Should Life Insurance Companies Invest in Hedge Funds? 1

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1 Should Life Insurance Companies Invest in Hedge Funds? 1 Thomas Berry-Stölzle 2 Hendrik Kläver 3 Shen Qiu 4 July The authors thank James M. Carson, Robert E. Hoyt, Harris Schlesinger and Heinrich R. Schradin for helpful comments and suggestions on this paper. Thomas Berry-Stölzle gratefully acknowledges financial support from the AXA Colonia-Studienstiftung im Stifterverband für die Deutsche Wissenschaft. Please address all correspondence to Thomas Berry-Stölzle. 2 Terry College of Business, University of Georgia, 206 Brooks Hall, Athens, GA 30602, Phone: (706) , Fax: (706) , thomas@stoelzle.org 3 Institute for Insurance Science, University of Cologne, Albertus-Magnus-Platz, Köln, Germany 4 Institute for Insurance Science, University of Cologne, Albertus-Magnus-Platz, Köln, Germany

2 Should Life Insurance Companies Invest in Hedge Funds? Abstract This paper investigates whether it is advantageous for life insurance companies to invest in hedge funds. Using a simplified model of a life insurance company with cliquetstyle interest rate guarantees and three investment opportunities, we consider the effect of the insurer s liability structure on the decision on how to invest in hedge funds. We address specific characteristics of hedge fund returns in our analysis with a three dimensional AR(1) GARCH(1,1) asset model. Our analysis reveals first, that it is advantageous for life insurers to invest in hedge funds, and second, that financially weak insurers have a higher incentive to invest in hedge funds. (JEL classification: G11, G22) 1

3 1 Introduction Since the early 1990s, hedge funds have become increasingly popular with institutional investors as well as high net worth individuals. The assets under management in the hedge fund industry rose from approximately $50 billion in 1990 to approximately $1 trillion by the end of 2004 (Malkiel and Saha (2005)). Like mutual funds, hedge funds provide actively managed portfolios in publicly traded assets. The main difference between hedge funds and mutual funds is that hedge funds are free from the regulatory controls stipulated by the Investment Company Act of 1940 (Brown and Goetzmann (2003)). Therefore hedge funds can freely choose the type of securities in which they invest as well as the type of positions they take. For example, hedge funds can invest in derivative securities, sell short or take on leveraged or undiversified positions (see, e.g., Fung and Hsieh (1997), Liang (2000)). Such non-standard investment strategies have the potential to generate risk return profiles different from traditional asset classes. Hence, an investment in hedge funds results in diversification benefits, making them an interesting investment alternative for life insurance companies. In 2004, the German Insurance Authority (BaFin) lifted the ban on hedge funds investments for German life insurance companies. Life insurers are now allowed to invest up to five per cent of their assets in hedge funds, using the opening clause up to ten percent. In addition, insurers can file an application with the Insurance Authority to increase the maximum hedge fund holding by another five percent to a total of 15 percent. The goal of this research is to provide answers to the timely question, whether or not life insurance companies should invest in hedge funds. If yes, how much they should invest in hedge funds, and how the insurance company s liability structure affects any potential advantage in such an investment in hedge funds. We address these questions with a simulation model of a life insurance company offering a with profit life insurance contract. With profit 2

4 contracts offer a cliquet-style (or year-by-year) minimum interest rate guarantee which may be specific to the German market, however, our model results are applicable to any other products or markets fulfilling the assumptions. Since cliquet-style interest rate guarantees restrict the possible asset management strategies of a life insurance company more than guarantees based on the benefits at maturity, our results are the limiting case and, hence, a benchmark for all other types of life insurance products and company structures. Kling, Richter, and Ruß (2007) developed a simulation model of a life insurance company offering a product with a cliquet-style guarantee. However, their model insurer can only invest in two asset classes for which both price processes follow geometric Brownian motions. Empirical studies of hedge funds returns found negative or positive skewness, larger kurtosis than those of the Normal distribution, and positive autocorrelation (Agarwal and Naik (2001), Brooks and Kat (2002), Fung and Hsieh (1997), Fung and Hsieh (2001), Martin (2001), Popova, Morton, and Popova (2003)). Therefore a Brownian motion is not adequate for modelling hedge funds returns. We extend the Kling, Richter, and Ruß (2007) simulation model on the asset side incorporating three correlated AR(1) GARCH(1,1) processes, and an objective function for optimizing the insurance company s asset allocation. Within this model framework we use Monte Carlo Simulations to examine different asset allocation strategies under various parameter settings. Our model of a life insurance company takes a balance sheet perspective and differentiates between the book value and market value of assets; the difference between the two is named hidden reserves. The model contains parameters for the capital holding of the company, the guaranteed interest rate, the percentage of asset return - based on book values - which has to be credited to the policyholders account, as well as parameters capturing restrictions on the creation of hidden reserves, and management decisions rules about the interest rate the management would like to credit to the policyholders account for competitive reasons. The outputs of our simulation model are 3

5 the shortfall probability of the insurance company and the expected return of the asset portfolio. This set-up allows us to vary the asset allocation and calculate Markowitz efficient frontiers for given parameter sets. Assuming that the insurer maximizes the expected return of its investments for a given ruin probability, we can analyze the relationship between the different model parameters and the percentage of hedge funds in the optimal asset allocation. We calibrate our model to the German stock index (DAX), the German bond index (REXP) and the Hedge Funds Indices of the Center for International Securities and Derivatives Markets (CISDM), and correct the estimated hedge fund returns for biases inherent in the hedge fund database (see, e.g., Liang (2000)). Our simulation analysis, based on the estimated parameters, provides the following four results: First, it is advantageous for life insurance companies to invest in hedge funds. For all parameter settings the insurer can substantially increase the expected portfolio return for a given shortfall probability by investing in hedge funds. Hedge funds help to reduce the volatility of the overall asset portfolio which is especially beneficially for life insurance companies with cliquet-style interest rate guarantees. Second, the higher the interest rate guarantee is, the less capital an insurer holds, and the less possibilities the accounting system provides to smooth the book values of the investment portfolio, the more incentives an insurance company has to diversity its portfolio by investing in hedge funds. For such companies the difference between the expected asset return with and without the possibility of investing in hedge funds is higher than for the insurer in our base case. Third, the higher the interest rate guarantee, the fewer capital an insurance company holds, and the more restrictive the accounting system is, the bigger is the optimal hedge fund holding in the company s investment portfolio. Fourth, if our model insurer tolerates a ruin probability of five per cent, an investment in event-driven multi-strategy hedge funds leads to the highest expected returns. Our results are of interest not only to life insurance and investment companies, but also to insurance reg- 4

6 ulators. The fact that weakly capitalized insurers have a strong incentive to invest in hedge funds based on a standard optimization model raises the question whether such insurers would be able to bear the risks associated with their hedge fund holding in an extreme market crash. Moral hazard could encourage very weak insurers to increase their hedge fund holdings substantially in a go-for broke behavior. The remainder of this paper is organized as follows. In Section 2, we present the model. Section 3 describes how we calibrate our model. In Section 4, we present the empirical results and discuss their implications for the asset management of insurance companies as well as for insurance regulators. The final section concludes and puts forth additional research topics. 2 Model In this section, we introduce our model which is a selective simplification of a standard German life insurance company. Our model builds on the basic Kling, Richter, and Ruß (2007) framework of a life insurance company offering a product with a cliquetstyle guarantee. We extend this framework incorporating a more realistic model for the investment portfolio of the insurance company. Our model takes a balance sheet perspective and differentiates between the market value of the insurance company s assets A t and the book value of its assets. The difference between these two is named the hidden reserves R t. The index t = 1,..., T denotes the time period. The technical reserves L t of the insurance company reflect their liabilities. These liabilities result from the insurers obligations to their policyholders. Therefore, we will also refer to L t as the book value of the policyholders account. If we assume that the value of the technical reserves L t correspond to the book value of the insurer s 5

7 assets at time t = 1,..., T then the simple balance sheet described in Figure 1 represents the financial situation of our insurance company at time t. Since the reserve account R t is given by R t = A t L t, the financial situation of our insurance company is determined by the development of its assets A t over time as well as the development of the product dependent liabilities. Note that the reserve account R t may not only consist of the hidden asset valuation reserves but include the insurer s capital and other additional components as well. However, we will refer to R t as the hidden reserves or the reserves. - Please insert FIGURE 1 approximately here The Insurance Company s Assets Our model insurance company is invested in a mix of three different assets: a stock, a bond and a hedge fund. Let q 1, q 2 and q 3 denote the percentage of assets the insurance company invests in the three different investment instruments, and let S 1,t, S 2,t and S 3,t, t = 1,..., T denote the price processes of the stock, bond and hedge fund, respectively. If we set the prices of the three assets equal to one at time t = 1 the value of the insurance company s assets at time t = 2,..., T is determined by A t = A 1 (q 1 S 1,t + q 2 S 2,t + q 3 S 3,t ). (1) The development of the three asset price processes S 1,t, S 2,t and S 3,t over time is determined by three correlated AR(1) GARCH(1,1) processes of the corresponding stan- 6

8 dardized continuous rates of returns. Let r i,t = ln( Si,t S i,t 1 ) E(ln( S i,t S i,t 1 )) σ(ln( S i,t S i,t 1 )) (2) denote the standardized continuous rate of return of asset i = 1, 2, 3. We model r i,t as an AR(1) process r i,t = a i,0 + a i,1 r i,t 1 + Z i,t (3) where Z t follows a GARCH(1,1) process with conditional variance h ii,t = α 0 + α 1 Z 2 i,t 1 + β 1 h ii,t 1 (4) and conditional covariance for i j h ij,t = ρ ij α 0 + α 1 Z i,t 1 Z j,t 1 + β 1 h ij,t 1. (5) Then H t := ( h 11,t h 12,t h 13,t h 12,t h 22,t h 23,t ) (6) h 13,t h 23,t h 33,t is the conditional covariance matrix, and H t is positive definite for all t N if H 0 is positive definite. This follows from a more general result of Engle and Kroner (1995). Our model is easy to simulate. Let ε t = (ε 1t, ε 2t, ε 3t ) be independent and identically N (0, I 3 )-distributed random vectors for all t Z; here I n is the n-dimensional identity matrix. Let (Z 1t, Z 2t, Z 3t ) = C t ε t, where C t is a positive definite matrix satisfying C 2 t = H t, i.e. C t is a root of H t. Then each of the time series (Z 1t ) t Z, (Z 2t ) t Z and (Z 3t ) t Z follows a GARCH(1,1) process. (Z it ) t Z, i = 1, 2, 3, are weakly stationary if and 7

9 only if α 1 + β 1 < 1. For every t and i j the unconditional correlation between Z it and Z jt is ρ ij. This procedure allows us to generate three correlated GARCH(1,1) processes which are then used as input for equation (3). However, the conditional correlations h ij,t hii,t h jj,t for i j depend on t. For a more general approach and details concerning multivariate GARCH, see Engle and Kroner (1995). If the insurance company pays dividends to equity holders at time t the insurance company s overall assets get reduced by the dividend payments D t. Let A t value of the insurance company s assets before dividend payments and A + t denote the the value of the company s assets after dividend payments. Then the company s assets are recursively given by and A t = A + t 1 (q 1 S 1,t 1 + q 2 S 2,t 1 + q 3 S 3,t 1 ) (q 1S 1,t + q 2 S 2,t + q 3 S 3,t ) (7) A + t = A t D t. (8) 2.2 The Insurance Company s Liabilities The development of an insurance company s liabilities over time depend on multiple factors like the products the company sells, the legal environment the company operates in, the accounting rules applicable as well as specific management decisions. To model such a complex process, we need a number od simplifying assumptions. We follow the approach of Kling, Richter, and Ruß (2007) and only focus on a standard German life insurance company. Thus, our model includes the main characteristics of the regulatory and legal framework in Germany and the German accounting system. However, we argue 8

10 that the restrictive German environment with its cliquet-style interest rate guarantees provides a limiting case for other countries. Thus, the results from our model are of interest to insurers in other operating environments as well. Furthermore, our model insurer offers only one simple product: a single-premium term-fix insurance contract. The premium P is paid up front at time t = 0, and the benefit is paid at the end of the fixed contract period T regardless of whether the insured person is still alive or not. Therefore, there are no mortality effects considered in our model. The benefit of this contract only depends on the development of the policyholders account over time and is given by P L T L 0. Ignoring mortality effects in a model of a life insurance company might seem to be inappropriate. However, if we assume that the insurance company s new business compensates for mortality and policy surrenders the liabilities resulting from a term-fix contract in combination with the corresponding assets provide a good approximation for a whole insurance company. In Germany, there is a cap on the rate of return which may be used to discount technical reserves. 1 Almost all life insurance products sold on the German market use the maximum rate of return to discount the reserves. Since surrender values of life insurance products are legally required to be close to the value of the technical reserves, policyholders have a year by year guarantee of earning at least this rate of return on their account value. Currently, the guaranteed interest rate is 2.25%. However, these cliquetstyle interest rate guarantees are given for the whole contract period and cannot be changed retrospectively. 2 Therefore, most contracts in the portfolio of a typical German life insurance company are still entitled to 2.75%, 3.25% or even 4% interest per year. In our model, here, g denotes the guaranteed interest rate. Furthermore, at least 90% of the investment returns on a book value basis have to be 1 Verordnung über Rechnungsgrundlagen für die Deckungsrückstellung (DeckRV). 2 2 Abs. 2 DeckRV. 9

11 credited to the policyholders account. 3 Since historically the guaranteed interest rate was much smaller than the market interest rates, this legal requirement was introduced to give policyholders a fair share of the surplus earned with their money. In our model δ represents such a minimum participation rate. The minimum participation is based on book values, and, hence, not every rise in the market value of the assets rises the insurer s obligations towards its policyholders. The management of the insurance company has some discretion whether to use certain accounting rules like the lower of cost and market principle which avoids showing rises in asset values on the balance sheet. However, there are restrictions to which extend a company can redirect investment gains into hidden reserves. In our model at least the fraction y of rises in market values has to be shown on the balance sheet. Thus, the insurance company has to credit at least the amount δ y (A t A + t 1) to the policyholders account. There are no restrictions on how the return on assets which exceeds the interest rate guarantees to policyholders is distributed. Usually German insurers credit a smoothed rate of interest to the policyholders account. This interest rate is typically much higher than the guaranteed rate. Insurers usually hold their overall interest rate as constant as possible to signal stability and low risk compared to other investment instruments as for example mutual funds. To model such a management decisions rule explicitly, we assume that the insurance company s management monitors the company s reserve quota x t = R t L t, and takes actions keep x t within a pre-specified range [a, b]. As long as a x t b the insurer credits a target rate of interest z > g to the policyholder s account. Thus, the value of the policyholders account L t = (1+z)L t 1 = (1+g)L t 1 +(z g)l t 1 increases by the guaranteed interest rate and by a fraction of the company s surplus. We further assume that the insurance company pays a fraction d of any surplus credited 3 1 Abs. 2 of the Verordnung über die Mindestbeteiligung in der Lebensversicherung (ZRQuotenV) for policies sold after July 29, 1994, and 81c Abs. 2 VAG in combination with 4 ZRQuotenV for policies sold on July 29, 1994 or before. 10

12 to the policyholders as dividends to its shareholders. If the reserve quota x t does not stay within the range [a, b], the insurer first reduces the target interest rate z as well as the dividend payments. If the insurer has not enough investment income and hidden reserves to provide the interest rate guarantees it is bankrupt. More precisely, at every point in time t = 1,..., T we distinguish between the following five cases: 1. Case 1: The insurance company does not have enough assets to provide the interest rate guarantees: A t < (1+g)L t 1. If this happens for one t = 1,..., T, the company is bankrupt. 2. Case 2: The insurance company has enough assets to fulfill its interest rate guarantees but cannot meet the management goal to hold the reserve quote within the range [a, b]: (1 + g)l t 1 A t (1 + a)(1 + g)l t 1. In this case the insurer credits only the guaranteed interest rate to the policyholders account and does not pay dividends to its shareholders: L t = (1 + g)l t 1 and D t = Case 3: If the insurance company credits the target rate of interest z to the policyholders account, the reserve quota would fall below a; but if the insurer credits only the guaranteed rate g to the policyholders account, the reserve quota would be higher than a: (1 + a)(1 + g)l t 1 < A t < [(1 + a)(1 + z) + d(z g)]l t 1. In this case, the insurance company credits an interest rate u (g, z) to the policyholders account such that the resulting reserve quota x t = a. Then the insurer s liabilities rise to L t = A t +d(1+g)l t 1 1+a+d ; and the insurer pays dividends D t = d A t (1+a)(1+g)L t 1 1+a+d. 4. Case 4: If the insurer credits the target rate of interest z to the policyholders account, the reserve quota x t stays in the range pre-specified [a, b]: [(1 + a)(1 + z) + d(z g)]l t 1 A t [(1 + b)(1 + z) + d(z g)]l t 1. In this case, the insurer 11

13 credits the target interest rate z to the policyholders account, and distributes a fraction of its surplus to the shareholders: L t = (1+z)L t 1 and D t = d(z g)l t Case 5: The insurance company does extremely well. If the company credits the target interest rate z to the policyholders account, there would be more capital in the company than desired by the management (x t > b): A t > [(1 + b)(1 + z) + d(z g)]l t 1. In this case, the insurance company credits a higher interest rate v (z, ) to the policyholders account such that the resulting reserve quota x t = b. Then the insurer s liabilities rise to L t = A t +d(1+g)l t 1 ; and the insurer 1+b+d pays dividends D t = d A t (1+b)(1+g)L t 1 1+b+d. To ensure that our model insurance company does not violate the legally required minimum participation of policyholders on book value investment returns, we explicitly check in the cases 2, 3, 4 and 5 whether the company credits at least δ y (A t A + t 1) to the policyholder s account. If the insurance company violated this condition and L t L t 1 < δ y (A t A + t 1) we will enforce the minimum participation by setting L t = L t 1 + δ y (A t A + t 1) and adjusting the dividend payment D t = d[δ y (A t A + t 1) gl t 1 ]. 2.3 Optimal Asset Allocation The goal of this paper is to analyze whether a life insurance company should invest in hedge funds, or in other words whether the optimal asset allocation of this life insurance company includes hedge funds. To address this research question, we have to determine the optimal asset allocation of our model insurance company. The model described so far, determines among other things wether the insurance company goes bankrupt. If we use this model in a Monte Carlo simulation, we can 12

14 calculate the shortfall probability of the company. For a given asset allocation and given asset scenarios, we can also calculate the expected rate of return of the insurance company s investment portfolio. Therefore, we can interpret the whole model as a function ( shortfallprobability E(portf olioreturn) ) = f(assetallocation, assetscenarios, others) (9) of the asset allocation, the asset scenarios and the parameters reflecting the product characteristics and the company s operating environment. Since this function returns a risk as well as a return measure, we can derive a Markowitz efficient frontier assuming that short sales are not allowed. The insurer can then choose a portfolio from the efficient set according to its risk preferences. Note, that we do not allow our model insurance company to re-balance its portfolio. Hence, the asset allocations considered in this model are static in nature. 3 Data and Model Calibration In our numerical analysis, we calibrate the model to empirically reasonable parameter values. We then examine the effect of different insurance product characteristics and operating environments on the insurer s optimal asset allocation. Our main interest, here, is whether there will be hedge funds in the optimal asset allocation. - Please insert TABLES 1 and 2 approximately here - We use monthly return data from the German stock index (DAX), the German bond index (REXP) and the CISDM hedge fund strategy indices from the Center for Inter- 13

15 national Securities and Derivatives Markets (CISDM) for the months 01/1992 through 12/2005 to calibrate our model. Table 1 presents a description of the investment strategies underlying the various CISDM hedge fund style indices. More precisely, for one set of simulations we use the DAX, the REXP and one of the hedge funds indices, respectively. For each of these three empirical time series we then calculate the mean and standard deviation and standardize the time series by subtracting the mean and dividing by the standard deviation. For each pair of time series i j {1, 2, 3} we calculate the correlation coefficient ρ i,j. Table 2 presents the annualized means, standard deviations and correlation coefficients of the return time series. We then estimate the AR(1) parameter a i,1 separately for each time series i = 1, 2, 3 and determine one pair of GARCH(1,1) parameters α 1 and β 1 for all three time series due to the fact that in the described model all time series must have the same parameters. Since the estimated GARCH parameter values for the REXP are very small, we set α 1 and β 1 to the average of the parameter estimates for the DAX and the hedge funds time series. The GARCH processes are weakly stationary if α 1 + β 1 < 1 which was the case with all our triples consisting of DAX, REXP and one of the CISDM hedge fund strategy indices. We calculate the parameter α 0 = 1 α 1 β 1 to ensure that the unconditional variance Var(Z it ) is equal to one. Table 3 summarizes the AR(1) GARCH(1,1) parameters of our asset model. - Please insert TABLE 3 approximately here - In the simulation procedure of the time series we first simulate the GARCH series. As starting values we simulate a 3-dimensional normal distribution with the given correlation coefficients. The further values of the time series are determined by calculating 14

16 the conditional covariance matrix H t, deriving the root C t and multiplying it with normally distributed innovations ε t. Starting from these GARCH processes, we build the AR processes, multiply the standard deviation and add the mean to each observation (see Section 2.1 for details). This results in a set of simulated series of the DAX, REXP and hedge funds. We use the last step of our simulation procedure - adding the mean of the return time series to the simulated processes - to fine tune the model. Since the estimated mean return for bonds based on the data is 6.8% per year which seems to be too high compared to the current low interest rates, we reduce the expected return for bonds to 5%. We also address the well documented biases present in hedge fund databases (see, e.g., Ackermann, McEnally, and Ravenscraft (1999), Brown, Goetzmann, and Ibbotson (1999), Liang (1999), and Fung and Hsieh (2000)) at this step in the simulation process. The most prominent bias is the survivorship bias which results from the fact that hedge funds can choose voluntarily whether they report their return data to a database. Thus, poor performing hedge funds will probably stop reporting their returns. In their empirical analysis of hedge fund returns, Fung and Hsieh (2000) find a survivorship bias of 3%. We, hence, subtract 3% from the estimated annual hedge fund returns and calibrate our simulated time series accordingly. 4 Numerical Results and Discussion In the following numerical analysis, we will focus on the effect of different parameter settings on Markowitz efficient portfolios. We are specifically interested in how the hedge fund position in efficient investment portfolio depends on the insurance company s liability structure and whether there are types of hedge funds which are more suitable for insurance companies than others. 15

17 4.1 Efficient Portfolios With and Without Hedge Funds Let us first focus on the basic question whether life insurance companies should invest in hedge funds at all. To address this question we calculated a Markowitz efficient frontiers for a same insurance company which only invests in stocks and bonds, and compare it to the efficient frontier of the same company adding hedge funds as a third investment alternative. We set the parameters of our model insurance company to the values specified in Table 4. All calculations in this paper are based on 10,000 independent Monte Carlo simulations unless otherwise stated. - Please insert FIGURE 2 approximately here - Figure 2 shows the two efficient frontiers with and without Convertible Arbitrage hedge funds as a possible investment opportunity. Obviously, the efficient frontier which is calculated with hedge funds as an investment alternative lies above the one calculated without hedge funds. Therefore, adding hedge funds to the investment universe of our model insurance company increases the expected rate of return of all efficient portfolio substantially. If we have a closer look on the asset allocations of these efficient portfolios we find that they include a huge hedge fund position. The portfolio with the highest rate of return for a zero percent shortfall probability consists of 6.14% stocks, 2.91% bonds and 90.95% hedge funds. If we move along the efficient frontier towards higher shortfall probabilities the hedge fund position decreases and the stock position increases whereas the bond position hardly changes. For example the efficient asset allocation for a shortfall probability of 4.05% includes 28.00% stocks, 1.56% bonds and 70.44% hedge funds. The main reason of these high hedge fund percentages in efficient asset allocations is the fact that hedge funds reduce the overall volatility of the investment portfolio substantially. 16

18 Hedge fund returns have on average a relatively low standard deviation. In addition, the correlation between hedge fund returns and stock market returns is relatively low and the correlation between hedge fund returns and bond returns is even negative in some cases (see Table 2). If we compare, for example, the standard deviations of the two efficient portfolios with 7.5% expected rate of return we find that the standard deviation of the portfolio with hedge funds is 3.28% which is much lower than the 10.11% of the portfolio without hedge funds. The difference between the insurer s shortfall probability is even more dramatic. A portfolio standard deviation of 10.11% in the case without hedge funds results in a shortfall probability of 27.71%. If the insurance company invests in hedge funds, however, it does not default in any one of the scenarios. These results indicate that life insurance companies with cliquet-style interest rate guarantees should not hold volatile asset portfolios. The inclusion of hedge funds is beneficial to further diversity the asset portfolio and, hence, to reduce the overall volatility of the insurer s investments. Thus, the answer to our question whether life insurance companies should invest in hedge funds is yes, and this result is robust to the selection of any of the CISDM hedge fund style indices. At least on the surface, the huge hedge fund holdings suggested by our model seem to contradict conventional asset management practices. However, one has to keep in mind that our model only reflects normal market conditions. Extreme events like stock or bond market crashes are not captured by our model. Especially since our model only considers correlations between the three asset processes, but the dependence structure in extreme market situations is probably completely different and not as favorable as the collapse of the hedge fund Long Term Capital Management demonstrates. Therefore, we recommend that insurance companies should determine limits for their hedge fund investments. Such limits should reflect extreme market conditions as well as the fact that the hedge fund market is still relatively small and cannot grow unlimitedly. Especially 17

19 the growth potential of hedge funds looking for arbitrage opportunities is limited by an efficient capital market. Given the results from our model, limits for hedge fund positions in the asset allocation will probably be binding in any optimization. 4.2 Efficient Portfolios Including Different Types of Hedge Funds In this section we analyze whether there are types of hedge funds which are more suitable for insurance companies than others. Following our recommendation we assume that the model insurance company sets a limit of 5% for hedge funds in its asset allocation. Given this constraint, we calculate the efficient frontiers for all 9 hedge fund styles. Table 5 shows the ranking of the hedge fund styles according to its expected rate of return for a given shortfall probability of 5%. We find that the Event Driven hedge funds perform best and the Global Macro hedge fund perform worst. The difference in the expected rate of return of these two extreme cases is 0.47%. Thus, insurance companies should choose carefully the hedge funds in which they invest. - Please insert TABLE 5 approximately here The Liability Structure s Influence on Efficient Portfolios The previous analysis kept the parameters of the insurance company s liability structure constant. We now vary some of the parameters individually and examine the effect on the fraction of hedge funds in the efficient asset allocations. In a first step, we vary the guaranteed rate of interest and assume that our model insurance company guarantees g = 4% instead of g = 2.75%. Graph 2 of Figure 3 shows 18

20 the efficient frontiers with and without hedge funds for the base case as well as for the case with a 4% interest rate guarantee. In both cases, the efficient frontier with hedge funds lies above the one calculated without hedge funds. If we have a closer look on the asset allocations of the efficient portfolios for the case of the higher guarantee we find that they include a huge hedge fund position as well. The portfolio with the highest rate of return for a zero percent shortfall probability consists of 1.36% stocks, 1.06% bonds and 97.58% hedge funds. If we move along the efficient frontier towards higher shortfall probabilities the hedge fund position decreases and the stock position increases, for example the efficient asset allocation for a shortfall probability of 3% includes 24.00% stocks, 0.71% bonds and 75.29% hedge funds. Therefore, the fraction of hedge funds in efficient portfolios is higher for insurance companies with higher interest rate guarantees. The intuition behind this result is that companies with higher cliquet-style guarantees should further reduce the volatility of its asset portfolio. However, the most interesting result is that the difference between the efficient frontiers with and without hedge funds is greater for insurers with higher interest rate guarantees. Thus, such insurance companies can gain more from adding hedge funds to their investment universe and, hence, have a greater incentive to invest in hedge funds. - Please insert FIGURE 3 approximately here - Second, we reduce the insurance company s initial reserve quota x 0 to 10% instead of the 20% in the base case. This means that our model insurance company is now much less capitalized. Graph 3 of Figure 3 shows the efficient frontiers with and without hedge funds for the less capitalized insurance company as well as for the reference company. Once again, for the less capitalized insurer, the efficient frontier with hedge funds lies 19

21 above the one calculated without hedge funds. The asset allocations of the efficient portfolios still includes a huge hedge fund position, the pattern of the asset allocations, however, is now slightly different. The lowest possible shortfall probability is 0.03%, and the corresponding portfolio with the highest rate of return consists of 1.78% stocks, 23.28% bonds and 74.94% hedge funds. If we move along the efficient frontier towards higher shortfall probabilities the bond position decreases and the hedge fund position increases, for example the efficient asset allocation for a shortfall probability of 0.16% includes 3.00% stocks, 0.19% bonds and 96.82% hedge funds. Once the bond position is close to zero, the stock position increases and the hedge fund position decreases again. For example, the asset allocation for a shortfall probability of 3.87% consists of 21.00% stocks, 1.13% bonds and 77.87% hedge funds. For shortfall probability above 0.1%, however, the fraction of hedge funds in efficient portfolios is higher than in the base case. We also find that the difference between the efficient frontiers with and without hedge funds is greater for less capitalized insurers. In other words less capitalized insurers can gain more from investing in hedge funds and, hence, have a greater incentive to do it. Third, we analyze how a change in the accounting system towards a full fair value oriented accounting approach affects an insurance companies liability structure and how the resulting structure affects the company s decisions to invest in hedge funds. By setting the accounting system parameter in our model to y = 0.95, our model insurer is forced to show 95% of all rises in the market value of its assets on its balance sheet. Such a close to full fair value accounting system restricts the possibilities of the management to redirect investment gains into hidden reserves, and, hence, reduces the financial buffer of the insurance company. Note that y = 0.5 in the base case. Graph 4 of Figure 3 shows the efficient frontiers with and without hedge funds for the two different accounting environments. Let us focus on the fair value accounting case. The efficient frontier with 20

22 hedge funds lies above the one calculated without hedge funds. The asset allocations of the efficient portfolios include a huge hedge fund position, the pattern of the asset allocations is different from the base case, but similar to the case of a less capitalized insurance company. The lowest possible shortfall probability is 0.01%, and the portfolio with the highest rate of return given this shortfall probability consists of 0.16% stocks, 28.84% bonds and 71.00% hedge funds. If we move along the efficient frontier towards higher shortfall probabilities the bond position decreases and the hedge fund position increases. The efficient asset allocation for a shortfall probability of 0.03%, for example, includes 1.00% stocks, 0.03% bonds and 98.97% hedge funds. Once the bond position is close to zero, the stock position increases and the hedge fund position decreases again. For example, the asset allocation for a shortfall probability of 3.58% consists of 15.00% stocks, 0.82% bonds and 84.18% hedge funds. For all shortfall probabilities above 0.2% the fraction of hedge funds in efficient portfolios is higher than in the base case. In analogy to the previous cases, we find that the difference between the efficient frontiers with and without hedge funds is greater for insurers companies under a fair value accounting environment. In other words such an environment creates greater incentives to invest in hedge funds. In summary, insurance companies with higher interest rate guarantees and lower financial reserves should invest a bigger proportion of their assets in hedge funds. Such companies also have a greater incentive to invest in hedge funds. Adding hedge funds as a possible investment alternative increases the expected portfolio return for these insurance companies much more than for comparable companies with lower interest rate guarantees and lower financial reserves. This result is especially important for insurance regulators. Higher interest rate guarantees increase the liabilities of an insurance companies, and, hence, ceteris paribus decrease the financial strength of an insurance company. Therefore financially weak insurers have strong incentives to invest in hedge funds. Based on a 21

23 model under normal market conditions their hedge fund position should be quite high. But such insurers probably do not have the capital to deal with extreme market events, and a huge hedge fund holding might exacerbate their situation. Moral hazard could encourage very weak insurers to increase their hedge fund holdings substantially in a gofor broke behavior. Thus, insurance regulator should monitor the hedge fund positions of financially weak insurers closely. 5 Conclusion In this paper we study whether it is advantageous for life insurance companies to invest in hedge funds. To address this research question, we extend the Kling, Richter, and Ruß (2007) model of a standard German life insurance company with cliquet-style interest rate guarantees by incorporating three correlated AR(1) GARCH(1,1) processes, and an objective function for optimizing the insurer s asset allocation. Within this model framework we use Monte Carlo Simulations to examine different asset allocations under various parameter settings. While our results are obtained with the assumption of restrictive cliquet-style guarantees, we argue that our results are the limiting case and, hence, a benchmark for all other types of life insurance products and company structures. We calibrate our model to the German stock index (DAX), the German bond index (REXP) and the Hedge Funds Indices of the Center for International Securities and Derivatives Markets (CISDM), and correct the estimated hedge fund returns for biases inherent in hedge fund databases. Our simulation analysis provides the following four results: First, it is advantageous for life insurance companies to invest in hedge funds since hedge funds help to reduce the volatility of the overall asset portfolio. Second, the higher the interest rate guarantee is, the less capital an insurer holds, and the less possibilities the accounting system provides to smooth the book values of the investment 22

24 portfolio, the more incentives an insurance company has to diversity its portfolio by investing in hedge funds. Third, the higher the interest rate guarantee, the fewer capital an insurance company holds, and the more restrictive the accounting system is, the bigger is the optimal hedge fund holding in the company s investment portfolio. Fourth, if our model insurer tolerates a ruin probability of five per cent, an investment in eventdriven multi-strategy hedge funds leads to the highest expected returns. Our paper also contributes to the literature on asset models by introducing a simple simulation procedure for the correlated AR(1) GARCH(1,1) processes. However, our model only covers normal market conditions. Extreme events, where one would expect tail dependencies between asset price processes, are left for future research. References Ackermann, C., R. McEnally, and D. Ravenscraft, 1999, The Performance of Hedge Funds: Risk, Return, and Incentives, Journal of Finance, 54, Agarwal, V., and N. Y. Naik, 2001, Risk and Portfolio Decisions Involving Hedge Funds, Technical Report, London Business School. Brooks, C., and H. M. Kat, 2002, The Statistical Properties of Hedge Fund Index Returns and Their Implications for investors, Journal of Alternative Investments, 5, Brown, S., and W. N. Goetzmann, 2003, Hedge Funds with Style, Journal of Portfolio Management, 29, Brown, S. J., W. N. Goetzmann, and R. Ibbotson, 1999, Offshore Hedge Funds: Survival and Performance, Journal of Business, 72, Engle, R., and K. Kroner, 1995, Multivariate Simultaneous Generalized ARCH, Econometric Theory, 11, Fung, W., and D. A. Hsieh, 1997, Empirical Characteristics of Dynamic Trading Strategies: The Case of Hedge Funds, Review of Financial Studies, 10,

25 Fung, W., and D. A. Hsieh, 2000, Performance Characteristics of Hedge Funds and CTA Funds: Natural Versus Spurious Biases, Journal of Financial and Quantitative Analysis, 35, Fung, W., and D. A. Hsieh, 2001, The Risk in Hedge Funds Strategies: Theory and Evidence from Trend Followers, Review of Financial Studies, 14, Kling, A., A. Richter, and J. Ruß, 2007, The Interaction of Guarantees, Surplus Distribution, and Asset Allocation in with Profit Life Insurance Policies, Insurance: Mathematics and Economics, 40, Liang, B., 1999, On the Performance of Hedge Funds, Financial Analysts Journal, 55, Liang, B., 2000, Hedge Funds: The Living and the Dead, Journal of Financial and Quantitative Analysis, 35, Malkiel, B. G., and A. Saha, 2005, Hedge Funds: Risk and Return, Financial Analysts Journal, 61, Martin, G., 2001, Making Sense of Hedge Fund Returns: A New Approach, in E. Acar (ed.), Added Value in Financial Institutions: Risk or Return?. pp , Prentice Hall, Englewoods Cliffs. Popova, I., D. P. Morton, and E. Popova, 2003, Optimal Hedge Fund Allocation with Asymmetric Preferences and Distributions, Technical Report, University of Texas. 24

26 Table 1: Hedge Fund Styles Strategies Group Type of Hedge Fund Strategies Description Relative Value Equity Market Neutral Index Take long positions in undervalued equities and offsetting short positions in overvalued equities within the context of minimizing the market exposure in the underlying equity market traded. Convertible Arbitrage Index Take long positions in convertible securities and try to hedge those positions by selling short the underlying common stock. The goal is to minimize the market exposure. Event Driven Distressed Securities Index Take Positions in the securities of companies where the price has been or is expected to be affected by a distressed situation, such as an announcement of reorganization due to financial or business difficulties. The goal is to make use of event driven return opportunities that may independent of general market movements. Event Driven Multi-Strategy Index Attempt to predict the outcome of corporate events like liquidations, spin-offs and others transactions and take the necessary position to make a profit. The goal is to make use of event driven return opportunities that may independent of general market movements. Merger Arbitrage Index Concentrate on companies that are the subject of mergers, tender offers or exchange offers in order to use event driven return opportunities. Opportunistic Emerging Markets Index Invest in equities or sovereign debt of companies located in emerging or developing economies. The underlying market exposure will not be systematically eliminated or minimized. Equity Long/Short Index Take long and short equity positions varying from net long to net short, depending if the market is bullish or bearish. No systematical eliminating or minimizing of the underlying market exposure. Global Macro Index Employ opportunistically long and short multiple financial and non-financial assets using systematic trend following models. No systematical eliminating or minimizing of the underlying market exposure. Total Equal Weighted Hedge Fund Index Represents the overall composition of the hedge fund universe. Note: This classification of hedge fund styles is based on the CISDM hedge fund strategy indices from the Center for International Securities and Derivatives Markets (CISDM). Source: CISDM. 25

27 Table 2: Descriptive Statistics and Correlations of Asset Return Time Series Mean Std. Dev. Correlation Correlation (annual.) (annual.) with DAX with REXP Convertible Arbitrage Distressed Securities Event Driven Equity Long/Short Emerging Markets Equity Market Neutral Global Macro Merger Arbitrage Equal Weighted DAX REXP Note: The reported statistics are based on monthly log return time series. We calculate the annualized mean by multiplying the empirical mean with 12, and we calculate the annualized standard deviation by multiplying the empirical standard deviation with 12. Note that the mean hedge fund returns reported in this table are the raw estimates which are not adjusted for biases present in hedge fund databases. For our simulation, we subtract 3% from each of these values. Furthermore, we do nor use the estimated REXP return, but set the annual return of bonds in our model equal to 5%. 26

28 Table 3: Parameter Values for Asset Processes AR(1) GARCH(1,1) a i,0 a i,1 α 0 α 1 β 1 Convertible Arbitrage Distressed Securities Event Driven Equity Long/Short Emerging Markets Equity Market Neutral Global Macro Merger Arbitrage Equal Weighted DAX REXP Note: To simulate the AR(1) processes, we use the estimated parameters obtained from the monthly time series data. The reported values in columns two and three of this table are rounded. The GARCH(1,1) parameters reported in this table, however, are exactly the ones used in our simulations. 27

29 Table 4: Base Case Parameter Settings x 0 = 20% Initial reserve quota of the insurance company at time t = 0. g = 2.75% Minimum interest rate guarantee. a = 5% Trigger for management intervention. If the reserve quota x t < a the management cuts the target interest rate and takes other actions to bring x t within the pre-specified range [a, b]. b = 30% Trigger for management intervention. The management intervenes if x t > b. δ = 90% Minimum participation rate. At least the fraction δ of the investment returns on a book value basis have to be credited to the policyholders account. d = 3% Dividend factor. y = 50% Accounting system parameter. The fraction y of rises in the market value of assets is reflected in the book values. T = 10 Number of time periods (years) considered. Note: The parameter values of our base case are the same Kling, Richter, and Ruß (2007) use in their reference case. Thus, our results are directly comparable to their results. 28

30 Table 5: Ranking of Hedge Fund Styles Rank Hedge Fund Style Rate of Asset Allocation (in%) Return (in%) DAX REXP HF 1 Event Driven Equal Weighted Equity Long/Short Distressed Securities Merger Arbitrage Convertible Arbitrage Emerging Markets Equity Market Neutral Global Macro Note: The presented ranking is based on the assumptions that first, the insurance company can only invest up to 5% in hedge funds, and second, the insurance company is not willing to accept a shortfall probability above 5%. 29