Life Settlements and Viaticals

Transcription

1 Life Settlements and Viaticals Paul Wilmott, Wilmott Associates, This is taken from the Second Edition of Paul Wilmott On Quantitative Finance, published by John Wiley & Sons 26 In this chapter life sex death 1 Introduction And now for something completelymorbid Life settlements and viaticals are contracts associated with death The two expressions can be used interchangeably, although viatical is often associated with a life with an already diagnosed terminal illness With these contracts expiration means precisely that Life settlements are a secondary market for the life insurance policies held by individuals These individuals may, typically later in life, want to sell their policy The common reasons are they can no longer afford to pay the life-insurance premium, as part of estate planning, they need the money to pay for medical treatment, they no longer need the life insurance The policy is usually worth a lot more than its surrender value Many of these life insurance policies are then usually packaged together and sold as one product To the quant, the question is how to model and price, and hedge, individual policies and portfolios of policies 2 Life Expectancy In estimating the chances of an individual dying at different ages you typically have non-specific and specific pieces of information The first comes from actuarial tables for populations as a whole Such tables may be specific to the type of person selling their policy, in the sense that it may refer to their sex, whether or not they smoke, the illnesses they suffer from But it won t be specific to the individual person For that sort of information the policy purchaser will require the individual to be examined by an approved life expectancy provider This will be a medical examiner ME who will examine the policyholder to determine a life expectancy for that individual Although termed a life expectancy LE the number given by the ME will actually be an estimate for the median of the person s life, and not its expected value or average The ME will often include in their estimation an allowance for expected new cures for terminal illnesses Figure 1 shows a typical LE certificate from a LE provider 21 Sex One of the dominant factors affecting LE is sex No, not how often you get it, although I m sure than has an impact, but your gender As every man knows, their life expectancy is invariably shorter than that for a woman, all things being equal American males born now have a LE of 74 years, American females 795 Japanese woman have a LE of 83 years, the longest in the world In the US, men smoke more cigarettes and drink more alcohol than women Men are three times as likely as women to die from accidents and four times more likely to be murdered 22 Health The World Health Organization WHO recently designed a new system of measuring life expectancy that takes into account disability, and so measures healthy lifespan Under this system Japan still comes top of the LE table, with an average healthy lifespan of 75 years Sierra Leone comes in bottom with 26 years Under this system the US ranks 24th Christopher Murray, Director of WHO s Global Programme on Evidence for Health Policy, said, Basically, you die earlier and spend more time disabled if you re an American rather than a member of most other advanced countries The WHO s reasons for the low ranking of the US are, amongst others, as follows Some ethnic groups and the inner city poor have poor health, more like that of the poor from a developing country rather than a rich country HIV causes a higher proportion of death and disability to young and middle-aged Americans than in most other advanced countries 56 Wilmott magazine

2 sex, smoking or non smoking, and various health factors A section of such a table is shown in Figure 2 The numbers in the cells are probabilities We will use such information but convert it into a form that we are already used to, that of the probability of default Thus we shall interpret death as default, with no prospect of recovery 4 Death Seen as Default Let us introduce some simple notation, pa will be the probability of dying at age a More precisely, the probability of dying between ages a and a + da is +da a ps ds, or, for small da, approximately pa da This function we will get from actuarial tables In practice the function will vary from person to person, we will deal with this later It will also be a function of time, not just age The probability of a 6-year old dying before their 61st birthday is different now from what it was 5 years ago So we should write pa, t Again, we ll come back to this point later on A typical such pa is shown in Figure 3 If we write Pa; a as the probability of still being alive at age a given that you were definitely alive at age a then so that dp da = pap, Pa; a = exp a ps ds Figure 1: Life expectancy certificate High incidence of cancer related to tobacco High coronary heart disease rate High levels of violence, especially homicides, compared to other industrial countries 3 Actuarial Tables Actuaries quantify probability of death in their famous tables These tables give probabilities of dying within the year, say, according to a person s age, See Figure 4 The median, or life expectancy, is that a such that exp a ps ds = 1 2 The probability density function for age at death is dp = pa exp da a ps ds See Figure 5 And so, the expected age at death is a apa exp a ps ds da ^ Wilmott magazine 57

3 5 Pricing a Single Policy And so to pricing of a single life policy Here is an example 7 Principal $1,, Monthly premiums $2,83 Policy purchase price $375, Life expectancy 52 years This means that the policy holder is aged 7 when the policy is sold It was sold for $375, The purchaser must take over payment of the monthly insurance premiums $2,83 while the insured is still alive On his/her death the purchaser will receive $1,, The policy holder has a life expectancy of 52 years To value this policy I have taken a published table of mortality rates, and adjusted them to give approximately the correct LE Of course, this can be done in any number of ways Figure 2: A section of an actuarial table Adjust1 = 2 Adjust2 = 25 If > 7 Then ProbabilityOfDeath = 1 If > 75 Then ProbabilityOfDeath = 2 If > 8 Then ProbabilityOfDeath = 36 If > 85 Then ProbabilityOfDeath = 7 If > 9 Then ProbabilityOfDeath = 1 If > 95 Then ProbabilityOfDeath = 13 If > 1 Then ProbabilityOfDeath = 19 If > 15 Then ProbabilityOfDeath = 2 If > 11 Then ProbabilityOfDeath = 1 ProbabilityOfDeath = ProbabilityOfDeath + Adjust1 + Adjust pa Pa;a Figure 3: Pa Figure 4: Pa;a 58 Wilmott magazine

4 6 5 3 PDF of PVs of total payments PDF Figure 5: Probability function for age at death -$3, -$2, -$1, $ $1, $2, $3, $4, $5, Figure 7: Probability distribution of present value of all cashflows, 1, Monte Carlo simulations $6, 12 PDF of death 7, PV of cashflows 1 6, 5, 8 4, 3, 6 2, Figure 6: Probability function for age at death, 1, Monte Carlo simulations The resulting distribution of ages at death is given in Figure 6 This plot and all results that follow are based on simple Monte Carlo simulations of ages at death, not on any analytic calculations The standard deviation of ages at death is 358 Associated with this policy are various cashflows as far as the purchaser is concerned: The purchase payment; The monthly premiums; The final principal For each of the simulations of death we can present value all of these cashflows and so get a distribution of present values This is shown in Figure 7, using a 3% interest rate The mean is $31,12, 1, , -2, -3, -4, Figure 8: Present value of all cashflows versus age at death the standard deviation $157,41 So, on average you will almost double your money if you buy this policy for the $375, In Figure 8 you can see how this present value varies with age at death After 84 years you start to lose money on this investment 51 Internal Rate of Return Most investors ie purchasers in these products do not think in terms of present value Instead they like to work in terms of internal rate of return, or yield So for a policy such as the above, rather than know the ^ Wilmott magazine 59

5 Probability Probability Density Function PDF for IRR % 1% 2% 3% 4% 5% 6% 7% 8% 9% 1% Figure 9: Probability distribution of IRR IRR PDF of PVs of total payments, grouped $ $1, $2, $3, $4, $5, $6, 45 4 PDF of IRR, grouped % 1% 2% 3% 4% 5% 6% 7% Figure 1: Probability distributions, five policies expected PV of all cashflows, they d like to know the expected IRR Figure 9 shows the distribution of IRR The average is 257% and the standard deviation is 365% One of the problems with using the IRR as a measure of the value of this policy is that the calculation is very sensitive to age of death, especially at the short end When someone dies immediately after the policy has been purchased you will find the IRR is infinite So IRR is not necessarily that informative 6 Pricing Portfolios People invest in these portfolios by buying many, dozens or hundreds, at a time This reduces exposure to individual lives The process of pricing, calculating risks and IRRs is exactly the same for a portfolio of many policies as it is for a single policy, except that you must simulate all of the underlying lives simultaneously At least we can reasonably assume that deaths are not correlated Let s crunch the numbers for many policies, all identical to the one analyzed above In practice, of course, the policies and actuarial tables will vary from policy to policy But this is just a detail needed in programming, not in concept In Figures 1 12 are the probability density functions for the present value of all cashflows and for the IRRs, with number of identical policies being five, 2 and 1 The statistics for these portfolios are as follows No of policies SD of PV $ Av IRR % SD of IRR % 1 157, , , , , , $ $5, $1, 61 Extension Risk PDF of PVs of total payments, grouped $15, $2, $25, PDF of IRR, grouped $3, $35, $4, $45, % 5% 1% 15% 2% 25% 3% Figure 11: Probability distributions, 2 policies If you buy one of these policies or portfolios, the last thing you want if for the insured to have a long life Obviously, each life is going to be difficult to forecast, but when you look at portfolios you hope that some form of 6 Wilmott magazine

6 PDF of PVs of total payments, grouped $ $5, $1, $15, $2, $25, $3, $35, $4, PDF of IRR, grouped % 2% 4% 6% 8% 1% 12% 14% 16% 18% 2% Figure 12: Probability distributions, 1 policies the Central Limit Theorem will be working One of your major exposures, however, is to a systematic extension to lifespans generally, so called extension risk The unexpected cure for otherwise terminal illnesses, for example On a positive note, positive for the owner of the contract if not the insureds, there are factors which may systematically decrease lifespans Factors systematically increasing lifespans: Health care improvements Health education improvement Improvements in safety Improvements in nutrition Therapeutic advances Genetic research Early detection Improved access to medical care Globalization -more research and advances Environmental changes Shifting demographics Diet Telecommuting Cell phone usage Technology Factors systematically decreasing lifespans: Catastrophes Terrorist activities War Globalization -prone to more diseases Natural disasters Nuclear accidents Global warming Outbreak of disease Stress Drug-resistant germs Pollution Smoking Side effects of new drugs Shifting demographics Increasing wealth disparity Cell phone usage Lack of sleep Source: Klein, 22 It appears that lately the probability of dying at any age is decreasing by a factor of about 2% per annum So that if the probability of dying at age a today is p then the probability of someone aged a next year, dying next year is 98 p Probability of death is not time homogeneous And this is where the more general function pa, t comes in, it is the hazard rate for a person aged a at time t For most people in most countries this function is slowly decreasing with time as health generally improves However, there are some exceptions to this even in developed countries 7 Summary This is a subject that is becoming popular with quants, and as such we should expect a little bit more in the way of sophistication of the modeling in the future This will be driven in the main by the increasing amount of securitization of these life products, portfolios of policies lumped together and sold as a package Further Reading See Stephen Jay Gould s The median isn t the message 1985 about his diagnosis of and fight against abdominal mesothelioma See Klein 22 for a discussion of the future of mortality forecasting Appendix: The of Quants Here s a fun bit of mathematics, not totally unrelated to the above, and which shows you a little bit more of the mathematics of life and death It concerns the probability distribution of the age of quants and the age at which people first become a quant Of course, the same idea can be applied to other problems as well Suppose Na, t is the number of quants in the world who are aged a at time t There is a steady stream of newbies into the business, so that in a time step dt a number na dt of people of age a become quants I have kept this independent of time for simplicity 1 The increment in number of quants aged a in a time step dt is caused by newcomers joining, and quants aging an amount dt as well So From Taylor series we get Na, t + dt = Na dt, t + na dt N t + N ^ a = na Wilmott magazine 61

7 The general solution of this is Na, t = f a t + nτ dτ, where f is an arbitrary function Let s suppose that at time t = there were no quants, Na, = This means we can find f and we end up with Na, t = nτ dτ Further suppose that all quants retire at age a R then the probability density function for the age of quants is just Pa, t = R Na, t = Na, t da R nτ dτ nτ dτ da After a long time this settles down to the steady-state distribution P a = R nτ dτ a R τ nτ dτ If we know na we can find P a or vice versa One of the unrealistic bits in the above is that all quants retire at the same age Suppose instead that a proportion of quants retire or change job, or die, etc And let s make that the coefficient of proportionality a function of age, αa With N and n having the same meanings as before we now get N t + N = na αa N a The general solution of this is Na, t = f a t e αs ds + e αs ds τ nτ e αs ds dτ, where f is again an arbitrary function The solution having Na, = is Na, t = e αs ds τ nτ e αs ds dτ We can then scale this to find the probability density function The steady-state limit is e a αs ds τ nτ e αs ds dτ P a = e a αs ds τ nτ e αs ds dτ da In the simple case that α is constant, ᾱ, this becomes ᾱe ᾱa nτ dτ P a = nτ dτ FOOTNOTE & REFERENCES 1 I have also been a bit loose with the definition of N, it should also contain some bucket size Gould, SJ 1985 The median is not the message Discover June 4 42 Klein, A 22 Determining the future of mortality Contingencies September/ October W 62 Wilmott magazine