Deriving Conditional Retirement Annuity Values

Why Condition on Retirement?

A pension valuation may first consider whether a member reaches retirement and then consider payments after retirement. These are distinct questions. A conditional retirement-annuity value is the expected present value of the pension payments, given that the member has retired at the specified age. It uses survival after retirement; it does not, by itself, include the probability that an active member retires.

This distinction is useful when a pension is valued under competing exits. Earlier material on competing exits can determine the probability and timing of retirement. Once retirement at a specified age is given, the annuity calculation starts from that retirement date and conditions on the retiree being alive then. The calculations below use annual payments and an effective annual interest rate.

Definition: Suppose a person retires at age $x_r$. Write ${}_kp_{x_r}$ for the probability that the person survives for $k$ years after that age, conditional on being alive at age $x_r$. For an annual pension of 1 payable at each year-end while the retiree is alive, the conditional value at retirement is the expected present value of those payments, using the post-retirement survival probabilities and discount factors.

Assume the annual effective interest rate is $i$, so $v=1/(1+i)$. If the annual pension amount is $B$, payments are made in arrears (at the end of each year), and the pension stops on death, its value for at most $n$ years is

B a_{\overline{n}|x_r} = B\sum_{k=1}^{n}v^k\,{}_kp_{x_r}.

Here $a_{\overline{n}|x_r}$ is defined by this sum: it is the value at retirement of a temporary annual annuity-immediate, conditional on retirement at age $x_r$. If payments continue for life, the corresponding value is

B a_{x_r} = B\sum_{k=1}^{\infty}v^k\,{}_kp_{x_r},

provided the series converges. In either case, the $k$-th payment is included only if the retiree survives to the end of year $k$. A payment made immediately at retirement changes the timing: the corresponding annuity-due value for at most $n$ annual payments is

B\ddot{a}_{\overline{n}|x_r} = B\sum_{k=0}^{n-1}v^k\,{}_kp_{x_r}, \qquad {}_0p_{x_r}=1.

These formulas assume the stated payment dates, an annual effective interest rate, and payments that stop at death. They do not directly value a pension with a guaranteed payment period, a survivor benefit after the retiree’s death, or a different payment frequency; those features change which payments are made and when.

Building the Survival Probabilities

If annual post-retirement survival probabilities are given by year, multiply them to obtain survival over multiple years. For example, if the probability of surviving the first year is $p_1$, and the conditional probability of surviving the second year given survival through the first is $p_2$, then ${}_2p_{x_r}=p_1p_2$. In general, with $p_j$ the conditional probability of surviving post-retirement year $j$,

{}_kp_{x_r}=\prod_{j=1}^{k}p_j.

The product follows from the multiplication rule for conditional probabilities: to survive $k$ years, the retiree must survive year 1, then year 2 given survival through year 1, and so on. If a single annual survival probability $p$ applies in every year, then ${}_kp_{x_r}=p^k$. This constant-survival assumption is a simplifying model, not a general property of mortality.

Fix the valuation date.
For a conditional value, take the retirement date as time zero. Do not include the probability of reaching retirement in this value.

Set the payment dates.
For payments in arrears, the first possible payment is at time 1. For payments in advance, the first payment is at time 0.

Calculate survival to each payment date.
Use the probability of being alive when that payment is due, conditional on being alive at retirement.

Discount and add.
Multiply each possible payment by its survival probability and discount factor, then sum the expected present values.

Worked Values

Worked Example: A Three-Year Pension in Arrears

A hypothetical pension pays $10,000 at each of the first three year-ends after retirement, provided the retiree is alive on the payment date. The effective annual interest rate is $i=10\%$. The conditional probabilities of surviving each post-retirement year are 0.90, 0.90, and 0.90. Find the value at retirement, rounded to cents.

Method. This is a three-year temporary annuity-immediate. First find cumulative survival: ${}_1p_{x_r}=0.90$, ${}_2p_{x_r}=0.90(0.90)=0.81$, and ${}_3p_{x_r}=0.81(0.90)=0.729$. The discount factor is $v=1/1.10=10/11$.

\begin{aligned} \text{Value} &=10{,}000\left[0.90v+0.81v^2+0.729v^3\right]\\ &=10{,}000\left[\frac{9}{11}+\frac{81}{121}+\frac{729}{1331}\right]\\ &=10{,}000\left[\frac{1089+891+729}{1331}\right]\\ &=10{,}000\left[\frac{2709}{1331}\right]\\ &=\$20{,}353.12. \end{aligned}

The result is below the undiscounted total of $30,000 because payments may not be made if death occurs before a payment date, and because future payments are discounted. Common mistake avoided: using the one-year survival probability for all three payments. The second and third payments require survival through two and three years, respectively.

Worked Example: An Annuity-Due With Two Survival Intervals

A hypothetical pension pays $10,000 immediately on retirement and again at each of the next two anniversaries while the retiree is alive. The annual effective interest rate is $i=5\%$, and conditional annual survival probabilities are 0.96 and 0.92 for the first and second post-retirement years. Find the value at retirement, rounded to cents.

Method. The immediate payment is certain conditional on retirement, so its survival factor is ${}_0p_{x_r}=1$. The year-1 survival probability is 0.96. Survival to the second anniversary is $0.96(0.92)=0.8832$. With $v=1/1.05=20/21$, the three payment terms are

\begin{aligned} \text{Value} &=10{,}000\left[1+0.96v+0.8832v^2\right]\\ &=10{,}000\left[1+\frac{19.2}{21}+\frac{353.28}{441}\right]\\ &=10{,}000\left[1+0.9142857143+0.8010884354\right]\\ &=10{,}000(2.7153741497)\\ &=\$27{,}153.74. \end{aligned}

The payment at retirement has neither a discount factor below 1 nor a mortality adjustment: it is paid immediately, given retirement. Common mistake avoided: treating the first payment as one year after retirement. That would incorrectly replace the initial 1 in the bracket with a discounted, survival-weighted term.

Worked Example: Whole-Life Value and the Retirement Probability

In a hypothetical model, a pension pays $8,000 at each year-end for as long as the retiree is alive. After retirement, the conditional probability of surviving each year is 0.90, and the annual effective interest rate is $i=5\%$. Find the conditional value at retirement. Then suppose the probability of retiring exactly five years from the current valuation date is 0.20. Find the current expected present-value contribution from that retirement event, assuming no other payment is associated with it.

Conditional value at retirement. Under the stated constant-survival assumption, ${}_kp_{x_r}=0.90^k$. Thus each term in the annuity-immediate series has ratio $0.90v=0.90/1.05=6/7$. The infinite geometric series converges because $6/7<1$, and

\begin{aligned} a_{x_r} &=\sum_{k=1}^{\infty}\left(\frac{6}{7}\right)^k =\frac{6/7}{1-6/7} =6,\\ \text{Conditional value at retirement} &=8{,}000(6)=\$48{,}000.00. \end{aligned}

Contribution at the current date. The five-year discount factor is $v^5=(20/21)^5=0.7835261665$. The retirement event has probability 0.20, so its probability-weighted, discounted contribution is

0.20(0.7835261665)(\$48{,}000)=\$7{,}521.85.

The $48,000 is the annuity value conditional on retirement; the $7,521.85 is the contribution at an earlier valuation date for the specified event. Common mistake avoided: multiplying the conditional annuity value by 0.20 and calling that the value at retirement. The retirement probability belongs in a value taken before the retirement event, not in the conditional value itself.

Why the Formula Is an Expected Present Value

For a finite annuity-immediate, define the payment at time $k$ to be $B$ if the retiree is alive at that time and zero otherwise. Its present value is therefore $Bv^k$ times an indicator that equals 1 when the retiree survives to time $k$, and 0 otherwise. Taking the conditional expectation given survival to retirement gives

E[\text{payment at time }k\text{, discounted}\mid\text{alive at retirement}] = Bv^k\,{}_kp_{x_r}.

The present value of the whole pension is the sum of the discounted payments. By linearity of expectation, the expected present value is the sum of the expected present values of those payments. Hence

E[\text{present value}\mid\text{alive at retirement}] = \sum_{k=1}^{n}Bv^k\,{}_kp_{x_r} = B\sum_{k=1}^{n}v^k\,{}_kp_{x_r}.

This argument does not require payment indicators at different dates to be independent. Survival to a later payment date already implies survival to each earlier date. For an annuity-due, the same argument includes the time-zero payment, which is certain under the conditioning, and shifts the remaining payment dates accordingly.

Timing and Conditioning: The Main Checks

Before applying a formula, state which event makes each payment payable. For a pension paid while the retiree is alive, a payment at year-end $k$ is weighted by survival to that date, ${}_kp_{x_r}$, not by the probability of dying during the year. Then match that date to its discount factor $v^k$. In particular, annuity-immediate payments start at $k=1$, while an annuity-due includes $k=0$.

Feature	Payment in arrears	Payment in advance
First payment date	One year after retirement	At retirement
First survival factor	${}_1p_{x_r}$	${}_0p_{x_r}=1$
First discount factor	$v$	1
Later payment at time $k$	$Bv^k{}_kp_{x_r}$	$Bv^k{}_kp_{x_r}$, with the due schedule ending one year earlier for the same number of payments

A second check is whether the requested value is conditional or unconditional. If the question asks for the value at retirement, use the post-retirement annuity formula without multiplying by the probability of retirement. If it asks for a value at an earlier date, the appropriate retirement probability and discounting to that date may also be required. The result is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: At a specified retirement age, value each annual payment by multiplying its amount by the probability of being alive on its payment date and by the discount factor for that date. Condition on retirement first; include the probability of reaching retirement only when valuing from an earlier date.

Check Your Understanding

For each problem, identify the payment timing and use survival conditional on being alive at retirement.

A hypothetical pension pays $5,000 at the first two year-ends after retirement if the retiree is alive. The annual effective interest rate is 4%, and conditional annual survival probabilities are 0.80 and 0.75. Find the value at retirement, to cents. (Answer: $6,619.82.)
A hypothetical pension pays $6,000 immediately at retirement and at the next two anniversaries while the retiree is alive. The annual effective interest rate is 5%, and the first two annual survival probabilities are 0.90 and 0.80. Find the value at retirement, to cents. (Answer: $15,061.22.)
A hypothetical whole-life pension pays $1,000 at each year-end while the retiree is alive. Annual conditional survival is 0.92 in every year, and $i=4\%$. Find the conditional value at retirement, to cents. (Answer: $7,666.67.)
A hypothetical pension pays $4,000 at each of the first three year-ends while the retiree is alive. Annual conditional survival is 0.90 in each year, and $i=5\%$. Find the value at retirement, to cents. (Answer: $8,886.30.)
A hypothetical retirement event has probability 0.25 of occurring exactly four years from now. Conditional on retirement, the pension value at that date is $30,000. If $i=5\%$, find the current probability-weighted present-value contribution from this event, to cents. (Answer: $6,170.27.)

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