Choosing the Correct Interval for Deferred Cover

Why the Coverage Interval Matters

A deferred death benefit does not cover every death from the issue date. It pays only if death occurs after a specified waiting period, and a deferred term benefit also stops covering deaths after a specified duration. The first valuation task is therefore not to substitute values into an integral: it is to identify exactly when the contract pays.

For example, “a four-year term beginning after a two-year deferment” describes coverage during years 2 through 6, not years 2 through 4. A mistaken endpoint changes both the probability of payment and the present value. This tutorial uses continuous death benefits, for which the payment is made at the time of death.

Definition: Let $T_x$ be the future lifetime of a life aged $x$. A benefit deferred for $m$ years and then payable for a further $n$ years is paid if $m \leq T_x < m+n$. This is a deferred term benefit. A benefit deferred for $m$ years and payable for the rest of life is paid if $T_x \geq m$. This is a deferred whole-life benefit. In both cases, the benefit is paid at death.

The wording “after a deferment of $m$ years” sets the start of coverage at time $m$. If cover lasts a further $n$ years, its end is at time $m+n$. The length of the covered interval is $n$, not $m+n$. For continuous models with a density of death times, including or excluding either exact endpoint does not change the expected value: the probability of death at one specified instant is zero. The interval notation makes the contract’s intended timing clear.

Valuing a Deferred Death Benefit

Suppose the benefit amount is $B$, payable immediately at death. Let $\delta$ be the constant force of interest, so a payment at time $t$ has present-value factor $e^{-\delta t}$. Write ${}_tp_x$ for the probability of surviving $t$ years from age $x$, and $\mu_{x+t}$ for the force of mortality at attained age $x+t$. The density of future lifetime at time $t$ is ${}_tp_x\mu_{x+t}$. Thus, the value is obtained by discounting the benefit and weighting it by the probability density of death during the covered interval.

Theorem: Under a continuous future-lifetime model, a benefit $B$ payable at death only during the interval from $m$ to $m+n$ has expected present value $$ B\int_m^{m+n} e^{-\delta t}\,{}_tp_x\mu_{x+t}\,dt. $$ If the benefit is payable at death after a deferment of $m$ years for the rest of life, the expected present value is $$ B\int_m^\infty e^{-\delta t}\,{}_tp_x\mu_{x+t}\,dt. $$

These formulas assume a continuous death-time distribution and a benefit paid at death. They do not describe a benefit payable at a fixed maturity date, or a model with a positive probability of death at a particular instant. In a discrete-year model, the death-period convention and payment timing must be handled explicitly.

Why the Integral Uses the Coverage Interval

The justification follows directly from the possible death times. If death occurs at time $t$, the present value of the benefit is $Be^{-\delta t}$ when $t$ falls within the covered interval, and zero otherwise. Multiplying by the density of death and integrating over all possible death times gives the expected present value. The zero-payment period before deferment is therefore left out of the integral.

Translate the wording.
If coverage begins after $m$ years and lasts $n$ further years, the covered times are from $m$ to $m+n$.

Specify death-time probabilities.
At time $t$, the density of death is ${}_tp_x\mu_{x+t}$.

Discount payments at their actual times.
A death benefit paid at time $t$ is discounted by $e^{-\delta t}$, not by a factor based only on the length of the covered term.

Integrate over covered times only.
Multiply the discounted benefit by the death density and integrate from the coverage start to the coverage end, or to infinity for deferred whole-life cover.

This reasoning also shows why deaths before the deferment ends do not generate a claim: their present value is zero under the contract, so those times are not included in the valuation integral.

Constant-Force Model: A Useful Simplification

For worked calculations, assume a constant force of mortality $\mu$, so ${}_tp_x=e^{-\mu t}$ and the death density is $\mu e^{-\mu t}$. With constant force of interest $\delta$, the integrand becomes $B\mu e^{-(\delta+\mu)t}$. Integrating gives a direct expression for a deferred term benefit.

B\int_m^{m+n}\mu e^{-(\delta+\mu)t}\,dt = B\frac{\mu}{\delta+\mu} \left(e^{-(\delta+\mu)m}-e^{-(\delta+\mu)(m+n)}\right).

For deferred whole-life cover, the upper endpoint is infinity. Since $e^{-(\delta+\mu)t}$ tends to zero as $t$ tends to infinity when $\delta+\mu>0$, its value is $B\frac{\mu}{\delta+\mu}e^{-(\delta+\mu)m}$. These simplified formulas require constant forces over the relevant ages and times; if mortality or interest varies, use the integral with the appropriate functions instead.

Worked Example: A Deferred Four-Year Term Benefit

A hypothetical 40-year-old has a continuous death benefit of $50,000 if death occurs after a two-year deferment but within the following four years. Assume a constant force of mortality $\mu=0.05$ and a constant force of interest $\delta=0.05$. The corresponding effective annual interest rate is $i=e^{0.05}-1=0.05127$. Find the expected present value, rounded to the nearest cent.

Method. “After a two-year deferment” means coverage starts at time 2. The benefit lasts a further four years, so it ends at time $2+4=6$. The covered interval is $[2,6)$.

The combined exponential rate is $\delta+\mu=0.05+0.05=0.10$, and the coefficient is $\frac{\mu}{\delta+\mu}=\frac{0.05}{0.10}=0.5$. Therefore,

50{,}000(0.5)\left(e^{-0.10(2)}-e^{-0.10(6)}\right) = 25{,}000\left(e^{-0.2}-e^{-0.6}\right).

Using $e^{-0.2}=0.818730753$ and $e^{-0.6}=0.548811636$, the difference is $0.269919117$. Thus, the expected present value is $25{,}000(0.269919117)=\$6{,}747.98$, rounded to the nearest cent.

Check. The probability of death in the covered interval is $e^{-0.1(2)}-e^{-0.1(6)}=0.269919117$, which is below 1. The expected present value is also below the benefit amount. The interval choice avoids the mistake of ending cover at year 4.

Worked Example: Deferred Whole-Life Cover

A hypothetical policy pays $100,000 immediately at death, but only if the insured survives a five-year deferment. Assume a constant force of mortality $\mu=0.07$ and a constant force of interest $\delta=0.03$, equivalent to an effective annual rate $i=e^{0.03}-1=0.03045$. Calculate the expected present value to the nearest cent.

Method. This is deferred whole-life cover: the covered interval starts at time 5 and continues for the rest of life. Under the constant-force model, $\delta+\mu=0.10$, and $\frac{\mu}{\delta+\mu}=\frac{0.07}{0.10}=0.7$.

100{,}000\int_5^\infty 0.07e^{-0.10t}\,dt = 100{,}000(0.7)e^{-0.10(5)}.

Since $e^{-0.5}=0.606530660$, the value is $100{,}000(0.7)(0.606530660)=\$42{,}457.15$, rounded to the nearest cent.

Check. The probability of surviving the deferment is $e^{-0.07(5)}=e^{-0.35}=0.704688$. The expected present value is less than $100{,}000(0.704688)$, as expected because the benefit is also discounted after the deferment. The calculation avoids treating deferred whole-life cover as if its benefit were paid at year 5.

Worked Example: Correcting a Misread Coverage End

A hypothetical insurer issues a continuous death benefit of $80,000, payable after a three-year deferment for the following four years. Assume $\mu=0.07$ and $\delta=0.03$, so the effective annual interest rate is $i=e^{0.03}-1=0.03045$. Find the correct expected present value and compare it with the value from incorrectly ending coverage at year 4. Round both values to the nearest cent.

Method. The deferment is three years, and the four-year coverage period follows it. Therefore, the correct covered interval is $[3,7)$, not $[3,4)$. Here $\delta+\mu=0.10$ and $\frac{\mu}{\delta+\mu}=0.7$.

\text{Correct value} = 80{,}000(0.7)\left(e^{-0.3}-e^{-0.7}\right).

Using $e^{-0.3}=0.740818221$ and $e^{-0.7}=0.496585304$, the difference is $0.244232917$. Thus, the correct value is $56{,}000(0.244232917)=\$13{,}676.04$, rounded to the nearest cent.

Common mistake. Ending cover at year 4 uses only the interval $[3,4)$, even though the policy provides four years of cover after the deferment. That incorrect calculation is

80{,}000(0.7)\left(e^{-0.3}-e^{-0.4}\right) = 56{,}000(0.740818221-0.670320046) = \$3{,}947.90.

The incorrect value is smaller because it omits deaths during years 4 through 7. The correct value includes the full four-year coverage period. This comparison is a useful check: changing the covered interval from $[3,4)$ to $[3,7)$ must increase the expected value.

Common Interval Errors

The main challenge is translating contract language into time coordinates before calculating. The following distinctions help prevent errors.

Wording	Covered death times	Interpretation
Deferred $m$ years, then covered for $n$ years	$[m,m+n)$	The term duration begins after the deferment.
Covered only during the first $n$ years after issue	$[0,n)$	There is no deferment.
Deferred $m$ years, then covered for life	$[m,\infty)$	There is no finite coverage endpoint.
Covered from year $m$ to year $r$	$[m,r)$	The duration is $r-m$, not $r$.

A common trap is to confuse the deferment period with the coverage duration. The deferment determines where the interval starts; the stated term duration determines how far beyond that start the interval ends. Another trap is to discount from the start of coverage instead of from the policy issue date. A benefit paid at time $t$ is discounted for $t$ years, even when coverage begins later.

The exact treatment of an endpoint matters in models that assign positive probability to death at a particular time. Under the continuous density assumptions used here, a single endpoint has probability zero, so including or excluding that endpoint does not alter the expected present value. It does not, however, make a wrongly chosen interval correct: confusing year 4 with year 7 changes an entire range of covered death times.

Key takeaway: First express the covered death times as an interval. For a deferred $n$-year term beginning after $m$ years, integrate from $m$ to $m+n$; for deferred whole-life cover, integrate from $m$ to infinity. Discount each payment from the issue date to the actual death time.

Check Your Understanding

For each problem, identify the covered interval before calculating. Round monetary answers to the nearest cent and probabilities to four decimal places.

A hypothetical continuous benefit of $50,000 is payable after a one-year deferment for the next three years. Assume constant forces $\mu=0.06$ and $\delta=0.04$. What is its expected present value? (Answer: $7,035.52)
A $100,000 benefit is payable at death if death occurs after a five-year deferment, with coverage continuing for life. Assume constant forces $\mu=0.07$ and $\delta=0.03$. What is the expected present value? (Answer: $42,457.15)
A $90,000 continuous death benefit covers the interval from year 3 to year 8. Assume constant forces $\mu=0.04$ and $\delta=0.06$. What is its expected present value? (Answer: $10,493.61)
Under a constant force of mortality $\mu=0.20$, what is the probability of death during the interval from year 2 to year 5? (Answer: 0.3024)
A hypothetical $75,000 benefit is payable at death after a 1.5-year deferment for a further 2.5 years. Assume constant forces $\mu=0.04$ and $\delta=0.06$. What is its expected present value? (Answer: $5,711.64)

The interval-first method is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: translate the benefit wording, identify when a payment can occur, and only then apply the continuous valuation formula.

Wording	Covered death times	Interpretation
Deferred \(m\) years, then covered for \(n\) years	\([m,m+n)\)	The term duration begins after the deferment.
Covered only during the first \(n\) years after issue	\([0,n)\)	There is no deferment.
Deferred \(m\) years, then covered for life	\([m,\infty)\)	There is no finite coverage endpoint.
Covered from year \(m\) to year \(r\)	\([m,r)\)	The duration is \(r-m\), not \(r\).

What You'll Learn