Closed-Form Continuous Values Under Constant Force

Why Constant Forces Give Closed Forms

A continuous life annuity pays at a steady rate while the insured is alive. Its value depends on two effects at each future time: the payment must be discounted for interest, and the insured must survive long enough for it to be paid. Under constant forces of interest and mortality, both effects are exponential. Their product is also exponential, so the expected present value can be evaluated directly rather than approximated.

The central quantity is the sum of the forces. It captures the rate at which a future payment is reduced by both discounting and the possibility that it will not be made. The same integral handles temporary, deferred, and whole-life payment periods; only the limits of integration change.

Definition: Suppose the force of interest is constant at $\delta$, and the force of mortality at age $x$ is constant at $\mu$ for all future durations under consideration. Then ${}_tp_x=e^{-\mu t}$ for $t\geq 0$. Define $\alpha=\delta+\mu$. A unit-rate continuous life annuity pays at rate 1 per year while the life is alive; its expected present value over a specified time interval is the integral of the discount factor times the probability of survival to each payment time.

Here, $\delta\geq 0$ and $\mu\geq 0$. A payment rate $c$ means that the amount paid over a short interval of length $dt$ is $c\,dt$, provided the life is then alive. A temporary annuity stops at its stated term, a deferred annuity starts at its deferment time only if the life has survived, and a whole-life annuity continues for as long as the life survives.

The Expected Present-Value Integral

At time $t$, a unit-rate annuity contributes $dt$ if the life is alive. Its present value at time 0 is $e^{-\delta t}dt$. The probability that this payment is made is ${}_tp_x=e^{-\mu t}$. Thus its expected present-value contribution is $e^{-\delta t}e^{-\mu t}dt=e^{-\alpha t}dt$.

For an interval from time $r$ to time $s$, where $0\leq r<s$, the value is therefore:

\int_r^s e^{-\alpha t}\,dt = \begin{cases} \dfrac{e^{-\alpha r}-e^{-\alpha s}}{\alpha}, & \alpha>0,\\[6pt] s-r, & \alpha=0. \end{cases}

The $\alpha=0$ case is the limit of the expression as $\alpha$ tends to zero. Under the stated nonnegative-force assumptions it requires both forces to be zero: there is neither discounting nor mortality, so each unit of time contributes one unit of value.

Result: For a continuous annuity paying at rate $c$ from time $r$ to time $s$, while the life is alive, multiply the interval integral by $c$. If the payment period is temporary, $r=0$ and $s=n$. If it is deferred, $r=m$ and $s=m+n$. A whole-life annuity uses $s=\infty$.

Temporary Continuous Annuities

For a unit-rate temporary continuous annuity payable from time 0 for $n$ years, the payment interval is $[0,n]$. When $\alpha>0$, its value is:

\int_0^n e^{-\alpha t}\,dt = \frac{1-e^{-\alpha n}}{\alpha}.

For a payment rate $c$, the value is $c(1-e^{-\alpha n})/\alpha$. If $\alpha=0$, the value is $cn$. These formulas assume the annuity stops at time $n$; survival beyond $n$ does not generate further payments.

Worked Example: A Temporary Annuity With a Specified Payment Rate

A hypothetical contract pays continuously at a rate of $1,200 per year while a person aged $x$ is alive, for at most 8 years. Assume a constant force of interest $\delta=0.03$ and a constant force of mortality $\mu=0.07$. Find the expected present value at time 0, rounded to the nearest cent.

Method. Use the temporary-annuity integral. The combined force is $\alpha=\delta+\mu=0.03+0.07=0.10$. The term is $n=8$, and the payment rate is $c=1200$.

The unit-rate value is:

\int_0^8 e^{-0.10t}\,dt = \frac{1-e^{-0.80}}{0.10} = \frac{1-0.4493289641}{0.10} = 5.506710359.

Multiplying by the payment rate gives:

$$ 1200(5.506710359)=6608.052431. $$

The expected present value is $6,608.05, rounded to two decimal places. As a check, the value is below the maximum undiscounted payment of $1200\times 8=\$9,600$, as expected when payments are discounted and may stop at death. The method avoids treating the 8-year payment stream as certain.

Deferred Continuous Annuities

For a unit-rate annuity deferred $m$ years and payable for at most $n$ years thereafter, payments can occur between times $m$ and $m+n$. The expected present value at time 0 is the integral over that interval:

\int_m^{m+n} e^{-\alpha t}\,dt = \frac{e^{-\alpha m}-e^{-\alpha(m+n)}}{\alpha} = e^{-\alpha m}\frac{1-e^{-\alpha n}}{\alpha}, \qquad \alpha>0.

This expression has two useful interpretations. The factor $e^{-\alpha m}=e^{-\delta m}{}_mp_x$ accounts for both discounting and survival to the start of payments. The remaining factor is the value, at time $m$ conditional on survival to that time, of a temporary annuity lasting $n$ years. If the rate is $c$, multiply the expression by $c$.

Worked Example: A Deferred Four-Year Payment Period

A hypothetical unit-rate continuous annuity is deferred for 3 years and then payable for 4 years while the insured is alive. Assume $\delta=0.04$ and $\mu=0.06$, both constant. Find its expected present value at time 0, rounded to six decimal places.

Method. The combined force is $\alpha=0.04+0.06=0.10$. Payments occur from time $3$ through time $7$, so the integral limits are 3 and 7:

\int_3^7 e^{-0.10t}\,dt = \frac{e^{-0.30}-e^{-0.70}}{0.10}.

Using $e^{-0.30}=0.7408182207$ and $e^{-0.70}=0.4965853038$:

\frac{0.7408182207-0.4965853038}{0.10} = \frac{0.2442329169}{0.10} = 2.442329169.

The expected present value is 2.442329, rounded to six decimal places. As a check, the four-year payment period has a value below 4, the value it would have if every unit payment were certain and undiscounted.

Common mistake to avoid. Using only the four-year term and calculating $(1-e^{-0.40})/0.10$ incorrectly places the payment period at time 0. The deferred payments occur at times 3 through 7; the correct expression includes both the deferment and its survival and discount effects.

Whole-Life and Deferred Whole-Life Values

A whole-life continuous annuity has payment interval $[0,\infty)$. When $\alpha>0$, its expected present value is the improper integral:

\int_0^\infty e^{-\alpha t}\,dt = \lim_{b\to\infty}\frac{1-e^{-\alpha b}}{\alpha} = \frac{1}{\alpha},

because $e^{-\alpha b}$ tends to zero as $b$ tends to infinity. A unit-rate whole-life annuity therefore has value $1/(\delta+\mu)$, and a rate-$c$ annuity has value $c/(\delta+\mu)$. If $\alpha=0$, the integral over an infinite interval diverges, so there is no finite expected present value.

If the whole-life annuity is deferred $m$ years, its value at time 0 is:

\int_m^\infty e^{-\alpha t}\,dt = \frac{e^{-\alpha m}}{\alpha}, \qquad \alpha>0.

The deferment factor is $e^{-\alpha m}=e^{-\delta m}{}_mp_x$. It must not be replaced by $e^{-\delta m}$ alone: the annuity pays only if the insured survives to the deferment time.

Worked Example: A Deferred Whole-Life Annuity

A hypothetical benefit pays continuously at a rate of $900 per year for life, beginning 5 years from now, provided the insured is alive then. Assume a constant force of interest $\delta=0.05$ and a constant force of mortality $\mu=0.03$. Find the expected present value at time 0 to the nearest cent.

Method. This is a deferred whole-life annuity with $m=5$, $c=900$, and $\alpha=0.05+0.03=0.08$. The value is:

900\frac{e^{-0.08(5)}}{0.08} = 900\frac{e^{-0.40}}{0.08}.

Since $e^{-0.40}=0.6703200460$, the unit-rate value is:

\frac{0.6703200460}{0.08}=8.379000575.

Multiplying by $900 per year gives $900(8.379000575)=7541.100518$. The expected present value is $7,541.10, rounded to two decimal places. It is less than the corresponding immediate whole-life value $900/0.08=\$11,250$, consistent with the additional delay and survival condition.

Why the Combined-Force Formula Holds

The formula follows from the expected value of the payment stream, not from treating death as an independent discount rate added by convention. Let $I_t$ equal 1 if the life is alive at time $t$ and 0 otherwise. A unit-rate temporary annuity's random present value is the integral of its discounted payments while alive. Taking expectations gives:

E\left[\int_0^n e^{-\delta t}I_t\,dt\right] = \int_0^n e^{-\delta t}E[I_t]\,dt = \int_0^n e^{-\delta t}{}_tp_x\,dt = \int_0^n e^{-(\delta+\mu)t}\,dt.

The exchange of expectation and integration is valid here because the integrand is nonnegative and the interval is finite. For a whole-life term, the corresponding nonnegative improper integral gives the finite value when $\alpha>0$, and diverges when $\alpha=0$. This derivation also shows what must change if the forces are not constant: the survival and discount factors can no longer generally be combined into this single constant-exponent form.

Specify the forces.
Confirm that interest and mortality are constant over the full interval, and calculate $\alpha=\delta+\mu$.

Set the payment interval.
Use $[0,n]$ for temporary, $[m,m+n]$ for deferred temporary, and $[m,\infty)$ for deferred whole-life payments.

Integrate and scale.
Evaluate the exponential integral, then multiply by the payment rate if it is not 1.

Check the interpretation.
Confirm that deferment, survival, and the finite or infinite term are all reflected in the limits and factors.

Common Pitfalls and Interpretation

The most important practical decision is to translate the contract into the correct payment times before integrating. A deferment changes the start of the interval; a term limits its end. Once those limits are correct, the combined-force integral is direct.

Omitting mortality. The factor $e^{-\delta t}$ alone values certain payments, not payments contingent on survival. For a life annuity, include ${}_tp_x=e^{-\mu t}$.
Omitting interest during deferment. The time-0 value includes discounting from each payment time, not just from the first payment date.
Using the wrong endpoint. A deferment of $m$ followed by $n$ years of payments means the upper endpoint is $m+n$, not $n$.
Using an infinite-term formula for a temporary annuity. The expression $1/\alpha$ applies only when payments continue for life from the start (or after the stated deferment), not when a finite term is specified.

These values are useful for valuing continuous payment benefits and for checking that a proposed answer has a sensible scale. They are exact under the stated constant-force assumptions; when forces vary over time, the simple closed forms need not apply.

Check Your Understanding

For each problem, assume the stated forces apply throughout the relevant future period. Values are expected present values at time 0.

A unit-rate continuous annuity is payable for 5 years while the life is alive. If $\delta=0.02$ and $\mu=0.08$, find its value to six decimal places. (Answer: 3.934693)
A unit-rate continuous whole-life annuity begins immediately. If $\delta=0.04$ and $\mu=0.06$, find its value. (Answer: 10)
A unit-rate annuity is deferred for 2 years and then payable for 3 years. If $\delta=0.03$ and $\mu=0.07$, find its value to six decimal places. (Answer: 2.122001)
A hypothetical whole-life benefit pays continuously at $900 per year immediately. If $\delta=0.05$ and $\mu=0.03$, find its expected present value to the nearest cent. (Answer: $11,250.00)
A hypothetical temporary benefit pays continuously at $2,500 per year for at most 4 years while the insured is alive. If $\delta=0.01$ and $\mu=0.04$, find its expected present value to the nearest cent. (Answer: $9,063.46)

What You'll Learn