Deferred Whole-Life Insurance Values

Why Defer a Whole-Life Benefit?

A whole-life policy ordinarily pays a death benefit whenever the insured dies. A deferred whole-life insurance benefit is different: it pays only if death occurs after a specified deferment period. For instance, a hypothetical contract might provide no death benefit during its first five years, then pay a fixed amount at death for the rest of the insured’s life. If death occurs during those first five years, the contract pays nothing.

The distinction is about when a death qualifies for payment, not when the benefit is paid. In this tutorial the benefit is paid immediately at death, provided death occurs after the deferment period. The earlier material on continuous death-benefit present values supplies the discounting framework; here the key task is to restrict valuation to qualifying death times.

Definition: Let $T_x$ be the future lifetime of a life aged $x$, and let $m$ be a fixed deferment period. A unit deferred whole-life benefit pays 1 at time $T_x$ if $T_x>m$, and pays 0 if $T_x\leq m$. At force of interest $\delta$, its present value is $e^{-\delta T_x}$ when $T_x>m$, and 0 otherwise. For a benefit amount $b$, multiply each payment and its present value by $b$.

The Present-Value Integral

Assume a continuous future-lifetime model with survival probability ${}_tp_x=P(T_x>t)$ and force of mortality $\mu_{x+t}$ at duration $t$. The density of death at time $t$ is ${}_tp_x\mu_{x+t}$: the life must first survive to $t$, then die in the next small interval at the rate described by the mortality force. A payment at time $t$ is discounted by $e^{-\delta t}$.

Thus, only death times $t>m$ contribute to the expected present value (EPV). For a fixed benefit amount $b$, the general valuation formula is:

\operatorname{EPV} =b\int_m^\infty e^{-\delta t}\,{}_tp_x\,\mu_{x+t}\,dt.

If the mortality model has a finite ultimate age, the integral can equivalently stop at the maximum possible future lifetime; survival beyond that age is zero. The formula assumes a benefit paid immediately at death, a constant force of interest $\delta$, and the stated mortality model. It does not value a benefit paid at the end of a policy year, or a payment made simply because the life survives to time $m$.

Key result: The lower limit $m$ enforces the deferment condition. The survival probability and mortality force describe the chance and timing of death at each eligible time; the discount factor values the resulting payment at time 0.

Constant-Force Mortality

A useful special case assumes a constant force of mortality $\mu$ for all future durations, so ${}_tp_x=e^{-\mu t}$ and $\mu_{x+t}=\mu$. Under this model, the death-time density is $\mu e^{-\mu t}$. If the force of interest is also constant, the deferred whole-life EPV has a simple closed form.

\begin{aligned} \operatorname{EPV} &=b\int_m^\infty e^{-\delta t}e^{-\mu t}\mu\,dt\\ &=b\mu\int_m^\infty e^{-(\delta+\mu)t}\,dt\\ &=b\frac{\mu}{\delta+\mu}e^{-(\delta+\mu)m}. \end{aligned}

The final equality follows by integrating the exponential: for $\delta+\mu>0$, an antiderivative of $e^{-(\delta+\mu)t}$ is $-e^{-(\delta+\mu)t}/(\delta+\mu)$. Evaluating from $m$ to infinity gives $e^{-(\delta+\mu)m}/(\delta+\mu)$. The result requires a constant mortality force after issue and a constant force of interest. If mortality varies by duration, the constant-force expression does not apply; use the integral with the actual survival function and mortality force.

When the given interest assumption is an effective annual rate $i$, convert it to a force of interest using $\delta=\ln(1+i)$. Using $i$ directly in an exponential discount factor would mix an annual effective rate with continuous discounting.

Worked Example: Constant Forces and a Four-Year Deferment

A hypothetical policy issued to a life aged 50 pays 1 immediately at death, but only if death occurs more than four years after issue. Assume a constant force of mortality $\mu=0.04$ per year and a constant force of interest $\delta=0.06$ per year. Find the EPV.

Method. Under the constant-force model, use $b\mu e^{-(\delta+\mu)m}/(\delta+\mu)$. The death benefit is one unit, so $b=1$, and the deferment is $m=4$.

First add the forces in the exponent and denominator:

\delta+\mu=0.06+0.04=0.10.

Next calculate the exponential term and the leading ratio:

e^{-(\delta+\mu)m}=e^{-0.10(4)}=e^{-0.4}=0.670320, \qquad \frac{\mu}{\delta+\mu}=\frac{0.04}{0.10}=0.4.

Therefore,

\operatorname{EPV}=1(0.4)(0.670320)=0.268128.

To four decimal places, the EPV is 0.2681 per unit of benefit. The result is positive and less than the corresponding immediate whole-life value $0.4$, as expected: the deferred contract removes some death payments and all surviving payments are discounted.

Common mistake avoided. Using $0.4$ as the answer would value a whole-life benefit with no deferment. The exponential factor $e^{-0.4}$ is essential because only deaths after year 4 qualify.

Valuing Piecewise Mortality Forces

A mortality force need not be constant throughout the life of the policy. If it changes at a known duration, construct the survival probability from the forces over the successive intervals, then integrate over eligible death times. It is important to retain the survival accumulated before the deferment ends: a death after $m$ is possible only if the life survived to $m$.

The next calculation illustrates the integral directly. It does not assume that a single constant mortality force applies both before and after the deferment period.

Worked Example: Deferment Ends at a Mortality-Force Change

A hypothetical unit benefit pays immediately at death only when death occurs after year 3. Assume $\delta=0.04$ per year. The force of mortality is $0.02$ per year during the first three years and $0.08$ per year thereafter. Find the EPV.

Method. For $t\geq3$, survival to time $t$ requires surviving the first three years and then the additional $t-3$ years. Under the stated piecewise-constant forces,

{}_tp_x=e^{-0.02(3)}e^{-0.08(t-3)},\qquad \mu_{x+t}=0.08,\quad t\geq3.

Substitute these quantities into the integral with lower limit 3:

\begin{aligned} \operatorname{EPV} &=\int_3^\infty e^{-0.04t} e^{-0.02(3)}e^{-0.08(t-3)}(0.08)\,dt\\ &=0.08e^{-0.06}e^{-0.12} \int_3^\infty e^{-0.12(t-3)}\,dt\\ &=0.08e^{-0.18} \int_0^\infty e^{-0.12s}\,ds. \end{aligned}

In the last step, set $s=t-3$. The remaining integral is $1/0.12$, so:

\operatorname{EPV} =0.08e^{-0.18}\frac{1}{0.12} =\frac{2}{3}(0.835270) =0.556847.

To four decimal places, the EPV is 0.5568 per unit of benefit. The factor $e^{-0.06}$ accounts for survival through the first three years, while $e^{-0.12}$ accounts for discounting to year 3. The remaining integral values deaths after year 3 using the post-deferment mortality and interest forces.

Common mistake avoided. Applying the post-year-3 mortality force to the entire period would incorrectly replace the specified first-three-year force. The survival term must reflect each interval actually survived.

Effective Annual Interest and a Stated Benefit Amount

A problem may give an effective annual rate rather than a force of interest. For continuous payment at death, first convert the annual rate $i$ to $\delta=\ln(1+i)$, then use the continuous discount factor. Also distinguish the EPV per unit from the EPV in dollars: the latter includes the benefit amount.

Worked Example: Converting Interest and Scaling the Benefit

A hypothetical deferred whole-life policy pays $100,000 immediately at death if death occurs after ten years. Assume a constant force of mortality $\mu=0.04$ per year and an effective annual interest rate $i=0.05$. Find the EPV in dollars.

Method. Convert the effective annual rate to a force, evaluate the constant-force formula with $m=10$, and multiply by the benefit amount $b=100{,}000$.

The force of interest is:

\delta=\ln(1.05)=0.0487902.

The combined force and the ratio in the formula are:

\delta+\mu=0.0487902+0.04=0.0887902, \qquad \frac{\mu}{\delta+\mu} =\frac{0.04}{0.0887902} \approx0.450500.

The deferment factor is:

e^{-(\delta+\mu)m}=e^{-0.0887902(10)} =e^{-0.887902}\approx0.411518.

The EPV per dollar of benefit is $0.450500(0.411518)\approx0.185389$. Therefore:

\operatorname{EPV}=100{,}000(0.185389)\approx \$18{,}538.92.

Rounded to the nearest cent, the EPV is $18,538.92. As a check, it is less than the $100,000 face amount, and the ten-year deferment reduces the value below that of an otherwise identical immediate whole-life benefit.

Common mistake avoided. Substituting $i=0.05$ for $\delta$ in $e^{-\delta t}$ does not use the stated effective annual rate correctly. The required conversion is $\delta=\ln(1+i)$.

Timing Checks and Exam Traps

A reliable setup begins by identifying the payment time and the qualifying event. For deferred whole-life insurance, the payment time is the time of death, and the qualifying event is death after the deferment period. This gives a lower integration limit of $m$, not 0, and no separate payment at time $m$ for survival.

The following checks help identify errors before a calculation is finalized:

Check the contract wording. “Pays after a deferment period” means deaths during that period produce no benefit. It does not mean a benefit is paid at the end of the deferment to those who survive.
Check the time units. The deferment, mortality force, and interest force must use consistent time units. If $m$ is measured in years, forces stated per year belong in the exponent.
Check the model. The closed form uses constant forces after issue. For changing mortality, use the specified survival and force within the integral.
Check the payment timing. Immediate payment at death uses continuous discounting to $T_x$. A benefit paid at a policy-year end requires a different payment-time valuation.
Check the scale. The formula with $b=1$ gives value per unit. Multiply by the stated benefit amount to obtain a dollar value.

For a constant-force model with positive $\mu$, an increase in $m$ reduces the EPV through the factor $e^{-(\delta+\mu)m}$. This is a useful directional check: postponing the start of coverage cannot increase the value of an otherwise unchanged death benefit.

Key takeaway: For a continuous deferred whole-life benefit, integrate discounted death probabilities only over times after the deferment. Under constant forces, the unit-benefit EPV is $\frac{\mu}{\delta+\mu}e^{-(\delta+\mu)m}$; use the direct integral when the mortality force varies.

Check Your Understanding

For each problem, assume the benefit is paid immediately at death only if death occurs after the stated deferment. Give present values per unit unless a benefit amount is specified.

A hypothetical life has constant force of mortality $0.03$ and force of interest $0.07$. The deferment is 2 years. Find the unit-benefit EPV, to four decimal places. (Answer: 0.2456)
Under constant forces $\mu=0.05$ and $\delta=0.05$, a benefit is deferred for 6 years. Find the unit-benefit EPV, to four decimal places. (Answer: 0.2744)
The force of interest is $0.03$. The mortality force is $0.01$ for the first 2 years and $0.05$ thereafter. For a unit benefit deferred for 2 years, find the EPV to four decimal places. (Answer: 0.5769)
A unit benefit is deferred for 5 years. The mortality force is constantly $0.06$, and the effective annual interest rate is $0.04$. Convert the rate and find the EPV to four decimal places. (Answer: 0.3682)
A $250,000 benefit is deferred for 3 years. Assume constant forces $\mu=0.06$ and $\delta=0.04$. Find the EPV in dollars, rounded to the nearest cent. (Answer: $111,122.73)

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