4. Life Insurance Payments A life insurance premium must take into account the following factors 1. The amount to be paid upon the death of the insured person and its present value. 2. The distribution of the remaining lifetime of this individual given his/her present age (and possibly other factors). 3. Additional costs (administration, taxes etc.) We only consider the first two factors, i.e. we will calculate the so called net premium. 1 / 38
Mortality Tables The distribution of an individual s remaining lifetime is estimated by the use of mortality tables. These tables are based on a set of individuals who were born in a set period, a so called cohort. The most important information in these tables are the mortality rates at a given age t (normally measured in years). These are given by the probability that an individual dies by age t + 1 given that he/she is alive at age t. 2 / 38
Mortality Tables Let N t be the number of individuals in the cohort who are alive at age t. Then the number of individuals who are alive at age t who die before the age of t + 1 is given by M t, where M t = N t N t+1 Dividing this by the number of individuals still alive at age t, we obtain an estimate of the mortality rate at age t m t = N t N t+1 N t. The survival rate at age t is given by s t = 1 m t. 3 / 38
Mortality Tables Age Cohort Size Mortality Mortality Rate 28319 40 9 377 225 28 319 9377225 = 0.003020 30758 41 9 348 906 30 758 9348906 = 0.003290 33173 42 9 318 148 33 173 9318148 = 0.003560 35933 43 9 284 975 35 933 9284975 = 0.003870 38753 44 9 249 042 38 753 9249042 = 0.004190 41907 45 9 210 289 41 907 9210289 = 0.004550 4 / 38
Mortality Tables An estimate of the probability of surviving from age t to age t + k, σ t,t+k, is given by the proportion of the individuals who survive till age t, who are still alive at age t + k. Thus σ t,t+k = N t+k N t. Hence, the probability of dying between the ages of t and t + k is given by µ t,t+k, where µ t,t+k = 1 σ t,t+k. 5 / 38
Mortality Tables It should be noted that mortality rates are lower now than in the past and thus the mortality rates estimated from such tables tend to overestimate the mortality rate of the present cohort of individuals of a given age. This means that the price of life insurance based on these tables will also tend to be overestimated. Insurance companies often use trends in the mortality rates to estimate the mortality rate of the present cohort (not considered in this course). It should be noted that if a medical test is necessary before an insurance contract is made, then the mortality rate of those who have just insured themselves is lower than those of the same age who insured themselves some time ago. 6 / 38
Calculation of a One-off Premium We consider a policy that will pay a stated nominal amount if the insured person dies within a certain period. The fair price of such a premium is given by the amount of money that must be invested now at the current interest rate, in order to cover the expected cost of the payment upon the death of the insured person. It is assumed that the premium is paid at the beginning of the year and insurance claims are paid at the end of the year. Insurance companies usually assume a low interest rate due to risk aversion. If the interest rate is greater, then they make a profit. 7 / 38
Calculation of a One-off Premium for a One-year Period Suppose that the person being insured is of age t, the amount to be paid on his/her death is K and the interest rate is 100R%. The appropriate premium, P is K P = Km t 1 + R. Note that 1+R is the amount that needs to be invested to cover the claim and m t is the probability that a claim will be made. 8 / 38
Example 4.1 Suppose that a person of age 40 wishes to insure his/her life for the coming year. The payoff to be made in the case of death is $100 000. The interest rate is 4% per annum and the mortality rates are assumed to be those given in the table above. Calculate the appropriate insurance premium. 9 / 38
Example 4.1 10 / 38
Calculation of a One-off Premium for a Longer Period Suppose the person being insured is of age t, then the probability that he/she dies in the i-th year of the policy, p i (i 2) is given by p i = σ t,t+i 1 m t+i 1. Note that this results from the fact that to die in the i-th year of the policy (i.e. between the age of t + i 1 and t + i), this individual must first survive from age until age t + i 1 and then die in the following year. Note that the probability of death in the first year, p 1, is equal to m t. 11 / 38
Calculation of a One-off Premium for a Longer Period Suppose that the person being insured is of age t, the amount to be paid on his/her death is K and the interest rate is 100R%. Also, the period of cover is assumed to be k years. The appropriate premium for such a policy is the sum of k terms, V 1, V 1,..., V k, where V i = Kp i (1 + R) i. Note that V i is the part of the premium that covers the expected costs from the insured person dying during the i-th year he/she is covered. 12 / 38
Example 4.2 Suppose that a person of age 40 wishes to insure his/her life for the coming five years. The payoff to be made in the case of death is $100 000. The interest rate is 4% per annum and the mortality rates are assumed to be those given in the table above. Calculate i) the probability of death in each year of the policy ii) the appropriate insurance premium. 13 / 38
Example 4.2 14 / 38
Example 4.2 15 / 38
Example 4.2 16 / 38
Example 4.2 17 / 38
Example 4.2 18 / 38
Simplification Based on a Uniform Mortality Rate It may be reasonable to assume that the mortality rate over the period of insurance is constant. Suppose that the annual mortality rate (i.e. the probability that somebody dies before age t + 1 given that they survive to age t) is λ. It follows that the probability that an individual dies in the i-th year of the policy is p i = (1 λ) i 1 λ. The corresponding present investment required is K (1+R) i. The first part of this probability is the probability that the individual survives the first i 1 years, the second the probability that the individual then dies in the i-th year. 19 / 38
Simplification Based on a Uniform Mortality Rate In this case, the formula for the price based on a one-off payment is given by P = k i=1 p i K (1 + R) i = k i=1 (1 λ) i 1 λk (1 + R) i Hence P = λk 1 λ k i=1 ( ) 1 λ i 1 + R 20 / 38
Simplification Based on a Uniform Mortality Rate This is a geometric sequence, where the initial term is c = λk 1+R. The ratio is r = 1 λ 1+R It follows that and the number of terms is k. P = λk 1 + R 1 r k 1 r, 21 / 38
Example 4.3 Assuming the annual mortality rate between the ages of 40 and 45 is 0.0036, calculate the appropriate price for the policy considered in Example 4.2. 22 / 38
Example 4.3 23 / 38
Example 4.3 24 / 38
Mixed Life Insurance/Pension Scheme A policy in which a fixed nominal amount K is paid on the death or retirement at age T of an individual can be calculated in a similar way. In this case it is assumed that the mortality rate at age T 1 is 1, since the amount K is always paid at age T if the individual survives to age T 1. For convenience, it will be assumed that for younger individuals the mortality rate is constant. 25 / 38
Mixed Life Insurance/Pension Scheme Suppose there are T years until an individual retires. The probability that the insurance payment is made in the final year is simply the probability that the insured individual survives the first T 1 years of the policy. In this case, the required initial investment is probability of this event is (1 λ) T 1. K (1+R) T and the In all the other cases, the probability of payment in the i-th year is p i = (1 λ) i 1 λ, i = 1, 2,... T 1, and the required investment is (as in the simplified model above). K (1+R) i 26 / 38
Mixed Life Insurance/Pension Scheme The cost of the premium for such a scheme is thus P = where r = 1 λ 1+R. λk 1 + R 1 r T 1 K(1 λ)t 1 + 1 r (1 + R) T, The first part is the cost of life insurance over the period before retirement. The second part is the cost for obtaining the payment upon retirement. 27 / 38
Example 4.4 Suppose an individual of age 40 buys an insurance policy which guarantees a payoff of $50,000 on retirement (at age 65) or death (whichever occurs first). Assuming a constant annual mortality rate of 0.005 and an interest rate of 3%, find the fair price of such a policy. 28 / 38
Example 4.4 29 / 38
Example 4.4 30 / 38
Regular Premiums Here, we assume that insurance covers T years, the mortality rate λ is constant and the interest rate is 100R%. Premiums are paid annually at the beginning of the year. The amount to be paid on death is K. This payment is made at the end of the year. The formula defining the appropriate premium is overly simple due to the assumptions, but the method of deriving this result illustrates the general approach. 31 / 38
Fundamental Equation of Life Insurance The fundamental equation of life insurance says Expected present value of premiums = Expected present value of claims. The expected present value of claims under the assumptions made was derived above T p i K V C = (1 + R) i = λk ( ) 1 r k, 1 + R 1 r i=1 where p i is the probability of death in year i and r = 1 λ 1+R. 32 / 38
Fundamental Equation of Life Insurance The premium P is paid at the beginning of each year that the insured individual remains alive (up to and including year T ). The insured individual is still alive at the beginning of year i with probability (1 λ) i 1 (the probability he/she survives the first i 1 years). The present value of the premium paid at the beginning of year i is P (1+R) i 1. 33 / 38
Fundamental Equation of Life Insurance It follows that the expected present value of premiums is given by V P = T P i=1 ( 1 λ 1 + R ) ( ) i 1 1 r k = P. 1 r From the fundamental equation of life insurance V P = V C P = λk 1 + R 34 / 38
Fundamental Equation of Life Insurance It can be seen that this is exactly the same as the case of insurance for one year with λ = m t. This makes sense, since one can think of such a policy as a sequence of one-year policies. In the real world, the premiums are constant, but the mortality rate is generally increasing. Using the above approach to derive the appropriate premium, initially the premium is larger than the one corresponding to the present mortality rate. This is compensated for in later years, since the premium would then be smaller than that corresponding to the present mortality rate. 35 / 38
Example 4.5 Calculate the appropriate life insurance premium to be paid annually when the annual mortality rate is 0.006, the interest rate is 5% and the amount to be paid when the insured person dies is $300 000. 36 / 38
Example 4.5 37 / 38
Example 4.5 38 / 38