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1. Risk Analysis & Modelling Lecture 8: Life Insurance Models

2. www.angelfire.com/linux/riskanalysisRiskCourseHQ@Hotmail.com

3. Life Insurance • The use of Statistics and Mathematics in the Modelling of Life Insurance dates back to the 17th Century • Life Insurance is an Insurance Policy which pays a claim in the event of the Insured dying • In the event of death a payment known as the Death Benefit is paid to the Beneficiary • We will examine standard risk models relating to two types of Life Insurance: Whole Life and Term Life • Term Life is an insurance policy lasting for a specified period of time • Whole Life is an insurance policy lasting for the entire life of the insured – the claim is certain but the timing is not

4. Term Life • Term Life is similar in nature to Non-Life Insurance • It insurers the Policyholder against death over a period of time known as the Term • If the Insured dies during the Term of the Policy a Death Benefit payment is made by the Life Insurance Company to the Beneficiary of the Policy • From a statistical modelling perspective, there are two fundamental differences between Life and Non-Life Insurance: there can only be one claim per policy (you can only die once) and the value of the claim or death benefit is fixed

5. Life vs Non-Life Insurance Non-Life Frequency Severity Model Severity of Claims: How large are the claims? Frequency of Claims per Policy: How Many Claims? Binomial Life Model Pay the death benefit Yes, Probability = X Does the Insured Die? No, Probability = 1 – X

6. Life Tables • Central to Risk Modelling in Life Insurance is the estimation of the probability of death or mortality over a period of time • Standard Life Tables give mortality statistics for an individual of an exact age and sex • We will start off by looking at Static Life Tables which are based on the entirely on the current Mortality Rates • The Life Tables we will use to build our models today are provided by the Government Actuary’s Department (GAD) and are publicly available at: http://www.ons.gov.uk/ons/index.html

7. Structure of Life Tables • Life Tables are usually structured as follows: qx is the Probability of Dying, or the probability of a person of a given age dying within the next year

8. px = 1 - qx • Directly related to the qx is the probability of survival (px) which is related to the death probability by: lx is the Number of Survivors at a given age from an initial sample of 100,000 individuals born at the same time (for example, 97661 or 97.66% out of an initial sample of 100,000 will still be alive by 40) dx is the number of deaths per year from the same hypothetical sample of 100,000 individuals ex is the average life expectancy for a person of the given age

9. Question 1 • The “Static Life Table” in your Excel workbook states that the lx value for males aged 65 is 86992.6 • This means that out of a sample of 100,000 newly born male babies only 86992.6 will still be alive on their 65th birthday based on current mortality rates • Given this information calculate the probability of a newly born male surviving to his 65th birthday (the proportion still alive at 65) • Using this, then calculate the probability of a newly born male child dying before his 65th birthday

10. Question 2 • The “Static Life Table” in your Excel workbook states that the lx value for males aged 40 is 97661.1 and the lx value for males aged 70 is 80981.9 • So only 80981.9 out of 97661.1 aged 40 are still alive at age 70 • Using this information calculate the probability of a male aged 40 still being alive on their 70th birthday • Then calculate the probability of a male aged 40 dying before their 70th birthday

11. Calculating the Probability of Death • The probability of an individual surviving n years after their xth birthday is: • This formula measures the probability of survival by looking at the ratio of the number alive at year x + n and the number alive at year x (of the initial 100,000) • The probability of dying over the same period is simply one minus the probability of survival:

12. For example, using the information in our static life table we can calculate the probability of an individual dying within the 4 years following their 40th birthday • First let us calculate the probability of survival • We can then calculate the probability of death as

13. Interpolating Survival Values • The Life Table only provides lx (Number of Survivors) values for whole years • This means we can only use it to calculate probabilities of death over an entire year after a birthday • For example, we cannot directly calculate the probability of someone who is 41 years and 6 months of age dying within the next 15 months • In order to calculate the probability of death we need to interpolate or imply the values for lx in between the values we are given – or imply the lx value for 41 years and 6 months (41.5 years) from the values for 41 years and 42 years

14. Uniform Distribution of Deaths (UDD) • A simple and popular method of calculating values for lx between those given in the Life Table is to assume deaths occur uniformly over the year • This assumption means that deaths from the initial sample of 100,000 are spread out evenly over the year • This assumption will mean that the lx values will decrease in a straight line between two adjacent values in the Life Table: • Where lx* is a value some proportion of the way (a) between lxt and lxt+1

15. Applying this formula to the earlier example, to calculate the value for lx for individuals 41.5 years (41 years and 6 months): • 1.25 years (15 months) later the individual will be 42.75 the value for lx at this age is: • Therefore the probability of death for an individual of 41.5 years of age dying within the next 1.25 years is:

16. Linear Interpolation of LX LX Values from Life Table 97422.5 97205.8 The Uniform Distribution of Deaths assumption means that LX decreases in a straight line between values in the Life Table 42.75 41.5

17. Mortality VBA Functions • To help us build our model we are going to use 2 VBA functions that automate the calculation of the LX and QX values • This function takes an LX range in the LXColumn Parameter, and the Age (integer or decimal) and returns the linear interpolated value: Public Function LX(LXColumn As Range, Age) NearestIntegerAge = Int(Age) LX = LXColumn.Cells(NearestIntegerAge + 1, 1) + (Age - NearestIntegerAge) * (LXColumn.Cells(NearestIntegerAge + 2, 1) - LXColumn.Cells(NearestIntegerAge + 1, 1)) End Function • This function returns the probability of death for an individual of a specified Age within a term specified by Years, notice this function uses the LX function specified above: Public Function QX(LXColumn As Range, Age, Years) LX1 = LX(LXColumn, Age) LX2 = LX(LXColumn, Age + Years) QX = 1 - (LX2 / LX1) End Function

18. Survival Distribution Function & Life Tables • Although it is not immediately obvious the Life Table is just a way of representing the CDF of the random life expectancy of an individual • To see this we can calculate the probability of a new born child surviving beyond an age t (Survival Distribution Function SDF) by taking: • From this we can calculate the probability of an individual dying at or before time t, or the life expectancy being less than t:

19. Survival Distribution Function (SDF) SDF 91% chance of surviving beyond 60th birthday (probability that lifespan will be greater than 60) CDF 9% chance of dying at or before 60th birthday (probability lifespan will be less than 60)

20. Unsystematic Mortality Risk and the Bernoulli Distribution • One of the risks that an Insurer faces on a portfolio of Term Life policies is that the number of claims (or deaths) will be larger than expected purely by chance – sometimes called Unsystematic Mortality Risk • Claims on each Term Life policy in the portfolio follow a Bernoulli distribution – a discrete distribution which can just take two values 1 (death) or 0 (survival) • The distribution is entirely defined by the probability of observing the value 1 (or death) qx, the value of observing a 0 (or survival) is then 1 – qx • There are only two possibilities – you either die over the period or you survive

21. Revision: Random Bernoulli Trials Values the CDF If rand() > 1 – qx then 1 otherwise 0 rand()

22. Random Death and Claim in Excel Formula • If we know the probability of death over a period (qx) then we can simulate whether or not a death occurs (1) or does not occur using the formula: =IF(rand() > 1 – qx, 1, 0) • Equally, if we know the death benefit that will be claimed in the event of death (C) we can simulate the random claim: =IF(rand() > 1 – qx, C, 0) • This will give a value of C if a death and claim occurs otherwise it will give a value of 0 (no claim)

23. Simulating Aggregate Claims • On the “Aggregate Claim” sheet we simulate the Total or Aggregate Claim over a year on a portfolio of 2000 Term Life policies • The technique we will use is almost identical to that which we used for the non-life portfolio • Using the Bernoulli Distribution we will simulate the level of claims for each policy, sum the individual claims to obtain a simulated Aggregate Claim and then sort a sample of 1000 simulated Aggregate Claims to obtain an estimate of the Empirical CDF

24. Aggregate Claims CDF In this sample 99% of the time the aggregate claim will be less than or equal to 3,000,000 (PML99%)

25. Catastrophic Mortality Events • Catastrophic Mortality Events increase the level of Mortality and influence the Aggregate Claims Distribution • The 1918 Influenza epidemic is often cited as a Catastrophic Mortality Event – increasing the rate of mortality qx by an average of 0.5% across all ages • This increase in mortality is called the Epidemic Mortality Rate • The risk of this happening is an example of a Systematic Mortality Risk (Systematic Risks are not diversifiable through the law of large numbers) • Epidemics of this magnitude are believed to occur on average once every 100 years (Mean Return Period) - a 1% probability of occurring in any given year • This probability of an epidemic occurring is called the Epidemic Rate

26. We can add this morality CAT risk to our simulation by simulating the event of an epidemic (using a Bernoulli random variable with probability of 1%) and in the event of this occurring increasing all morality rates by 0.5%: • Where qx is the death probability derived from the mortality table, b is a Bernoulli random number with P = 0.01 and qx’ is the adjusted death probability we expect to observe in the event of an Influenza Epidemic

27. Effect of Epidemic on QX Values Upward shift in QX values caused by Epidemic

28. Aggregate Claims CDF with CAT Risk The CAT Mortality Event Extends the Upper Tail of the Aggregate Claims Distribution In 99% of the time the aggregate claim will be less than 3,650,000 (PML99%)

29. Average Aggregate Claims • We could also use our simulated, random sample of 1000 Aggregate Claims to estimate the Average Aggregate Claim • We should all get a value around 1,300,000 however the answer is random – sometimes it is higher and sometimes it is lower • We can think as our answers from the Monte Carlo Simulation as giving us a sample from an unknown underlying distribution whose Mean is unknown the true value of 1,308,424 • The problem is we do not know how far away our answer is from the true answer or how much error we have in our answer…

30. Distribution of Answers We can think of our answers from the Monte Carlo Simulation as being random samples from an unknown distribution whose unknown Mean value is the true answer 1308424

31. Averaging Answers • One thought that might have crossed your mind is that we could take an average of the different answers • This would effectively give us a larger sample and the volatility of our answer would decrease (accuracy increase) • The relationship between the volatility of an average and the volatility of the independent random variables making up that average can be calculated as:

32. In this case sx is the standard deviation of each answer provided by a single simulation and sA is the standard deviation of an average of N answers This formula tells us that if we want to halve the standard error we must quadruple the number of simulations (N = 4) So if we go from 1000 to 4000 (averaging four simulations of 1000 is equivalent to running 4000 simulations once - see appendix) we expect to double the accuracy (halve the standard deviation) If we want to increase the accuracy by a factor of 10 we must increase the number of simulations by a factor of 100

33. Normally Distributed Answers • If our estimate is made up of the average (or sum) of a large number of answers derived from smaller simulations we can state that this average will be Normally Distributed due to the Central Limit Theorem • Even if we do not know the distribution of the individual answers the distribution of their average is approximated by the Normal Distribution • We can then use the Normal Distribution to place probabilistic limits on the error from the Monte Carlo Simulation • Our answer will be within 3 standard deviations of the true answer (mean) 99.74% of the time and 4 standard deviations away from the true answer 99.994% of the time

34. Normally Distributed Answers 99.99% of the time a Normally Distributed Random Variable will be within 4 Standard Deviations of its Mean -4 Std Dev +4 Std Dev True Answer

35. Let E be that Maximum Error we are willing to have in our “average” answer with 99.99% certainty, let s be the standard deviation of the answer from simulations of size X: • Rearranging we can see that • Using this formula we can see how many simulations of this size we have to average to achieve the desired level of accuracy with a 99.99% confidence level

36. Monte Carlo Accuracy Vs Simulation Size As the number of simulations increase the accuracy of the Monte Carlo Simulation increases but at a decreasing rate (square root N) 100 Simulations 1000 Simulations

37. Whole Life Insurance • Whole Life Insurance is a form of Permanent Life Insurance which lasts for the entire lifespan of the Policy Holder • Whole Life policies differ from Term Life and Non-Life Policies in that a claim is certain – the Insured will eventually die • The uncertainty is when the death will occur • Whole Life Insurance can be looked at as a combination of Insurance and Saving, in which a pool of funds are built up over time to pay out the inevitable claim • We will start by looking at the simplest form of Whole Life policy which is a Non-Profit (Non-Participating) policy for which the Premiums and Death Benefits are fixed

38. Cohorts and Cohort Life Tables • Since a Whole Life Policy can span for a number of decades it is important to take into account long term trends in Mortality • The Static Life table we used in our Term Life model only took into account current Mortality Rates • A Cohort is a group of individuals of approximately the same Age and Sex and whose Mortality Rates are similar • A Cohort Life Table gives estimates of the projected future mortality of the Cohort as they age, taking into account future trends in mortality • We will be using a Cohort Life Table from the Government Actuary’s Department for a Cohort of Males aged 25 found on the “Cohort Life Table” Sheet which gives the forecasts for the future Morality Rates for a group of individuals age 25 years today

39. Cohort Vs Static LX Values The LX Values for a Cohort of 25 year old Males diverges due to forecasts in decreasing mortality rates in the future

40. Our Example Cohort • We will model a specific cohort made up of 5000 individuals aged exactly 25 years purchasing Whole Life Insurance in the current year with a face value of £100,000 • The Life Insurance policy will pay £100,000 in the event of the individual’s death or when they reach 100 years of age (maturity of the policy) • The lifespan of the cohort is 75 years • Every year the policy holders pay a fixed premium • The Premium is to be paid annually at the start of the year and any Death Benefit is paid at the end of the year • The Premiums paid by the Policy Holders accumulate in the Premium Reserve and the Death Benefits payments are paid from this reserve

41. Payments Over Time Premium Reserve Year Start Premium Reserve Year End Premium Reserve From Previous Year End Remaining Premium Reserve Carried Forward Death Benefits - = New Premium Payments Remaining Policyholders Year End Deaths During Year Policyholders Year Start

42. How Much Should The Annual Premium Be? • The first question we will answer is how much the Annual Premium on this Whole Life Policy should be • The Minimum Premium Level that the Insurer will charge will be at that level which just covers the Expected Payments – the Premium Reserve is just enough to make the final payment in 75 years time • Using our Cohort Life Table for males aged 25 we can forecast the expected Mortality Rates and Cash Flows in and out of the Premium Reserve • On the “Non-Stochastic 1” Spreadsheet we the expected behaviour of the Premium Reserve over the next 75 years

43. Pattern of Payments • The Mortality Rate is not constant between years and rises as the policy holders age • Early on in the lifespan of the Cohort most of the individuals are still alive so the premium income is at its highest value, as people die and their Premium payments cease this will decline • The Death Benefit payments will increase as the Mortality Rate increases but will later decline as very few people are left alive in the Cohort

44. Minimum Premium Level Accumulated premium reserve to pay future death benefits Premium Income Declines Accumulated premium just sufficient to pay final death benefit at the end of the 75 year period Variable death benefit payments Year 0 Year 75

45. Premium Reserve Evolution Premium Reserve grows as Premium Income outweighs Death Benefit payments The reserve declines as payments to policy holders outstrip premium income Annual premium has been set such that the reserve falls to zero with final payment

46. Premium Reserve Per Policy As payment becomes certain the reserve per policy approaches 100,000

47. Extension: Interest on Premium Reserve • The Life Insurer builds up a large amount of capital over the 75 year life span of the Cohort • So far we have been assuming that this Capital is just sitting there • In reality this Capital would be earning Interest Income • We will assume that a fixed rate of interest is earned over the 75 year term at 5.5% on the Premium Reserve at the End of the Year

48. Additional Income From Interest Premium Income Interest Income earned on premium reserve Death Benefit Payments Year 0 Year 75

49. Extension: Policy Surrenders • To Increase the marketability of Whole Life Policies they can often be cashed in or Surrendered • Even if we did not allow people to Surrender their policies we would expect some individuals to just stop paying their Premium and drop out of the Cohort or Lapse • The amount the Policy Holder receives for their policy is the Surrender Value • The proportion of policy holders surrendering their policies in a given year is measured by the Surrender Rate (or Ratio): • We will assume that the Surrender Value is simply a percentage of the Premium Reserve per Policy

50. Policy Surrenders Policy Holders Year Start Policies Surrendered and Receive Surrender Value of Policy from Premium Reserve Policyholders Die and Receive Death Benefit from Premium Reserve Remaining Policy Holders Year End