Deferred Endowment Insurance

What Makes an Endowment Deferred?

An endowment insurance contract can provide a benefit on death during a specified term and a benefit at maturity if the insured survives. A deferred endowment adds a waiting period: there is no benefit during the initial deferment, and the insurance coverage begins only after that period. Correct valuation therefore requires distinguishing three events: death during deferment, death during the covered term, and survival to maturity.

Suppose the deferment lasts $m$ years and the covered term lasts a further $n$ years. The death benefit is paid immediately at death if death occurs after the deferment and no later than the end of the covered term. The maturity benefit is paid at time $m+n$ if the insured is alive then. The contract pays nothing if death occurs during the deferment. Benefits and timings are hypothetical in the examples below.

Definition: A deferred endowment with deferment $m$, covered term $n$, death benefit $B$, and maturity benefit $C$ pays $B$ at the moment of death if death occurs in the covered interval $(m,m+n]$, and pays $C$ at time $m+n$ if the insured survives to that time. It pays no benefit for death during the deferment. If benefits are payable at the endpoint on death or survival, the events are mutually exclusive; under a continuous lifetime model, the probability of death at one exact instant is zero.

Let $T_x$ be the future lifetime of a life aged $x$, and let ${}_tp_x=P(T_x>t)$. Write $\mu_{x+t}$ for the force of mortality at duration $t$, and let $\delta$ be a constant force of interest. The present value of a payment $Z$ at time $t$ is $Ze^{-\delta t}$. For a nonconstant force of interest, the discount factor to time $t$ would instead be $e^{-\int_0^t\delta_s\,ds}$; the constant-force formulas below assume $\delta$ is constant.

General Expected Present Value

The death-benefit contribution is an integral over the covered interval. At time $t$, the probability density of death is ${}_tp_x\mu_{x+t}$: the life must survive to $t$, then die at the instantaneous rate $\mu_{x+t}$. Discounting the benefit at death and integrating over the times at which it may be paid gives the first term below. The maturity payment is made only if the life survives the full deferment and covered term.

\operatorname{EPV} = B\int_m^{m+n} e^{-\delta t}\,{}_tp_x\,\mu_{x+t}\,dt + C e^{-\delta(m+n)}\,{}_{m+n}p_x.

This expression assumes a unit benefit amount $B$ is paid at death during coverage and a unit maturity amount $C$ is paid at the end of the term, with no other payments. It handles varying mortality forces directly, provided the survival probabilities and mortality force in the integrand correspond to the same lifetime model.

Derivation: On a small interval from $t$ to $t+dt$, the probability of death is approximately ${}_tp_x\mu_{x+t}\,dt$. The discounted death-benefit contribution is therefore approximately $B e^{-\delta t}{}_tp_x\mu_{x+t}\,dt$. Summing these contributions over $m\le t\le m+n$ and taking the limit gives the integral. Separately, survival to $m+n$ has probability ${}_{m+n}p_x$, and the payment at that time has present value $Ce^{-\delta(m+n)}$. Adding the two mutually exclusive payment values gives the stated EPV.

The formula also makes the deferment explicit: the death integral starts at $m$, not at zero, and the maturity payment is discounted and survival-weighted from issue. Equivalently, one can condition on survival to the end of the deferment. If the interest force is constant, factor $e^{-\delta m}{}_mp_x$ from both terms, leaving the value at time $m$ of an $n$-year endowment for a life then aged $x+m$:

\operatorname{EPV} = e^{-\delta m}{}_mp_x \left[ B\int_0^n e^{-\delta s}\,{}_sp_{x+m}\,\mu_{x+m+s}\,ds + C e^{-\delta n}{}_np_{x+m} \right].

This factorization follows from ${}_{m+s}p_x={}_mp_x\,{}_sp_{x+m}$ and from splitting the discount factor $e^{-\delta(m+s)}$ into $e^{-\delta m}e^{-\delta s}$. It is useful when the deferment and covered-period assumptions are described separately. It does not imply that the contract pays anything during the deferment: the factor represents the probability and discounting of reaching the point at which coverage begins.

Constant Forces During the Deferment and Coverage

If the force of mortality is a constant $\mu$ from issue through maturity, then ${}_tp_x=e^{-\mu t}$. For constant $\delta$, the discounted density of death at time $t$ is $\mu e^{-(\delta+\mu)t}$. Integrating this density over the covered interval gives the deferred death-benefit factor. The maturity factor is the probability of survival to $m+n$ multiplied by the discount factor to that time.

\operatorname{EPV} = B\frac{\mu}{\delta+\mu} \left(e^{-(\delta+\mu)m}-e^{-(\delta+\mu)(m+n)}\right) + C e^{-(\delta+\mu)(m+n)}.

This simplified expression requires constant mortality and interest forces over the entire interval from issue to maturity. It cannot be used unchanged if either force varies with duration. If the forces are piecewise constant, split the death integral at each change point and use the survival and discount accumulated up to that point.

Worked Example: Constant Forces and Two Benefit Components

A hypothetical contract has a two-year deferment followed by a three-year covered term. It pays $B=\$50{,}000$ on covered-period death and $C=\$70{,}000$ at maturity. Assume a constant force of interest $\delta=0.04$ and a constant force of mortality $\mu=0.06$ from issue through year 5. Find the EPV, rounded to the nearest cent.

Method. Use the constant-force formula, with $m=2$, $n=3$, and $\delta+\mu=0.10$. The covered-period death factor is:

\frac{0.06}{0.10}\left(e^{-0.10(2)}-e^{-0.10(5)}\right) = 0.6\left(e^{-0.2}-e^{-0.5}\right) = 0.6(0.8187307531-0.6065306597) = 0.1273200560.

Thus the death-benefit EPV is $50{,}000(0.1273200560)=\$6{,}366.0028\ldots$. The maturity factor is $e^{-0.10(5)}=e^{-0.5}=0.6065306597$, so the maturity-benefit EPV is $70{,}000(0.6065306597)=\$42{,}457.1462\ldots$.

Adding the unrounded components gives $\$6{,}366.0028\ldots+\$42{,}457.1462\ldots=\$48{,}823.1490\ldots$, or \$48,823.15 to the nearest cent. The check is that the death payment is valued only for deaths in years 2 through 5, while the maturity payment is weighted by survival and discounting through all five years. The common mistake avoided is treating the contract as if coverage began at issue.

Piecewise-Constant Mortality

When mortality changes by duration, the survival probability to a time $t$ is determined by the accumulated force: ${}_tp_x=e^{-\int_0^t\mu_{x+s}\,ds}$. On a covered interval where both $\delta$ and $\mu$ are constant, the death-benefit integral can be evaluated as an exponential integral. If the mortality force changes at an interval boundary, first calculate the discount and survival accumulated to that boundary, then integrate using the new force.

Locate each time interval.
Mark the deferment endpoint, maturity time, and every point where the mortality or interest force changes.

Accumulate survival and discounting.
At the start of each covered segment, include the probability of surviving from issue to that segment and the discount factor to its start.

Value death payments segment by segment.
Integrate the discounted death density on each covered segment and multiply by $B$. Do not include any segment within the deferment.

Value maturity separately.
Multiply $C$ by the probability of surviving to maturity and the discount factor to maturity, then add it to the death-benefit EPV.

Worked Example: Deferment With a Change in Mortality

A hypothetical policy has a one-year deferment and a two-year covered term. The death benefit is $\$40{,}000$, and the maturity benefit is $\$50{,}000$. The force of interest is constant at $\delta=0.02$. The force of mortality is $0.04$ during year 1 and $0.10$ during years 2 and 3. Find the EPV, rounded to the nearest cent.

Method. Death benefits are payable only during years 2 and 3, when the mortality force is $0.10$. By the start of year 2, the accumulated interest-plus-mortality force is $(0.02+0.04)(1)=0.06$, so the combined discount-and-survival factor is $e^{-0.06}$. During the two covered years, the combined force is $0.02+0.10=0.12$. The death-benefit factor is therefore:

e^{-0.06}\frac{0.10}{0.12}\left(1-e^{-0.12(2)}\right) = e^{-0.06}\frac{0.10}{0.12}(1-e^{-0.24}) = 0.167455260\ldots

The death-benefit EPV is $40{,}000(0.167455260\ldots)=\$6{,}698.2104\ldots$. For maturity, survival and discounting accumulate over all three years: the exponent is $0.02(3)+0.04(1)+0.10(2)=0.30$. Thus the maturity-benefit EPV is $50{,}000e^{-0.30}=\$37{,}040.9110\ldots$.

The total is $\$6{,}698.2104\ldots+\$37{,}040.9110\ldots=\$43{,}739.1214\ldots$, or \$43,739.12 to the nearest cent. As a check, the survival and discount factor to the start of coverage appears in the death value, while the maturity value includes the additional two covered years. A common mistake is to apply the year-1 mortality force to the covered death integral; year 1 is the deferment, not a covered interval.

Solving for a Benefit and Avoiding Timing Errors

The same components can be used to solve for an unknown benefit. In particular, if the target EPV and death benefit are given, subtract the death-benefit EPV from the target and divide by the maturity factor. The maturity factor must include survival as well as discounting; dividing by a discount factor alone would value a payment even on paths where the insured has died before maturity.

Worked Example: Finding the Maturity Benefit

A hypothetical deferred endowment has a two-year deferment and a three-year covered term. The death benefit is $\$50{,}000$, and the desired EPV is $\$40{,}000$. Assume constant forces $\delta=0.04$ and $\mu=0.06$ from issue through maturity. Find the maturity benefit $C$, rounded to the nearest cent.

Method. The combined force is $\delta+\mu=0.10$. The death-benefit factor is:

\frac{0.06}{0.10}(e^{-0.2}-e^{-0.5}) = 0.6(0.8187307531-0.6065306597) = 0.1273200560.

The death-benefit EPV is $50{,}000(0.1273200560)=\$6{,}366.0028\ldots$. The maturity factor is $e^{-0.10(5)}=e^{-0.5}=0.6065306597$. Therefore:

C=\frac{40{,}000-6{,}366.0028\ldots}{0.6065306597} =\$55{,}453.0866\ldots

Hence $C=$\$55,453.09 to the nearest cent. Substitution checks the result: the maturity payment contributes $\$40{,}000-\$6{,}366.0028\ldots=\$33{,}633.9972\ldots$, and adding the death-benefit EPV returns the target $\$40{,}000$, apart from rounding. The common mistake avoided is dividing by $e^{-0.5}$ before subtracting the death-benefit EPV, rather than solving the linear EPV equation.

A useful timing check is to write the payment rule in words before writing an integral. The lower limit of the death-benefit integral is the end of the deferment; the upper limit is maturity. The maturity benefit is a single payment at maturity, contingent on survival to that point. If the contract instead paid on death during the deferment, it would be a different coverage design and would require an additional integral.

Component	Payment condition	Valuation feature
Covered-period death benefit	Death after deferment and by maturity	Integrate discounted death density over the covered interval
Maturity benefit	Survival to the end of the covered term	Discount the benefit at maturity and multiply by survival probability
Death during deferment	Occurs before coverage begins	No payment under the deferred endowment defined here

Key takeaway: A deferred endowment EPV is the sum of a death-benefit value over the covered interval and a survival-contingent maturity value. Deferment affects both components through survival and discounting before coverage starts, but it creates no benefit for death during the waiting period.

Check Your Understanding

For each problem, state the payment timing before calculating. Round monetary answers to the nearest cent.

A hypothetical contract has a one-year deferment and a two-year covered term. It pays $\$10{,}000$ on covered-period death and $\$15{,}000$ at maturity. Assume constant forces $\delta=0.03$ and $\mu=0.07$ from issue to maturity. Find the EPV. (Answer: $\$12{,}260.41$)
A hypothetical contract has a two-year deferment followed by one year of coverage. It pays $\$25{,}000$ on covered-period death and $\$40{,}000$ at maturity. Assume constant forces $\delta=0.04$ and $\mu=0.06$. Find the EPV. (Answer: $\$30{,}801.42$)
A hypothetical contract has a two-year deferment followed by a three-year covered term. With $\delta=0.04$ and $\mu=0.06$ constant, the death benefit is $\$50{,}000$. What maturity benefit gives an EPV of $\$40{,}000$? (Answer: $\$55{,}453.09$)
A hypothetical contract has a two-year deferment and a one-year covered term. The death benefit is $\$20{,}000$, and the maturity benefit is $\$30{,}000$. Assume constant forces $\delta=0.05$ and $\mu=0.05$ from issue through maturity. Find the EPV. (Answer: $\$23{,}003.67$)

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