Deferred Whole-Life Recursion During the Waiting Period

Why the Waiting Period Changes the Recursion

A whole-life policy normally pays a death benefit whenever death occurs while the policy is in force. A deferred whole-life policy has a waiting period: if the insured dies during that period, no benefit is paid. This absence of a payment is the key to its recursion. During the waiting period, the value advances to the next year only if the insured survives, so the recursion contains a survival probability but no death-benefit term.

The focus here is the discrete annual model. Assume a unit benefit is payable at the end of the year of death, but only if death occurs after the deferral period. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. Let $q_x$ be the probability that a life aged $x$ dies within one year, and let $p_x=1-q_x$. Assume the mortality probabilities and interest rate used in a calculation are appropriate to the ages and years being valued.

Definition: Write $D_{x,n}={}_{n|}A_x$ for the expected present value of a unit whole-life death benefit deferred $n$ years for a life aged $x$. No benefit is paid if death occurs during the first $n$ years. If the life survives those $n$ years, the benefit is payable at the end of the year of death. The ordinary whole-life value is the boundary case $D_{x,0}=A_x$.

This definition describes the value at the start of the deferral period, conditional on being alive at age $x$. It does not include a refund or any other payment for death during the waiting period. The recursion below therefore applies to this benefit design, not to a contract that returns premiums or pays a partial benefit on an early death.

The One-Year Recursion During Deferral

Suppose at least one full year remains before the benefit becomes payable. During the first year, either the insured dies or survives. If death occurs, it is within the waiting period and the contract pays zero. If the insured survives, the value at the end of the year is the value of the same deferred whole-life benefit at age $x+1$, with one fewer year remaining to defer.

Theorem: For an integer $n\geq 1$, under the annual discrete model, $$ D_{x,n}=v p_x D_{x+1,n-1}. $$ In particular, when one year remains, $$ D_{x,1}=v p_x A_{x+1}. $$

The equation can be justified directly from the possible outcomes during the first year. A death in that year has probability $q_x$ and produces a payment of zero. Survival has probability $p_x$; conditional on survival, the future value is $D_{x+1,n-1}$, discounted one year to age $x$. Thus the expected value is $v(0\cdot q_x+p_xD_{x+1,n-1})$, which is exactly $v p_xD_{x+1,n-1}$. This is why the recursion does not include the usual whole-life death-benefit term $v q_x$.

Check the deferral remaining.
If $n\geq 1$, a death in the coming year receives no benefit under the stated contract.

Move to the next age only on survival.
Use $p_x$ and reduce the remaining deferral from $n$ to $n-1$.

Discount the future value one year.
Multiply by $v$, then use the appropriate value at age $x+1$.

Use the boundary at eligibility.
When no deferral remains, the value is ordinary whole-life insurance: $D_{x,0}=A_x$.

Applying the one-year relation repeatedly gives the value after any integer deferral period:

D_{x,n}=v^n\,{}_np_x\,A_{x+n}, \qquad {}_np_x=\prod_{j=0}^{n-1}p_{x+j}.

Here ${}_np_x$ is the probability of surviving all $n$ years, and $A_{x+n}$ is the whole-life value when the insured reaches age $x+n$. The factor $v^n$ discounts that value back $n$ years. This expression is also the pure endowment factor $v^n{}_np_x$ multiplied by the whole-life value at the end of the deferral. It relies on annual discrete benefit timing and the stated zero-benefit condition during deferral; it is not the formula for a contract that pays during the waiting period.

Worked Calculations

Worked Example: Constant Mortality During a Three-Year Deferral

A hypothetical policy pays 1 at the end of the year of death, but only after a three-year waiting period. The effective annual interest rate is $i=0.05$, and the annual mortality probability is $q=0.10$ at every future age. Calculate the ordinary whole-life value and the deferred values for one, two, and three years. Use the annual discrete model.

Method and boundary value. First find $A_x$. With constant mortality, $p=1-q=0.90$, and the whole-life recursion is $A_x=v(q+pA_{x+1})$. Since mortality and the whole-life value are the same at every age, write $A_x=A_{x+1}=A$. Also $v=1/1.05=20/21$. Then

A=\frac{20}{21}\left(0.10+0.90A\right), \qquad 21A=2+18A, \qquad A=\frac{2}{3}.

One year deferred. The one-year survival-discount factor is $vp=(20/21)(0.90)=6/7$. Therefore $D_{x,1}=(6/7)(2/3)=4/7=0.5714286$.

Two years deferred. Apply the same factor a second time: $D_{x,2}=(6/7)(4/7)=24/49=0.4897959$.

Three years deferred. Apply it once more: $D_{x,3}=(6/7)(24/49)=144/343=0.4198251$. Rounded to four decimal places, the three deferred values are $0.5714$, $0.4898$, and $0.4198$.

Check. Each added year requires another year of survival and discounting, so the value decreases by a factor of $6/7$ at each step. The calculation also avoids treating a death during the waiting period as a claim.

Worked Example: Changing Mortality by Age

A hypothetical unit-benefit contract has a two-year deferral. The effective annual interest rate is $i=0.05$, $q_x=0.10$, $q_{x+1}=0.20$, and the ordinary whole-life value at age $x+2$ is $A_{x+2}=0.60$. Calculate $D_{x,2}$, showing the intermediate value after one year. The supplied value $A_{x+2}$ summarizes mortality beyond age $x+2$.

Method. The survival probabilities are $p_x=0.90$ and $p_{x+1}=0.80$; the discount factor is $v=1/1.05=20/21$. At age $x+1$, one year remains to defer, so

D_{x+1,1}=v p_{x+1}A_{x+2} =\frac{20}{21}(0.80)(0.60) =\frac{16}{35} =0.4571429.

Now apply the recursion at age $x$:

D_{x,2}=v p_xD_{x+1,1} =\frac{20}{21}(0.90)\left(\frac{16}{35}\right) =\frac{96}{245} =0.3918367.

Rounded to four decimal places, $D_{x,2}=0.3918$. As a check, the product form gives $v^2p_xp_{x+1}A_{x+2}$, which uses both annual survival probabilities, two years of discounting, and the whole-life value at the end of the deferral. It gives the same result.

Common mistake avoided. Using $q_x=0.10$ as a first-year benefit contribution would be wrong: death in that first year occurs before eligibility, so its benefit is zero.

Worked Example: Inferring Mortality From a Deferred Value

At age $x$, a hypothetical one-year deferred whole-life policy has value $D_{x,1}=0.45$. The effective annual interest rate is $i=0.04$, and the ordinary whole-life value at age $x+1$ is $A_{x+1}=0.60$. Find $p_x$ and $q_x$, then verify the recursion.

Method. Use $D_{x,1}=vp_xA_{x+1}$ and solve for $p_x$. Since $v=1/1.04$,

p_x=\frac{D_{x,1}}{vA_{x+1}} =\frac{0.45(1.04)}{0.60} =\frac{0.468}{0.60} =0.78.

Because $p_x+q_x=1$, $q_x=1-0.78=0.22$. To check, substitute the survival probability back into the recursion:

v p_xA_{x+1} =\frac{1}{1.04}(0.78)(0.60) =\frac{0.468}{1.04} =0.45.

The result is consistent with the stated deferred value. The method avoids solving an ordinary whole-life recursion at age $x$, which is not needed: the deferred value depends on survival at age $x$ and the supplied whole-life value at age $x+1$.

Boundary Cases and Exam Traps

The most important boundary is $n=1$. At the start of that year, benefits are still unavailable. If the insured survives to age $x+1$, the waiting period is over and the contract has ordinary whole-life value $A_{x+1}$. Thus $D_{x,1}=vp_xA_{x+1}$, not $v(q_x+p_xA_{x+1})$. The latter expression includes a death payment in the year when none is payable.

For $n=0$, however, there is no waiting period. The value is $A_x$, and the ordinary whole-life recursion can include a death benefit in the coming year. Do not apply the waiting-period recursion with $n=0$ as though death in the first year still received zero; that would contradict the definition of ordinary whole-life insurance.

Another common error is to use the same age-specific survival probability in every year without a stated constant-mortality assumption. When mortality varies by age, use $p_x$, then $p_{x+1}$, and so on. Similarly, a deferral of $n$ years requires $n$ survival factors and $n$ years of discounting, not $n-1$.

Key takeaway: While a deferred whole-life benefit is still unavailable, its one-year value is $v p_x$ times the value at the next age with one fewer year remaining. Death during that year contributes zero; ordinary whole-life value applies only once the deferral has ended.

Check Your Understanding

For each problem, use the annual discrete model and the stated unit-benefit design. Round numerical values to four decimal places unless the answer is exact.

A policy has one year of deferral, $i=0.04$, $q_x=0.10$, and $A_{x+1}=0.50$. Find $D_{x,1}$. (Answer: $0.4327$)
A two-year deferred policy has $i=0.05$, $p_x=0.95$, $p_{x+1}=0.90$, and $A_{x+2}=0.40$. Find $D_{x,2}$. (Answer: $76/245=0.3102$)
A one-year deferred value is $D_{x,1}=0.48$, the effective annual interest rate is $i=0.03$, and $A_{x+1}=0.80$. Find $p_x$ and $q_x$. (Answer: $p_x=0.618,\ q_x=0.382$)
Assume $q=0.08$ at every future age and $i=0.04$. Find the value of a three-year deferred unit whole-life benefit. (Answer: $(23/26)^3(2/3)=12167/26364=0.4615$)
A two-year deferred policy has $i=0.05$, $p_x=0.80$, $p_{x+1}=0.75$, and $A_{x+2}=0.56$. Calculate the value at age $x+1$ with one year left to defer, then calculate the value at age $x$. (Answer: $D_{x+1,1}=0.4000,\ D_{x,2}=0.3048$)

What You'll Learn