7
Policy values
7.3 Policies with annual cash flows 177 during the term of a policy, not just at inception. We therefore extend the future loss random variable definition, in net and gross versions. Consider a policy which is still in forcetyears after it was issued. The present value of future net loss random variable is denotedLnt and the present value of gross future loss random variable is denotedLgt, where
Lnt =Present value, at timet, of future benefits
−Present value, at timet, of future net premiums and
Lgt =Present value, at timet, of future benefits +Present value, at timet, of future expenses
−Present value, at timet, of future gross premiums.
We drop thenorgsuperscript where it is clear from the context which is meant.
Note that the future loss random variableLt is defined only if the contract is still in forcetyears after issue.
The example below will help establish some ideas. The important features of this example for our present purposes are that premiums are payable annually and the sum insured is payable at the end of the year of death, so that all cash flows are at the start or end of each year.
Example 7.1Consider a 20-year endowment policy purchased by a life aged 50. Level premiums are payable annually throughout the term of the policy and the sum insured, $500 000, is payable at the end of the year of death or at the end of the term, whichever is sooner.
The basis used by the insurance company for all calculations is the Standard Select Survival Model, 5% per year interest and no allowance for expenses.
(a) Show that the annual net premium,P, calculated using the equivalence principle, is $15 114.33.
(b) Calculate E[Lnt]fort=10 andt=11, in both cases just before the premium due at timetis paid.
Solution 7.1(a) You should check that the following values are correct for this survival model at 5% per year interest:
¨
a[50]:20 =12.8456 and A[50]:20 =0.38830.
The equation of value forPis
Pa¨[50]:20 −500 000A[50]:20 =0, (7.1)
giving
P=500 000A[50]:20
¨
a[50]:20 =$15 114.33.
(b) Ln10 is the present value of the future net loss 10 years after the policy was purchased,assuming the policyholder is still alive at that time. The policyholder will then be aged 60 and the select period for the survival model, two years, will have expired eight years ago. The present value at that time of the future benefits is 500 000vmin(K60+1,10)and the present value of the future premiums isPa¨min(K
60+1,10). Hence, the formulae for Ln10andLn11are
Ln10=500 000vmin(K60+1,10)−Pa¨min(K
60+1,10)
and
Ln11=500 000vmin(K61+1,9)−Pa¨min(K
61+1,9) . Taking expectations and using the annuity values
¨
a60:10 =7.9555 and a¨61:9 =7.3282 we have
E[Ln10] =500 000A60:10 −Pa¨60:10 =$190 339 and
E[Ln11] =500 000A61:9 −Pa¨61:9 =$214 757.
2 We are now going to look at Example7.1in a little more detail. At the time when the policy is issued, att =0, the future loss random variable,Ln0, is given by
Ln0=500 000vmin(K[50]+1,20)−Pa¨min(K
[50]+1,20) .
Since the premium is calculated using the equivalence principle, we know that E[Ln0] =0, which is equivalent to equation (7.1). That is, at the time the policy is issued, the expected value of the present value of the loss on the contract is zero, so that, in expectation, the future premiums (from time 0) are exactly sufficient to provide the future benefits.
7.3 Policies with annual cash flows 179 Consider the financial position of the insurer at time 10 with respect to this policy. The policyholder may have died before time 10. If so, the sum insured will have been paid and no more premiums will be received. In this case the insurer no longer has any liability with respect to this policy. Now suppose the policyholder is still alive at time 10. In this case the calculation in part (b) shows that the future loss random variable,Ln10, has a positive expected value ($190 339) so that future premiums (from time 10) arenotexpected to be sufficient to provide the future benefits. For the insurer to be in a financially sound position at time 10, it should hold an amount of at least $190 339 in its assets so that, together with future premiums from time 10, it can expect to provide the future benefits.
Speaking generally, when a policy is issued the future premiums should be expected to be sufficient to pay for the future benefits and expenses. (If not, the premium should be increased!) However, it is usually the case that for a policy which is still in forcetyears after being issued, the future premiums (from time t) are not expected to be sufficient to pay for the future benefits and expenses.
The amount needed to cover this shortfall is called thepolicy value for the policy at timet.
The insurer should be able to build up its assets during the course of the policy because, with a regular level premium and an increasing level of risk, the premium in each of the early years is more than sufficient to pay the expected benefits in that year, given that the life has survived to the start of the year. For example, in the first year the premium of $15 114.33 is greater than the EPV of the benefit the insurer will pay in that year, 500 000vq[50] =$492.04. In fact, for the endowment insurance policy studied in Example7.1, for each year except the last the premium exceeds the EPV of the benefits, that is
P>500 000vq[50]+t fort=0, 1,. . ., 18.
The final year is different because
P=15 114.33<500 000v=476 190.
Note that if the policyholder is alive at the start of the final year, the sum insured will be paid at the end of the year whether or not the policyholder survives the year.
Figure7.1shows the excess of the premium over the EPV of the benefit payable at the end of the year for each year of this policy.
Figure 7.2shows the corresponding values for a 20-year term insurance issued to (50). The sum insured is $500 000, level annual premiums are payable throughout the term and all calculations use the same basis as in Example7.1.
The pattern is similar in that there is a positive surplus in the early years which