How can actuaries prepare themselves best for future products and risk structures. The products examined in Chapter 12 contain financial guarantees embedded in life-contingent benefits.
Summary
Background
The calculations in Halley's article bear a remarkable similarity to some of the work still used by actuaries in the pensions and life insurance fields. However, many changes have taken place since the first long-term policy at the end of the eighteenth century.
Life insurance and annuity contracts .1 Introduction
Whole life insurance pays out a lump sum upon the policyholder's death, when it happens. When an annuity depends on the beneficiary's survival, it is called an 'annuity'.
Other insurance contracts
Last Survivor Annuity A last survivor annuity is similar to the joint lifetime annuity, except that payments continue as long as at least one of the lives survives, and stops upon the couple's second death. Upon death of the insured life, the annuitant, if the annuitant is still alive, receives an annuity for the rest of his or her life.
Pension benefits
The formula can be interpreted as a pension benefit of, for example, 2% of the final average salary for each year of service. The fixed benefit is financed by contributions paid by the employer and (usually) the employee over the working life of the employee.
Mutual and proprietary insurers
Typical problems
The actuary may consider the possibility that the policyholder may decide to terminate the contract early. Every year or so, the actuary must determine how much money the insurer or pension plan should hold to ensure that future liabilities will be covered with a reasonably high probability.
Notes and further reading
Exercises
Summary
The future lifetime random variable
This individual could have died before reaching old age – the probability of this was Pr[T0 We can also relate the power of the mortality function at any age x+t,t > 0 to the lifetime distribution of Tx. We can also use equation (2.9) to develop a formula for Sx(t) in terms of the power of the mortality function. This is called the delayed mortality probability because it is the probability that death will occur in a given interval after a delayed period. We can drop the subscript if its value is 1, so that px represents the probability that(x) survives to at least agex+1. For more realistic models, such as Gompertz's, we can easily calculate the values using Excel or other suitable software. Although in principle we must estimate an infinite sum, the survival probability at some age will be sufficiently small that we can treat it as an effective limiting age. The analysis of mortality data is a large topic and is beyond the scope of this book. For more general distributions, the quantity af0(x)/S0(x), which actuaries call the force of mortality at age, is known as the hazard rate in survival analysis and the failure rate in reliability theory. Use the table to determine the age last birthday at which a life that is currently 70 years old is most likely to die. Hint: You will need to use numerical integration for parts (b) and (c). d) Ise◦x is always a non-increasing function of x. Summary For anyx≥x0, we can interpret lx+t as the expected number of survivors to agedx+to oflxondependent individuals agedx. Then the number of survivors to age x+ is a binomial random variable, say Lt, with parameterslxandtpx. The left side does not depend on us, which means that the density function is a constant for 0≤s<1, which also follows from the assumption of the uniform distribution for Rx. A second fractional age assumption is that the mortality force is constant across integer ages. A final point about Table 3.2 is that we compared the three national life tables using the value of the probability of death in one year, qx, instead of the mortality force, µx. This is because µx values are not published for any age for the US Life Tables. Given that people who buy term insurance are likely to be among the better paid in the population, we have an explanation for much of the difference between the values in Table 3.3. The CMI (Table A2) shows mortality force values based on data from men in the UK who purchased whole life or endowment policies. These are similar to those shown in Table 3.3 for term insurance policy holders and therefore much lower than the values for the whole population. This explains the differences between the mortality force values at age 56 in Table 3.4. The selected period for the survival model for this group is one year, since if they survive for a year after being 'selected', their future mortality depends only on their current age. For CMI (table A14), which relates to policyholders, it is five years, as mentioned above; for CMI (Table A2), which concerns whole life and endowment policyholders, the chosen period is two years. This is an important feature of many survival models based on data from, and therefore applicable to, life insurance policyholders. In the next section, we introduce notation and develop some formulas for selected survival models. This shows that within the ultimate part of the model we can interpret as the expected number of survivors to age out of the lives currently at age x( Use this table to calculate. a) the probability that a person who is currently 75 years old and has just been selected will survive to age 85. For those who have lived in country A for at least five years, the death rate at each age is 50% higher than that from US Life Tables, 2002, Women, at the same age. Summary For developing present value functions, it is generally easier, mathematically, to work in continuous time. We work in continuous time primarily because mathematical development is more transparent, more complete, and more flexible. In the case of an annuity, a continuous benefit of say $1 per year will be paid in infinite units of $dtin every interval(t,t+dt). In financial mathematics and corporate financing contexts, and especially if the interest rate is assumed to be risk-free, the common notation for the continuously compounded interest rate is r. Likewise, the second moment (approximately zero) of the present value of the death benefit is the same. In the situation where the death benefit is payable at the end of the year of death, the cash value of the benefit is the same. If the policy term ends after a fixed term of one year, the EPV is for the death benefit. Example 4.8 Consider a term insurance policy issued to (x) under which the death benefit is paid at the end of the year of death. This in turn means that the EPV of an insurance benefit that is payable immediately upon death, for example A¯x, can be written as an integral where the integrand can be evaluated directly, as follows. Functions like A¯x can still be evaluated numerically, but since the integrand must be evaluated numerically, the procedure can be a bit more complicated. Under a term policy issued to a life agedx, let Y denote the present value of a unit sum assured payable at the time of death within that year. Let Z2 denote the present value of a whole life insurance benefit, issued for the same life. Summary Introduction Review of annuities-certain Recall that, for integer n, Using equation (5.1), the random variable of the present value for the series of annuity payments, for example Y, can be written as. The difference between the formulations for the xin equations (5.7) and (5.8) is that in equation (5.7) the addition is taken over the possible payment dates, and in equation (5.8) the addition is taken over the possible death years. Using this formulation for the random variable Y, we can also directly derive the variance of the current annuity present value from the variance of the current insurance benefit. The second approach is to use the sum (here an integral) of the product of the amount paid in each infinitesimal interval(t,t+dt), the discount factor for the interval, and the probability that the payment is made. For annuities payable 1/monthly in arrears, we can use a comparison with the 1/monthly annuity due. The payments, associated probabilities and discount factors for the 1/monthly term annuity due are shown in the timeline diagram in Figure 5.7. Summing the EPVs of the individual payments for the deferred lifetime annuity gives 5.26). Suppose we have a table of the values of the entire annual income obligation, saya¨x, together with the life table function lx, and we need the term annuity value ¨x:n. The expected present value of the guaranteed annuity liability for one year is noted. 5.32) Figure 5.9 shows the timeline for the full annuity of the guaranteed unit for one year. We can derive similar results for guaranteed benefits payable 1/month; for example, a guaranteed monthly pension for years has EPV. The notation for the EPV of this annuity is (I¯a)¯ x if it is a whole life annuity, and (I¯a)¯ x:n if it is a long-term annuity. Derive an expression for the EPV of this benefit and simplify as much as possible. Since we ignore second and higher order derivatives, the right-hand side of (5.40) becomes. The right-hand side of equation (5.43) gives the first three terms of Woolhouse's formula, and this is the basis of our actuarial approaches. In this case, approximations based on Woolhouse's formula are superior, provided the three-term version is used. Using Woolhouse's formula only requires the 100-row integer age table, and the accuracy up to age 100 is excellent, using the exact or approximate values for µx. We also note that the approximation based on the UDD assumption is good at younger ages, with some deterioration at older ages. The approximations we have developed also apply to selected survival models, so that e.g. Notes and further reading Calculate the following, assuming an interest rate of 6% per annum:. e) the standard deviation of the present value of a four-year annuity obligation, with an annual payment of $1,000, payable for a selected age of 41 and. Lives are supposed to be independent. a) the current expected value of the total exit from the annuities,. Summary Preliminaries The table of selected and final lifetimes, at integer ages, for this model is shown in Table 3.7 and values of Ax at an effective interest rate of 5%. Since both terms of the random variable depend on the future lifetime of the same life [60], they are clearly dependent. Given an appropriate survival model along with assumptions about future interest rates and, for gross premiums, costs, the insurer can then determine an allocation for the present value of the future loss. Annuity benefit X per annum is paid monthly in advance by agex+for the remainder of (x)'s life. a) Write the net future loss random variable in terms of lifetime random variables for [x]. Usually, calculating the premium does not require the identification of the future loss random variable. The equivalence principle applied to gross premiums and benefits states that the EPV of the gross future loss random variable must equal zero. The insurer incurs an initial charge of $2000 plus 50% of the first premium and a renewal charge of 2.5% of each subsequent premium. Since the death benefit t)if the policyholder dies in the policy year, the EPV of the death benefit is If the policyholder survives to age 55, there is one extra bonus payment, and the present value of the future loss is Our goal is to calculate P, but P does not appear explicitly in either of the last two equations. The initial costs, incurred when the policy is issued, amount to 15% of the total premiums for the first year. Expect an initial fee of $2,000 plus 40% of the first premium, and a renewal fee of 2% of the second and. Since the initial cost is 1000 plus 50% of the first premium, i.e. 121P, we can write the EPV of the cost as. In addition to the premium principles described in this chapter, there is an additional important method of calculating premiums. We have omitted it in this work because we find that it has no particular advantage in practice. Assume that the death benefit is paid at the end of the year of death. a) Write down the expression for the gross future loss random variable. The death benefit is paid at the end of the year of death. a) Let L0 denote the gross future loss random variable for this policy. where P is the gross annual premium, Z1=. b). Summary Assumptions Level premiums are payable annually throughout the policy term and the sum assured is payable at the end of the year of death or at the end of the term, whichever is earlier. Figure 7.1 shows the excess of the premium over the EPV of the benefit payable at the end of the year for each year of this policy.The force of mortality
Actuarial notation
Curtate future lifetime .1 K x and e x
Notes and further reading
Exercises
Life tables
Fractional age assumptions
National life tables
Survival models for life insurance policyholders
Life insurance underwriting
Select and ultimate survival models
Notation and formulae for select survival models
Select life tables
Exercises
Introduction
Assumptions
Valuation of insurance benefits .1 Whole life insurance: the continuous case, A¯ x
Variable insurance benefits
Notes and further reading
Exercises
Annual life annuities
Annuities payable continuously .1 Whole life continuous annuity
Annuities payable m times per year .1 Introduction
Comparison of annuities by payment frequency
Guaranteed annuities
Increasing annuities
Evaluating annuity functions
Numerical illustrations
Functions for select lives
Exercises
Assumptions
The present value of future loss random variable
The equivalence principle .1 Net premiums
Gross premium calculation
Profit
The portfolio percentile premium principle
Extra risks
Notes and further reading
Exercises
Policies with annual cash flows .1 The future loss random variable
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