Integrated Pension Valuation with Competing Exits

Introduction

A pension valuation with several possible exits is not a collection of separate one-cause calculations. In each year, a member who is active at the start may retire, withdraw, become disabled, die, or remain active. The value depends both on which event occurs and on the member’s age, salary, and service when it occurs. A reliable calculation must keep those pieces on the same time line.

This tutorial develops an integrated valuation method for hypothetical pension benefits. It uses annual competing-decrement probabilities conditional on being active at the beginning of each year. Exit benefits are paid at the end of the year of exit unless stated otherwise. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: In a multiple-decrement model, $q_j(t)$ is the probability that an active member at time $t$ exits during the next year from cause $j$. The causes are mutually exclusive, so $p(t)=1-\sum_j q_j(t)$ is the probability of remaining active through the year. The probability of exiting from cause $j$ in a later year is conditional on surviving active through every preceding year.

Set the Valuation Basis Before Calculating

For a discrete annual model, let $B_{k,j}$ be the value, at the end of year $k$, of the benefit triggered by cause $j$. This may be a cash lump sum or a retirement pension’s conversion value at the date the pension commences. Let $P_{k-1}$ be the probability that the member is still active at the beginning of year $k$. With an effective annual interest rate $i$, the expected present value is

\operatorname{EPV} =\sum_{k\geq 1} v^k P_{k-1}\sum_j q_j(k)B_{k,j}, \qquad v=\frac{1}{1+i}, \qquad P_{k-1}=\prod_{r=1}^{k-1}\left(1-\sum_jq_j(r)\right).

The inner sum is the expected benefit conditional on being active at the beginning of year $k$. The factor $P_{k-1}$ converts it to an unconditional expected benefit. The factor $v^k$ discounts a payment made at the end of that year to the valuation date.

If retirement triggers an annual pension, first determine the pension amount at retirement using the plan formula. Its conversion value is that amount multiplied by an annuity factor appropriate to the retirement age and payment pattern. That conversion value is measured at the retirement date. Do not discount it from retirement to the valuation date until applying the year-of-exit discount factor.

Worked Valuation: Four Exit Causes Over Two Years

Worked Example: A Two-Year Integrated Pension Package

A hypothetical member is age 60 with 10 years of service and annual salary of $50,000 at the valuation date. Salary increases by 4% at the start of each year. The annual retirement pension is 1.5% of salary at retirement multiplied by completed service. The pension conversion factors are 10.0 at age 61 and 9.6 at age 62. They are annuity-due factors, valued at the retirement date on the plan’s post-retirement basis.

At the end of a year, withdrawal pays $2,000 plus $500 per completed service year. Disability pays a pension equal to 20% of salary at exit, with conversion factors 8.0 at age 61 and 7.7 at age 62. Death pays twice salary at exit. Assume annual interest $i=5\%$. The multiple-decrement probabilities, conditional on being active at the start of each year, are:

Year 1: retirement 0.10, withdrawal 0.08, disability 0.04, death 0.02. Year 2: retirement 0.12, withdrawal 0.06, disability 0.05, death 0.03. All exit benefits are valued at the end of the exit year.

Step 1: Determine the active-member probabilities. In year 1, the total decrement probability is $0.10+0.08+0.04+0.02=0.24$, so the probability of remaining active is $1-0.24=0.76$. Thus $P_0=1$ and $P_1=0.76$. The year 2 probabilities are conditional on being active at its start.

Step 2: Calculate the year 1 benefits. At exit the member is age 61, has 11 years of service, and salary $50{,}000(1.04)=\$52{,}000$. The annual retirement pension is $0.015(52{,}000)(11)=\$8{,}580$, giving a retirement conversion value of $8{,}580(10.0)=\$85{,}800$. The withdrawal benefit is $2{,}000+500(11)=\$7{,}500$. The disability conversion value is $0.20(52{,}000)(8.0)=\$83{,}200$. The death benefit is $2(52{,}000)=\$104{,}000$.

The expected benefit conditional on being active at the start of year 1 is $0.10(85{,}800)+0.08(7{,}500)+0.04(83{,}200)+0.02(104{,}000)=\$14{,}588$. Its present value is $14{,}588/1.05=\$13{,}893.33$.

Step 3: Calculate the year 2 benefits. At exit the member is age 62, has 12 years of service, and salary $52{,}000(1.04)=\$54{,}080$. The annual retirement pension is $0.015(54{,}080)(12)=\$9{,}734.40$; its conversion value is $9{,}734.40(9.6)=\$93{,}450.24$. The withdrawal benefit is $2{,}000+500(12)=\$8{,}000$. The disability conversion value is $0.20(54{,}080)(7.7)=\$83{,}283.20$. The death benefit is $2(54{,}080)=\$108{,}160$.

Conditional on being active at the start of year 2, the expected benefit is $0.12(93{,}450.24)+0.06(8{,}000)+0.05(83{,}283.20)+0.03(108{,}160)=\$19{,}102.9888$. The unconditional probability weight includes $P_1=0.76$. Thus the year 2 present value is $0.76(19{,}102.9888)/(1.05)^2=\$13{,}168.50$.

Step 4: Add the discounted years. The total is $\$13{,}893.33+\$13{,}168.50=\$27{,}061.83$, rounded to the nearest cent. This is the value at the initial date of the stated hypothetical benefits over the two-year active period.

Check. Each year’s cause probabilities sum to at most one: 0.24 in year 1 and 0.26 in year 2. The year 2 amount is weighted by 0.76 because only that fraction is expected to reach year 2 still active.

Why the Probability and Discount Factors Appear

The formula follows by conditioning on the year and cause of exit. For year $k$, the probability of reaching its start active is $P_{k-1}$. Given that event, the probability of exit from cause $j$ in that year is $q_j(k)$. Therefore the joint probability of reaching year $k$ active and then exiting from cause $j$ is $P_{k-1}q_j(k)$. The benefit is paid at time $k$, so its present value in that event is $v^kB_{k,j}$. Multiplying probability by present value and summing over all years and causes gives the EPV.

P(\text{exit from }j\text{ in year }k) =P(\text{active at start of }k)\, P(\text{exit from }j\mid\text{active at start of }k) =P_{k-1}q_j(k).

The product form for $P_{k-1}$ follows for the same reason: reaching year $k$ active requires remaining active in each earlier year, and the probability of remaining active in year $r$, conditional on being active at its start, is $1-\sum_j q_j(r)$. This is a sequential conditional-probability calculation, not an assumption that the annual exit events are independent.

Worked Valuation: Recomputing Benefits From Salary and Service

Worked Example: Age- and Service-Sensitive Exit Values

A hypothetical member starts at age 58 with 7 years of service and salary $60,000. Salary increases by 3% annually. Retirement pays an annual pension of 1.4% of exit salary per service year; the conversion factors are 10.2 at age 59 and 9.9 at age 60. Withdrawal pays $1,200 plus $400 per service year. Disability pays 18% of exit salary as an annual pension, with conversion factors 7.5 at age 59 and 7.2 at age 60. Death pays 1.5 times exit salary. The annual interest rate is 4%.

The multiple-decrement probabilities are: year 1, retirement 0.09, withdrawal 0.05, disability 0.02, death 0.01; year 2, retirement 0.11, withdrawal 0.04, disability 0.03, death 0.02. Benefits are paid or valued at the end of the exit year. Find the two-year EPV.

Year 1 amounts. Salary is $60{,}000(1.03)=\$61{,}800$ and service is 8 years. Retirement pension is $0.014(61{,}800)(8)=\$6{,}921.60$, so its conversion value is $6{,}921.60(10.2)=\$70{,}600.32$. Withdrawal is $1{,}200+400(8)=\$4{,}400$. Disability is $0.18(61{,}800)(7.5)=\$83{,}430$, and death pays $1.5(61{,}800)=\$92{,}700$.

The year 1 conditional expected benefit is $0.09(70{,}600.32)+0.05(4{,}400)+0.02(83{,}430)+0.01(92{,}700)=\$9{,}169.6288$. The year 1 decrement total is 0.17, so the probability of reaching year 2 active is $1-0.17=0.83$.

Year 2 amounts. Salary is $61{,}800(1.03)=\$63{,}654$, and service is 9 years. Retirement pension is $0.014(63{,}654)(9)=\$8{,}020.404$, giving conversion value $8{,}020.404(9.9)=\$79{,}402.00$. Withdrawal is $1{,}200+400(9)=\$4{,}800$. Disability is $0.18(63{,}654)(7.2)=\$82{,}495.584$, and death pays $1.5(63{,}654)=\$95{,}481$.

The year 2 conditional expected benefit is $0.11(79{,}402.00)+0.04(4{,}800)+0.03(82{,}495.584)+0.02(95{,}481)=\$13{,}310.70752$. Therefore $\operatorname{EPV}=9{,}169.6288/1.04+0.83(13{,}310.70752)/(1.04)^2 =\$8{,}816.95+\$10{,}214.39=\$19{,}031.34$, rounded to the nearest cent.

Method and check. Recalculate each benefit at the age, salary, and service at the relevant exit, then weight by the conditional decrement probabilities. The exit probabilities sum to 0.17 and 0.20 in the two years, both valid totals. This avoids incorrectly using year 1 salary or service for a year 2 exit.

Common Exam Trap: Forgetting Survival to a Later Year

In the first worked example, the year 2 conditional expected benefit is $19,102.9888. It is not yet an unconditional year 2 value: 24% of members exit in year 1. The correct year 2 term is $0.76(19{,}102.9888)/(1.05)^2=\$13{,}168.50$.

A common wrong calculation omits the 0.76 and uses $19{,}102.9888/(1.05)^2=\$17{,}326.97$. This treats every member as active at the start of year 2. It overstates the total EPV as $\$13{,}893.33+\$17{,}326.97=\$31{,}220.30$, rather than $27,061.83. The correction is to multiply each later-year conditional expectation by the probability of reaching that year active.

A related basis error is to use a probability for a cause that is conditional on the member remaining exposed to that cause alone as if it were a multiple-decrement probability. The annual cause probabilities used together in the formula must refer to the same basis and conditioning event: active at the start of the year with all competing causes operating.

Worked Valuation: Reconcile Cause Totals Before Discounting

Worked Example: A Two-Year Cash-Value Check

A hypothetical pension plan assigns end-of-year exit values that already reflect the member’s age and service at exit. In year 1, the values for retirement, withdrawal, disability, and death are respectively $50,000, $4,000, $30,000, and $40,000. In year 2 they are $55,000, $5,000, $32,000, and $45,000. The annual multiple-decrement probabilities are $(0.10,0.05,0.03,0.02)$ in year 1 and $(0.12,0.04,0.02,0.02)$ in year 2. Assume $i=5\%$, and every stated value is paid at the end of the exit year.

Year 1. The conditional expected exit value is $0.10(50{,}000)+0.05(4{,}000)+0.03(30{,}000)+0.02(40{,}000)=\$6{,}900$. Its present value is $6{,}900/1.05=\$6{,}571.43$. The total decrement probability is 0.20, so the probability of reaching year 2 active is 0.80.

Year 2. Conditional on being active at its start, the expected exit value is $0.12(55{,}000)+0.04(5{,}000)+0.02(32{,}000)+0.02(45{,}000)=\$8{,}340$. The unconditional discounted value is $0.80(8{,}340)/(1.05)^2=\$6{,}051.70$.

Total and check. The EPV is $\$6{,}571.43+\$6{,}051.70=\$12{,}623.13$, rounded to the nearest cent. The year 2 contribution is lower than its conditional value because it is discounted and only 80% of members reach that year active. The method also verifies that the year 2 decrement probabilities are used conditionally, not as unconditional population probabilities.

Valuation Checklist

Before accepting an integrated pension value, check the event definitions and the timeline as carefully as the arithmetic. A conversion factor for a retirement pension belongs at the retirement date; the discount factor then moves that value to the valuation date. A benefit paid at year-end uses $v^k$, whereas a benefit paid at another time requires a discount factor consistent with that payment time and the stated fractional-year assumptions.

Fix the basis.
Confirm that all cause probabilities are multiple-decrement probabilities conditional on being active at the same time point.

Advance member data.
For each exit year, update attained age, service, and salary according to the stated rules.

Value each cause’s benefit.
Apply the relevant benefit formula and any conversion factor at the benefit’s stated valuation date.

Weight and discount.
Multiply by survival active to the year, the conditional cause probability, and the discount factor for the payment date.

Reconcile.
Check that cause probabilities sum to no more than one in each year and that every later-year term includes prior active survival.

Check Your Understanding

For each problem, use the stated competing-decrement basis and payment timing.

A member has one-year exit probabilities of 0.08 for retirement, 0.05 for withdrawal, 0.03 for disability, and 0.02 for death. The corresponding end-year values are $40,000, $6,000, $30,000, and $50,000. At $i=5\%$, find the one-year EPV. (Answer: $5,142.86)
At zero interest, year 1 retirement and death probabilities are 0.10 and 0.04; year 2 probabilities are 0.15 and 0.05, conditional on being active at the start of year 2. Retirement pays $60,000 and death pays $30,000 at each year-end. Find the two-year EPV. (Answer: $16,230.00)
A member starts with 6 years of service. Salary is $48,000 and grows 5% at the start of each year. Retirement pension is 1.25% of exit salary per service year, and the annuity-due conversion factor at age 62 is 9.5. The year 1 total decrement probability is 0.20; the year 2 retirement probability is 0.10 conditional on being active at its start. At $i=4\%$, what is the EPV of the year 2 retirement benefit only? (Answer: $3,718.49)
Year 1 retirement, withdrawal, disability, and death probabilities are 0.12, 0.06, 0.02, and 0.04. The year 2 death probability is 0.03 conditional on being active at the start of year 2. A year 2 death benefit is $100,000, paid at year-end. At $i=5\%$, calculate the EPV of this benefit. (Answer: $2,068.03)

What You'll Learn