data.
Recently, the problem of optimal investment for an insurer has attracted a lot of attention, due to the fact that the insurer is allowed to invest in financial markets in practice. In the meantime, this is also a very interesting portfolio selection problem in the finance theory. Some important early work is done by Browne , where the risk process is approximated by a Brownian motion with drift and the stock price is modeled by a geometric Brownian motion. Under these assumptions, when the interest rate of a risk-free bond is zero, Browne  shows that minimizing the ruin probability and maximizing the expected constant absolute risk aversion (CARA) utility of the terminal wealth produce the same type of strategy. In Hipp and Plum , the classical Cramér-Lundberg model is adopted for the risk reserve and the insurer can invest in a risky asset to minimize the ruin probability. However, the interest rate of the bond in their model is implicitly assumed to be zero. Liu and Yang  extend the model of Hipp and Plum  to incorporate a nonzero interest rate. But in this case, a closed-form solution cannot be obtained. Azcue and Muler  consider the borrowing constraints for an optimal investment problem of an insurer, and some numerical solutions to minimize the ruin probability are obtained. In the literature, the optimal investment problem for an insurer is usually studied via the stochastic control theory; see  for details. By this approach, Yang and Zhang  use a jump-diffusion process to model the risk process and obtain the optimal investment strategy for CARA utility maximization. The martingale approach, which has been widely used in mathematical finance, is applied to investigate the optimal investment problem for an insurer by Wang et al. . In their paper, the risk process is modeled by a Lévy process, the security market containing a bond and a stock is described by the standard Black-Scholes model, and the optimal strategies are worked out explicitly for the mean-variance criteria and the CARA utility. Zhou  studies a model similar to that in , where the prices of assets are described by the Lévy processes, and the optimal strategy is obtained explicitly for the CARA utility.
In most of these papers, the optimal investment for an insurer is considered in a complete market. But the real market is always incomplete; that is, the number of risky assets (stocks) is strictly less than the dimension of the underlying Brownian motion. Hence, it is necessary to find an optimal strategy in an incomplete market. However, the traditional martingale method cannot be used directly in an incomplete market. To overcome the problem of incompleteness, many researchers have developed different ways to handle the problem. For instance, Karatzas et al.  complete the market by introducing additional fictitious stocks and then making them untradable. Zhang  provides an easier method by directly making the dimension of the underlying Brownian motion equal to the number of available stocks.
The optimal investment problem for an insurer in an incomplete market is studied in the present paper. For this case, the traditional martingale method is problematic. Thus, we first complete the market via the approach proposed by Zhang  so that the martingale method can be used. In addition, only one stock is considered in [7, 8]. But an insurer will invest in multiple stocks to decrease risk. Thus in order to make our model more practical, we consider a financial market of one bond and stocks. Moreover, the insurer’s objective in this paper is to maximize the utility of the discounted value of his/her terminal wealth, which is different from that in most works. When the risk process is modeled by a Lévy process and the security market is described by the standard Black-Scholes model, the optimal strategies are obtained explicitly for the quadratic and CARA utilities.
An optimal strategy of CARA utility maximizing in an incomplete market is also obtained in Wang . The result is general but not unique and cannot be expressed concretely in . Whereas our corresponding optimal strategy is explicit and concrete.
This paper is structured as follows. In Section 2 the problem is formulated. The incomplete market is transformed into a complete one in Section 3. The optimal portfolios for quadratic and CARA utility functions are worked out explicitly in Section 4. Section 5 presents some remarks and computational results for optimal strategies in a complete market and an incomplete market. Section 6 contains some conclusions.
2. Formulation of the Problem
The insurer can invest in a financial market consisting of one riskless bond with price given by and stocks with prices satisfying where is the interest rate and is a -dimensional Brownian motion defined on a filtered complete probability space with the component Brownian motions being independent. denotes the appreciation rate vector, is the volatility matrix, and define as a fixed and finite time horizon with . In addition, the matrix is assumed to have the full row rank. As in , under these assumptions, we restrict ourselves to the situation where that is, the number of stocks in the market is smaller than or equal to the dimension of the underlying Brownian motion. This financial market is complete when , while incompleteness arises if . In this paper, we consider the insurer’s investment problem in an incomplete market.
Suppose the initial reserve is and the premium rate is a constant ; the risk process of the insurer is modeled by where is a compound Poisson process defined on , is a homogeneous Poisson process with intensity and represents the number of claims occurring in the time interval , is the size of the th claim. Thus the compound Poisson process represents the cumulative amount of claims in the time interval . The claims’ sizes are assumed to be an i.i.d. sequence with a common cumulative distribution function satisfying and . Furthermore, it is usually assumed that , and the -dimensional Brownian motion are mutually independent. If denotes the compensated compound Poisson process , where is the mean of , then the risk process (3) can be rewritten as As in , we generalize the above risk process model to a stochastic cash flow, which is still denoted by and satisfies where is a 1-dimensional compensated pure jump Lévy process defined on , is a constant, is the usual augmentation of the natural filtration of with , and and are mutually independent. Let denote the jump measure of . Its dual predictable projection has the form with and . To avoid some tedious technical arguments, we assume throughout this paper. Under this assumption, it is well known that is square integrable and has the following Lévy decomposition property: For details, refer to Cont and Tankov . Further, note that for the risk process model (3), we have .
The insurer is allowed to invest in those stocks as well as in the bond. A trading strategy can be expressed in terms of the dollar amount invested in each asset. Let , be the dollar amount of the th stock in the investment portfolio at time , and let and . To be mathematically rigorous, for each , is an -predictable process.
Definition 1. A trading strategy is called admissible if .
The set of all admissible trading strategies is denoted by . For each and an initial capital , the wealth process of the insurer satisfies the following dynamics: where is an vector. Wang et al.  consider an optimal investment problem similar to (7) in the one stock case and solve it elegantly by the martingale method in a complete market. However, in an incomplete market, the martingale method will be problematic. Thus, we will use the method proposed by Zhang  to transform our incomplete market into a complete one.
3. Transformation from an Incomplete Market to a Complete One
Step 1. Reducing the dimension of the Brownian motion.
For each , let . Then the volatility matrix in the incomplete market can be written as . Define where denotes the Euclidean norm. Expressing the stock prices in terms of , it gives where is an -dimensional Brownian motion. But the component Brownian motions are no longer independent. Specifically, one can consult Exercise 4.15 in  for the following result: where and represents the inner product of and . It follows from the Cauchy-Schwarz inequality that . Under the assumption that the volatility matrix has the full row rank, we have
Step 2. Creating independent component Brownian motions from the correlated ones.
Let be the matrix generated by the correlation coefficients of the correlated -dimensional Brownian motion with The matrix is nonsingular, symmetric, and positively semi-definite. So, there exists a nonsingular matrix such that . It can be shown that there exist independent Brownian motions such that For details, refer to Exercise 4.16 in . So far, we have derived a complete market with stocks and -independent component Brownian motions. The -stocks in the incomplete market can now be re-written as The volatility matrix, denoted by , under the completed market is given by Put Then With the help of and , we transfer the model (7) to the following one in the completed market which is equivalent to Define which represents the discounted value of .
Suppose that the insurer has a utility function of his wealth and he/she aims to maximize the expected utility of his/her discounted terminal wealth, that is, The utility function is strictly concave and continuously differentiable on ; thus there exists at most a unique optimal discounted terminal wealth for the company.
Furthermore, for any and , we have from the concavity of . according to (24) we claim the following proposition.
Proposition 2. If there exists a strategy such that then is the optimal trading strategy.
Remark 3. Such sufficient conditions to the optimal investment are well known in the martingale approach; refer to  and others.
By (22), (25) is equivalent to that Thus, problem (23) can be solved by applying (26) in terms of the concrete utility functions.
It is well known that maximizing the expected quadratic utility is equivalent to finding a mean-variance efficient strategy while CARA utility is commonly used. Thus, we will work out the optimal strategies explicitly for these two utilities in the next section.
4. Optimal Strategies for Different Utility Functions
First, we introduce some notations and a martingale representation theorem that will be used. Let be the predictable -algebra on , which is generated by all left-continuous and -adapted processes, and . Let (resp., ) be the set of all -predictable, -valued processes such that a.s. . Let be the set of all -measurable, -valued functions defined on such that is locally integrable and increasing, and also a.s. for all , where is the indicator function. Moreover, let be the set of all -measurable, -valued functions defined on such that . Finally, denotes the set of all -adapted processes with Càdlàg paths such that . It is well known that every square-integrable martingale belongs to , and it is easy to see that if then .
One can consult Proposition 9.4 in  for the following result.
Lemma 4 (Martingale Representation). For any local (resp., square-integrable) martingale , there exist some with , and such that for all .
4.1. Quadratic Utility
In this subsection, we consider problem (23) for the quadratic utility function , where is a parameter. In case that there is only one stock and one bond in a complete market, a closed-form solution is obtained by Wang et al. . Motivated by their work, we show that a closed-form solution can also be obtained when there are several stocks in an incomplete market. Since , condition (26) can be written as Put Then and a.s. for any stopping time a.s.
Lemma 5. Let . Then satisfies condition (29) if and only if there exists a such that solves the following forward-backward stochastic differential equation (FBSDE) for .
Proof. Suppose satisfies condition (29). From (22) we have solving the forward SDE in (31) for and it is clear that is a square-integrable martingale. For any stopping time , let , whose th component is and the other components are all 0. Then . Denote by the th row vector of for ; that is, . For any positive integer , substituting into (29), we have that
is constant over all stopping times a.s., which implies that
Since is a square-integrable martingale, by Lemma 4, there are with and such that
By Itô’s formula,
which together with (33) implies for each . Thus, it is easy to see that
By (34), solves the backward SDE in (31) for , and hence solves FBSDE (31).
Conversely, suppose that there exists such that solves FBSDE (31). It is easy to verify that for any , by Itô’s formula, is a local martingale, where Furthermore, for any , we have . Hence, which implies that the family is uniformly integrable and hence is a martingale. Thus we have for any , implying that satisfies condition (29).
In what follows, we will solve FBSDE (31) by two steps.
Step 1. Conjecturing the form of solution.
Put where is a nonrandom Lebesgue-integrable function to be determined. If solves FBSDE (31), then by Itô’s formula, which implies
Let solve FBSDE (31). Then . Comparing -term -term, respectively, in (43) with those in (22) and noting (6), it is reasonable to conjecture that is, Note that by (19), and are both nonsingular, . Hence, (45) can be re-written as Substituting (44) and (46) into (43), we have where the second equality follows from the forward SDE in (31) and the third equality holds owing to . If we take then (47) is reduced as and FBSDE (31) is solved.
Step 2. Verifying the conjecture in Step 1.
Let Then the following SDE has a solution of the form where Furthermore, let and is obtained through the forward SDE in (31). Then by the procedure same as that in (47), it is easy to verify that Thus, solves FBSDE (31).
Finally, by Proposition 2, Lemma 5, and arguments in Steps 1−2, we have the following theorem.
4.2. CARA Utility
In this subsection, we consider problem (23) for the CARA utility function , where . The procedure is similar to what is done in . Since , condition (26) can be written as Similar to the case of quadratic utility, in what follows, we split the procedure into three steps.
Step 1. Conjecturing the form of that satisfies condition (57).
Put and for all . Then a.s. for any stopping time a.s. Let be a probability measure on such that .
For any stopping time a.s., let , whose th component is and the other components are all 0. Then . As in Section 4.1, denote by the th row vector of for ; that is, . For any positive integer , substituting into (57), we have being constant over all stopping times a.s., which implies that Since is a martingale, then is a local martingale. By Lemma 4, there exist some satisfying and such that that is, Furthermore, by the Doléans-Dade exponential formula, we have By Girsanov’s Theorem, we know that is an -dimensional Brownian motion under , which together with (60) implies , . Thus, it is easy to see that that is,
On the other hand, by (22), we have In order to have (58), comparing -term and -term, respectively, in (64) with those in (67) and taking (66) into account, it is reasonable to conjecture that is,
Step 2. Verifying that in (64) satisfies (58), where and are defined in (69), and is defined in (66).
Substituting (69) into (67), we have where For and defined as above, it is easy to see that is a martingale. Substituting (66) and (69) into (64), we have where is a constant. Since is a martingale, we have and hence Finally, by (70)–(74), we have which is just what we desired.
Step 3. Verifying that in (69) satisfies condition (57).
It is clear that . For any , by Girsanov’s Theorem, we can see that is a local martingale under . Furthermore, for any , we have . Hence which implies that the family is uniformly integrable under . Hence is a martingale under . Thus we have for any , which implies (57).
By Proposition 2 and arguments in steps 5–7, we can claim the following theorem.
5. Remarks and Computational Results
In this section, we present some remarks and conduct computational experiments for the optimal investment problems of this paper according to the real market data. The objective of these computational tests is to contrast the insurer’s optimal strategies in an incomplete market with those in a complete market.
According to Theorems 6 and 7, the optimal strategy of CARA utility is independent of the claim process, while that of quadratic utility is related to the claim process. This is due to the specific nature of the CARA utility.
5.1. Quadratic Utility
For quadratic utility, as shown in (51)–(55), is a left-continuous process with a jump at any time , where is the claim size at time . From (79) the effect of claims on the efficient strategy of quadratic utility can be interpreted as follows.(i)When no claim occurs at time , the wealth process is continuous, and so is .(ii)When a claim occurs at time , the wealth process has a jump , which is just the corresponding claim size; the optimal strategy also has a jump and the jump size on each stock is the corresponding element of .
In terms of (79), we can consider the transaction cost of realizing the optimal strategy of quadratic utility. According to , the transaction cost is quantified by the portfolio weight turnover for a portfolio , that is, where denotes the 1-norm. Since transaction cost is an important aspect of any investment strategy, we are interested in comparing the costs of implementing the optimal strategies in an incomplete market with those in a complete market.
Suppose there are 12 underline Brownian motions, 12 stocks for investment in a complete market, and 10 stocks in an incomplete market. The universe of assets are chosen from the 12 industry categories of finance.cn.yahoo.com in 2009 (see Tables 1 and 2). The appreciation rate vector is estimated through the mean rate of return of each stock. The covariance matrix of the rate of return on 12 stocks is estimated based on the real market data. In a complete market, the volatility matrix is calculated from such that . Next, the volatility matrix of the 10 stocks in an incomplete market is obtained by eliminating corresponding row of . Let the risk-free rate and the claim size is distributed according to the exponential distribution with parameter 1; that is, . Let . With exponential claim size randomly generated and calculated from (13) and (18), Figure 1 plots the transaction cost in an incomplete market and in a complete market, respectively.
We easily observe that the optimal strategies in an incomplete market incurs lower transaction cost than those in a complete market. Because there are fewer stocks in an incomplete market, than in a complete market, the number of stocks whose investment amount will be changed is smaller in an incomplete market. Thus the transaction cost for fewer stocks is low.
5.2. CARA Utility
For the optimal strategy of CARA utility, (69) shows that the discounted amount held in each stock is fixed. A similar investment strategy for CARA utility in an incomplete market is also obtained in Wang . However, the expression of the optimal strategy in  contains a g-inverse of matrix , which is not unique. In our results, replaces the g-inverse matrix, which makes the optimal strategy explicit and deterministic. Due to the determinacy and nonrandomness of the optimal strategy for CARA utility, we can analyze the differences between complete market and incomplete market conveniently from the results of CARA utility.
The experimental procedure is the same as in Section 5.1 and all the parameters are set as in Section 5.1 except . Let represent one year and the insurer rebalances his portfolio once a month. Then and there are 12 investment periods.
In Figure 2, we compare the average total dollar amount invested in stocks in an incomplete market with that in a complete market as the risk aversion coefficient ranges from 1 to 100. The average total dollar amount invested in stocks for a sequence of portfolios with stocks is defined as From Figure 2 we find that in an incomplete market, the insurer will invest more money in stocks (risky assets). The practical number of stocks in an incomplete market is less than the ideal number of stocks. For example, in our experiments, there is only 10 stocks for investment while the ideal number of attainable stocks is 12 in the incomplete market. Thus an insurer will invest more in the only 10 stocks. An investor is venturesome in an incomplete market. Also we see that the total amount invested in stocks decreases as increases. Since is the absolute risk aversion coefficient, this result is consistent with intuition.
Figure 3 is a plot of the relative difference of the optimal strategy in an incomplete market with respect to that in a complete market; that is, it plots as a function of . The above denote the average total dollar amount invested in stocks in an incomplete market and in a complete market, respectively. It is obvious that this difference is independent of the risk aversion coefficient but only related to the difference of the market structure. No matter what the risk aversion coefficients are, the effects of the incomplete market on the corresponding optimal strategies are the same.
Given a sequence of portfolios , we quantify its average transaction cost by Figure 4 plots the average transaction costs of 12 periods optimal strategies for CARA utility in the two different markets. Same as the quadratic utility case, the cost of implementing the optimal investment in an incomplete market is lower than that in a complete market.
In this paper, we consider the optimal investment problem for an insurer in an incomplete market. After transforming the incomplete market into a complete one, we solve the problem via the martingale approach. From the computational experiments, we find that an insurer’s optimal investment strategy in an incomplete market incurs lower transaction cost than the one in a complete market. The amount invested in stocks in an incomplete market is more than that in a complete market. For CARA utility, the computational results also show that the incomplete market produces the same effects on the optimal strategies of insurers with different risk aversion coefficients.
The authors are very grateful to reviewers for their suggestions and this research was supported by the National Natural Science Foundation of China (Grant no. 11201335) and the Research Projects of the Social Science and Humanity on Young Fund of the Ministry of Education (Grant no. 11YJC910007). J. Cao would like to acknowledge the financial support from Auckland University of Technology during his sabbatical leave from July to December 2009 and thank the Department of Mathematics at Tianjin University for hospitality to host him in October 2009.
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