Deferred Whole-Life Insurance Factorization

Why Factor Deferred Whole-Life Insurance?

A deferred whole-life policy does not pay if death occurs during its deferment period. If the insured survives that period, the policy then provides whole-life coverage. Valuing it directly means accounting for both parts of this timing: the chance of reaching the end of deferment and the present value of the benefits paid afterward.

The factorization makes those parts explicit. It separates the discounting and survival needed to reach the future valuation age from the whole-life insurance value at that age. This is useful both for calculating a deferred benefit and for checking that the probability of payment has been handled correctly.

Definition: A discrete $m$-year deferred whole-life insurance on a life aged $x$, with unit benefit, pays 1 at the end of the year of death if the insured survives $m$ years and then dies at some later time. It pays nothing if death occurs within the first $m$ years. Write its actuarial present value as ${}_{m|}A_x$. The whole-life insurance value $A_x$ is for unit benefit payable at the end of the year of death.

Assume $m$ is a nonnegative integer, benefits are paid at the end of the year of death, and the effective annual interest rate is $i$, with $v=1/(1+i)$. The survival probability for $m$ years is ${}_mp_x$. Mortality after the deferment period is valued using the whole-life insurance factor $A_{x+m}$ for a life then aged $x+m$.

The factorization is

{}_{m|}A_x = v^m\,{}_mp_x\,A_{x+m}.

This is for unit benefit. If the benefit is a fixed amount $B$, its actuarial present value is $B\,{}_{m|}A_x$. The formula assumes the same interest basis is used for the deferral and the insurance value after deferral. It is a discrete insurance formula; it does not describe a policy paying continuously at the moment of death.

Why the Factorization Holds

The key is to condition on survival to time $m$. If the life does not survive that long, the deferred policy pays nothing. If the life does survive, the remaining benefit pattern is exactly that of whole-life insurance at age $x+m$, viewed at time $m$. Its value at time 0 must therefore be multiplied by both the probability of reaching time $m$ and the discount factor for $m$ years.

The sum over possible years of death makes this argument precise. Let $k$ count years lived after the deferment period. A payment in year $m+k+1$ requires survival to year $m+k$, followed by death in the next year. For $k=0,1,2,\ldots$, its probability is ${}_mp_x\,{}_kp_{x+m}\,q_{x+m+k}$, where $q_{x+m+k}$ is the one-year death probability at that age. Thus

{}_{m|}A_x =\sum_{k=0}^{\infty}v^{m+k+1}{}_mp_x\,{}_kp_{x+m}\,q_{x+m+k} =v^m{}_mp_x\sum_{k=0}^{\infty}v^{k+1}{}_kp_{x+m}\,q_{x+m+k} =v^m{}_mp_x A_{x+m}.

The first equality uses the probability of survival through the deferment period and then survival for $k$ more years before death. The second factors out terms common to every possible post-deferment year of death. The remaining sum is the definition of $A_{x+m}$. This also explains why the conditional whole-life value is evaluated at age $x+m$, not at age $x$.

Worked Calculations

Worked Example: Constant Mortality After Deferral

A hypothetical policy provides a unit benefit payable at the end of the year of death, but only if death occurs after a three-year deferment. The effective annual interest rate is $i=20\%$, and the one-year death probability is $q=0.20$ at every age and in every future year. Calculate the deferred whole-life insurance value and verify it by summing deferred death benefits directly.

Method. First calculate the whole-life factor at the age reached after deferment, then apply the factorization. Under the stated constant mortality model, the one-year survival probability is $p=1-q=0.80$. The discount factor is $v=1/1.20=5/6$.

The whole-life factor is the geometric sum

A_{x+3}=\sum_{k=0}^{\infty}v^{k+1}p^kq =\frac{vq}{1-vp} =\frac{(5/6)(0.20)}{1-(5/6)(0.80)} =\frac{1/6}{1/3} =0.50.

The probability of surviving the three-year deferment is ${}_3p_x=0.80^3=0.512$, and the discount factor to the end of the deferment is $v^3=(5/6)^3=125/216$. Therefore

{}_{3|}A_x =\frac{125}{216}(0.512)(0.50) =\frac{125}{216}\frac{64}{125}\frac12 =\frac{4}{27} \approx 0.1481.

For a direct check, the probability of death in year 4 is $0.80^3(0.20)=0.1024$. The discounted value of that payment is $(5/6)^4(0.1024)=4/81$. Each later year's discounted death-payment term is $vp=(5/6)(0.80)=2/3$ times the previous term. The sum is therefore $(4/81)/(1-2/3)=4/27$, agreeing with the factorized value. The common mistake avoided here is valuing the post-deferment insurance at age $x$ instead of age $x+3$.

Worked Example: Age-Specific Mortality and a Direct Check

A hypothetical unit-benefit policy has a two-year deferment. The effective annual interest rate is $i=25\%$. The one-year death probabilities at ages $x$ and $x+1$ are $0.10$ and $0.20$, respectively. At age $x+2$, the death probability is $0.20$; at age $x+3$, it is $0.25$; and at age $x+4$, death is certain within one year. Find the deferred whole-life insurance value.

Method. Calculate survival through the two-year deferment, calculate $A_{x+2}$ from the post-deferment mortality probabilities, and then apply the factorization. Here $v=1/1.25=0.80$. Survival through the deferment is ${}_2p_x=(1-0.10)(1-0.20)=0.90(0.80)=0.72$.

Starting at age $x+2$, the possible death payments are at the end of years 1, 2, and 3 after that age. Their probabilities are $0.20$, $0.80(0.25)=0.20$, and $0.80(0.75)(1)=0.60$, respectively. Hence

A_{x+2} =(0.80)(0.20)+(0.80)^2(0.20)+(0.80)^3(0.60) =0.1600+0.1280+0.3072 =0.5952.

The deferment discount factor is $v^2=0.64$. Therefore

{}_{2|}A_x =(0.64)(0.72)(0.5952) =(0.4608)(0.5952) =0.27426816 \approx 0.2743.

To check, the probabilities of death in years 3, 4, and 5 from age $x$ are $0.72(0.20)=0.144$, $0.72(0.80)(0.25)=0.144$, and $0.72(0.80)(0.75)=0.432$. Discounting each to time 0 gives $(0.80)^3(0.144)=0.073728$, $(0.80)^4(0.144)=0.0589824$, and $(0.80)^5(0.432)=0.14155776$. Their sum is $0.27426816$, which matches the factorization. The calculation includes the possibility of death in year 3, but correctly excludes deaths in years 1 and 2 from the benefit.

Worked Example: A Stated Benefit and a Deferment Survival Trap

A hypothetical insurer considers a \$25,000 whole-life benefit deferred for two years. The effective annual interest rate is $25\%$. The one-year survival probabilities at ages $x$ and $x+1$ are $0.95$ and $0.90$. After age $x+2$, the one-year death probabilities are $0.10$ at age $x+2$, $0.20$ at age $x+3$, and $1$ at age $x+4$. Find the actuarial present value of the benefit.

Method. Find the unit-benefit deferred value first, and multiply by \$25,000 only after the insurance factor has been calculated. The two-year survival probability is ${}_2p_x=0.95(0.90)=0.855$, and $v=0.80$.

At age $x+2$, the probabilities of death in years 1, 2, and 3 are $0.10$, $0.90(0.20)=0.18$, and $0.90(0.80)(1)=0.72$. Thus

A_{x+2} =(0.80)(0.10)+(0.80)^2(0.18)+(0.80)^3(0.72) =0.0800+0.1152+0.36864 =0.56384.

The two-year deferral factor is $v^2{}_2p_x=(0.64)(0.855)=0.5472$. The unit-benefit value is \[ {}_{2|}A_x=(0.5472)(0.56384)=0.308533248. \] Scaling by the stated benefit gives

\$25{,}000(0.308533248)=\$7{,}713.3312\approx \$7{,}713.33.

As a direct check, the probabilities of payment in years 3, 4, and 5 are $0.855(0.10)=0.0855$, $0.855(0.90)(0.20)=0.1539$, and $0.855(0.90)(0.80)=0.6156$. Their discounted sum is $(0.512)(0.0855)+(0.4096)(0.1539)+(0.32768)(0.6156)=0.043776+0.06303744+0.201719808=0.308533248$. Multiplying by \$25,000 gives the same answer. The trap avoided is multiplying by the chance of surviving to age $x+2$ but forgetting to discount the value at age $x+2$ by $v^2$.

Interpreting the Factors and Avoiding Errors

Each part of $v^m{}_mp_xA_{x+m}$ has a distinct role: $v^m$ discounts a value at time $m$ to time 0, ${}_mp_x$ accounts for the condition that the policy is in force after deferment, and $A_{x+m}$ values the covered future deaths conditional on reaching age $x+m$. The survival probability is not included again inside $A_{x+m}$, because that factor is conditional on being alive at the start of the post-deferment period.

For example, a tempting but incorrect calculation is ${}_mp_xA_{x+m}$, omitting $v^m$. It accounts for reaching the deferment date but leaves the conditional insurance value at time $m$, rather than time 0. Another incorrect calculation is $v^m{}_mp_xA_x$: it uses the wrong age for post-deferment mortality. A useful check is to write the first possible payment time explicitly: for an $m$-year deferment, the earliest covered death benefit is paid at the end of year $m+1$, not at the end of year $m$.

The factorization also clarifies the boundary case $m=0$. Since $v^0=1$ and ${}_0p_x=1$, it reduces to $A_x$, as it should. For positive $m$, it does not include a pure endowment payment at time $m$: survival to that time only establishes eligibility for later whole-life coverage. The identity must be adjusted if the contract pays a benefit during the deferment period, includes a survival benefit at the end of deferment, uses a different benefit amount after deferment, or pays at a different time than the end of the year of death.

Key takeaway: For discrete unit-benefit whole-life insurance deferred $m$ years, first reach age $x+m$ by surviving and discounting, then value whole-life coverage at that age: ${}_{m|}A_x=v^m{}_mp_xA_{x+m}$. Keep the survival, discount, and conditional insurance factors separate.

Check Your Understanding

For each problem, use end-of-year-of-death benefits and the stated annual effective interest rate. Answers are rounded only where indicated.

A unit-benefit policy is deferred two years. The annual effective interest rate is $10\%$, and the death probability is $0.10$ every year at every age. Calculate the deferred whole-life value. (Answer: $0.3347$)
A unit-benefit policy is deferred one year at $i=20\%$. The one-year survival probability to the deferment date is $0.90$, and $A_{x+1}=0.50$. Calculate its value. (Answer: $0.3750$)
A unit-benefit policy is deferred two years at $i=25\%$. Survival probabilities for the first two years are $0.75$ and $0.80$. After deferment, the death probability is $0.25$ in the first year and $1$ in the second. Calculate ${}_{2|}A_x$. (Answer: $0.26112$)
At $i=10\%$, a three-year deferred policy has ${}_3p_x=0.729$ and $A_{x+3}=0.40$. Find the unit-benefit value, rounded to four decimal places. (Answer: $0.2192$)
A hypothetical \$12,000 benefit is deferred two years. At $i=5\%$, the probability of surviving the deferment is $0.90$, and the whole-life factor at the deferred age is $0.30$. Find the present value in dollars, rounded to the nearest cent. (Answer: \$2,938.78)

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