Types of Insurance & Life Annuity Contracts

There are three types of contracts to consider: insurance, life annuities, and endowments. More complicated kinds of contracts — which we do not discuss in detail — can be obtained by combining (superposing or subtracting) these in various ways. A further possibility, which we address in Chapter 10, is

to restrict payments to some further contingency (e.g., death-benefits only under specified cause-of-death).

In what follows, we adopt several uniform notations and assumptions.

Let x denote the initial age of the holder of the insurance, life annuity, or endowment contract, and assume for convenience that the contract is initiated on the holder’s birthday. Fix a nonrandom effective (i.e., APR) interest rate i , and retain the notation v = (1 + i)⁻¹, together with the other notations previously discussed for annuities of nonrandom duration.

Next, denote by m the number of payment-periods per year, all times being measured from the date of policy initiation. Thus, for given m, insurance will pay off at the end of the fraction 1/m of a year during which death occurs, and life-annuities pay regularly m times per year until the annuitant dies. The term or duration n of the contract will always be assumed to be an integer multiple of 1/m. Note that policy durations are all measured from policy initiation, and therefore are exactly x smaller than the exact age of the policyholder at termination.

The random exact age at which the policyholder dies is denoted by T , and all of the contracts under discussion have the property that T is the only random variable upon which either the amount or time of payment can depend. We assume further that the payment amount depends on the time T of death only through the attained age Tm measured in multiples of 1/m year. As before, the survival function of T is denoted S(t), and the density either f (t). The probabilities of the various possible occurrences under the policy are therefore calculated using the conditional probability distribution of T given that T ≥ x, which has density f(t)/S(x) at all times t ≥ x.

Define from the random variable T the related discrete random variable T_m = [T m]

m = age at beginning of 1 m

th of year of death

which for integer initial age x is equal to x + k/m whenever x + k/m ≤ T < x + (k + 1)/m. Observe that the probability mass function of this random variable is given by

P (T_m = x + k

= P (T ≥ x + k m

¯

¯

¯ T ≥ x) · P (T < x + k + 1

m

¯

¯

¯ T ≥ x + k

m) (4.1)

= _k/mp_x · 1/mq_x+k/m

As has been mentioned previously, a key issue in understanding the special nature of life insurances and annuities with multiple payment periods is to understand how to calculate or interpolate these probabilities from the prob-abilities jpy (for integers j, y) which can be deduced or estimated from life-tables.

An Insurance contract is an agreement to pay a face amount — perhaps modified by a specified function of the time until death — if the insured, a life aged x, dies at any time during a specified period, the term of the policy, with payment to be made at the end of the 1/m year within which the death occurs. Usually the payment will simply be the face amount F (0), but for example in decreasing term policies the payment will be F (0) · (1 − ^k−1_nm) if death occurs within the kth successive fraction 1/m year of the policy, where n is the term. (The insurance is said to be a whole-life policy if n = ∞, and a term insurance otherwise.) The general form of this contract, for a specified term n ≤ ∞, payment-amount function F (·), and number m of possible payment-periods per year, is to

pay F (T − x) at time T^m− x + _m¹ following policy initiation, if death occurs at T between x and x + n.

The present value of the insurance company’s payment under the contract is evidently

½ F (T − x) v^Tm−x+1/m if x ≤ T < x + n

0 otherwise (4.2)

The simplest and most common case of this contract and formula arise when the face-amount F (0) is the constant amount paid whenever a death within the term occurs. Then the payment is F (0), with present value F (0) v−x+ ([mT ]+1)/m, if x ≤ T < x + n, and both the payment and present value are 0 otherwise. In this case, with F (0) ≡ 1, the net single premium has the standard notation A^(m)1_x:ne. In the further special case where m = 1,

the superscript m is dropped, and the net single premium is denoted A¹_x:ne. Similarly, when the insurance is whole-life (n = ∞), the subscript n and bracket ne are dropped.

A Life Annuity contract is an agreement to pay a scheduled payment to the policyholder at every interval 1/m of a year while the annuitant is alive, up to a maximum number of nm payments. Again the payment amounts are ordinarily constant, but in principle any nonrandom time-dependent schedule of payments F (k/m) can be used, where F (s) is a fixed function and s ranges over multiples of 1/m. In this general setting, the life annuity contract requires the insurer to

pay an amount F (k/m) at each time k/m ≤ T − x, up to a maximum of nm payments.

To avoid ambiguity, we adopt the convention that in the finite-term life annuities, either F (0) = 0 or F (n) = 0. As in the case of annuities certain (i.e., the nonrandom annuities discussed within the theory of interest), we refer to life annuities with first payment at time 0 as (life) annuities-due and to those with first payment at time 1/m (and therefore last payment at time n in the case of a finite term n over which the annuitant survives) as (life) annuities-immediate. The present value of the insurance company’s payment under the life annuity contract is

(Tm−x)m

X

k=0

F (k/m) v^k/m (4.3)

Here the situation is definitely simpler in the case where the payment amounts F (k/m) are level or constant, for then the life-annuity-due payment stream becomes an annuity-due certain (the kind discussed previously under the Theory of Interest) as soon as the random variable T is fixed. Indeed, if we replace F (k/m) by 1/m for k = 0, 1, . . . , nm − 1, and by 0 for larger indices k, then the present value in equation (4.3) is ¨a^(m)min(Tm+1/m, n)e, and its expected present value (= net single premium) is denoted ¨a^(m)x:ne.

In the case of temporary life annuities-immediate, which have payments commencing at time 1/m and continuing at intervals 1/m either until death or for a total of nm payments, the expected-present value notation

is a^(m)x:ne. However, unlike the case of annuities-certain (i.e., nonrandom-duration annuities), one cannot simply multiply the present value of the life annuity-due for fixed T by the discount-factor v^1/m in order to obtain the corresponding present value for the life annuity-immediate with the same term n. The difference arises because the payment streams (for the life annuity-due deferred 1/m year and the life-annuity immediate) end at the same time rather than with the same number of payments when death occurs before time n. The correct conversion-formula is obtained by treating the life annuity-immediate of term n as paying, in all circumstances, a present value of 1/m (equal to the cash payment at policy initiation) less than the life annuity-due with term n + 1/m. Taking expectations leads to the formula

a^(m)x:ne = ¨a^(m)_x:n+1/me − 1/m (4.4)

In both types of life annuities, the superscripts ^(m) are dropped from the net single premium notations when m = 1, and the subscript n is dropped when n = ∞.

The third major type of insurance contract is the Endowment, which pays a contractual face amount F (0) at the end of n policy years if the policyholder initially aged x survives to age x + n. This contract is the simplest, since neither the amount nor the time of payment is uncertain. The pure endowment contract commits the insurer to

pay an amount F (0) at time n if T ≥ x + n

The present value of the pure endowment contract payment is

F (0) vⁿ if T ≥ x + n, 0 otherwise (4.5) The net single premium or expected present value for a pure endowment contract with face amount F (0) = 1 is denoted A_x:ne¹ or nEx and is evidently equal to

A_x:ne¹ = nEx = vⁿnpx (4.6) The other contract frequently referred to in beginning actuarial texts is the Endowment Insurance, which for a life aged x and term n is simply the sum of the pure endowment and the term insurance, both with term n and the same face amount 1. Here the contract calls for the insurer to

pay $1 at time Tm+ _m¹ if T < n, and at time n if T ≥ n

The present value of this contract has the form vⁿ on the event [T ≥ n]

and the form v^Tm−x+1/m on the complementary event [T < n]. Note that T_m+ 1/m ≤ n whenever T < n. Thus, in both cases, the present value is given by

v^{min(Tm−x+1/m, n)} (4.7)

The expected present value of the unit endowment insurance is denoted A^(m)x:ne. Observe (for example in equation (4.10) below) that the notations for the net single premium of the term insurance and of the pure endowment are intended to be mnemonic, respectively denoting the parts of the endowment insurance determined by the expiration of life — and therefore positioning the superscript 1 above the x — and by the expiration of the fixed term, with the superscript 1 in the latter case positioned above the n.

Another example of an insurance contract which does not need separate treatment, because it is built up simply from the contracts already described, is the n-year deferred insurance. This policy pays a constant face amount at the end of the time-interval 1/m of death, but only if death occurs after time n , i.e., after age x + n for a new policyholder aged precisely x. When the face amount is 1, the contractual payout is precisely the difference between the unit whole-life insurance and the n-year unit term insurance, and the formula for the net single premium is

A^(m)_x − A^(m)1x:ne (4.8)

Since this insurance pays a benefit only if the insured survives at least n years, it can alternatively be viewed as an endowment with benefit equal to a whole life insurance to the insured after n years (then aged x + n) if the insured lives that long. With this interpretation, the n-year deferred insurance has net single premium = nEx · A^x+n. This expected present value must therefore be equal to (4.8), providing the identity:

A^(m)_x − A^(m)1x:ne = vⁿ npx · A^x+n (4.9)