Whole-Life Continuous Annuity Values

Why Continuous Whole-Life Annuities Need a Survival Integral

A whole-life annuity pays for as long as the annuitant is alive. When payments are continuous, a payment is not made at one yearly date; instead, benefits accrue at a specified rate over time. The expected present value must therefore account for both discounting and the chance that the annuitant is alive at each instant.

Let $T_x$ denote the future lifetime of a person currently aged $x$, and let ${}_tp_x=P(T_x>t)$ be the probability of surviving for at least $t$ more years. Assume a constant force of interest $\delta$, so a payment at time $t$ has discount factor $e^{-\delta t}$. A unit continuous whole-life annuity pays at rate 1 per year while the annuitant is alive.

Definition: The expected present value of a unit continuous whole-life annuity for a life aged $x$ is $$ \bar{a}_x=E\left[\int_0^{T_x}e^{-\delta t}\,dt\right] =\int_0^\infty e^{-\delta t}{}_tp_x\,dt. $$ A level annuity paying continuously at rate $c$ per year has value $c\bar{a}_x$. The integral is finite when the discounted survival probabilities have a finite integral; this holds, for example, when $\delta>0$.

The notation $\bar{a}_x$ distinguishes continuous payments from annuity-due or annuity-immediate payments made at discrete dates. The rate $c$ is an amount per year, not an amount paid once per year. For example, a rate of $1,200 per year means that $100 accrues over each month, but the valuation is performed continuously rather than as twelve monthly payments.

Why the Survival-Function Formula Holds

For a particular future lifetime $T_x$, the present value is the accumulated discounted payment rate from time zero up to death. To take its expectation, view that payment at each time $t$ as present only if the annuitant is alive then. The indicator of survival to $t$ is $1_{\{T_x>t\}}$, whose expectation is ${}_tp_x$.

E\left[\int_0^\infty e^{-\delta t}1_{\{T_x>t\}}\,dt\right] =\int_0^\infty e^{-\delta t}E[1_{\{T_x>t\}}]\,dt =\int_0^\infty e^{-\delta t}{}_tp_x\,dt.

The exchange of expectation and integration is valid because the integrand is nonnegative. At the instant of death, whether a payment is treated as ending immediately before or at death does not change the integral: a single instant has zero duration. The result assumes a level rate paid only while alive, with no guaranteed payment period or benefit after death.

Using a Force of Mortality to Obtain Survival

If the force of mortality at duration $t$ is $\mu_{x+t}$, then the survival probability from time zero to duration $t$ is

{}_tp_x=\exp\left(-\int_0^t\mu_{x+s}\,ds\right).

This is an unconditional probability from the original valuation date. If the force changes at a duration $r$, survival after $r$ must still include the probability of having survived to $r$. Multiplying that probability by conditional survival after $r$ prevents a common breakpoint error.

Specify the benefit and interest basis.
Identify the continuous payment rate $c$ and the force of interest $\delta$.

Write unconditional survival.
Use the given survival function or derive it from the force of mortality, including survival through every earlier interval.

Set up the valuation integral.
Multiply survival at time $t$ by $e^{-\delta t}$, and integrate over the whole lifetime. Split the integral at any change in the survival model.

Scale and check.
Multiply the unit value by $c$. Confirm that survival is 1 at time zero, is continuous at any breakpoint, and does not increase with time.

Worked Valuations

Worked Example: Constant Force of Mortality

A hypothetical contract pays continuously at a rate of $2,000 per year while a life aged $x$ survives. The force of interest is $\delta=0.06$, and the force of mortality is constant at $\mu_{x+t}=0.04$ for all $t\geq0$. Find the expected present value. The equivalent effective annual interest rate is $e^{0.06}-1\approx6.1837\%$.

Method. A constant force of mortality $\mu$ gives ${}_tp_x=e^{-\mu t}$. Substituting this survival function into the whole-life continuous annuity integral gives

\bar{a}_x =\int_0^\infty e^{-0.06t}e^{-0.04t}\,dt =\int_0^\infty e^{-0.10t}\,dt.

For $k>0$, $\int_0^\infty e^{-kt}\,dt=1/k$, since the antiderivative is $-e^{-kt}/k$ and $e^{-kt}$ tends to zero as $t$ tends to infinity. Therefore,

\bar{a}_x=\frac{1}{0.10}=10.

The contract pays at rate $2,000, so its expected present value is $2{,}000(10)=\$20{,}000.00$. Check: discounting and mortality combine in the integrand as $e^{-(0.06+0.04)t}$; the resulting unit value is 10 years of payment rate, giving a dollar value of $20,000.00. The common mistake avoided is to use the mortality rate alone as the denominator, which would omit interest discounting.

Worked Example: Mortality Force Changes After Two Years

A hypothetical annuity pays continuously at a rate of $750 per year for life. The force of interest is $\delta=0.03$. The force of mortality is 0.04 during the first two years and 0.10 thereafter. Find the expected present value.

Method. For $0\leq t\leq2$, survival is ${}_tp_x=e^{-0.04t}$. For $t\geq2$, first account for surviving the initial two years, then apply the later force to the remaining duration:

{}_tp_x=e^{-0.04(2)}e^{-0.10(t-2)} =e^{-0.08-0.10t+0.20} =e^{0.12-0.10t},\qquad t\geq2.

At the breakpoint, this expression gives $e^{0.12-0.20}=e^{-0.08}$, exactly the survival probability at two years. The unit annuity value is the sum of the two duration segments:

\begin{aligned} \bar{a}_x &=\int_0^2 e^{-0.03t}e^{-0.04t}\,dt +\int_2^\infty e^{-0.03t}e^{0.12-0.10t}\,dt\\ &=\frac{1-e^{-0.14}}{0.07} +e^{0.12}\frac{e^{-0.13(2)}}{0.13}\\ &=\frac{1-e^{-0.14}}{0.07}+\frac{e^{-0.14}}{0.13}\\ &\approx1.86631093+6.68737104\\ &\approx8.55368197. \end{aligned}

The annuity’s expected present value is $750(8.55368197\ldots)=\$6{,}415.26$, rounded to the nearest cent. Check: the second segment uses the unconditional survival probability, including survival through the first two years. The common mistake avoided is using $e^{-0.10(t-2)}$ alone after year two; that is conditional survival from year two and incorrectly resets survival to 1 at the breakpoint.

Worked Example: A Mixture of Two Mortality Groups

A hypothetical portfolio contains two groups of annuitants. A randomly selected member belongs to group 1 with probability 0.60 and group 2 with probability 0.40. Their constant forces of mortality are 0.03 and 0.08, respectively. The force of interest is $\delta=0.02$, and the annuity pays continuously at a rate of $1,250 per year. Find the expected present value for a randomly selected member.

Method. Conditional on group 1, survival to time $t$ is $e^{-0.03t}$; conditional on group 2, it is $e^{-0.08t}$. The unconditional survival probability is the probability-weighted sum:

{}_tp_x=0.60e^{-0.03t}+0.40e^{-0.08t}.

Insert this mixture into the valuation integral and integrate each component:

\begin{aligned} \bar{a}_x &=\int_0^\infty e^{-0.02t} \left(0.60e^{-0.03t}+0.40e^{-0.08t}\right)\,dt\\ &=0.60\int_0^\infty e^{-0.05t}\,dt +0.40\int_0^\infty e^{-0.10t}\,dt\\ &=0.60\left(\frac{1}{0.05}\right) +0.40\left(\frac{1}{0.10}\right)\\ &=0.60(20)+0.40(10)=16. \end{aligned}

Thus the expected present value is $1{,}250(16)=\$20{,}000.00$. Check: the mixture survival is 1 at time zero and is a weighted average of valid survival functions. The common mistake avoided is replacing the two mortality forces by their weighted average and then treating that average as the force for the whole portfolio; in general, a mixture of exponential survival functions is not a single exponential survival function.

Common Checks and Exam Traps

The survival function and the payment rate are both essential to the valuation. A correctly integrated discount factor cannot compensate for a survival function that has been restarted at a breakpoint, and a correct unit annuity value is not yet the value of a contract whose payment rate differs from 1.

Restarting survival at a breakpoint. For $t\geq r$, use survival to $r$ multiplied by conditional survival from $r$ to $t$. The unconditional survival probability must not jump back to 1.
Omitting discounting. The integrand is $e^{-\delta t}{}_tp_x$, not survival alone, unless the force of interest is zero.
Confusing a rate with an instalment. A continuous rate of $c$ dollars per year means the annuity value is $c\bar{a}_x$. It does not mean that $c$ is paid once at the end of each year.
Using the wrong mortality basis for a mixture. Average survival probabilities using the group weights before integrating, or integrate each conditional survival function and then weight the values. Do not assume that the weighted average force gives the mixture survival.
Forgetting the whole-life horizon. The upper limit is infinity. A finite upper limit would value a temporary annuity instead.

The survival-integral approach is useful because it applies directly to any specified survival function for which the integral can be evaluated. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus: candidates should be able to translate a continuous payment stream into an expected present value, preserve the correct survival probabilities, and scale the unit value to the stated payment rate.

Key takeaway: For level payments at rate $c$, value a continuous whole-life annuity by integrating $c e^{-\delta t}{}_tp_x$ from zero to infinity. When mortality changes, build unconditional survival correctly before integrating.

Check Your Understanding

For each question, use the survival-integral method. Unless stated otherwise, payments are continuous and cease at death.

A hypothetical life has constant force of mortality 0.05 and force of interest 0.05. The annuity pays at a rate of $1,000 per year. Find its expected present value. (Answer: $10,000.00)
The force of mortality is 0.02 for the first year and 0.06 thereafter. The force of interest is 0.04. Find the unit whole-life continuous annuity value, to four decimal places. (Answer: 10.3882)
A portfolio has two groups, with weights 0.25 and 0.75 and constant mortality forces 0.04 and 0.09, respectively. If $\delta=0.01$, find the unit annuity value. (Answer: 12.5000)
A life has constant force of mortality 0.07, and the force of interest is 0.03. A hypothetical annuity pays continuously at a rate of $300 per year. Find its expected present value. (Answer: $3,000.00)

What You'll Learn