Deriving Expected Values for Vesting Benefits

Introduction

A vested benefit may be payable only after a member meets an eligibility condition, and later payments may depend on the member remaining alive. Its expected present value therefore reflects two distinct questions: what is the probability that the member is eligible for the benefit, and how likely is that eligible member to survive to each payment date? Combining probabilities without specifying what they are conditional on can produce an incorrect value, even when each probability looks reasonable in isolation.

This tutorial derives the valuation for a fixed vesting date and annual payments contingent on survival. It emphasizes the population to which each survival probability applies. The calculations use hypothetical benefit designs and data; the techniques are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Fix a valuation date 0 and a vesting date $m$. Let $E_m$ be the event that the member is eligible for the vested benefit and alive at time $m$. Write $p_E=P(E_m)$. For $k\geq 0$, define \[ {}_kp_E=P(\text{alive at time }m+k\mid E_m). \] In particular, ${}_0p_E=1$. These are survival probabilities for the cohort eligible and alive at time $m$, not automatically the probabilities for the original population.

Building the Expected Present Value

Assume a deterministic effective annual interest rate $i$, with $v=1/(1+i)$. Suppose the vested benefit pays $B_k$ at time $m+k$ if the member is alive then, for $k=0,1,\ldots,n$. A payment at $k=0$ is made at the vesting date, while $k\geq 1$ payments occur later. The benefit amounts $B_k$ are fixed once the member is eligible.

A payment at time $m+k$ is made precisely when both $E_m$ occurs and the member survives to that payment date. By the definition of conditional probability, $P(E_m\text{ and alive at }m+k)=p_E\,{}_kp_E$. Discounting each possible payment to time 0 and using linearity of expectation gives the expected present value:

\operatorname{EPV} =\sum_{k=0}^{n}B_kv^{m+k}P(E_m\text{ and alive at }m+k) =p_E\sum_{k=0}^{n}B_kv^{m+k}{}_kp_E.

If the same annual pension $B$ is payable at times $m+1,\ldots,m+n$, but not at vesting, the formula becomes

\operatorname{EPV} =p_EB\sum_{k=1}^{n}v^{m+k}{}_kp_E.

The survival terms can be constructed from sequential conditional probabilities. Let $r_j=P(\text{alive at }m+j\mid E_m,\text{ alive at }m+j-1)$. Then, by repeated conditioning, ${}_kp_E=\prod_{j=1}^{k}r_j$. Each $r_j$ must describe survival for members in the eligible cohort who have survived to the beginning of that interval. A marginal survival rate for a broader population need not equal this conditional rate.

Why the Formula Holds

The formula follows from valuing each payment as a random cash flow. For a payment of $B_k$ at time $m+k$, define an indicator $I_k$ that equals 1 if the member is eligible at time $m$ and alive at time $m+k$, and 0 otherwise. The present value of that payment is $B_kv^{m+k}I_k$. Its expected value is $B_kv^{m+k}E[I_k]$, and the expectation of an indicator is the probability of its event.

E[B_kv^{m+k}I_k] =B_kv^{m+k}P(E_m\text{ and alive at }m+k) =B_kv^{m+k}p_E{}_kp_E.

Adding these expectations over the payment dates gives the stated EPV; this step does not require the payment indicators at different dates to be independent. Survival to a later date already implies survival to the earlier payment dates in this model. What does matter is that the conditional survival probabilities used in ${}_kp_E$ match the eligible cohort.

Specify eligibility.
Define the event that makes the benefit vested, including whether it requires the member to be alive at the vesting date. Determine its probability $p_E$.

Set out payment dates and amounts.
For each date, state the amount payable and whether payment is conditional on survival at that date.

Use conditional survival for the vested cohort.
Find ${}_kp_E$ directly, or multiply sequential rates conditional on eligibility and survival to the previous date.

Discount and sum.
Multiply each payment by its payment probability and discount factor, then add the terms.

Worked Valuations

Worked Example: Three Payments After Vesting

A hypothetical plan values benefits at time 0 using an effective annual interest rate of 5%. At time 5, the probability that a member is both eligible and alive is 0.60. The plan pays $10,000 at each of times 6, 7, and 8 if the member is alive on that date. Among members eligible and alive at time 5, the one-year survival probabilities are 0.90 from time 5 to 6, 0.80 from time 6 to 7, conditional on eligibility at time 5 and survival to time 6, and 0.75 from time 7 to 8, conditional on eligibility at time 5 and survival to time 7. Calculate the EPV.

Method. Find the probability of eligibility and survival to each payment date by multiplying the stated cohort-conditional rates, then discount each expected payment. The stated conditional basis is essential: each successive rate applies to the eligible cohort that survived to the start of that interval.

The probability of eligibility and survival to time 6 is $0.60(0.90)=0.54$. To time 7 it is $0.54(0.80)=0.432$. To time 8 it is $0.432(0.75)=0.324$. At 5%, the discount factors are $v^6=1/1.05^6=0.7462154$, $v^7=0.7106813$, and $v^8=0.6768394$.

The time-6 payment contributes $10{,}000(0.54)(0.7462154)=\$4{,}029.56$. The time-7 payment contributes $10{,}000(0.432)(0.7106813)=\$3{,}070.14$. The time-8 payment contributes $10{,}000(0.324)(0.6768394)=\$2{,}192.96$.

Therefore the EPV is $4{,}029.56+3{,}070.14+2{,}192.96=\$9{,}292.66$, rounded to the nearest cent. As a check, the joint probabilities decrease across the dates, $0.54$, $0.432$, and $0.324$, as survival requires.

Common mistake avoided: using a survival rate for the general population in place of one explicitly conditional on eligibility. The inputs here specify the eligible cohort at every step, so the products are justified.

Worked Example: Deferred Pension With Cohort-Conditional Survival

A hypothetical pension is valued at time 0 with effective annual interest of 5%. The probability that a member is eligible and alive at time 2 is 0.40. An annual pension of $6,000 is paid at times 3, 4, and 5 if the member is alive on each payment date. Conditional on being eligible and alive at time 2, the probability of surviving to time 3 is 0.90. Conditional on eligibility and survival to time 3, the probability of surviving to time 4 is 0.80; conditional on eligibility and survival to time 4, the probability of surviving to time 5 is 0.75. Find the EPV.

Method. This is a deferred life-contingent pension. The payment probabilities are found from the eligibility probability and the sequence of survival rates specified for the eligible cohort.

The probabilities of eligibility and being alive at times 3, 4, and 5 are, respectively, $0.40(0.90)=0.36$, $0.36(0.80)=0.288$, and $0.288(0.75)=0.216$. At 5%, $v^3=0.8638376$, $v^4=0.8227025$, and $v^5=0.7835262$.

The discounted contributions are $6{,}000(0.36)(0.8638376)=\$1{,}865.89$, $6{,}000(0.288)(0.8227025)=\$1{,}421.63$, and $6{,}000(0.216)(0.7835262)=\$1{,}015.45$. Their sum is $\$4{,}302.97$, rounded to the nearest cent.

Check. Conditional on eligibility at time 2, the payment probabilities are 0.90, 0.72, and 0.54. Multiplying each by 0.40 gives the joint probabilities used above. This verifies that eligibility has been applied once, rather than omitted or counted twice.

Common mistake avoided: treating the probability of being alive at time 5 as though it were an unconditional population probability. Here it is derived specifically by following the members eligible at time 2.

Worked Example: Two Eligible Groups With Different Survival

A hypothetical plan has two mutually exclusive eligibility groups at time 4. The probability of being eligible and alive in group A is 0.25; for group B it is 0.35. Group A receives $9,000 annually at times 5 and 6 if alive. Its conditional survival rates are 0.90 from time 4 to 5 and 0.85 from time 5 to 6, with the second rate conditional on group A eligibility and survival to time 5. Group B receives $12,000 at those same dates; its corresponding conditional rates are 0.80 and 0.75, each conditional on group B eligibility and survival to the start of the interval. At an effective annual interest rate of 4%, calculate the total EPV.

Method. Value each group using its own eligibility probability and conditional survival rates, then add the group values. The groups are mutually exclusive, so their payment expectations can be added without overlap.

For group A, the joint probabilities of eligibility and survival to times 5 and 6 are $0.25(0.90)=0.225$ and $0.225(0.85)=0.19125$. For group B, they are $0.35(0.80)=0.28$ and $0.28(0.75)=0.21$. At 4%, $v^5=0.8219271$ and $v^6=0.7903145$.

Group A contributes $9{,}000(0.225)(0.8219271)=\$1{,}664.40$ at time 5 and $9{,}000(0.19125)(0.7903145)=\$1{,}360.33$ at time 6, for a total of $\$3{,}024.73$. Group B contributes $12{,}000(0.28)(0.8219271)=\$2{,}761.68$ at time 5 and $12{,}000(0.21)(0.7903145)=\$1{,}991.59$ at time 6, for a total of $\$4{,}753.27$.

The combined EPV is $\$3{,}024.73+\$4{,}753.27=\$7{,}778.00$, rounded to the nearest cent. The four terms account for both payment dates in both groups; no group’s survival experience is substituted for the other’s.

Common mistake avoided: pooling the groups and applying one averaged survival rate without also preserving the different benefit amounts and eligibility probabilities. Valuing each group on its own conditional basis keeps the probability and benefit attached to the correct cohort.

Assumptions and Limits of the Calculation

The formula is deliberately precise about what must be known. The probability $p_E$ must refer to the event that makes the benefit available, as defined in the problem. If eligibility at time $m$ requires the member to have survived to that date, the probability should be joint: it is the probability of being eligible and alive then. If a problem instead supplies the probability of eligibility conditional on being alive at time $m$, that conditional probability must be combined with the probability of being alive at time $m$ to obtain the joint probability.

Likewise, later survival data must apply to the vested cohort. A population-wide probability of survival over an interval does not, by itself, determine survival for members eligible at the start of that interval. Eligibility may be associated with employment history, health, or other characteristics that also affect mortality. A general-population rate can be used only when the problem supplies an appropriate assumption—for example, that survival is unaffected by eligibility—or otherwise establishes that the rate applies to the eligible cohort.

The calculation also assumes the stated benefit amounts and payment conditions are sufficient to identify the cash flows. If the benefit amount varies by the member’s circumstances, or payment is conditional on something other than survival, the EPV must reflect those conditions as well. The simple formula does not price an unspecified benefit rule. Interest is assumed deterministic at the stated effective annual rate, and all cash flows and time points must use consistent units.

Common Exam Traps

Mixing conditional bases. Multiplying a vesting probability by an unconditional survival rate assumes the rate also applies to vested members. That assumption must be stated or given.
Using the wrong starting cohort. A rate from time $m+1$ to $m+2$ must be conditional on eligibility and survival through time $m+1$, if that is the cohort being valued.
Forgetting that eligibility can include survival. Check whether the supplied vesting probability is conditional on being alive or already joint with survival.
Discounting from the wrong date. A payment at time $m+k$ is discounted by $v^{m+k}$ to time 0, not merely by $v^k$.
Including a payment at vesting when none is specified. A pension beginning one year after vesting has no time-$m$ payment; its sum starts at $k=1$.

Key takeaway: Value each payment using the probability that the member is eligible at vesting and survives to that payment date. Survival after vesting must be conditional on the eligible cohort, unless an explicit assumption justifies using another basis.

Check Your Understanding

For each problem, identify the appropriate eligibility and survival probabilities before discounting. Money answers are rounded to the nearest cent.

A hypothetical plan has an effective annual interest rate of 5%. The probability of being eligible and alive at time 3 is 0.45. It pays $20,000 at time 4 if the member is alive, and the probability of surviving from time 3 to 4 conditional on eligibility and survival at time 3 is 0.80. Find the EPV. (Answer: $5,923.46)
At 4% effective annual interest, a benefit of $5,000 is paid at times 3, 4, and 5 if alive. The probability of eligibility and survival at time 2 is 0.50. Conditional survival rates for the eligible cohort are 0.90 from time 2 to 3, 0.80 from time 3 to 4, and 0.75 from time 4 to 5. Find the EPV. (Answer: $4,648.49)
At 3% effective annual interest, a hypothetical vested pension pays $10,000 at times 5 and 6 if alive. The probability of eligibility and survival at time 4 is 0.30. For eligible members, conditional survival is 0.90 from time 4 to 5 and 0.80 from time 5 to 6. Find the EPV. (Answer: $4,138.01)
Two mutually exclusive eligibility groups are present at time 4. Group A has probability 0.20 of being eligible and alive, receives $15,000 at time 6 if alive, and has conditional survival probability 0.80 from time 4 to 6. Group B has probability 0.30 of being eligible and alive, receives $10,000 at time 6 if alive, and has conditional survival probability 0.90 from time 4 to 6. At 5% effective annual interest, find the combined EPV. (Answer: $3,805.70)
A plan has probability 0.40 of eligibility and survival at time 2. Survival from time 2 to 3 conditional on eligibility is 0.90. The only other information supplied is that the overall population survival probability from time 3 to 4 is 0.80. A vested payment is due at time 4 if alive. Is its payment probability determined? State what additional information or assumption is needed. (Answer: No; the survival probability from time 3 to 4 conditional on eligibility and survival at time 3, or an assumption that survival over that interval is unaffected by eligibility, is required.)

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