Constant-Force Whole-Life Insurance Values

Why Constant Forces Give a Closed Form

A continuous whole-life insurance benefit is paid at the moment of death, whenever death occurs. Its present value therefore depends on two forces acting over the time until death: mortality determines the payment time, and interest determines how much that payment is worth at issue. When both forces are constant, the payment-time distribution and discount factor combine in a way that makes the expected present value an elementary integral.

The result is useful both as a direct valuation formula and as a check on calculations based on the general integral for continuous whole-life insurance. The assumptions matter: the formula below applies when the mortality force remains constant for all future durations and the interest force is constant over the same period. It is not a general formula for age-varying mortality or changing interest.

Definition: Consider a life aged $x$. Assume the force of mortality is constant at $\mu_x>0$ for every future duration, so that ${}_tp_x=e^{-\mu_x t}$ for $t\geq 0$. Assume a constant force of interest $\delta\geq 0$, so a payment at time $t$ has discount factor $e^{-\delta t}$. A continuous whole-life insurance paying 1 at the moment of death has present value random variable $e^{-\delta T_x}$, where $T_x$ is the future lifetime.

Under this model, the probability of surviving at least $t$ years is ${}_tp_x=e^{-\mu_x t}$. Consequently, the probability of death within $t$ years is ${}_tq_x=1-e^{-\mu_x t}$. The future lifetime has density $f_{T_x}(t)=\mu_xe^{-\mu_x t}$ for $t\geq 0$. These expressions describe an exponential future-lifetime distribution; they follow from assuming the same mortality force at every future duration.

The Closed-Form Whole-Life Value

Let $\bar A_x$ denote the expected present value of a unit benefit payable immediately upon death. The general continuous valuation is the expected discounted benefit, integrated over all possible death times. Substituting the constant-force density gives a closed form.

\bar A_x =E\left[e^{-\delta T_x}\right] =\int_0^\infty e^{-\delta t}\mu_xe^{-\mu_x t}\,dt =\frac{\mu_x}{\mu_x+\delta}.

The integral converges because $\mu_x+\delta>0$, which holds under the stated assumptions. A level benefit of $B$ dollars at death has expected present value $B\bar A_x$. This is a net single-benefit present value: it is not, by itself, a gross premium and does not include expenses, profit provisions, or a premium payment pattern.

Theorem: With constant force of mortality $\mu_x>0$ and constant force of interest $\delta\geq 0$, the expected present value of a continuous whole-life insurance paying $B$ at death is $\displaystyle B\bar A_x=B\frac{\mu_x}{\mu_x+\delta}$. The formula assumes the full benefit is paid at the instant of death and that both forces remain constant for the whole future lifetime.

Why the Integral Reduces to a Ratio

The discount factor and the density each contribute an exponential term. Multiplying them adds their exponents: $e^{-\delta t}e^{-\mu_x t}=e^{-(\mu_x+\delta)t}$. Thus the expected present value is the mortality force multiplied by the integral of a single exponential over all nonnegative times.

\begin{aligned} \bar A_x &=\int_0^\infty \mu_x e^{-(\mu_x+\delta)t}\,dt\\ &=\mu_x\left[-\frac{e^{-(\mu_x+\delta)t}}{\mu_x+\delta}\right]_{0}^{\infty}\\ &=\mu_x\left(0+\frac{1}{\mu_x+\delta}\right) =\frac{\mu_x}{\mu_x+\delta}. \end{aligned}

The endpoint at infinity contributes zero because $\mu_x+\delta>0$, so $e^{-(\mu_x+\delta)t}$ tends to zero. The endpoint at zero contributes 1 inside the brackets. The closed form is therefore a consequence of integrating the joint effect of the death-time density and discounting, not an assumption about the average time of death.

Worked Applications

Worked Example: Unit Benefit Under Two Constant Forces

A hypothetical policy on a life aged 45 pays 1 immediately at death. Suppose the force of mortality is $\mu_{45}=0.04$ at every future duration and the force of interest is $\delta=0.06$. Find the expected present value. Also state the equivalent effective annual interest rate.

Method and assumptions. This is continuous whole-life insurance with a unit benefit. The forces are constant, so use $\bar A_{45}=\mu_{45}/(\mu_{45}+\delta)$. The equivalent effective annual rate is $i=e^\delta-1$.

First calculate the equivalent annual rate:

i=e^{0.06}-1=1.0618365-1=0.0618365,

or about $6.184\%$. Next calculate the expected present value:

\bar A_{45} =\frac{0.04}{0.04+0.06} =\frac{0.04}{0.10} =0.40.

Check. The result is between 0 and 1, as expected for a positive unit benefit discounted at a positive force of interest. If interest were removed by setting $\delta=0$, the value would be 1 because the benefit would eventually be paid without discounting. The common mistake avoided here is treating the force $\delta=0.06$ as an effective annual rate; its equivalent annual rate is $e^{0.06}-1$, not $0.06$.

Worked Example: Convert an Annual Rate and Value a Dollar Benefit

A hypothetical continuous whole-life contract pays \$200,000 at death. The force of mortality is constant at $\mu_x=0.03$, and the effective annual interest rate is $i=0.05$. Find the expected present value in dollars. Round the final dollar amount to the nearest cent.

Method and assumptions. The insurance benefit is paid at the moment of death, and mortality remains constant over the future lifetime. The given interest rate is effective annually, so it must be converted to a force before using the closed form: $\delta=\ln(1+i)$.

Convert the rate:

\delta=\ln(1.05)=0.048790164.

Now calculate the unit-benefit value, retaining the unrounded force:

\bar A_x =\frac{0.03}{0.03+0.048790164} =\frac{0.03}{0.078790164} \approx 0.3807582.

Scale by the benefit amount:

200{,}000(0.3807582)\approx \$76{,}151.64.

Answer: The expected present value is \$76,151.64. Check. The value is less than the undiscounted benefit of \$200,000. Using $i=0.05$ directly in place of $\delta$ would give a different result because the formula uses a force of interest, not an effective annual rate.

Worked Example: Infer the Mortality Force From a Target Value

A hypothetical continuous whole-life policy on a life aged 60 pays \$50,000 at death. Its expected present value is \$30,000, and the force of interest is $\delta=0.04$. Assuming constant mortality and interest forces, find the force of mortality.

Method and assumptions. Divide the dollar value by the benefit to obtain the unit-benefit value. Then rearrange $\bar A_x=\mu_x/(\mu_x+\delta)$ to solve for $\mu_x$.

The unit-benefit value is

\bar A_{60}=\frac{30{,}000}{50{,}000}=0.60.

Set this equal to the constant-force formula and solve:

\begin{aligned} 0.60&=\frac{\mu_{60}}{\mu_{60}+0.04},\\ 0.60(\mu_{60}+0.04)&=\mu_{60},\\ 0.60\mu_{60}+0.024&=\mu_{60},\\ 0.024&=0.40\mu_{60},\\ \mu_{60}&=\frac{0.024}{0.40}=0.06. \end{aligned}

Check. Substitution gives $0.06/(0.06+0.04)=0.60$, and multiplying by \$50,000 returns \$30,000. The inferred force is positive, as required by the model. A common algebra error is to divide by the target value rather than by $1-\bar A_x$; the rearranged identity is $\mu_x=\bar A_x\delta/(1-\bar A_x)$.

Interpreting and Checking the Formula

The ratio $\mu_x/(\mu_x+\delta)$ makes the competing effects transparent. Increasing the mortality force, while holding interest fixed, raises the value because the benefit is expected sooner. Increasing the force of interest, while holding mortality fixed, lowers the value because future payments receive heavier discounting. More formally, for positive $\mu_x$ and nonnegative $\delta$,

\frac{\partial \bar A_x}{\partial \mu_x} =\frac{\delta}{(\mu_x+\delta)^2}\geq 0, \qquad \frac{\partial \bar A_x}{\partial \delta} =-\frac{\mu_x}{(\mu_x+\delta)^2}<0.

These directions are useful checks, but they do not replace checking the assumptions or the arithmetic. Two boundary cases also help: when $\delta=0$, $\bar A_x=1$; and, for fixed positive $\delta$, the value tends to 0 as $\mu_x$ tends to 0. If both forces are positive, the value lies strictly between 0 and 1.

Change or case	Effect on $\bar A_x$	Reason
Increase $\mu_x$, with $\delta$ fixed	Increases	Death and payment tend to occur sooner
Increase $\delta$, with $\mu_x$ fixed	Decreases	More discounting applies to future payments
Set $\delta=0$	Equals 1	The eventual unit benefit is not discounted
Set $\mu_x=0$	The stated whole-life model no longer has a death payment	The formula was defined for $\mu_x>0$

A frequent exam trap is mixing annual effective rates with forces. If $i$ is the effective annual rate, the constant force that gives the same one-year accumulation is $\delta=\ln(1+i)$, since $e^\delta=1+i$. For example, with $i=0.06$ and $\mu_x=0.04$, an incorrect substitution of $0.06$ for $\delta$ gives $0.04/(0.04+0.06)=0.4000$. The correct conversion is $\delta=\ln(1.06)\approx0.0582689$, giving

\bar A_x =\frac{0.04}{0.04+0.0582689} \approx 0.4070.

The difference is not a rounding issue; it results from using the wrong interest-rate measure. Another important distinction is between a continuous benefit payable at the instant of death and a benefit payable at the end of the year of death. The latter has a different present-value expression and is not valued by this continuous-payment formula.

Key takeaway: Under constant force of mortality $\mu_x>0$ and constant force of interest $\delta\geq0$, a level benefit $B$ paid immediately at death has expected present value $B\mu_x/(\mu_x+\delta)$. Convert any effective annual rate using $\delta=\ln(1+i)$, and verify that the result responds sensibly to changes in mortality and interest.

Check Your Understanding

For each problem, use the constant-force continuous whole-life model and keep the stated payment timing in mind.

A hypothetical policy pays 1 at death. The constant force of mortality is $0.02$, and the constant force of interest is $0.03$. Find the expected present value. (Answer: $0.4000$)
A hypothetical \$150,000 benefit is paid immediately at death. The constant forces are $\mu_x=0.04$ and $\delta=0.06$. Find its expected present value in dollars. (Answer: \$60,000.00)
A unit benefit is payable at death, the constant mortality force is $0.05$, and the effective annual interest rate is $0.04$. Convert the interest rate to a force and find the expected present value to four decimal places. (Answer: $\delta=\ln(1.04)\approx0.03922$; $\bar A_x\approx0.5604$)
A continuous whole-life policy has a unit-benefit expected present value of $0.25$ and a constant force of interest of $0.03$. Find the constant force of mortality and verify the value by substitution. (Answer: $\mu_x=0.01$; $0.01/(0.01+0.03)=0.25$)
For a hypothetical \$80,000 whole-life benefit, the force of mortality is $0.08$ and the force of interest is $0.02$. Find the expected present value. Then state whether doubling both forces changes the value. (Answer: \$64,000.00; no change, because $0.16/(0.16+0.04)=0.80$)

Change or case	Effect on \(\bar A_x\)	Reason
Increase \(\mu_x\), with \(\delta\) fixed	Increases	Death and payment tend to occur sooner
Increase \(\delta\), with \(\mu_x\) fixed	Decreases	More discounting applies to future payments
Set \(\delta=0\)	Equals 1	The eventual unit benefit is not discounted
Set \(\mu_x=0\)	The stated whole-life model no longer has a death payment	The formula was defined for \(\mu_x>0\)

What You'll Learn