Life Settlement Portfolio Valuation settlement… · life settlement. The higher percentile setting...

Life Settlement Portfolio Valuation

MITACS Industrial Summer School 2008Simon Fraser University

August 8, 2008

Greg HamillDavid LinIris ShauJessica WuCheng You

With Huaxiong Huang, Tom Salisbury, and Phillip Poon

Life Settlement Portfolio Valuation page 1 of 23

Contents

1 Introduction 3

2 Estimating Dynamics of Mortality Table 52.1 Denotations and Assumptions . . . . . . . . . . . . . . . . . . . . 52.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Present Value Percentile Model 103.1 Model description and assumptions . . . . . . . . . . . . . . . . . 103.2 Detailed model establishment . . . . . . . . . . . . . . . . . . . . 113.3 Numerical simulation and analysis . . . . . . . . . . . . . . . . . 11

4 Options Comparison Model 134.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . 134.2 A Case Study Using the Option Comparison Model . . . . . . . . 16

5 Conclusion 22

1


Abstract

Life settlement, a resale of insurance policy, has been a growing indus-try in financial market, because of the appealing rate of return of investorand favorable tax treatment. It exists in three different forms, term life,whole life and universal life. We study the whole life settlement in thispaper. A whole life settlement can be treated as a Senior Life Settlementsinsurance, in which usually seniors who are over sixty sell their insurancepolicy to hedge fund; and therefore assume mortality risk, also named aslongevity risk in our paper, and liquidity risk exist in this industry. Usingthe discrete-time stochastic process model from Cairns, Blaske and Dowd(2006), we quantify longevity risk as the risk that the average insurer liveslonger than anticipated.

We then estimate the net present value of individual life settlementfor hedge fund; this analysis can be extended to the acceptable sellingprice of life settlement. Given the distribution of the net present valuewith sample size of one thousand, we pick up the value at five percentileas the acceptable price hedge fund bears. Based on a large sample size ofdistribution, the markup price at five percentile can represent a reasonableprice for life settlement in a general consideration.

Longevity risk and liquidity risk can be more efficiently managed bycomparing the hedge fund portfolios . We determine the net worth of lifesettlement between one portfolio with life settlement and one without.We set a benchmark for the proportion of times that the first hedge funddoes better than the second one since hedge funds want to ensure thelife settlements will yield a profit. Also, we include a probability arrayof liquidity. And the adjustment allows us to study different liquidityaffected the hedge fund.

2


1 Introduction

A life settlement is the selling of any high value policies to a hedge fund asthe sellers no longer have a need for the policy and can receive a significantlylarger amount for their policy than the insurance company would have paidfor surrenders. Life insurance companies are mostly opposed to life settlementsbecause when the owners sell their policy to a hedge fund, the hedge fundis guaranteed to receive the death benefit from the insurance company whenthe policyholder dies. Without the existence of hedge funds, the policyholderwould just sell the policy back to the insurance company for an amount far lessthan the death benefit. The idea of life settlements originated from the viaticalindustry which was first conceived in Europe in the 1880s and then venturedto the United States in 1989 during the rise of HIV cases, in which the ownerhas a life expectancy of less than two years. Since then, life settlements havegrown to include a wider variety of diversified portfolios, they are term-life,whole life and universal life settlements. Most hedge funds build a portfolio ofabout 5-50 policies; the policies chosen are based on the preference of the hedgefund. Usually, the sellers are above the age of sixty in the whole life settlementmarket. According to some literatures, in the past 15 years, average annualrate 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 sinceprior returns do not assure future returns. Although life settlements are startingto become increasingly common, hedge funds still do not have a precise wayto put an exact value of their investment at any given time. Hedge fundsneed a way to find the net present value of their investment at any time forpurposes of marking to market, as well as being able to divide the worth ofthe investment among each investor. In trying to calculate these values, hedgefunds face numerous issues including: longevity risk, since the policyholder maylive longer than expected, resulting in more premium payments and a longerwait time for the death benefit; illiquidity risk, since hedge fund may not easilyre-sell the insurance policies or may not have cash to pay for those investors whowant to withdraw. Also life settlement is still a incomplete market, there existsmarket uncertainty, for example, the size and number of buyers and sellers; It isalso an important factor because hedge funds can buy more policies and receivegreater death benefits if there is less competition. All the above factors shouldbe monitored because of their presence in life settlement.

Looking into the life expectancy of the general population, it keeps improvingin 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 from1976 to 2005, released by Canadian Human Mortality Database, the survivalprobability beyond 2005 of someone aged 60 at 2005 can be obtained. Hedgefund can make their decisions over this reference of lifetime model. With thisincomplete life settlement market, it is a way to use models that have similaritiesas life settlement with a consideration of possible risk factors. The models usedin this paper much fit the data, have good performance , and, importantly forour purpose, quantify the uncertainty surrounding future mortality, the discountfactor involving longevity and illiquidity.

Once a hedge fund buys the policy from a policyholder, the policyholderno longer has any commitment to the hedge fund. Because the hedge fund no

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longer has any contact with the policyholder, the policyholder’s state of healthat any given time is unknown (ie. the policyholder may be diagnosed with afatal disease the day after he or she sells their policy), so a policyholder’s lifeexpectancy is still unpredictable. We calculate longevity risk by using mortalitytables to estimate the life expectancy of the policyholder. For better predictions,we look at the physiological age of policy holders depending on gender andcurrent state of health. We even then, there are actuarial underwriters whogive a mortality rating for each individual. The mortality rating compares themortality of the individual to that of the population. Our next consideration isthe illiquidity risk. When hedge funds need to make any changes (ie there maybe entering and exiting investors at any given time) but the policyholders havenot died yet, so there is no money available. The hedge fund is stuck with avaluable asset but they are unable to use it unless they decide to sell or tradeit at a discount. Almost the entire life settlement market is illiquid becauseunfortunately, there is nothing the hedge fund can do until the policy holderdies.

The main problem now is incorporating all of these risks into the calculatingthe net present value of the portfolios. In approaching this problem, we figureout what the discount factors are. This paper is organized as follows. In section1 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 ofBritish Columbia from 1976 to 2005, released by Canadian Human MortalityDatabase (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 mortalitycurve which tells us the general improvements in life expectancy and which agesgroup of people get this improvement. In subsequence section, we focus on thedynamics of the survival probability. Because of the random variable term inour model, we then can get a confidence intervals of the distribution of thesurvival rate, in which provide us different percentiles to do further simulationin the sections below.

In section 3, we construct a Present Value Percentile Model. We study theappropriate present value, can also be defined as the price of life settlement, inorder to get a basic idea of how much hedge fund is willing to pay for buying lifesettlement. After obtaining a mortality table, we can further use it to estimatethe death year of insurers. We simulate a model by one thousand times and getnet death benefit hedge fund received from these one thousand people. By usinga risk-free interest rate as the discount factor, we find one thousand net presentvalues and then pick up a value at five percentile for example, which representsan acceptable value hedge fund spent on buying a life settlement contract. Thepercentile means how much hedge fund can tolerate the high payment to thelife settlement. The higher percentile setting point, the higher price hedge fundbear.

In section 4, we evaluate the net worth of life settlement by comparing twohedge funds, in which one involves both life settlement and bonds, anotherinvolves bonds only. This Options Comparison Model, not only various uncer-tainty, including longevity risk and liquidity risk, are incorporated; also it showsthe benchmark for the proportion of times of how life settlement performs. Forthe consistence, we introduce an idea of reinsurance in our simulation. Thebasic arrangement of reinsurance is that hedge fund pays for life settlement fee,

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premium and an additional amount of reinsurance fee. Reinsurer financiallyviable at the time the contract comes due, thus the risk of the insurance payingoff later than expected can be removed. The reason of incorporating reinsuranceis that we try to set the same due date of the premium payment for compari-son, and we choose twenty years in our case. In our process, we show the cashflow of hedge fund to demonstrate the longevity risk in life settlement. At thesame time, hedge fund can find when to liquidate its bonds in order to pay itsobligation.

Section 5 is conclusion. we compare price set by our two models and realprice 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-timestochastic process approach from Cairns, Blake and Dowd (2006).

2.1 Denotations and Assumptions

DenotationP (t, x) probability that an individual aged x in year t survives

until year t + 1Q(t, x) 1− P (t, x)m(t, x) dentral death rate for individuals aged x in year t,

approximated by q(t,x)1−q(t,x)/2

1

S(t) survival probability that an individual survives until year t,subject to S(t + 1) = S(t)(1−m(t, x))

Assumption1 Mortality table released by CHMD are unbiased estimates.2 survival probability in the starting year is equal to 1, i.e.S(0) = 1.

2.2 Mathematical Model

Based on Mortality Table of British Columbia from 1976 to 2005, released byCanadian Human Mortality Database (CHMD),2 we adopt the following modelto predict mortality rates above the age of 60 in future:(let t = 0 correspond tothe beginning of 1976)

log(Q(t, x)P (t, x)

) = A1(t) + A2(t)x (1)

t = 1, 2, · · ·x = 60, 61, · · · , 105

Using the real data to estimate A(t) = (A1(t), A2(t))T , t = 1, 2, · · · , 30,We find the fittings for each year are all quite good. In fact, among the 30

1For a full discussion, the reader is referred to Benjamin and Pollard (1993) or Bowers etal.(1986).

2It’s a public data source, with web link http://www.bdlc.umontreal.ca/chmd/prov/bco/bco.htm

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regressions, the least R2 is 0.9938. And we illustrate the fitting line in year2005 as follows.

60 70 80 90 100

−5

−4

−3

−2

−1

Figure 1:Fitting line of Model(1) in 2005

Age of cohort at 2005

log(

Q/P

)

These results show a clear trend in both series. The downward trend in A1(t)reflects general improvements in mortality over time at all ages. The increasingtrend in A2(t) means that the curve is getting slightly steeper over time: thatis, mortality improvements have been greater at lower ages. There were alsochanges in the trend and in the volatility of both series. To make forecastsof the future distribution of A(t) = (A1(t), A2(t))T , we will model A(t) as atwo-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 triangularmatrix and Z(t) is a 2-dimensional standard normal random variable.

These results show a clear trend in both series. The downward trend in A1(t)reflects general improvements in mortality over time at all ages. The increasingtrend in A2(t) means that the curve is getting slightly steeper over time: thatis, mortality improvements have been greater at lower ages. There were alsochanges in the trend and in the volatility of both series. To make forecastsof the future distribution of A(t) = (A1(t), A2(t))T , we will model A(t) as atwo-dimensional random walk with drift.

6


1975 1980 1985 1990 1995 2000 2005

−11

.0−

10.5

−10

.0−

9.5

Fiqure 2: Estimated A1 from 1976 to 2005

Year

A1

1975 1980 1985 1990 1995 2000 2005

0.08

50.

090

0.09

50.

100

0.10

5

Fiqure 3: Estimated A2 from 1976 to 2005

Year

A2

These results show a clear trend in both series. The downward trend in A1(t)

7


reflects general improvements in mortality over time at all ages. The increasingtrend in A2(t) means that the curve is getting slightly steeper over time: thatis, mortality improvements have been greater at lower ages. There were alsochanges in the trend and in the volatility of both series. To make forecastsof the future distribution of A(t) = (A1(t), A2(t))T , we will model A(t) as atwo-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 triangularmatrix and Z(t) is a 2-dimensional standard normal random variable.

Fitting the model, we find the estimates are:

µ =(−0.064562

0.000616

), and V = CCT =

(0.030168198 −0.00041833−0.00041833 0.00000596

)

Based on A(t), t = 1, 2, · · · , 30, we can simulate A(t), t = 31, 32, · · · , 130according to (2) to estimate the mortality table from 2006 to 2106.

In subsequent sections we will focus on the dynamics of the survival prob-ability, 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 beginningof 2005 from now on.

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Fiqure 4: Mean and 90% confidence interval of S(t)

t

S(t

)

8


By simulating dynamics of A(t) a large number of times, we calculate themean, 5th and 95th percentiles of the distribution of S(t) and show them infigure 4.

2.3 Model Verification

We already described the model above and estimated all the parameters, in-cluding A(t), t = 1, 2, · · · , 30, µ and C,in which to predict mortality table infuture. In this subsection, we show the effectiveness of our model.

We use the Life Table for British Columbia between 1976 and 1995 to fit ourmortality model (2) and then further predict the Mortality Table for year 1996to year 2005. 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 eachsimulation, we calculate mortality rate of a man aged 65 and another aged 85for each year from 1996 to 2005, and get the mean and 90% confidence intervalof both the mortality rates in each year. They are showed in Figure 5 andFigure 6, together with real mortality rate.

1996 1998 2000 2002 2004

0.01

00.

012

0.01

40.

016

0.01

80.

020

Fiqure 5: Estimated and real mortality rate for people aged 65

Year

qx

Mean90% Confidence IntervalReal Data

9


1996 1998 2000 2002 2004

0.06

0.08

0.10

0.12

0.14

Fiqure 6: Estimated and real mortality rate for people aged 85

Year

qx

Mean90% Confidence IntervalReal Data

To summarize the mortality tables we estimated, we find that, for a managed 65, the real mortality rate lies within the 90% confidence interval of ourmodel; 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 than3 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 mortalityrisk. Referring to Cairns, Blake and Dowd (2006), we quantify aggregate mor-tality risk as the risk that the average policy holder lives longer than is predictedby the model.

In this section, we use simulations to quantify the aggregate mortality riskfaced by an hedge fund buying life settlements from a single birth cohort. Forpeople whose life expectancy is longer than the expected, hedge funds sufferfrom the risk of getting their money back late i.e. longevity risk. This risk maycause insolvency problems so that hedge fund investors demand an aggregatemortality 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


single type from people in a single birth cohort who have population averagemortality and changes in mortality rates as predicted by (2). We also fit theforecast of mortality for males and females separately. Moreover, we assumethat 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 theannual survival probabilities of people in British Columbia, Canada. To be morespecific, we adopt simple linear regression to estimate the vectors A1 and A2 forthe latter stochastic process, based on the data of mortality tables from 1976to 2005. Then, we utilize the estimated vectors to fit into the stochastic modelto get

µ =(−0.064562

0.000616

), and V = CCT =

(0.030168198 −0.00041833−0.00041833 0.00000596

)

After we obtain the stochastic model, we predict the further 20 years survivalprobabilities for determining the time of cash flows. Next, we choose the specificrates of interest 5% to discount the resulting cohort premiums, and sum thepremiums to arrive at a present value. We use this present value incorporatingthe aggregate mortality risk for pricing. It can be expressed in the followingformula:

PV =T−1∑t=1

Ct

(1 + r)t+

S

(1 + r)T

where T is the year that a person dies, r is the discount rate and S is theinsurance value. The hedge fund makes a loss if, in a particular simulation, themortality draw it experiences results in premiums that exceed the settlements.As Friedberg & Webb (2007), we calculate the amounts by which the total cashflows in the future by the hedge fund on life settlement sold to a single birthcohort will exceed amounts forecast using our random walk model (2) at the95th and 99th percentiles of the distribution of present values, assuming that theonly source of variation is aggregate mortality risk. The distribution of presentvalues shows possible choices of the price. The percentile also means that themaximum probability that hedge funds accept losses i.e. loss probability. Thereason that we choose the percentile instead of the overall mean is that if wesimply choose the expectation of the distribution of present values, the hedgefund will have 50% chance to lose money. In order to eliminate the unhappysituation, we consider it from an angle of the maximum probability that hedgefunds 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 2004-2005 published by Life InsuranceSettlement Association (LISA), in 2005, the average amount of death bene-fit settled by each participant is ($3, 413, 014, 939/1, 746)= $1954762, averageproceeds paid to sellers is $349884.6, average amount of premium per year is$2573.04. We simply set, in our case, both the death benefit and premium oflife settlement to corresponding average values.

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• Case one: reinsurance in 20 years

Percentile Age 60 Age 65 Age 70 Age 75 Age 80 Age 851% 667008 667008 667008 667008 667008 7806625% 667008 667008 667008 667008 780662 960544

Table 1: Present Values, Male


Table 2: Present Values, Female

• Case two: without reinsurance


Table 3: Present Values, Male


Table 4: Present Values, Female

In reality, Hedge funds prefer to buy life insurances from unhealthy peopleso we construct the ratings to the health of different people. The rating meansa multiplier to the probability a person dies, which is larger than 1. In ourcase, we take the rating 120%, meaning that the probability that the persondies is 1.2 times of the probability that average people die. We also considerwomen and men separately because we observe that on average women outlivemen 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 reinsurethe life settlements. For men or women selling their life insurances before or atage 75, whenever loss probability is 1% or 5%, the price of the life settlementthat the hedge fund can offer are basically the same. This is because a largerproportion of people in BC before or at age 75 are expected to live longer than20 years so that the reinsurance company pays the value of the life insurancepolicy. Hence, the cash flows patterns are exactly the same.

Another case is that hedge funds do not reinsure life settlements. For men orwomen, when loss probability is 0.01 or 0.05, the present values rise increasinglyas the age of selling increases. It shows that as the person gets old, the valueof the life insurance increases because the person is more likely to die in thenear future and the hedge fund can get its return quickly. Comparing the twoloss probability, we can see that if hedge funds have to bear more probabilityof loss, they should price the life settlements higher. Moreover, comparing menand women, we can see that men’s life insurance is more valuable than women’sand the reason is that men’s longevity is shorter than women’s.

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In addition, we can see that as the age grows, the price differences for the99th and 95th percentile become larger, meaning that the variation of mortalityincreases as the age increases. In this case, we can claim that our pricing modelis more sensitive for old seniors than for young seniors.

By further examining the two cases, we can claim that when people selltheir insurance policies before age 80, the price with reinsurance is much higherthan that without reinsurance. This situation shows that when hedge funds donot have reinsurance the risk they bear is higher. Consequently, hedge fundsare 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 riskcan be hedged by reinsurance policies.

To sum up, we find the present values of future cash flows of life settlementsincorporating the aggregate mortality risk to form a certain distribution andchoose the 95th and 99th percentiles to represent potential losses arising fromaggregate 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, thisperson has Y% of living to year N+1) of an individual over time via the two fac-tor stochastic model (Cairns, Blake & Dowd (2006)) is necessary to accuratelyestimate mortality rates for one person given any life expectancy rating, thisprocess does not consider a group of people’s different death rates collectivelyand hence, does not give accurate estimates of the value of a group of life settle-ments. We will now introduce a method that a hedge fund or investor can use todetermine the value of a portfolio of life settlements. The method will have theadvantage of incorporating longevity risk (the uncertainty in mortality trendsof a homogeneous population) and liquidity risk (the danger that a hedge fundwill not be able to raise enough cash to meet immediate financial obligations).

Because hedge funds vary widely in both the nature of their investments andtheir return benchmarks, we can assume a simple hedge fund that is involved inonly one or two securities, mainly life settlements and bonds. One must realizethe beauty of our process because our approach can be applied to any hedgefund, 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 usesince it will illustrate the appropriateness of our methodology in evaluating lifesettlements while incorporating various uncertainties.

The process we have developed will compare the net worth of a hedge fundafter that is involved with both life settlements and bonds to the net valueof a hedge fund of equivalent initial wealth that only invests in bonds aftertwenty years. Twenty years is a realistic time frame since many hedge fundsbuy reinsurance on their life settlements in order to guarantee that they receivethe death benefit at some point in the near future (typically twenty or twentyfive years), and hence avoid the risk of paying premiums for a person who livesmuch longer than expected.

The first hedge fund we have created can divide its money between life set-tlements and bonds. We will arbitrarily pick values of the initial hedge fund

13


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 investmentsequally between 50 life settlements and bonds. Thus, at the start, we are es-timating the value of the portfolio of life settlements. For simplicity, we willassume that this hedge fund has life settlements of people who are of the samecohort 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 thetwenty year period. For the life settlements, negative cash flows (i.e. the premi-ums) will occur every year until simulations will accurately describe scenarioson the whole spectrum of mortality; some situations will have many people passaway much earlier or much later than expected. The bonds will only have apositive cash inflow that is based on the risk free interest rate of 6%. After 20years, we will assume that all existing life settlements will pay the death benefitdue to the reinsurance that the hedge fund buys on the policies. Hedge fundscan pay a fee, usually 3 - 5% of the death benefit, to a reinsurance companythat will pay the hedge fund the value of the insurance policy after some periodof time (typically 20 - 30 years) if the original seller of the policy has not diedyet. Hence, our assumption that the firm will receive the death benefit after 20years is realistic. The added benefit of simulating cash flows for only 20 years isthat we can limit the amount of longevity risk since we are not predicting toofar into the future. The predictions for mortality rates are very volatile after30 years and hence predicting too far into the future would yield results witha great deal of uncertainty. Of course, our model can predict mortality ratesdeep into the future; the problem is that the results would not be very reliableor accurate.

We can trace the wealth accumulation process of two hedge fund, W1, W2

in the following way.

W1(x, t, T0, n(t)) = W1(x, t− 1, T0, n(t− 1)) ∗ (1 + r)− n(t) ∗ α

+(n(t− 1)− n(t)) ∗ β (4)t = 1, 2, · · · , T

W2(t, T0) = W2(T0, T0) ∗ (1 + r)t−T0 (5)t = 1, 2, · · · , T

Here,

x = the amount of initial wealth put into life settlement marketT0 = the initial year

n(t) = the number of people that are still alive in year tr = bond rateα = premium of each life settlement in each yearβ = death benefit of each life insurance

W1(x, t, T0, n(t)) = wealth of hedge fund 1 in year tW2(t, T0) = wealth of hedge fund 2 in year t

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The complete simulation can be run 1,000 times or more in order to givea range of ending wealths that reliably describe the range of outcomes for thehedge fund. One must note that it is possible for the hedge fund to make hugesums of money in the simulation if people die much earlier than expected, orlose 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 premiumson the life settlements runs out, the firm will liquidate its bonds in order to payits obligations. This process is very realistic since a hedge fund in the real worldwill have to liquidate its other investments if it is losing money and needs topay off its outstanding debt.

The range of ending wealths for the first hedge fund is then compared to theending wealth of the second hedge fund, which is only involved with bonds. Thesecond hedge fund will have only positive cash flows and more importantly, itsending wealth after twenty years is exact, since it is not involved with any riskyassets that have variable payout. Based on the 1,000 or more simulations, onecan determine the proportion of outcomes where the first hedge fund fares betterthan the second, i.e. when the first hedge fund has a greater wealth than thesecond 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 thesecond will change as well. If we want to increase the proportion, then we mustlower the initial value of the life settlement portfolio.

To account for longevity risk, we will set a benchmark for the proportionof times that the first hedge fund does better than the second. A reasonablenumber is 95%, since hedge funds want to ensure that their investments willyield 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 tobe absolutely certain that investing in life settlements will turn a profit. Bysetting a benchmark, the hedge fund can eliminate a huge amount of longevityrisk 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 lifesettlements leads to a proportion that is below 95%, then the portfolio wasovervalued and one must lower the initial value of the portfolio. On the samethought, if the proportion is too high, then the life settlements are undervaluedand one should raise the initial value. After adjusting the initial value, onejust needs to simulate the cash flows of the first hedge fund again and see whatproportion arises and then adjust the initial value accordingly. The processmust repeat until the hedge fund determines a value of the portfolio that leadsto an acceptable proportion.

Theoretically, our idea can be formulated by

h(x, T0) = P (W1(x, T, T0, n(T )) > W2(T, T0)) (6)

x(T0, p) = {x : h(x, T0) = p} (7)

As shown below, h(x, T0) is always a non-increasing function of x. Then,x(T0, p) is either a unique point or a interval. Accordingly, the initial value setto the life settlement is x(T0, 0.95).

Returning to our simple example, if valuing the life settlements at $50,000lead the first hedge fund to have an ending wealth greater than the second only

15


75% of the time, then one could lower the initial value of the life settlementswhile raising the amount of money invested in the bonds. A reasonable stepwould 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 proportionarises.

Another important point of this process is that the final value of the lifesettlements will yield a price that includes a liquidity premium. By comparingthe ending wealths of the two hedge funds, one is comparing the effects of havingliquid and illiquid assets. The second hedge fund is a completely liquid firm sinceit can cash the bonds at any time. The first hedge fund represents a firm thatis involved with both liquid and illiquid assets. Life settlements are illiquidsince hedge funds do not trade these settlements between each other and onecannot receive the cash value of the settlement whenever it wants since a cashinflow is only possible once a person passes away. Thus, by setting a benchmarkproportion for how well one wants the first hedge fund to perform compared tothe second, he or she is considering the impacts that holding onto illiquid assetshas on the wealth of a firm as well as determining a price that compensates thefirm for keeping illiquid assets.

A drawback of our approach is that we do not consider when a hedge fundneeds to liquidate its assets in order to meet its financial obligations. Whilewe 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 iseasily overcome by making a probability array of liquidity, which would assignprobabilities for the event that a certain percentage of the hedge fund’s totalworth needs to be liquidated in order for the firm to pay its immediate debt. Thearray would assign these probabilities for every future year. For instance, wecould assume that in five years, the hedge fund will have a 5% chance that it needto 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 totalwealth due to a potential financial crisis in 2 years. The liquidity probabilitiescan be set according to a hedge fund’s preferences and what they believe thechances are of certain future events. Hence, our model can incorporate thevarying 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 includea probability array of liquidity, where the first hedge fund would not be ableto meet the liquidity demands. For example, if the first hedge fund needed toliquidate 70% of its worth, but 40% of its wealth is tied up in the illiquid lifesettlements, then the hedge fund would not meet its financial obligations. Thus,we can assume that the hedge fund would go under and stop functioning. Inthis instance, the second hedge fund would automatically perform better thanthe first hedge fund since the this hedge fund only has liquid bonds, which canbe always be used to meet liquidity demands. This adjustment to our modelallows us to study how the different liquidity demands affect the performanceof our two hypothetical hedge funds and thus gives us a more comprehensiveunderstanding 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 ahedge fund can use it to determine the future prices that it would be willing to

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pay for a portfolio of life settlements. Because our data only covers mortalityrates up to 2005, we will assume that the present year is 2005. Hence, if a hedgefund wishes to determine the price it would pay for a portfolio of life settlementsin 5 years, for instance, then we would want the price in 2010.

The portfolio we shall price shall consist of 20 life settlements. The individ-uals who sold those life insurance policies will be of the same cohort (male, age65) and have a life expectancy rating of 120%, i.e. these people will perish at1.2 times the rate as someone of the same demographics. The life settlementsare each worth $1,954,762 and the premium paid per year is $2,573.04. Wewill determine not only the price that the hedge fund would pay in 2005 (thecurrent year), but also the price five years from then, or in 2010. Since we wantto study the effects that illiquid assets have on the value of the hedge funds, wewill 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 employthe method discussed before. We shall start with a certain amount of moneyinvested in the life settlements and bonds for the first hedge fund, and thensimulate 2,000 times how the cash flows over the next 20 years affect the wealthof the first hedge fund. One can see how the total worth of the first hedgefund 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 mostas the years go by, with the exception of the last year where the reinsurancecreates a huge cash inflow for the hedge fund. The death benefits help to closethe range of ending wealths and limit the amount of longevity risk that oursimulations are exposed to. By arranging the ending wealths $13,450,000 inincreasing order. The smooth curve reflects that the trend that ending wealthsfor hedge fund one are not very erratic and further illustrate how the reinsurancelimits the effects of longevity risk.

After simulating various initial prices for the life settlements, we can deter-mine the price that will allow the first hedge fund to have a 95% chance ofperforming better than the second hedge fund. The relationship between theinitial life settlement investment and the probability that the first hedge fundoutperforms the second reveals a strong downward trend between $12,500,000.While this graph reveals a steep drop, one must realize that the dip occurswhen 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)20). Since the average ending wealth of the first hedge fundis about $160 million, the change in income is significant and will greatly affecthow the first hedge fund does compared to the second. Upon viewing a closerversion of the drop, one can see that the changes in the initial price is relativelysmall as compared to the change in the proportion of times that the first hedgefund does better than the second.

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2005 2010 2015 2020 2025

0.0e

+00

5.0e

+07

1.0e

+08

1.5e

+08

2.0e

+08

Fiqure 7: Wealth accumulation process of hedge fund 1

Year

Wea

lth

Mean90% Confidence Interval

0 500 1000 1500 2000

1.55

e+08

1.65

e+08

1.75

e+08

Fiqure 8: Ending wealth of 2000 simulations in ascending order

Case Number

Wea

lth

According to the results of the simulations, the price of the 20 life settlements

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should be between $13,820,000 and $13,970,000, so the price of each life settle-ment is between $691,000 and $698,500. Since we are considering a portfolio ofidentical life settlements, we can divide the total price to attain an individualprice for a life settlement.

0.2 0.4 0.6 0.8

0.0

0.2

0.4

0.6

0.8

1.0

Figure 9:h(x,T0), with T0 being 2005 and x percent of initial wealth

Percent of wealth put into Life Settlement

P(W

1 >

W2)

Before venturing further, one point must be clarified about the process offuture pricing. Under the model described in the previous section, a hedge fundcan determine what it would pay for a group of life settlements in the presenttime. This value is helpful, especially for a hedge fund that is considering in-vesting in life settlements, but is not the most important. We need to determinea 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 lifesettlement that a hedge fund would pay at some time in the future. The firstdifference between this method and the one before is that our simulations willstart from whatever year a hedge fund wants to determine the price it is willingto 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 com-pany. 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.

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13600000 13700000 13800000 13900000 14000000

0.93

0.94

0.95

0.96

0.97

Figure 10: h(x,T0) over x, with T0 being 2005 and x in dollars

price of 20 life settlement at 2005

P(W

1>W

2)

Because the process described in the previous section only gives the pricea hedge fund would be willing to pay in the starting year, we must shift thestarting year from 2005 to whatever year we want to determine a price for the lifesettlements. In order to start from a future year, we must adjust the mortalitytables for each individual up to that year. This correction is very simple since wecan use the stochastic process described earlier to evolve the mortality tables.Thus, in our new model, the mortality tables for each individual will changestarting from 2005, but the simulations for cash pay outs of the life settlementsand bonds will start at 2005 and end at 2025.

Another issue to consider is the number of life settlements that still need tobe cashed at that future year. In other words, a hedge fund at the present timemay want to price a portfolio of 20 life settlements, but in five years, the hedgefund cannot assume that all 20 sellers are still alive if it wishes to accuratelyprice the life settlements. In order to determine a reasonable estimate for thenumber 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 ourstochastic process, in order to determine an expected number of people thatwould die by that year. By people, we are referring to a homogeneous groupof individuals with equivalent health. For this example, we can determine thatin 5 years, we would expect between one and two people out of the group topass away. Hence, a hedge fund trying to determine the price it would pay fora group of life settlements in 2010 should only consider the price of 18 or 19 lifesettlements, rather than 20, since at least one settlement has most likely paid

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out its death benefit.The following table shows expected number of death in each year among

20 people aged 65 in 2005, according to 10000 simulations of our random walkmodel (2).

Year 2005 2006 2007 2008 2009 2010Death 0.280 0.324 0.370 0.388 0.376 0.404Year 2011 2012 2013 2014 2015 2016Death 0.412 0.494 0.468 0.568 0.578 0.544Year 2017 2018 2019 2020 2021 2022Death 0.614 0.688 0.680 0.680 0.792 0.724Year 2023 2024 2025 2026 2027 2028Death 0.728 0.796 0.822 0.912 0.856 0.772Year 2029 2030 2031 2032 2033 2034Death 0.756 0.724 0.636 0.588 0.588 0.528Year 2035 2036 2037 2038 2039 2040Death 0.400 0.330 0.268 0.216 0.200 0.132Year 2041Death 0.108

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 thefirst 5 years. So, for our example, we will assume that two settlements havebeen cashed before 2010.

Since the initial wealths for the first and second hedge fund in 2005 is dif-ferent 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 wealthis easy since it is only involved with bonds that have a constant growth, mainly6%. Its initial wealth in 2010 is $66,911,279. The first hedge fund’s wealth canbe determined as a simple average of the ending wealths that are determiningby running 100 simulations of the cash flows for the life settlements from 2005to 2010. 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 hasexactly two life settlements paid out. The initial wealth in 2010 for the firsthedge fund is $49,231,000.

Following in an equivalent manner as above, we can vary the initial price ofthe life settlement portfolio and determine which value will allow the first hedgefund to have a 95% probability of making more money than the second hedgefund. A graph of how the initial life settlement investment affects the probabilitythat the first hedge fund does better than the second makes apparent that theprice of the 18 life settlements in 2010, given the reinsurance in 2025, shouldbe between $15,700,000 and $15,750,000. Thus, the price of each settlement isbetween $872,222 and $875,000. When comparing the price the hedge fund iswilling to pay for each of the life settlements, they will notice that the price forthe life settlement in 2010 is higher. The reason for this trend is that in thefuture, all of the policy sellers are closer to death and the firm only needs towait 15 years, instead of 20, for reinsurance. Hence, the investors are facing lessrisk and they should be willing to pay more for the more stable asset.

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15600000 15800000 16000000 16200000

0.90

0.92

0.94

0.96

Figure 11: h(x,T0) over x, with T0 being 2010

price of 18 life settlement after 5 years

P(W

1>W

2)

While this example only considered a homogeneous portfolio of life settle-ments, one can easily adjust the model to include a more diverse portfolio. Inorder to do so, one must adjust the mortality tables for each individual in theportfolio 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 portfoliois then calculated using the aforementioned processes. One shortcoming of ourmodel is that we cannot determine the individual prices for each life settlementin 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 andDowd Model. Second, we establish Present Value Percentile Model to valuatethe portfolio of the life settlements. Third, we adopt Options Comparison Modelin comparing two hedge funds to calculate the portfolio.

By comparing the prices calculated by the two models, we can conclude thatfor men with the life expectancy rating 120% at age 65 selling their policies,after choosing the 95th percentile of the distribution of predicted prices, theprices in Present Value Percentile Model and Options Comparison Model arevery close, since they are $667008 and ($691000, $698500) respectively. In addi-tion, comparing different prices we set to life settlement under various condition

22


illustrate the reasonableness and extensibility of our models.The average price of each life settlement in 2005, according to Data Col-

lection Report 2004-2005 published by Life Insurance Settlement Association(LISA), is $349884.6. Thus, price set by our models is about twice as much asthe market price, which means, given a hedge fund’s profit target, the highestprice 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 Mortalitywith Parameter Uncertainty: Theory and Calibration, The Journal of Riskand Insurance, Vol.73, No.4, 687-718

[2] Friedberg and Webb, 2007, Life is Cheap: Using Mortality Bonds to HedgeAggregate Mortality Risk, The B.E. Jounal of Economic Analysis & Policy,Vol.7, Iss.1, Art.31

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