Deferred Whole-Life Insurance as a Pure Endowment Product

Why Deferred Whole-Life Insurance Has a Pure-Endowment Factor

A deferred whole-life policy provides insurance only after a specified waiting period. If the insured dies during that period, the policy pays no death benefit; if the insured survives the period, whole-life cover applies thereafter. Valuing the contract therefore requires two pieces: the value of reaching the start of cover, and the value of the insurance once cover has begun.

The first piece is the same survival-and-discount factor used for a pure endowment. The second is a whole-life insurance value at the age reached after the deferment. Their product gives the value at issue. This relation is useful both for calculating deferred insurance and for checking which mortality and interest factors belong in a calculation.

Definition: Consider a discrete unit-benefit whole-life insurance issued at age $x$, deferred for $n$ years. It pays 1 at the end of the year of death if the insured survives $n$ years and then dies. It pays nothing if death occurs during the $n$-year deferment. Its actuarial present value is denoted ${}_{n|}A_x$. The pure-endowment factor is ${}_nE_x=v^n{}_np_x$, where $v=1/(1+i)$ and ${}_np_x$ is the probability of surviving from age $x$ to age $x+n$.

The whole-life insurance value $A_{x+n}$ is evaluated at age $x+n$, conditional on being alive at that age. It represents the expected present value, at that time, of a unit benefit paid at the end of the future year of death. Thus, ${}_nE_x$ discounts for the $n$ years before cover begins and weights the value by the probability of surviving those years.

The Deferred Whole-Life Identity

For the discrete benefit timing just defined, the central identity is

Theorem: For an $n$-year deferred, unit-benefit, discrete whole-life insurance, $$ {}_{n|}A_x={}_nE_x A_{x+n}=v^n{}_np_x A_{x+n}. $$

This formula assumes the benefit is paid at the end of the year of death, no benefit is paid during the deferment, and the future mortality and interest assumptions used for $A_{x+n}$ are those applying after the insured reaches age $x+n$. The identity also assumes the usual life-contingency model in which, conditional on survival to age $x+n$, the future lifetime is valued using the mortality model from that age onward.

Identify the event that can generate a benefit.
A benefit is possible only if the insured survives $n$ years and then dies at some later time. Death during the deferment contributes zero.

Condition on survival to time $n$.
Given survival to age $x+n$, the insurance value measured at that time is $A_{x+n}$.

Value the condition of reaching time $n$.
The probability of survival is ${}_np_x$, and the discount factor from time $n$ to issue is $v^n$. Their product is ${}_nE_x$.

To justify the multiplication, let $Z$ be the present value at issue of the deferred insurance benefit. If the insured does not survive $n$ years, then $Z=0$. If the insured does survive, the benefit stream after time $n$, valued at time $n$, has conditional expected value $A_{x+n}$; its value at issue is $v^n$ times that amount. Taking expectations gives

E[Z]=P(\text{survive }n\text{ years})\,v^n A_{x+n} ={}_np_x\,v^n A_{x+n}.

This is a conditioning argument, not an assumption that survival and the future insurance value are independent. The future insurance value is conditional on being alive at time $n$, while the factor ${}_np_x$ accounts for the chance of reaching that point at all.

Calculating Deferred Values

A reliable calculation separates the three factors before multiplying: the discount factor $v^n$, the survival probability ${}_np_x$, and the future whole-life value $A_{x+n}$. In particular, ${}_np_x$ is a probability, not a discount factor, and it must not be omitted merely because a future whole-life value is supplied.

Worked Example: Recovering the Future Whole-Life Value

A hypothetical policy provides a unit benefit under a two-year deferred whole-life contract. Its actuarial present value is 0.18. The effective annual interest rate is 25%, and ${}_2p_x=0.80$. Find $A_{x+2}$.

Method. Use the deferred whole-life identity and solve for the whole-life factor at the age reached after the deferment. The given survival probability covers the full two years.

First, calculate $v$ and the two-year discount factor:

v=\frac{1}{1.25}=0.8, \qquad v^2=0.8^2=0.64.

The pure-endowment factor is

{}_2E_x=v^2{}_2p_x =0.64(0.80)=0.512.

Now divide the deferred value by this factor:

A_{x+2}=\frac{{}_{2|}A_x}{{}_2E_x} =\frac{0.18}{0.512} =0.3515625.

Check. Multiplying back gives $0.512(0.3515625)=0.18$, as required. The answer is $0.3515625$. A common mistake is to divide by the survival probability alone: $0.18/0.80=0.225$. That calculation omits the two-year discount factor, so it does not invert the pure-endowment factor.

Worked Example: Computing a Deferred Value from Given Factors

A hypothetical insurer values a unit-benefit whole-life policy deferred for two years. The effective annual interest rate is 10%, the one-year survival probabilities are ${}_1p_x=0.90$ and ${}_1p_{x+1}=0.80$, and the whole-life value at age $x+2$ is $A_{x+2}=0.35$. Find the value at age $x$.

Method. Multiply the pure-endowment factor for two years by the whole-life value at the age when cover starts. The two one-year survival probabilities multiply because survival for two years requires survival through each year.

The two-year survival probability is

{}_2p_x={}_1p_x\,{}_1p_{x+1} =0.90(0.80)=0.72.

At 10% interest, $v=1/1.10=10/11$, so

v^2=\left(\frac{10}{11}\right)^2 =\frac{100}{121} \approx 0.82644628099.

Therefore, the pure-endowment factor and deferred whole-life value are

{}_2E_x=v^2{}_2p_x =\frac{100}{121}(0.72) \approx 0.59504132231,

{}_{2|}A_x={}_2E_x A_{x+2} \approx 0.59504132231(0.35) \approx 0.20826446281.

Check. The result is less than $A_{x+2}=0.35$, as expected: the value at issue is reduced both by the chance of not surviving to the start of cover and by discounting for two years. Rounded only at the final step, the deferred value is 0.2083.

Finding the Future Whole-Life Factor from Mortality

Sometimes $A_{x+n}$ is not supplied and must be calculated from a specified mortality model. For example, suppose that at each future age the one-year death probability is the same $q$, the one-year survival probability is $p=1-q$, and the effective annual interest rate is $i$. For a discrete unit benefit paid at the end of the year of death, the probability of death in future policy year $k$ is $p^{k-1}q$. Its present-value contribution is $v^kp^{k-1}q$. Summing over all future policy years gives

A_y=\sum_{k=1}^{\infty}v^kp^{k-1}q =vq\sum_{k=1}^{\infty}(vp)^{k-1} =\frac{vq}{1-vp},

provided $vp<1$, so that the geometric series converges. This expression is specific to constant one-year mortality probabilities after age $y$; it should not be used when the future mortality probabilities vary by age. In that case, calculate the future whole-life value from the applicable age-specific mortality model.

Worked Example: Constant Future Mortality After the Deferment

A hypothetical policy has a two-year deferment. The effective annual interest rate is 25%; the survival probabilities during the deferment are ${}_1p_x=0.90$ and ${}_1p_{x+1}=0.80$. After age $x+2$, assume a constant annual death probability $q=0.25$, so the annual survival probability is $p=0.75$. Find the deferred whole-life value.

Method. First derive $A_{x+2}$ from the constant-mortality model, then multiply it by the two-year pure-endowment factor. The constant mortality assumption applies only after the deferment; the two given deferment-year survival probabilities are used as stated.

The discount factor is $v=1/1.25=0.8$. For the whole-life insurance beginning at age $x+2$,

A_{x+2} =\frac{vq}{1-vp} =\frac{(0.8)(0.25)}{1-(0.8)(0.75)} =\frac{0.20}{0.40} =0.50.

The two-year survival probability is

{}_2p_x=(0.90)(0.80)=0.72.

The pure-endowment factor is

{}_2E_x=v^2{}_2p_x =(0.8)^2(0.72) =(0.64)(0.72) =0.4608.

Thus,

{}_{2|}A_x={}_2E_x A_{x+2} =(0.4608)(0.50)=0.2304.

Check. The deferred value is below the future whole-life value of 0.50 because the contract pays nothing if death occurs in the first two years and because the later insurance value is discounted to issue. The calculation uses the age-specific deferment survival probabilities before applying the constant mortality model after age $x+2$.

Benefit Timing and Common Errors

The identity depends on the contract’s benefit timing and conditions. Here the benefit is paid at the end of the year of death, and the insurance begins only after the $n$-year deferment. A contract that pays a benefit at the end of the deferment, pays during the deferment, or uses a different benefit amount is not described by this definition without further adjustment.

Omitting survival. Discounting $A_{x+n}$ by $v^n$ alone ignores the possibility of death during the deferment. Use ${}_np_x$ as well.
Omitting discounting. Multiplying ${}_np_x$ by $A_{x+n}$ values the future insurance at time $n$, not at issue. Include $v^n$.
Using the wrong age. The future whole-life factor is $A_{x+n}$, conditional on reaching age $x+n$, not $A_x$.
Applying the wrong mortality assumption. Survival through the deferment and mortality after the deferment may be specified separately. Use each assumption for its stated time period.
Changing the benefit timing silently. The formula uses a discrete benefit at the end of the year of death. A continuous insurance value is not interchangeable with $A_{x+n}$.

Key takeaway: For discrete unit-benefit deferred whole-life insurance, first value the chance of reaching the start of cover as a pure endowment, ${}_nE_x=v^n{}_np_x$. Then multiply by the conditional whole-life value at that age: ${}_{n|}A_x={}_nE_xA_{x+n}$.

Check Your Understanding

For each problem, use the discrete end-of-year-of-death benefit convention. Round probabilities or actuarial values to four decimal places only when needed.

A hypothetical two-year deferred whole-life policy has $i=10\%$, ${}_1p_x=0.90$, ${}_1p_{x+1}=0.80$, and $A_{x+2}=0.30$. Find ${}_{2|}A_x$. (Answer: 0.1785)
At $i=0$, a three-year deferred whole-life value is 0.18 and ${}_3p_x=0.75$. Find $A_{x+3}$. (Answer: 0.24)
A policy is deferred for two years. Let $i=25\%$, ${}_1p_x=0.80$, ${}_1p_{x+1}=0.90$, and suppose the annual death probability after age $x+2$ is constantly $q=0.20$. Find ${}_{2|}A_x$. (Answer: 0.2048)
A hypothetical contract has a three-year deferment, $i=0$, survival probabilities of 0.90 in each of the three deferment years, and $A_{x+3}=0.40$. Find the deferred value. (Answer: 0.2916)
A two-year deferred whole-life value is 0.24. The effective annual interest rate is zero and the one-year survival probabilities during the deferment are 0.80 and 0.75. Find $A_{x+2}$, and state which factor makes the deferred value smaller than the future whole-life value. (Answer: $A_{x+2}=0.40$; the 0.60 probability of surviving the deferment is the reducing factor.)

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