Life Settlement Portfolio Valuation

Transcription

1 Life Settlement Portfolio Valuation MITACS Industrial Summer School 2008 Simon Fraser University August 8, 2008 Greg Hamill David Lin Iris Shau Jessica Wu Cheng You With Huaxiong Huang, Tom Salisbury, and Phillip Poon

2 Life Settlement Portfolio Valuation page 1 of 23 Contents 1 Introduction 3 2 Estimating Dynamics of Mortality Table Denotations and Assumptions Mathematical Model Model Verification Present Value Percentile Model Model description and assumptions Detailed model establishment Numerical simulation and analysis Options Comparison Model Description of the Model A Case Study Using the Option Comparison Model Conclusion 22 1

3 Life Settlement Portfolio Valuation page 2 of 23 Abstract Life settlement, a resale of insurance policy, has been a growing industry in financial market, because of the appealing rate of return of investor and favorable tax treatment. It exists in three different forms, term life, whole life and universal life. We study the whole life settlement in this paper. A whole life settlement can be treated as a Senior Life Settlements insurance, in which usually seniors who are over sixty sell their insurance policy to hedge fund; and therefore assume mortality risk, also named as longevity risk in our paper, and liquidity risk exist in this industry. Using the discrete-time stochastic process model from Cairns, Blaske and Dowd (2006), we quantify longevity risk as the risk that the average insurer lives longer than anticipated. We then estimate the net present value of individual life settlement for hedge fund; this analysis can be extended to the acceptable selling price of life settlement. Given the distribution of the net present value with sample size of one thousand, we pick up the value at five percentile as the acceptable price hedge fund bears. Based on a large sample size of distribution, the markup price at five percentile can represent a reasonable price for life settlement in a general consideration. Longevity risk and liquidity risk can be more efficiently managed by comparing the hedge fund portfolios. We determine the net worth of life settlement between one portfolio with life settlement and one without. We set a benchmark for the proportion of times that the first hedge fund does better than the second one since hedge funds want to ensure the life settlements will yield a profit. Also, we include a probability array of liquidity. And the adjustment allows us to study different liquidity affected the hedge fund. 2

4 Life Settlement Portfolio Valuation page 3 of 23 1 Introduction A life settlement is the selling of any high value policies to a hedge fund as the sellers no longer have a need for the policy and can receive a significantly larger amount for their policy than the insurance company would have paid for surrenders. Life insurance companies are mostly opposed to life settlements because when the owners sell their policy to a hedge fund, the hedge fund is guaranteed to receive the death benefit from the insurance company when the policyholder dies. Without the existence of hedge funds, the policyholder would just sell the policy back to the insurance company for an amount far less than the death benefit. The idea of life settlements originated from the viatical industry which was first conceived in Europe in the 1880s and then ventured to the United States in 1989 during the rise of HIV cases, in which the owner has a life expectancy of less than two years. Since then, life settlements have grown to include a wider variety of diversified portfolios, they are term-life, whole life and universal life settlements. Most hedge funds build a portfolio of about 5-50 policies; the policies chosen are based on the preference of the hedge fund. Usually, the sellers are above the age of sixty in the whole life settlement market. According to some literatures, in the past 15 years, average annual rate of return has been 15.82% for the life settlement transactions completed (policies matured and the investors paid in full). Thus, it is more important to evaluate the portfolio of life settlement since prior returns do not assure future returns. Although life settlements are starting to become increasingly common, hedge funds still do not have a precise way to put an exact value of their investment at any given time. Hedge funds need a way to find the net present value of their investment at any time for purposes of marking to market, as well as being able to divide the worth of the investment among each investor. In trying to calculate these values, hedge funds face numerous issues including: longevity risk, since the policyholder may live longer than expected, resulting in more premium payments and a longer wait time for the death benefit; illiquidity risk, since hedge fund may not easily re-sell the insurance policies or may not have cash to pay for those investors who want to withdraw. Also life settlement is still a incomplete market, there exists market uncertainty, for example, the size and number of buyers and sellers; It is also an important factor because hedge funds can buy more policies and receive greater death benefits if there is less competition. All the above factors should be monitored because of their presence in life settlement. Looking into the life expectancy of the general population, it keeps improving in each year. It is necessary to build a model to predict the future survival rate. Forecasting life expectancy based on Mortality Table of British Columbia from 1976 to 2005, released by Canadian Human Mortality Database, the survival probability beyond 2005 of someone aged 60 at 2005 can be obtained. Hedge fund can make their decisions over this reference of lifetime model. With this incomplete life settlement market, it is a way to use models that have similarities as life settlement with a consideration of possible risk factors. The models used in this paper much fit the data, have good performance, and, importantly for our purpose, quantify the uncertainty surrounding future mortality, the discount factor involving longevity and illiquidity. Once a hedge fund buys the policy from a policyholder, the policyholder no longer has any commitment to the hedge fund. Because the hedge fund no 3

5 Life Settlement Portfolio Valuation page 4 of 23 longer has any contact with the policyholder, the policyholder s state of health at any given time is unknown (ie. the policyholder may be diagnosed with a fatal disease the day after he or she sells their policy), so a policyholder s life expectancy is still unpredictable. We calculate longevity risk by using mortality tables to estimate the life expectancy of the policyholder. For better predictions, we look at the physiological age of policy holders depending on gender and current state of health. We even then, there are actuarial underwriters who give a mortality rating for each individual. The mortality rating compares the mortality of the individual to that of the population. Our next consideration is the illiquidity risk. When hedge funds need to make any changes (ie there may be entering and exiting investors at any given time) but the policyholders have not died yet, so there is no money available. The hedge fund is stuck with a valuable asset but they are unable to use it unless they decide to sell or trade it at a discount. Almost the entire life settlement market is illiquid because unfortunately, there is nothing the hedge fund can do until the policy holder dies. The main problem now is incorporating all of these risks into the calculating the net present value of the portfolios. In approaching this problem, we figure out what the discount factors are. This paper is organized as follows. In section 1 of the paper, we adopt a discrete-time stochastic process model from Cairns, Blaske and Dowd (2006). Stochastic models are important for risk measurement, they produce assessment of risk premium. By using the Mortality Table of British Columbia from 1976 to 2005, released by Canadian Human Mortality Database (CHMD), we construct a model as a a random walk with drift. Then, we try to quantify parameter uncertainty in order to get a future mortality curve which tells us the general improvements in life expectancy and which ages group of people get this improvement. In subsequence section, we focus on the dynamics of the survival probability. Because of the random variable term in our model, we then can get a confidence intervals of the distribution of the survival rate, in which provide us different percentiles to do further simulation in the sections below. In section 3, we construct a Present Value Percentile Model. We study the appropriate present value, can also be defined as the price of life settlement, in order to get a basic idea of how much hedge fund is willing to pay for buying life settlement. After obtaining a mortality table, we can further use it to estimate the death year of insurers. We simulate a model by one thousand times and get net death benefit hedge fund received from these one thousand people. By using a risk-free interest rate as the discount factor, we find one thousand net present values and then pick up a value at five percentile for example, which represents an acceptable value hedge fund spent on buying a life settlement contract. The percentile means how much hedge fund can tolerate the high payment to the life settlement. The higher percentile setting point, the higher price hedge fund bear. In section 4, we evaluate the net worth of life settlement by comparing two hedge funds, in which one involves both life settlement and bonds, another involves bonds only. This Options Comparison Model, not only various uncertainty, including longevity risk and liquidity risk, are incorporated; also it shows the benchmark for the proportion of times of how life settlement performs. For the consistence, we introduce an idea of reinsurance in our simulation. The basic arrangement of reinsurance is that hedge fund pays for life settlement fee, 4

6 Life Settlement Portfolio Valuation page 5 of 23 premium and an additional amount of reinsurance fee. Reinsurer financially viable at the time the contract comes due, thus the risk of the insurance paying off later than expected can be removed. The reason of incorporating reinsurance is that we try to set the same due date of the premium payment for comparison, and we choose twenty years in our case. In our process, we show the cash flow of hedge fund to demonstrate the longevity risk in life settlement. At the same time, hedge fund can find when to liquidate its bonds in order to pay its obligation. Section 5 is conclusion. we compare price set by our two models and real price in market to show the reasonableness of the models. 2 Estimating Dynamics of Mortality Table To estimate the mortality table in every year in future we adopt a discrete-time stochastic process approach from Cairns, Blake and Dowd (2006). 2.1 Denotations and Assumptions Denotation P (t, x) probability that an individual aged x in year t survives until year t + 1 Q(t, x) 1 P (t, x) m(t, x) dentral death rate for individuals aged x in year t, q(t,x) 1 approximated by 1 q(t,x)/2 S(t) survival probability that an individual survives until year t, subject to S(t + 1) = S(t)(1 m(t, x)) Assumption 1 Mortality table released by CHMD are unbiased estimates. 2 survival probability in the starting year is equal to 1, i.e.s(0) = Mathematical Model Based on Mortality Table of British Columbia from 1976 to 2005, released by Canadian Human Mortality Database (CHMD), 2 we adopt the following model to predict mortality rates above the age of 60 in future:(let t = 0 correspond to the beginning of 1976) Q(t, x) log( P (t, x) ) = A 1(t) + A 2 (t)x (1) t = 1, 2, x = 60, 61,, 105 Using the real data to estimate A(t) = (A 1 (t), A 2 (t)) T, t = 1, 2,, 30, We find the fittings for each year are all quite good. In fact, among the 30 1 For a full discussion, the reader is referred to Benjamin and Pollard (1993) or Bowers et al.(1986). 2 It s a public data source, with web link 5

7 Life Settlement Portfolio Valuation page 6 of 23 regressions, the least R 2 is as follows. And we illustrate the fitting line in year Figure 1:Fitting line of Model(1) in 2005 log(q/p) Age of cohort at 2005 These results show a clear trend in both series. The downward trend in A 1 (t) reflects general improvements in mortality over time at all ages. The increasing trend in A 2 (t) means that the curve is getting slightly steeper over time: that is, mortality improvements have been greater at lower ages. There were also changes in the trend and in the volatility of both series. To make forecasts of the future distribution of A(t) = (A 1 (t), A 2 (t)) T, we will model A(t) as a two-dimensional random walk with drift. A(t + 1) = A(t) + µ + C Z(t + 1) (2) where µ is a constant 2 1 vector, C is a constant 2 2 upper triangular matrix and Z(t) is a 2-dimensional standard normal random variable. These results show a clear trend in both series. The downward trend in A 1 (t) reflects general improvements in mortality over time at all ages. The increasing trend in A 2 (t) means that the curve is getting slightly steeper over time: that is, mortality improvements have been greater at lower ages. There were also changes in the trend and in the volatility of both series. To make forecasts of the future distribution of A(t) = (A 1 (t), A 2 (t)) T, we will model A(t) as a two-dimensional random walk with drift. 6

8 Life Settlement Portfolio Valuation page 7 of 23 Fiqure 2: Estimated A1 from 1976 to 2005 A Year Fiqure 3: Estimated A2 from 1976 to 2005 A Year These results show a clear trend in both series. The downward trend in A 1 (t) 7

9 Life Settlement Portfolio Valuation page 8 of 23 reflects general improvements in mortality over time at all ages. The increasing trend in A 2 (t) means that the curve is getting slightly steeper over time: that is, mortality improvements have been greater at lower ages. There were also changes in the trend and in the volatility of both series. To make forecasts of the future distribution of A(t) = (A 1 (t), A 2 (t)) T, we will model A(t) as a two-dimensional random walk with drift. A(t + 1) = A(t) + µ + C Z(t + 1) (3) where µ is a constant 2 1 vector, C is a constant 2 2 upper triangular matrix and Z(t) is a 2-dimensional standard normal random variable. ˆµ = Fitting the model, we find the estimates are: ( ) ( , and ˆV = ĈĈT = Based on A(t), t = 1, 2,, 30, we can simulate A(t), t = 31, 32,, 130 according to (2) to estimate the mortality table from 2006 to In subsequent sections we will focus on the dynamics of the survival probability, S(t), which can be calculated directly from estimated mortality table. Assuming the person is 60 in 2005, we set t = 0 to correspond to the beginning of 2005 from now on. ) Fiqure 4: Mean and 90% confidence interval of S(t) S(t) t 8

10 Life Settlement Portfolio Valuation page 9 of 23 By simulating dynamics of A(t) a large number of times, we calculate the mean, 5th and 95th percentiles of the distribution of S(t) and show them in figure Model Verification We already described the model above and estimated all the parameters, including A(t), t = 1, 2,, 30, µ and C,in which to predict mortality table in future. In this subsection, we show the effectiveness of our model. We use the Life Table for British Columbia between 1976 and 1995 to fit our mortality model (2) and then further predict the Mortality Table for year 1996 to year By comparing our result with the real data released by CHMD, we find that we did a quite good estimation. We simulate A(t), t = 21, 22,, 30 according to (2) for 1000 times. In each simulation, we calculate mortality rate of a man aged 65 and another aged 85 for each year from 1996 to 2005, and get the mean and 90% confidence interval of both the mortality rates in each year. They are showed in Figure 5 and Figure 6, together with real mortality rate. Fiqure 5: Estimated and real mortality rate for people aged 65 qx Mean 90% Confidence Interval Real Data Year 9

11 Life Settlement Portfolio Valuation page 10 of 23 Fiqure 6: Estimated and real mortality rate for people aged 85 qx Mean 90% Confidence Interval Real Data Year To summarize the mortality tables we estimated, we find that, for a man aged 65, the real mortality rate lies within the 90% confidence interval of our model; and for a man aged 85, basically, the actual value lies within the 90% confidence interval, though there exist some outliers with a small error less than 3 basic points. Thus, the prediction we get is reliable. 3 Present Value Percentile Model 3.1 Model description and assumptions Hedge funds purchase life settlements and therefore assume aggregate mortality risk. Referring to Cairns, Blake and Dowd (2006), we quantify aggregate mortality risk as the risk that the average policy holder lives longer than is predicted by the model. In this section, we use simulations to quantify the aggregate mortality risk faced by an hedge fund buying life settlements from a single birth cohort. For people whose life expectancy is longer than the expected, hedge funds suffer from the risk of getting their money back late i.e. longevity risk. This risk may cause insolvency problems so that hedge fund investors demand an aggregate mortality risk premium. To focus on aggregate mortality risk, we impose a number of simplifications. We assume that the hedge fund buys a large number of life settlements of a 10

12 Life Settlement Portfolio Valuation page 11 of 23 single type from people in a single birth cohort who have population average mortality and changes in mortality rates as predicted by (2). We also fit the forecast of mortality for males and females separately. Moreover, we assume that there are no other miscellaneous costs such as transaction fees. 3.2 Detailed model establishment Like the mortality model we mentioned above, we have already predicted the annual survival probabilities of people in British Columbia, Canada. To be more specific, we adopt simple linear regression to estimate the vectors A 1 and A 2 for the latter stochastic process, based on the data of mortality tables from 1976 to Then, we utilize the estimated vectors to fit into the stochastic model to get ˆµ = ( ), and ˆV = ĈĈT = ( After we obtain the stochastic model, we predict the further 20 years survival probabilities for determining the time of cash flows. Next, we choose the specific rates of interest 5% to discount the resulting cohort premiums, and sum the premiums to arrive at a present value. We use this present value incorporating the aggregate mortality risk for pricing. It can be expressed in the following formula: P V = T 1 t=1 C t (1 + r) t + S (1 + r) T where T is the year that a person dies, r is the discount rate and S is the insurance value. The hedge fund makes a loss if, in a particular simulation, the mortality draw it experiences results in premiums that exceed the settlements. As Friedberg & Webb (2007), we calculate the amounts by which the total cash flows in the future by the hedge fund on life settlement sold to a single birth cohort will exceed amounts forecast using our random walk model (2) at the 95th and 99th percentiles of the distribution of present values, assuming that the only source of variation is aggregate mortality risk. The distribution of present values shows possible choices of the price. The percentile also means that the maximum probability that hedge funds accept losses i.e. loss probability. The reason that we choose the percentile instead of the overall mean is that if we simply choose the expectation of the distribution of present values, the hedge fund will have 50% chance to lose money. In order to eliminate the unhappy situation, we consider it from an angle of the maximum probability that hedge funds can bear losses. 3.3 Numerical simulation and analysis The following are the simulated results of the price of the single life settlement. According to Data Collection Report published by Life Insurance Settlement Association (LISA), in 2005, the average amount of death benefit settled by each participant is ($3, 413, 014, 939/1, 746)= $ , average proceeds paid to sellers is $ , average amount of premium per year is $ We simply set, in our case, both the death benefit and premium of life settlement to corresponding average values. ) 11

13 Life Settlement Portfolio Valuation page 12 of 23 Case one: reinsurance in 20 years Percentile Age 60 Age 65 Age 70 Age 75 Age 80 Age 85 1% % Table 1: Present Values, Male Percentile Age 60 Age 65 Age 70 Age 75 Age 80 Age 85 1% % Case two: without reinsurance Table 2: Present Values, Female Percentile Age 60 Age 65 Age 70 Age 75 Age 80 Age 85 1% % Table 3: Present Values, Male Percentile Age 60 Age 65 Age 70 Age 75 Age 80 Age 85 1% % Table 4: Present Values, Female In reality, Hedge funds prefer to buy life insurances from unhealthy people so we construct the ratings to the health of different people. The rating means a multiplier to the probability a person dies, which is larger than 1. In our case, we take the rating 120%, meaning that the probability that the person dies is 1.2 times of the probability that average people die. We also consider women and men separately because we observe that on average women outlive men from the mortality table. Since hedge funds have the options to reinsure the life settlements or not, we consider two cases about the problem. One case is that hedge funds reinsure the life settlements. For men or women selling their life insurances before or at age 75, whenever loss probability is 1% or 5%, the price of the life settlement that the hedge fund can offer are basically the same. This is because a larger proportion of people in BC before or at age 75 are expected to live longer than 20 years so that the reinsurance company pays the value of the life insurance policy. Hence, the cash flows patterns are exactly the same. Another case is that hedge funds do not reinsure life settlements. For men or women, when loss probability is 0.01 or 0.05, the present values rise increasingly as the age of selling increases. It shows that as the person gets old, the value of the life insurance increases because the person is more likely to die in the near future and the hedge fund can get its return quickly. Comparing the two loss probability, we can see that if hedge funds have to bear more probability of loss, they should price the life settlements higher. Moreover, comparing men and women, we can see that men s life insurance is more valuable than women s and the reason is that men s longevity is shorter than women s. 12

14 Life Settlement Portfolio Valuation page 13 of 23 In addition, we can see that as the age grows, the price differences for the 99th and 95th percentile become larger, meaning that the variation of mortality increases as the age increases. In this case, we can claim that our pricing model is more sensitive for old seniors than for young seniors. By further examining the two cases, we can claim that when people sell their insurance policies before age 80, the price with reinsurance is much higher than that without reinsurance. This situation shows that when hedge funds do not have reinsurance the risk they bear is higher. Consequently, hedge funds are reluctant to pay more for the life insurance policy so the price is lower. However, the difference between the two cases is so large that most of the risk can be hedged by reinsurance policies. To sum up, we find the present values of future cash flows of life settlements incorporating the aggregate mortality risk to form a certain distribution and choose the 95th and 99th percentiles to represent potential losses arising from aggregate mortality risk, under further considerations. 4 Options Comparison Model 4.1 Description of the Model While evolving the mortality conditional probabilities (e.g. given year N, this person has Y% of living to year N+1) of an individual over time via the two factor stochastic model (Cairns, Blake & Dowd (2006)) is necessary to accurately estimate mortality rates for one person given any life expectancy rating, this process does not consider a group of people s different death rates collectively and hence, does not give accurate estimates of the value of a group of life settlements. We will now introduce a method that a hedge fund or investor can use to determine the value of a portfolio of life settlements. The method will have the advantage of incorporating longevity risk (the uncertainty in mortality trends of a homogeneous population) and liquidity risk (the danger that a hedge fund will not be able to raise enough cash to meet immediate financial obligations). Because hedge funds vary widely in both the nature of their investments and their return benchmarks, we can assume a simple hedge fund that is involved in only one or two securities, mainly life settlements and bonds. One must realize the beauty of our process because our approach can be applied to any hedge fund, no matter how diverse its portfolio is. For instance, one can substitute bonds for stocks or bonds and stocks. A simple example is natural to use since it will illustrate the appropriateness of our methodology in evaluating life settlements while incorporating various uncertainties. The process we have developed will compare the net worth of a hedge fund after that is involved with both life settlements and bonds to the net value of a hedge fund of equivalent initial wealth that only invests in bonds after twenty years. Twenty years is a realistic time frame since many hedge funds buy reinsurance on their life settlements in order to guarantee that they receive the death benefit at some point in the near future (typically twenty or twenty five years), and hence avoid the risk of paying premiums for a person who lives much longer than expected. The first hedge fund we have created can divide its money between life settlements and bonds. We will arbitrarily pick values of the initial hedge fund 13

15 Life Settlement Portfolio Valuation page 14 of 23 worth and divide its money between the two securities. As a starting example, one could assume the hedge fund is worth $100,000 and splits its investments equally between 50 life settlements and bonds. Thus, at the start, we are estimating the value of the portfolio of life settlements. For simplicity, we will assume that this hedge fund has life settlements of people who are of the same cohort and health status. Based on the expected mortality rates determined by our stochastic model, we can simulate the hedge funds cash inflows and outflows for every year of the twenty year period. For the life settlements, negative cash flows (i.e. the premiums) will occur every year until simulations will accurately describe scenarios on the whole spectrum of mortality; some situations will have many people pass away much earlier or much later than expected. The bonds will only have a positive cash inflow that is based on the risk free interest rate of 6%. After 20 years, we will assume that all existing life settlements will pay the death benefit due to the reinsurance that the hedge fund buys on the policies. Hedge funds can pay a fee, usually 3-5% of the death benefit, to a reinsurance company that will pay the hedge fund the value of the insurance policy after some period of time (typically years) if the original seller of the policy has not died yet. Hence, our assumption that the firm will receive the death benefit after 20 years is realistic. The added benefit of simulating cash flows for only 20 years is that we can limit the amount of longevity risk since we are not predicting too far into the future. The predictions for mortality rates are very volatile after 30 years and hence predicting too far into the future would yield results with a great deal of uncertainty. Of course, our model can predict mortality rates deep into the future; the problem is that the results would not be very reliable or accurate. We can trace the wealth accumulation process of two hedge fund, W 1, W 2 in the following way. W 1 (x, t, T 0, n(t)) = W 1 (x, t 1, T 0, n(t 1)) (1 + r) n(t) α +(n(t 1) n(t)) β (4) t = 1, 2,, T W 2 (t, T 0 ) = W 2 (T 0, T 0 ) (1 + r) t T0 (5) t = 1, 2,, T Here, x = the amount of initial wealth put into life settlement market T 0 = the initial year n(t) = the number of people that are still alive in year t r = bond rate α = premium of each life settlement in each year β = death benefit of each life insurance W 1 (x, t, T 0, n(t)) = wealth of hedge fund 1 in year t W 2 (t, T 0 ) = wealth of hedge fund 2 in year t 14

16 Life Settlement Portfolio Valuation page 15 of 23 The complete simulation can be run 1,000 times or more in order to give a range of ending wealths that reliably describe the range of outcomes for the hedge fund. One must note that it is possible for the hedge fund to make huge sums of money in the simulation if people die much earlier than expected, or lose just as much and end up with a negative return if people live for too long. In the event that the hedge fund s money that is set aside to pay the premiums on the life settlements runs out, the firm will liquidate its bonds in order to pay its obligations. This process is very realistic since a hedge fund in the real world will have to liquidate its other investments if it is losing money and needs to pay off its outstanding debt. The range of ending wealths for the first hedge fund is then compared to the ending wealth of the second hedge fund, which is only involved with bonds. The second hedge fund will have only positive cash flows and more importantly, its ending wealth after twenty years is exact, since it is not involved with any risky assets that have variable payout. Based on the 1,000 or more simulations, one can determine the proportion of outcomes where the first hedge fund fares better than the second, i.e. when the first hedge fund has a greater wealth than the second after twenty years. As we change the initial value of the life settlements, the proportion of times where the first hedge fund s wealth is greater than the second will change as well. If we want to increase the proportion, then we must lower the initial value of the life settlement portfolio. To account for longevity risk, we will set a benchmark for the proportion of times that the first hedge fund does better than the second. A reasonable number is 95%, since hedge funds want to ensure that their investments will yield a profit for them in the long run, given any circumstances. Of course, a more conservative hedge fund could set a higher percentage if it wants to be absolutely certain that investing in life settlements will turn a profit. By setting a benchmark, the hedge fund can eliminate a huge amount of longevity risk since the 1,000 or more simulations accurately reflect population trends. An important aspect of the benchmark is that if the initial value of the life settlements leads to a proportion that is below 95%, then the portfolio was overvalued and one must lower the initial value of the portfolio. On the same thought, if the proportion is too high, then the life settlements are undervalued and one should raise the initial value. After adjusting the initial value, one just needs to simulate the cash flows of the first hedge fund again and see what proportion arises and then adjust the initial value accordingly. The process must repeat until the hedge fund determines a value of the portfolio that leads to an acceptable proportion. Theoretically, our idea can be formulated by h(x, T 0 ) = P (W 1 (x, T, T 0, n(t )) > W 2 (T, T 0 )) (6) x(t 0, p) = {x : h(x, T 0 ) = p} (7) As shown below, h(x, T 0 ) is always a non-increasing function of x. Then, x(t 0, p) is either a unique point or a interval. Accordingly, the initial value set to the life settlement is x(t 0, 0.95). Returning to our simple example, if valuing the life settlements at $50,000 lead the first hedge fund to have an ending wealth greater than the second only 15

17 Life Settlement Portfolio Valuation page 16 of 23 75% of the time, then one could lower the initial value of the life settlements while raising the amount of money invested in the bonds. A reasonable step would be to value the life settlements portfolio at $40,000 and the bonds at $60,000 and then run the 1,000 or more simulations to see what proportion arises. Another important point of this process is that the final value of the life settlements will yield a price that includes a liquidity premium. By comparing the ending wealths of the two hedge funds, one is comparing the effects of having liquid and illiquid assets. The second hedge fund is a completely liquid firm since it can cash the bonds at any time. The first hedge fund represents a firm that is involved with both liquid and illiquid assets. Life settlements are illiquid since hedge funds do not trade these settlements between each other and one cannot receive the cash value of the settlement whenever it wants since a cash inflow is only possible once a person passes away. Thus, by setting a benchmark proportion for how well one wants the first hedge fund to perform compared to the second, he or she is considering the impacts that holding onto illiquid assets has on the wealth of a firm as well as determining a price that compensates the firm for keeping illiquid assets. A drawback of our approach is that we do not consider when a hedge fund needs to liquidate its assets in order to meet its financial obligations. While we can study the effects of having illiquid assets in a hedge fund s portfolio, we are limited in studying the full implications of liquidity. This problem is easily overcome by making a probability array of liquidity, which would assign probabilities for the event that a certain percentage of the hedge fund s total worth needs to be liquidated in order for the firm to pay its immediate debt. The array would assign these probabilities for every future year. For instance, we could assume that in five years, the hedge fund will have a 5% chance that it need to liquidate 20% of its total wealth due to investors leaving. As another example, the hedge fund will have a 35% chance that it will have to sell off 50% of its total wealth due to a potential financial crisis in 2 years. The liquidity probabilities can be set according to a hedge fund s preferences and what they believe the chances are of certain future events. Hence, our model can incorporate the varying levels of liquidity that a hedge fund may face in the future. Due to the set up of our model, a scenario can exist, if we were to include a probability array of liquidity, where the first hedge fund would not be able to meet the liquidity demands. For example, if the first hedge fund needed to liquidate 70% of its worth, but 40% of its wealth is tied up in the illiquid life settlements, then the hedge fund would not meet its financial obligations. Thus, we can assume that the hedge fund would go under and stop functioning. In this instance, the second hedge fund would automatically perform better than the first hedge fund since the this hedge fund only has liquid bonds, which can be always be used to meet liquidity demands. This adjustment to our model allows us to study how the different liquidity demands affect the performance of our two hypothetical hedge funds and thus gives us a more comprehensive understanding of how liquidity affects the pricing of life settlements. 4.2 A Case Study Using the Option Comparison Model We will now apply our model to a concrete example in order to illustrate how a hedge fund can use it to determine the future prices that it would be willing to 16

18 Life Settlement Portfolio Valuation page 17 of 23 pay for a portfolio of life settlements. Because our data only covers mortality rates up to 2005, we will assume that the present year is Hence, if a hedge fund wishes to determine the price it would pay for a portfolio of life settlements in 5 years, for instance, then we would want the price in The portfolio we shall price shall consist of 20 life settlements. The individuals who sold those life insurance policies will be of the same cohort (male, age 65) and have a life expectancy rating of 120%, i.e. these people will perish at 1.2 times the rate as someone of the same demographics. The life settlements are each worth $1,954,762 and the premium paid per year is $2, We will determine not only the price that the hedge fund would pay in 2005 (the current year), but also the price five years from then, or in Since we want to study the effects that illiquid assets have on the value of the hedge funds, we will assume that both the first and second hedge fund have an initial wealth of $50,000,000. In determining the price the hedge fund would pay in 2005, we employ the method discussed before. We shall start with a certain amount of money invested in the life settlements and bonds for the first hedge fund, and then simulate 2,000 times how the cash flows over the next 20 years affect the wealth of the first hedge fund. One can see how the total worth of the first hedge fund varies over the 2,000 simulations for an initial life settlement investment of $13,450,000. An important feature of this graph is that the wealths vary most as the years go by, with the exception of the last year where the reinsurance creates a huge cash inflow for the hedge fund. The death benefits help to close the range of ending wealths and limit the amount of longevity risk that our simulations are exposed to. By arranging the ending wealths $13,450,000 in increasing order. The smooth curve reflects that the trend that ending wealths for hedge fund one are not very erratic and further illustrate how the reinsurance limits the effects of longevity risk. After simulating various initial prices for the life settlements, we can determine the price that will allow the first hedge fund to have a 95% chance of performing better than the second hedge fund. The relationship between the initial life settlement investment and the probability that the first hedge fund outperforms the second reveals a strong downward trend between $12,500,000. While this graph reveals a steep drop, one must realize that the dip occurs when we range the initial investments in life settlements from $12.5 million to $25 million. The change of $12.5 million, which would be invested in bonds, would affect the final wealth of the first hedge fund by roughly $40 million (= $12.5M (1.06) 2 0). Since the average ending wealth of the first hedge fund is about $160 million, the change in income is significant and will greatly affect how the first hedge fund does compared to the second. Upon viewing a closer version of the drop, one can see that the changes in the initial price is relatively small as compared to the change in the proportion of times that the first hedge fund does better than the second. 17

19 Life Settlement Portfolio Valuation page 18 of 23 Fiqure 7: Wealth accumulation process of hedge fund 1 Wealth 0.0e e e e e+08 Mean 90% Confidence Interval Year Fiqure 8: Ending wealth of 2000 simulations in ascending order Wealth 1.55e e e Case Number According to the results of the simulations, the price of the 20 life settlements 18

20 Life Settlement Portfolio Valuation page 19 of 23 should be between $13,820,000 and $13,970,000, so the price of each life settlement is between $691,000 and $698,500. Since we are considering a portfolio of identical life settlements, we can divide the total price to attain an individual price for a life settlement. Figure 9:h(x,T0), with T0 being 2005 and x percent of initial wealth P(W1 > W2) Percent of wealth put into Life Settlement Before venturing further, one point must be clarified about the process of future pricing. Under the model described in the previous section, a hedge fund can determine what it would pay for a group of life settlements in the present time. This value is helpful, especially for a hedge fund that is considering investing in life settlements, but is not the most important. We need to determine a method for pricing a portfolio of life settlements at some point in the future. The process that we described before can be used to determine a price of a life settlement that a hedge fund would pay at some time in the future. The first difference between this method and the one before is that our simulations will start from whatever year a hedge fund wants to determine the price it is willing to pay in that year. The simulations will run from that start year until 2025, which is when the fund will receive the death benefits from the reinsurance company. Remember that we are assuming after 20 years from the present year, i.e. 2005, the hedge fund will receive the face value of the insurance policies. 19

21 Life Settlement Portfolio Valuation page 20 of 23 Figure 10: h(x,t0) over x, with T0 being 2005 and x in dollars P(W1>W2) price of 20 life settlement at 2005 Because the process described in the previous section only gives the price a hedge fund would be willing to pay in the starting year, we must shift the starting year from 2005 to whatever year we want to determine a price for the life settlements. In order to start from a future year, we must adjust the mortality tables for each individual up to that year. This correction is very simple since we can use the stochastic process described earlier to evolve the mortality tables. Thus, in our new model, the mortality tables for each individual will change starting from 2005, but the simulations for cash pay outs of the life settlements and bonds will start at 2005 and end at Another issue to consider is the number of life settlements that still need to be cashed at that future year. In other words, a hedge fund at the present time may want to price a portfolio of 20 life settlements, but in five years, the hedge fund cannot assume that all 20 sellers are still alive if it wishes to accurately price the life settlements. In order to determine a reasonable estimate for the number of people that have died by that future year (2010 in this example), one can use the mortality curve of an individual, which is determined by our stochastic process, in order to determine an expected number of people that would die by that year. By people, we are referring to a homogeneous group of individuals with equivalent health. For this example, we can determine that in 5 years, we would expect between one and two people out of the group to pass away. Hence, a hedge fund trying to determine the price it would pay for a group of life settlements in 2010 should only consider the price of 18 or 19 life settlements, rather than 20, since at least one settlement has most likely paid 20

22 Life Settlement Portfolio Valuation page 21 of 23 out its death benefit. The following table shows expected number of death in each year among 20 people aged 65 in 2005, according to simulations of our random walk model (2). Year Death Year Death Year Death Year Death Year Death Year Death Year 2041 Death Table 5: Number of people out of 20 we expect to die each year As shown above, about 1.7 out of 20 people are expected to die within the first 5 years. So, for our example, we will assume that two settlements have been cashed before Since the initial wealths for the first and second hedge fund in 2005 is different than in 2010 due to the varying cash flows, we must adjust the initial $50,000,000 accordingly. For the second hedge fund, adjusting the initial wealth is easy since it is only involved with bonds that have a constant growth, mainly 6%. Its initial wealth in 2010 is $66,911,279. The first hedge fund s wealth can be determined as a simple average of the ending wealths that are determining by running 100 simulations of the cash flows for the life settlements from 2005 to A simulation is only accepted if exactly two people die in this period; otherwise, that simulation is rejected and the program runs again until it has exactly two life settlements paid out. The initial wealth in 2010 for the first hedge fund is $49,231,000. Following in an equivalent manner as above, we can vary the initial price of the life settlement portfolio and determine which value will allow the first hedge fund to have a 95% probability of making more money than the second hedge fund. A graph of how the initial life settlement investment affects the probability that the first hedge fund does better than the second makes apparent that the price of the 18 life settlements in 2010, given the reinsurance in 2025, should be between $15,700,000 and $15,750,000. Thus, the price of each settlement is between $872,222 and $875,000. When comparing the price the hedge fund is willing to pay for each of the life settlements, they will notice that the price for the life settlement in 2010 is higher. The reason for this trend is that in the future, all of the policy sellers are closer to death and the firm only needs to wait 15 years, instead of 20, for reinsurance. Hence, the investors are facing less risk and they should be willing to pay more for the more stable asset. 21

23 Life Settlement Portfolio Valuation page 22 of 23 Figure 11: h(x,t0) over x, with T0 being 2010 P(W1>W2) price of 18 life settlement after 5 years While this example only considered a homogeneous portfolio of life settlements, one can easily adjust the model to include a more diverse portfolio. In order to do so, one must adjust the mortality tables for each individual in the portfolio and then simulate the cash flows in an equivalent manner as above. The present and future price that a hedge fund would pay for this entire portfolio is then calculated using the aforementioned processes. One shortcoming of our model is that we cannot determine the individual prices for each life settlement in the portfolio. 5 Conclusion Basically, what we have done in pricing the life settlements is as follows. First, we simulate the mortality table through the model built by Cairns, Blake and Dowd Model. Second, we establish Present Value Percentile Model to valuate the portfolio of the life settlements. Third, we adopt Options Comparison Model in comparing two hedge funds to calculate the portfolio. By comparing the prices calculated by the two models, we can conclude that for men with the life expectancy rating 120% at age 65 selling their policies, after choosing the 95th percentile of the distribution of predicted prices, the prices in Present Value Percentile Model and Options Comparison Model are very close, since they are $ and ($691000, $698500) respectively. In addition, comparing different prices we set to life settlement under various condition 22

24 Life Settlement Portfolio Valuation page 23 of 23 illustrate the reasonableness and extensibility of our models. The average price of each life settlement in 2005, according to Data Collection Report published by Life Insurance Settlement Association (LISA), is $ Thus, price set by our models is about twice as much as the market price, which means, given a hedge fund s profit target, the highest price the hedge fund can accept is about twice as much as market average price. This point confirms that life settlement market has appealing rate of return. References [1] Cairns, Blake and Dowd, 2006, A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration, The Journal of Risk and Insurance, Vol.73, No.4, [2] Friedberg and Webb, 2007, Life is Cheap: Using Mortality Bonds to Hedge Aggregate Mortality Risk, The B.E. Jounal of Economic Analysis & Policy, Vol.7, Iss.1, Art.31 23