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Risk Analysis & Modelling
Lecture 7: Life Insurance Models
Life InsuranceThe use of Statistics and Mathematics to Model the risks faced by Life Insurance Companies dates back to the 17th Century Life Insurance is an Insurance Policy which pays a claim in the event of the Insured dyingIn the event of death a payment known as the Death Benefit is paid to the BeneficiaryWe will examine Risk Models for two types of Life Insurance: Whole Life and Term LifeTerm Life is an insurance policy lasting for a specified period of timeWhole Life is an insurance policy lasting for the entire life of the insured – the claim is certain but the timing is not
Term LifeTerm Life is similar in nature to Non-Life InsuranceIt insurers the Policyholder against death over a period of time known as the TermIf 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 PolicyFrom 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
Life vs Non-Life Insurance
0
0.02
0.04
0.06
0.08
0.1
0.12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Number Claims Per Week
Pro
bab
ility
0
0.0005
0.001
0.0015
0.002
0.0025
0 500 1000 1500 2000 2500
Claim Size
Pro
bab
ility
Den
isty
Non-Life Frequency Severity Model
Frequency of Claims per Policy: How Many Claims?
Severity of Claims: How large are the claims?
Binomial Life Model
Does the Insured Die?
Yes, Probability = X
No, Probability = 1 – X
Pay the death benefit
Life TablesThe primary input into any Life Insurance Risk Model is the estimation of the probability of death or mortality over a period of timeStandard Life Tables give mortality statistics for an individual of an exact age and genderWe 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
Structure of Life Tables
Life Tables are usually structured as follows:
qx is the probability of a person of a given age dying within the next year
Age qx lx dx ex40 0.001590 97661.1 155.3 40.90
41 0.001710 97505.8 166.7 40.00
42 0.001826 97339.1 177.7 39.00
43 0.001937 97161.4 188.2 38.10
44 0.002046 96973.2 198.4 37.20
45 0.002154 96774.8 208.4 36.30
46 0.002260 96566.3 218.2 35.30
47 0.002376 96348.1 228.9 34.40
Directly related to the qx is px which gives the probability of the individual surviving over the next year
px = 1 - qxlx 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
Question 1
The “Static Life Table” in your Excel workbook states that the lx value for males aged exactly 65 is 86992.6
This means that out of a sample of 100,000 newly born males 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
Question 2
The “Static Life Table” in your Excel workbook states that the lx value for males aged exactly 40 is 97661.1 and the lx value for males aged exactly 70 is 80981.9So only 80981.9 out of 97661.1 aged 40 are still alive at age 70Using this information calculate the probability of a male aged exactly 40 still being alive on their 70th birthdayThen calculate the probability of a male aged 40 dying before their 70th birthday
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 still alive at year x + n and the number initially alive at year x
The probability of dying over the same period is simply one minus the probability of survival:
x
nxxn l
lp
x
nxxnxn l
lpq 11
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
%7.0007.0993.01404 q
%3.991.97661
2.96973
40
44
40
440404
l
l
l
lp
Interpolating Survival ValuesThe Life Table only provides lx values for whole yearsThis means we can only use the Life Table to calculate the probability of death between the individuals birthdays which is rather limited!For example, we cannot directly calculate the probability of someone who is 41 years and 6 months of age dying within the next 15 monthsIn order make this type of calculation using the Life Table we need to interpolate or imply the in between values for lx – or imply the lx value for 41 years and 6 months (41.5 years) from the values for 41 years and 42 years…..
Uniform Distribution of Deaths (UDD)
Actuarial Science provides a series of techniques for interpolating the lx values provided in the Life Table called Graduation Theory
A simple and popular method for estimating lx values between those given in the Life Table is to assume that deaths will be evenly spread out (or occur uniformly) over the year – Uniform Distribution of Deaths
Graphically, 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 () between lxt and lxt+1
ttt lxlxlxlx 1.*
Applying this formula to the earlier example, to calculate the value for lx for individuals 41.5 years (41 years and 6 months) which is 0.5 or 50% of the way between 41 and 42:
1.25 years (15 months) later the individual will be 42.75 the value for lx at this age is (0.75 or 75% between 42 and 43):
Therefore the probability of death for an individual of 41.5 years of age dying within the next 1.25 years is:
5.974228.975051.97339*5.08.97505*5.0 4142415.41 lxlxlxlx
8.972051.973394.71619*75.01.97339*75.0 42434275.42 lxlxlxlx
%23.00023.09977.015.97422
8.9720511
5.41
75.425.4125.1
lx
lxq
Linear Interpolation of LX
41.5
97205.8
42.75
97422.5
LX Values from Life Table
The Uniform Distribution of Deaths assumption means that LX decreases in a straight line between values in the Life
Table
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 based on the Uniform Distribution of DeathsThis 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
Survival Distribution Function & Life Tables
Although it is not immediately obvious the Life Table is just a way of representing the CDF for 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 Function or SF) 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 or equal to t:
000,100)( tlxtSF
000,1001)(1)( tlxtSFtCDF
Survival Function (SF) and CDF of Life Expectancy
CDF
SF
9% chance of dying at or before 60th birthday (probability lifespan will be less than or equal to 60)
91% chance of surviving beyond 60th birthday (probability that
lifespan will be greater than 60)
Premium Calculation for Term Life Insurance
Just as with Non-Life insurance, a large part Term Life Policy is determined by the Expected Claim or Pure Risk PremiumFor example, if an individual of exactly 40 years of age wishes to purchase a life policy with a term of 4 years with a death benefit of £500,000 the expected claim is:
E[Z] = 4q40 * 500000 = 0.007 * 500000 = 3522
The Total Premium paid would have to be at least £3522 to cover the Expected LossTerm Life Premiums are normally paid for monthly, effectively the insured is buying a series of Term Life Policies lasting for one MonthThis calculation does not take into account costs or the timing of the payment, which as we will see is especially important for long term life contracts such a Whole Life Insurance
Review Question
Using the Static Life Table calculate the Pure Risk Premium if you wished to purchase a Term Life Policy for a 2 year period with a death benefit of £250,000
Use out QX VBA Function to calculate the probability you will die over the next 2 years!
Then multiple this probability by the death benefit to calculate the Pure Risk Premium.
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 RiskUnsystematic Mortality Risk can be reduced through Diversification of the Underwriting Portfolio (expanding the number of policies in the portfolio and reducing the risk through the Law of Large Numbers)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 and the value of observing a 0 (or survival): 1 – qxThere are only two possibilities – you either die over the period or you survive
Revision: Random Bernoulli Trials Values the CDF
0
0.2
0.4
0.6
0.8
1
0 1
rand()
If rand() > 1 – qx then 1 otherwise 0
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)
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 policiesThe technique we will use is almost identical to that which we used for the non-life portfolioUsing 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
Aggregate Claims CDF
In this sample 99% of the time the aggregate
claim will be less than or equal to 3,000,000
(PML99%)
Aggregate Loss HistogramOne way of reducing the Noise from the Monte Carlo Simulation when estimating the Aggregate Loss Distribution is to estimate a Histogram rather than a sorted Empirical CDFBy counting the number of simulated losses between values such as 1,000,000 and 1,250,000 a histogram of the losses can be obtainedBy summing the values in the Histogram the Cumulative Frequency Histogram can be calculated which would be equivalent to the Aggregate Claims CDFThe advantages of this approach is that the noise in the answer is reducedThe disadvantage of this approach is it will only give the probability of the loss falling within a range
Histogram of Simulated Losses
16% of simulated Aggregate Losses are between 1000000 and 1250000
The Cumulative Frequency Histogram gives the probability of the loss being less than or equal to a value
Catastrophic Mortality EventsCatastrophic Mortality Events increase the level of Mortality and influence the Aggregate Claims DistributionThe 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 agesThis increase in mortality is called the Epidemic Mortality RateThe risk of this happening is an example of a Systematic Mortality Risk (Systematic Risks are not diversifiable through the law of large numbers)Effectively the Influenza Epidemic are not diversifiable because they introduce high levels of correlation between the deaths occurring on the policies in the Underwriting PortfolioEpidemics 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
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% or 0.0005:
Where qx are the probabilities of death over the next year derived from the life 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
bqxqx~
*005.0'
Effect of Epidemic on QX Values
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0 20 40 60 80 100
QX QX with Catastrophe
Upward shift in QX values caused by Epidemic
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%)
Average Aggregate Claims
We could also use our simulated, random sample of 1000 Aggregate Claims to estimate the Average Aggregate ClaimWe should all get a value around 1,300,000 however the answer is random – sometimes it is higher and sometimes it is lowerWe 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 1,308,424The 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…
Distribution of Answers
1308424
We can think of our answers from the Monte Carlo Simulation as being random samples from an unknown
distribution whose unknown Mean is the true answer
Averaging AnswersOne thought that might have crossed your mind is that we could take an average of the different answersThis 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:
N
i
XN
A1
~1~
NX
A
In this case x is the standard deviation of each answer provided by a single simulation and 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) 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
Normally Distributed AnswersIf 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 should be close to Normally Distributed due to the Central Limit TheoremEven if we do not know the distribution of the individual answers the distribution of their average is approximated by the Normal DistributionWe can then use the Normal Distribution to place probabilistic limits on the error from the Monte Carlo SimulationOur 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
Normally Distributed Answers
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-3 -2 -1 0 1 2 3
Pro
ba
bil
ity
De
ns
ity
True Answer
-4 Std Dev +4 Std Dev
99.99% of the time a Normally Distributed Random Variable will be within 4 Standard Deviations
of its Mean
Let E be that Maximum Error we are willing to have in our “average” answer with 99.99% certainty, let 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
EN
*4
2
*4
EN
Monte Carlo Accuracy Vs Simulation Size
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0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Number Of Simulations
Re
lati
ve
Sta
nd
ard
De
via
tio
n 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
Whole Life InsuranceWhole Life Insurance is a form of Permanent Life Insurance which lasts for the entire lifespan of the Policy HolderWhole Life policies differ from Term Life and Non-Life Policies in that a claim is certain – the Insured will eventually dieThe uncertainty is when the death will occurWhole 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 claimWe 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
Our Example CohortWe 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 £100,000The 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 yearsEvery 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 yearThe Premiums paid by the Policy Holders accumulate in the Premium Reserve and the Death Benefits payments are paid from this reserveTo keep the number of calculations in our model to a minimum we exclude costs (profit margin, marketing costs and operational costs) but these could easily be added.
Cohorts and Cohort Life TablesSince a Whole Life Policy can span for a number of decades it is important to take into account long term trends in MortalityThe Static Life table we used in our Term Life model only took into account current Mortality RatesA Cohort is a group of individuals of approximately the same Age and Sex and whose Mortality Rates are similarA Cohort Life Table gives estimates of the projected future mortality of the Cohort as they age, taking into account future trends in mortalityWe 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
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
Payments Over Time
New PremiumPayments
Premium ReserveFrom Previous Year
End
Premium ReserveYear Start
Policyholders Year StartRemaining Policyholders
Year EndDeaths During Year
DeathBenefits
Remaining PremiumReserve Carried
Forward
Premium ReserveYear End
- =
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 beThe 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 to the remaining policyholders in 75 years timeUsing our Cohort Life Table for males aged 25 we can forecast the expected Mortality Rates and the Cash Flows in and out of the Premium ReserveOn the “Non-Stochastic 1” Spreadsheet we calculate the Expected cash flows for the Premium Reserve over the next 75 years
Premium Reserve Cash Flows
The Mortality Rate is not constant between years and rises as the policy holders ageEarly on in the lifespan of the Cohort most of the individuals are still alive so the Premium Income flowing into the Premium Reserve is at its highest As members of the Cohort die this Premium Income will declineThe Death Benefit payments will increase as the Mortality Rate increases but will later decline as very few people are left alive in the Cohort
Minimum Premium LevelPremium Income Declines
Death Benefit Payments
Accumulated premium reserve to pay future death benefits
Accumulated premium just sufficient to pay final death benefit at the end of the 75 year period, Premium Reserve is zero after final
paymentsYear 0 Year 75
Premium Reserve Evolution
0.00
20000000.00
40000000.00
60000000.00
80000000.00
100000000.00
120000000.00
140000000.00
160000000.00
180000000.00
200000000.00
0 10 20 30 40 50 60 70
Year Number
Pre
miu
m R
es
erv
e
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
Premium Reserve Per Policy
0.00
10000.00
20000.00
30000.00
40000.00
50000.00
60000.00
70000.00
80000.00
90000.00
100000.00
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76
As payment becomes certain the reserve per policy approaches 100,000
Extension: Interest on Premium Reserve
The Life Insurer builds up a large amount of capital over the 75 year life span of the CohortSo far we have been assuming that this Capital is just sitting thereIn reality this Capital would be earning Interest IncomeWe will assume that a fixed rate of interest is earned over the 75 year term at 5.5% on the Premium Reserve and that this interest is paid at the End of the Year
Additional Income From Interest
Year 0 Year 75
Interest Income earned on premium reserve
Death Benefit Payments
Premium Income
Extension: Policy SurrendersTo Increase the marketability of Whole Life Policies they can often be cashed in or SurrenderedEven if we did not allow people to Surrender their policies we would expect some individuals to just stop paying their Premium due to a change in circumstance and drop out of the Cohort or LapseThe amount the Policy Holder receives for their policy is the Surrender ValueThe proportion of individuals in the Cohort 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 and that Surrender Payments are made at the end of the year
Year ofStart at Policies ofNumber Total
Year During dSurrendere Policies ofNumber RateSurrender
Policy Surrenders
Policies Surrendered and Receive Surrender Value of Policy from Premium Reserve
Policyholders Die and Receive Death Benefit from Premium Reserve
Policy Holders Year Start
RemainingPolicy Holders Year End
Risk 1: Stochastic MortalitySo far our Whole Life model has assumed that the mortality of individuals in the Cohort follows a deterministic, non-random pattern – we have not modelled the Risk or Uncertainty of Underwriting this Cohort We will now use our Stochastic Mortality model to simulate the effect of a random number of deaths in each year and analyse the effect this has on the premium reserveFor each year we will simulate the number of deaths that occur by randomly determining if each individual in the cohort dies using Bernoulli random variablesSince the all individuals in the Cohort have approximately the same probability of death we will be able to simplify this task using the Binomial Distribution
Stochastic Mortality for the Cohort
1 1 1
0 0 0 0 0
Simulated Deaths For Cohort of 9 Identical Individuals, Total Deaths = 3
Random simulation of death (1) or survival (0), all have an equal probability of dying (P)
0
Life Insurance Frequency Distribution: Binomial DistributionThe sum of N Bernoulli Random Variables each with probability P follows a Binomial DistributionThe Binomial Distribution is the natural choice for the number or frequency of losses on a portfolio of Life Insurance Policies (or any class of Insurance for which there can only be one claim per policy)This Distribution is entirely determined by N which is the number of trials (or number of policies in the portfolio) and P which is the probability of success (or the probability of a claim/death)It should be noted that a Binomially Distributed Random Variable can take on values ranging from 0 (no claims) to N (every policy in the portfolio making a claim)The Spreadsheet comes with a function called RandomBinomial which will give us a random sample from the Binomial Distribution
Binomial Distribution where N = 9 and P = 0.3
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9
Sum of 9 Bernoulli Trials
Pro
ba
bili
ty
Probability that the sum of 9 Bernoulli Random Variables has a
value of 5 is 0.0735 or 7.35% (BINOMDIST(5,9,0.3,FALSE))
Risk 2 : Stochastic Investment Returns
We can also add the effect of random Investment Returns on the Premium Reserve
The simplest way to achieve this would be to assume that the return earned on the Premium Reserve is a Normally Distributed random variable with a certain Mean and Standard Deviation
By altering the mean and variance of the return we could model high or low risk investments
We will assume the investments have a low volatility with a standard deviation 0.5% about the average of 5.5%
Risk 3: Stochastic Surrender Rate
The number of individuals Surrendering their policy is also likely to be a randomWe will make the standard Actuarial assumption that the frequency or number of individuals surrendering their policy in a year follows a Poisson DistributionThis assumption means that the lapses or surrenders they follow a constant intensity Poisson ProcessThe Average Frequency will be determined by the Average Surrender Rate multiplied by the Number of Policies at the Start of the Year
Premium Reserve Volatility
What our model shows us is that the Premium Reserve becomes volatile or uncertain
When the Death Rate, Surrender Rate and Investment Income was fixed we could set the premium level which would allow the premium reserve to just decline to zero with the last payment
With random Deaths, Policy Surrenders and Investment Returns the Premium Reserve becomes extremely volatile
This makes pricing the premium on the policy more complicated
Premium Reserve Volatility
-5000000
0
5000000
10000000
15000000
20000000
25000000
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76
Year Number
Pre
miu
m R
eser
ve
Premium Reserve Path With Expected Death Rates and
Expected Investment Returns
Spread of Outcomes
Reducing The Risk By Increasing The Premium
The way the Life Insurer reduces the risk that the cohort will experience a shortfall in the Premium Reserve is to increase the Premium By setting the Premium level sufficiently high adverse shocks in mortality, investment returns and surrenders/lapses can be absorbedThe high Premium also means that the Life Insurer expects to have a larger Premium Reserve than is necessary to pay Death Benefits – it expects to have a SurplusThe higher Premium Levels can also make the Life Insurer uncompetitive
The Solution!
“Participating life insurance is built on a foundation of guaranteed values (i.e. basic premium, basic insurance coverage, guaranteed portion of cash values and
guaranteed portion of reduced paid-up values) that are determined using long-term assumptions for factors such as investment returns, mortality, expenses, lapses
and taxes.
If the actual results in the par account are collectively more favourable than the assumptions supporting the guaranteed values, earnings are generated in the par
account that become part of the par account surplus (retained earnings). Each year, the company distributes a portion of the earnings as approved by the Board
of Directors. Surplus is held in the par account to maintain the strength and stability of the par account into the future.”
Dealing With The Premium Reserve Surplus With Bonuses
By increasing the Premium so that the Premium Reserve is always sufficient to meet payments – even in adverse circumstances - we almost always have a profit or money left over after all the policies have been paid or a SurplusAt any point in time the Life Insurer can compare the current Premium Reserve level to the level it should be if the expected levels of Mortality, Investment Return and Surrenders occurIf the Premium Reserve is above this expected level it is in Surplus and the Insurer can afford to give some money back to policy holders in the form of a Bonus such as a Premium Rebate or Dividend
Defining Surplus
0
2000000
4000000
6000000
8000000
10000000
12000000
14000000
16000000
18000000
20000000
0 10 20 30 40 50 60 70
Year Number
Pre
miu
m R
es
erv
e
If the Premium Reserve is below its expected level then the additional premium charged
should return the reserve to its required level over time
If the reserve is greater than the expected level for the Premium
Reserve then a Bonus can be Paid
Deficit
Surplus Per Policy
We will use the Premium Reserve per Policy to assess whether there is a surplusThe Bonus Barrier will be defined as a percentage above the Expected Premium Reserve per PolicyIf the actual Premium Reserve is above this Bonus Barrier then any excess will be returned to Policy Holders in the form of a BonusIt is important to take any measure of Surplus on a Per Policy Basis since any Bonus will be paid on a Per Policy Basis
Bonus Barrier
0
2000
4000
6000
8000
10000
12000
0 10 20 30 40 50 60 70
Year Number
Pre
miu
m R
eser
ve P
er P
oli
cy
Expected Premium Reserve Per Policy Path
Bonus Barrier
If the reserve per policy is above the barrier then the excess above the barrier is paid
back to the policy holders
Surplus per Policy
What Our Model Shows UsWhat we observe is that through the use of the Bonus Barrier and a conservative Premium Level we can almost completely remove the risk of a Shortfall in the Premium ReserveThe explanation for this is simply that the additional Premium Income received from Policy Holders is more than enough to offset adverse movements in the level of death benefits, investment income and policy surrendersIn years in which these adverse movements are not observed the additional income is returned to the policy holder in the form of a Bonus or reduction in their PremiumThe Life Insurer could use this model to determine where to set the Bonus Barrier and the Premium
Offsetting Forces
Premium Reserve Level
Additional Premium Income Paid By Policy Holders
Adverse Additional Death Benefit and Below Average Investment Returns
The additional premium income is always enough to offset
additional payments
Systematic Mortality Risk
Our Whole Life Risk Model only measured Unsystematic Mortality RiskWe made the assumption that the long term forecasts of future Mortality Rates in the Cohort Life Table were accurateThe risk for a Whole Life Insurer is that the decline in future Mortality Rates are over estimated, leading to a Systematic Mortality RiskTo model this risk we have to look at the Stochastic Mortality Models which are outside the scope of the courseA (gentle!) introduction to a simple Stochastic Mortality Model is given in the Appendix
Appendix: Monte Carlo Simulation and the Efficient Frontier
We have seen that the Monte Carlo Simulation can used to make many calculationsIt can even be used to estimate the the Efficient Frontier!To do this we need to estimate random weights that add to 1For a portfolio of 3 assets the random weights would be take 3 rand() values and then divide them by their sum (see Monte Carlo Frontier Sheet)For each set of random portfolio weights the expected turns and standard deviation of return can be calculatedBy generating a 1000 or 10000 using VBA a “cloud” of portfolios can be generated the outline of which will form the efficient frontierSee spreadsheet Random Weights
Cloud of Random Portfolios
The outline of the cloud of random portfolios is the efficient frontier
Appendix: Life Insurance Risk Factors
The probability of mortality depends on more than just the age and sex of the insuredGenerally life insurers sub-divide policyholders into sub groups: Preferred Plus (lowest morality rates), Preferred, Standard and Sub Standard (highest mortality rates)Each of these groups have their own modified Mortality Tables In addition to sex and age, Medical Risk factors are used to determine which group a policyholder falls under such as Body Mass Index, hereditary risk of cardiovascular illness or Cancer, Cholesterol Levels, Blood PressureThere are separate categorizations for smokers and non-smokers (such as Preferred and Preferred Nicotine)Certain Medical Conditions such Alzheimers or a Recent Heart Attack can make an individual uninsurableBeing involved in high risk occupations or sporting activities will also effect the classification
Risk Factors & Classification
Blood Pressure
Body Mass Index
Hereditary Illness Determination ofRisk Classification
Preferred Plus Life
Table
Preferred Life
Table
Standard Life
Table
Sub Standard Life
Table
Cholesterol
Medical ConditionsUninsurable
Appendix: Stochastic MortalityWe have also assumed that the actual underlying average mortality rates will match the trend, there is a systematic mortality risk they will be above or below this trend
The morality table can be replace by an equation of the form:
eqxt=eb.t+a
Where eqx is the estimated death probability, a and b are parameters and t is the age of the individual and e is the natural number (2.718)We can fit this curve using a least squares regression which minimises the square of the difference between estimated and actual death probability by selecting a and b – this is an example of an exponential regression
2
1
n
iii eqxqxmin
Mortality Curve: e0.08*t-8.945
0.000000
0.020000
0.040000
0.060000
0.080000
0.100000
0.120000
0.140000
25 35 45 55 65 75 85
Age
Dea
th P
rob
abili
ty
Actual Mortality Table Data
Estimated Curve: e0.08*t-8.945
e0.08*79-8.945 = 7.2%
By fitting curves to different mortality tables over time we could obtain different estimated for the curve parameters a and bFrom this we could look at trends in a and b over time and by extrapolating these trends we could try to forecast future mortality rates using Time Series Analysis Methods – this is outside the scope of this courseWe could also look at fluctuations about the trend by adding random noise to the estimate
The noise factor allows us to simulate fluctuations about the trend and incorporate systematic mortality risk into our modelA more detailed look at a simple Stochastic Mortality Model can be found here: http://www.paulsweeting.com/library/2008sias.pdf
eqxt=eb.t+a+
