Continuously Decreasing Whole-Life Benefits

Why the Benefit Schedule Matters

A continuous whole-life policy pays at the moment of death, but the amount paid need not stay constant. In a decreasing-benefit design, the death benefit falls as time since issue increases. Its actuarial present value therefore depends on both the probability of death at each time and the benefit available at that time.

This tutorial considers a benefit that declines linearly from an initial amount to zero over a specified duration. After that duration the benefit remains zero. Thus, the policy is valued over the whole future lifetime, but deaths after the benefit has reached zero produce no payment. The benefit schedule is not the same as a level whole-life benefit, nor is it appropriate to replace the schedule by its average amount.

Definition: Let $T_x$ be the future lifetime of a life aged $x$, and let $B_0$ be the benefit at issue. For a specified duration $T>0$, a linearly decreasing benefit is $$ b(t)= \begin{cases} B_0\left(1-\dfrac{t}{T}\right), & 0\le t<T,\\ 0, & t\ge T. \end{cases} $$ The actuarial present value is the expected present value of the benefit paid at death: $E[b(T_x)e^{-\delta T_x}]$, where $\delta$ is the force of interest.

General Continuous Valuation

Write ${}_tp_x$ for the probability that a life aged $x$ survives at least $t$ years, and $\mu_{x+t}$ for the force of mortality at attained age $x+t$. The density of the future lifetime at time $t$ is ${}_tp_x\mu_{x+t}$. This follows because the life must survive to time $t$, then die in the next infinitesimal interval.

For any specified benefit function $b(t)$, the continuous death-benefit present value is therefore

E[b(T_x)e^{-\delta T_x}] = \int_0^\infty b(t)e^{-\delta t}{}_tp_x\mu_{x+t}\,dt.

For the decreasing schedule, the integrand is zero after $T$, so the valuation reduces to

\operatorname{APV} = \int_0^T B_0\left(1-\frac{t}{T}\right)e^{-\delta t}{}_tp_x\mu_{x+t}\,dt.

This integral applies with an age-dependent mortality force as written. A simple closed form results when mortality is constant at all future ages. Assume $T_x$ has an exponential distribution with force $\mu>0$, so ${}_tp_x=e^{-\mu t}$ and $\mu_{x+t}=\mu$. Define $k=\delta+\mu$. Then

\operatorname{APV} = B_0\mu\int_0^T\left(1-\frac{t}{T}\right)e^{-kt}\,dt = B_0\mu\left[ \frac{1}{k}-\frac{1-e^{-kT}}{Tk^2} \right].

The formula assumes a continuous death benefit, a linear decline to zero, constant force of mortality, and constant force of interest. It does not apply unchanged to a nonlinear schedule, a level benefit after $T$, or a nonconstant force of mortality. In those cases, use the general integral and the applicable survival model.

Deriving the Closed Form

The closed form comes from integrating the benefit schedule against the combined survival and discount factor. Since $e^{-\delta t}{}_tp_x=e^{-(\delta+\mu)t}=e^{-kt}$, split the integral into a constant-benefit part and a linearly weighted part:

\int_0^T\left(1-\frac{t}{T}\right)e^{-kt}\,dt = \int_0^T e^{-kt}\,dt -\frac{1}{T}\int_0^T te^{-kt}\,dt.

The first integral is $(1-e^{-kT})/k$. For the second, integration by parts with $u=t$ and $dv=e^{-kt}dt$ gives

\int_0^T te^{-kt}\,dt = \left[-\frac{te^{-kt}}{k}-\frac{e^{-kt}}{k^2}\right]_0^T = \frac{1-(1+kT)e^{-kT}}{k^2}.

Substituting these two results and simplifying yields

\frac{1-e^{-kT}}{k} - \frac{1-(1+kT)e^{-kT}}{Tk^2} = \frac{1}{k}-\frac{1-e^{-kT}}{Tk^2}.

Multiplying by $B_0\mu$, the death density factor and initial benefit, gives the stated actuarial present value. The factor $\mu$ is essential: the integral alone weights by survival and discounting, while the density of death at each time also includes the force of mortality.

Worked Example: Valuing a Benefit That Reaches Zero After Five Years

A hypothetical policy pays $ \$100{,}000(1-t/5)$ if death occurs at time $t$ years after issue for $0\le t<5$, and pays nothing for death at or after five years. Assume a constant force of mortality $\mu=0.10$ and force of interest $\delta=0.10$. The future lifetime is exponential with parameter $0.10$. Find the actuarial present value.

Method. Use the closed form for the linear schedule. Here $B_0=100{,}000$, $T=5$, and $k=\delta+\mu=0.10+0.10=0.20$. The corresponding effective annual interest rate is $i=e^{0.10}-1$, approximately $10.52\%$.

First calculate the integral factor:

\frac{1}{k}-\frac{1-e^{-kT}}{Tk^2} = \frac{1}{0.20} -\frac{1-e^{-(0.20)(5)}}{5(0.20)^2} = 5-\frac{1-e^{-1}}{0.20}.

Using $e^{-1}=0.3678794$, the numerator $1-e^{-1}=0.6321206$, and $0.6321206/0.20=3.1606028$. Thus the integral factor is $5-3.1606028=1.8393972$. Now multiply by $B_0\mu=100{,}000(0.10)=10{,}000$:

\operatorname{APV}=10{,}000(1.8393972)=\$18{,}393.97.

Check. A level whole-life benefit of $ \$100{,}000$ under the same assumptions would have value $100{,}000\mu/(\delta+\mu)=\$50{,}000$. The decreasing benefit is never greater than that level benefit and is zero after year five, so $ \$18{,}393.97$ is consistent with the schedule. The common mistake is to ignore the decline and use the level-benefit value.

Reading the Formula and the Benefit Schedule

The initial benefit $B_0$ and the duration $T$ fully determine this linear schedule. Its annual rate of decrease is $B_0/T$, so a benefit stated as $B_0-ct$ reaches zero at $T=B_0/c$, provided $B_0>0$ and $c>0$. The formula uses the duration until zero, not the number of years for which a level benefit would be paid.

The duration also affects how much of the benefit remains when death occurs. A shorter $T$ means a faster decrease and less benefit at any fixed positive time before the shorter zero date. A longer $T$ means a slower decrease, although the APV still depends on the full survival and discount pattern.

If mortality is not constant, the general integral remains valid but the closed form above generally does not. In that case, insert the appropriate ${}_tp_x$ and $\mu_{x+t}$ into the integrand and evaluate the integral analytically or numerically. Do not use a single constant $\mu$ unless the assumed model supports it.

Worked Example: Converting a Decline Rate Into a Zero-Benefit Duration

A hypothetical continuous insurance benefit starts at $ \$24{,}000$ and declines by $ \$6{,}000$ for each year since issue, reaching zero after four years. Assume constant force of mortality $\mu=0.25$, exponential future lifetime, and force of interest $\delta=0.25$. Find the APV. The effective annual interest rate is $e^{0.25}-1$, approximately $28.40\%$.

Method. The specified starting benefit and annual decline give $T=24{,}000/6{,}000=4$ years. This is the duration to zero, so use $B_0=24{,}000$ and $T=4$, rather than treating $ \$6{,}000$ as the benefit. Here $k=0.25+0.25=0.50$.

The integral factor is

\frac{1}{0.50} -\frac{1-e^{-(0.50)(4)}}{4(0.50)^2} = 2-\frac{1-e^{-2}}{1} = 2-(1-0.1353353) = 1.1353353.

The multiplier outside the integral is $B_0\mu=24{,}000(0.25)=6{,}000$. Therefore,

\operatorname{APV}=6{,}000(1.1353353)=\$6{,}812.01.

Check. The level whole-life value at the initial $ \$24{,}000$ benefit would be $24{,}000(0.25/0.50)=\$12{,}000$, which is greater than the decreasing-benefit value. A common error is to use the decline rate as the initial benefit or to confuse the four-year zero date with a constant four-year payment period.

Worked Example: A Larger Starting Benefit With a Longer Taper

A hypothetical policy pays $ \$90{,}000-\$15{,}000t$ at death for $0\le t<6$, and zero thereafter. Assume constant force of mortality $\mu=0.08$, exponential future lifetime, and force of interest $\delta=0.12$. Find the APV. The effective annual interest rate is $e^{0.12}-1$, approximately $12.75\%$.

Method. The initial benefit is $B_0=\$90{,}000$. The annual decline is $ \$15{,}000$, so $T=90{,}000/15{,}000=6$ years. The combined exponent parameter is $k=0.08+0.12=0.20$.

Substitute $kT=0.20(6)=1.2$ into the integral factor:

\frac{1}{0.20} -\frac{1-e^{-1.2}}{6(0.20)^2} = 5-\frac{1-0.3011942}{0.24} = 5-\frac{0.6988058}{0.24}.

The quotient is $0.6988058/0.24=2.9116908$, so the integral factor is $5-2.9116908=2.0883092$. The outside multiplier is $B_0\mu=90{,}000(0.08)=7{,}200$. Hence

\operatorname{APV}=7{,}200(2.0883092)=\$15{,}035.83.

Check. The corresponding level whole-life value at $ \$90{,}000$ would be $90{,}000(0.08/0.20)=\$36{,}000$, so the smaller tapered-benefit value is reasonable. The calculation also uses the six-year zero date, not the issue benefit as though it continued for life.

Common Mistake: Replacing the Schedule by an Average Benefit

A tempting shortcut is to take one-half of the initial benefit, because the schedule declines from $B_0$ to zero, and multiply that average by the value of level whole-life insurance. That does not generally work. The relevant benefit is the amount at the actual time of death, and death times are weighted by survival, mortality, and discounting. Moreover, deaths after $T$ receive no payment.

For the five-year example, that shortcut would use a level benefit of $ \$50{,}000$, giving $50{,}000(0.10/0.20)=\$25{,}000$. The correct value is $ \$18{,}393.97$, obtained by integrating the benefit at each possible death time. The shortcut applies a simple arithmetic average without accounting for the timing of death or the zero-benefit period.

Key takeaway: First write the benefit as a function of time and identify when it reaches zero. Then value the benefit at each possible death time using the death density and discount factor. Under constant forces, the closed form is $B_0\mu[1/k-(1-e^{-kT})/(Tk^2)]$, where $k=\delta+\mu$.

Check Your Understanding

For each problem, assume the stated force of mortality is constant at all future ages, the future lifetime is exponential with that parameter, and the force of interest is constant. Round monetary answers to the nearest cent.

A hypothetical policy pays $ \$40{,}000(1-t/8)$ before the benefit reaches zero at eight years. If $\mu=0.05$ and $\delta=0.05$, find its APV. (Answer: $ \$6{,}233.22$)
A benefit starts at $ \$75{,}000$ and decreases linearly to zero after three years. If $\mu=0.20$ and $\delta=0.10$, find its APV. (Answer: $ \$17{,}031.65$)
A benefit starts at $ \$120{,}000$ and decreases by $ \$20{,}000$ per year. If $\mu=0.10$ and $\delta=0.05$, determine the zero-benefit duration and the APV. (Answer: 6 years; $ \$27{,}250.64$)
A hypothetical contract pays $ \$50{,}000(1-t/5)$ until its benefit reaches zero. If $\mu=0.12$ and $\delta=0.08$, calculate the APV and state why using half the initial benefit as a level whole-life benefit is not generally valid. (Answer: $ \$11{,}036.38$; the actual benefit depends on time of death and is zero after year 5.)

What You'll Learn