Separating Active and Retired Mortality Bases

Why Two Mortality Bases May Be Needed

A deferred retirement pension depends on two different survival questions. First, does the member remain active and eligible until the retirement date? Second, after retirement, does the pensioner survive to each payment date? These questions can have different answers under the assumptions used for active members and retired pensioners.

The distinction matters even when both calculations use annual probabilities. Active-member survival may reflect remaining in active service without any competing exit, while post-retirement survival is governed by the mortality basis applicable to pensioners. Using one basis throughout can misstate the probability of receiving a pension payment.

Definition: Let ${}_np_x^{A}$ denote the probability that a member aged $x$ at the valuation date remains active and eligible for the specified retirement benefit for $n$ years. It accounts for the active-state survival condition in the model. Let ${}_kp_{x+n}^{R}$ denote the probability that a person who retires at age $x+n$ survives for $k$ further years under the retired-life mortality basis. The superscripts $A$ and $R$ distinguish the two bases; they are not interchangeable.

For a benefit payable only to a member who reaches a fixed retirement date $n$ years from now, the active basis applies up to that date. The retired basis applies from retirement onward. This setup assumes that the member retires at the specified date if still eligible and active, that the stated annual probabilities are conditional on being in the relevant state at the start of each year, and that pension payments occur at the ends of years after retirement.

Valuation Formula and Payment Timing

Suppose the pension pays $B_k$ at the end of the $k$th year after retirement, for $k=1,\ldots,m$. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. A payment at that date is made only if the member reaches retirement and then survives for $k$ years as a pensioner. Under the stated model, its expected present value at the valuation date is:

\operatorname{EPV}_0 = {}_np_x^{A} \sum_{k=1}^{m} B_k\,v^{n+k}\,{}_kp_{x+n}^{R}.

The exponent $n+k$ discounts from the payment date all the way back to time 0. The probability factor is the product of active survival to retirement and retired survival from retirement to the payment date. If the pension is payable for life, the sum extends over all possible payment years, subject to the relevant survival probabilities and payment terms.

It can also be useful to value the pension conditionally at the retirement date. This value excludes the probability of reaching retirement and the discounting from time 0 to retirement:

V_{\text{ret}} = \sum_{k=1}^{m} B_k\,v^k\,{}_kp_{x+n}^{R}, \qquad \operatorname{EPV}_0={}_np_x^{A}v^nV_{\text{ret}}.

These expressions assume the future pensioner’s mortality is described by the retired-life basis conditional on reaching retirement. If the benefit amount depends on the path taken before retirement, or if retirement can occur at several dates, the calculation must reflect those additional conditions rather than use this fixed-date formula unchanged.

Why the Probability Factors Multiply

The multiplication follows from conditioning on reaching retirement. Let $A$ be the event that the member remains active and eligible to time $n$, and let $R_k$ be the event that the retiree survives a further $k$ years. The payment at time $n+k$ requires both events. By the conditional-probability identity,

P(A\cap R_k)=P(A)P(R_k\mid A) ={}_np_x^{A}\,{}_kp_{x+n}^{R}.

The conditional term is evaluated using the retired basis because $R_k$ begins at retirement. Multiplying this joint payment probability by the amount $B_k$ and discount factor $v^{n+k}$, then summing over payment dates, gives the EPV formula. This argument does not require active and retired mortality probabilities to be equal; it requires that the retired survival assumption be the appropriate conditional basis after retirement.

Establish the retirement date.
Count the years from the valuation date to the specified retirement date. This determines the active period.

Find active survival to retirement.
Use the active-member basis to obtain the probability of remaining eligible and active through that period.

Value payments conditional on retirement.
For each payment, use retired survival to its payment date and discount for the years after retirement.

Bring the value back to time 0.
Multiply the conditional retirement value by active survival and discount to the retirement date, or use the full time-0 formula payment by payment.

Worked Valuations

Worked Example: A Two-Payment Pension After One Year

A hypothetical member is age 60 now and is eligible for a pension if still active at age 61, one year from now. The pension pays $5,000 at the end of the first year after retirement and $6,000 at the end of the second year after retirement. Use an effective annual interest rate of $25\%$. In the annual discrete model, the probability of remaining active for the year to retirement is $0.90$. Conditional on retirement, the probabilities of surviving the first and second years after retirement are $0.80$ and $0.70$, respectively. Find the EPV at time 0.

Method. First calculate the pension value at retirement using retired survival, then multiply by the probability of reaching retirement and discount one year. This avoids applying active-member survival to payments after retirement.

At retirement, the first payment is one year away, so its conditional value is:

\$5{,}000(0.80)v = \$5{,}000(0.80)\left(\frac{1}{1.25}\right) = \$3{,}200.

The second payment requires survival through both post-retirement years. Its conditional value is:

\$6{,}000(0.80)(0.70)v^2 = \$6{,}000(0.56)\left(\frac{1}{1.25^2}\right) = \$2{,}150.40.

Thus the value conditional on reaching retirement is $\$3{,}200+\$2{,}150.40=\$5{,}350.40$. The probability of reaching retirement is $0.90$, and the one-year discount factor is $v=0.80$. Therefore:

\operatorname{EPV}_0 = 0.90(0.80)(\$5{,}350.40) = \$3{,}852.288 \approx \boxed{\$3{,}852.29}.

As a direct check, the first payment contributes $5{,}000(0.90)(0.80)/1.25^2=\$2{,}304$, and the second contributes $6{,}000(0.90)(0.80)(0.70)/1.25^3=\$1{,}548.288$. Their sum is $\$3{,}852.288$, as above. The final dollar value is rounded to the nearest cent.

Worked Example: Quantifying the Error From Reusing Active Mortality

A hypothetical member has two years to a fixed retirement date. The annual probability of remaining active is $0.90$ in each of those two years. After retirement, the annual probability of surviving each year is $0.95$. A pension of $10,000 is paid at the end of each of the first two years after retirement, provided the pensioner is alive. Use an effective annual interest rate of $25\%$, with annual discrete survival and end-of-year payments. Calculate the correct time-0 EPV and compare it with the result from incorrectly continuing the active basis after retirement.

Method. Apply active survival only for the two pre-retirement years and retired survival for each post-retirement payment. Define the discounted annual factors $0.90/1.25=0.72$ for an active year and $0.95/1.25=0.76$ for a retired year.

The first payment is three years from time 0. Its factor is $0.72^2(0.76)$, and the second payment’s factor is $0.72^2(0.76)^2$. Therefore:

\begin{aligned} \operatorname{EPV}_{\text{correct}} &= \$10{,}000(0.72)^2\left(0.76+0.76^2\right)\\ &= \$10{,}000(0.5184)(0.76+0.5776)\\ &= \$10{,}000(0.69341184) = \$6{,}934.1184 \approx \boxed{\$6{,}934.12}. \end{aligned}

If active survival were incorrectly used for the two pension years as well, the factor would instead be $0.72^3+0.72^4$:

\$10{,}000\left(0.72^3+0.72^4\right) = \$10{,}000(0.373248+0.26873856) = \$6{,}419.8656 \approx \$6{,}419.87.

The incorrect basis understates the EPV by $\$6{,}934.1184-\$6{,}419.8656=\$514.2528$, or $\$514.25$ to the nearest cent. The discrepancy occurs because the calculation treated pensioner survival as if it followed the lower active-member survival probability.

Worked Example: Different Active Survival by Year and a Changing Pension

A hypothetical member will retire in two years if still active and eligible. The probability of active survival is $0.95$ in the first year and $0.90$ in the second year. After retirement, the probability of surviving the first pension year is $0.95$, and the conditional probability of surviving the second pension year is $0.90$. The pension pays $10,000 at the end of its first year and $12,000 at the end of its second year, if alive. Assume annual discrete transitions, an effective annual interest rate of $25\%$, and no other benefit conditions. Find the time-0 EPV.

Method. Construct each payment factor from the required survival path and discount period. In discounted-factor form, the active-year factors are $0.95/1.25=0.76$ and $0.90/1.25=0.72$. The first retired-year factor is also $0.95/1.25=0.76$, and the second is $0.90/1.25=0.72$.

The first payment requires survival through both active years and the first retired year. Its factor is:

$$ (0.76)(0.72)(0.76)=0.415872. $$

The second payment additionally requires survival through the second retired year. Its factor is:

$$ (0.76)(0.72)(0.76)(0.72)=0.29942784. $$

The EPV is therefore:

\begin{aligned} \operatorname{EPV}_0 &= \$10{,}000(0.415872)+\$12{,}000(0.29942784)\\ &= \$4{,}158.72+\$3{,}593.13408\\ &= \$7{,}751.85408 \approx \boxed{\$7{,}751.85}. \end{aligned}

The calculation uses the second-year active probability only before retirement and the second-year retired probability only after retirement. It also uses each benefit amount at its own payment date. The final value is rounded to the nearest cent.

Scope, Variations, and Exam Traps

Separating the bases is especially important when active membership and pensioner status have different assumed mortality. The active survival probability may be obtained from an active-state or competing-exit model; it represents the probability required by the benefit condition, not necessarily the probability of avoiding death alone. The retired survival probability begins at the retirement date and should be conditional on being a pensioner then.

The fixed-retirement formula is not itself a model for the probability of retiring at that date. If retirement can occur at multiple times, the value generally has contributions for each possible retirement date. Each contribution uses the probability of reaching that retirement date under the active model and then the appropriate retired survival from that date. Likewise, if a pension is payable on retirement rather than at a later year-end, its timing and survival condition must be adjusted; do not automatically include an extra year of retired survival.

Common mistake: continuing active mortality after retirement. This substitutes an active-state survival probability for retired-life survival. The second worked example shows the resulting understatement.
Common mistake: forgetting the pre-retirement discount period. A value calculated at retirement must be multiplied by $v^n$ when brought back $n$ years to time 0.
Common mistake: discounting only from retirement. In the time-0 formula, a payment $k$ years after retirement is discounted for $n+k$ years, not just $k$.
Common mistake: using cumulative retired survival incorrectly. If annual probabilities are given conditionally by year, multiply them to obtain survival through multiple years. For example, survival for two years is the first-year probability times the conditional second-year probability.
Common mistake: adding a retirement probability twice. If the active survival probability already represents remaining eligible and active until the fixed retirement date, do not multiply by a separate probability of being active again.

Key takeaway: For a fixed retirement date, use the active basis to determine the probability of reaching retirement and the retired basis to determine survival to each pension payment. Match each probability and discount factor to the time interval it covers.

Check Your Understanding

For each problem, use the stated annual discrete survival assumptions and end-of-year payment timing. Round monetary answers to the nearest cent.

A hypothetical member has one year until retirement. Active survival to retirement is $0.95$. A $1,000 pension payment is due at the end of the first year after retirement, conditional on retired survival of $0.98$ for that year. At $i=5\%$, find the time-0 EPV. (Answer: $844.44)
A hypothetical member has two years until retirement, with active survival probability $0.90$ in each year. The pension pays $4,000 at the end of its first year, and retired survival for that year is $0.80$. At $i=25\%$, find the time-0 EPV. (Answer: $1,327.10)
A member has one year until retirement, with active survival probability $0.90$. At $i=25\%$, a hypothetical pension pays $5,000 and $4,000 at the ends of its first and second years, respectively. Retired survival probabilities for the first and second years are $0.80$ and $0.75$. Find the time-0 EPV. (Answer: $3,409.92)
A member retires in two years if eligible. Active survival probabilities are $0.95$ and $0.90$; retired survival probabilities for the first and second pension years are $0.95$ and $0.90$. At $i=25\%$, the pension pays $10,000 and $12,000 at the ends of those two years. Find the conditional value at retirement and the time-0 EPV. (Answer: retirement value $14,166.40; time-0 EPV $7,751.85)
Use the assumptions in Question 4, but suppose an analyst applies active survival probability $0.90$ in both pre-retirement years and both pension years, with $i=25\%$ and two $10,000 pension payments. What is the correct EPV using the stated active survival of $0.90$ and retired survival of $0.95$ each year, and what is the analyst’s incorrectly calculated EPV? (Answer: correct $6,934.12; incorrect $6,419.87)

This separation of mortality bases is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

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