Disability Pension Values Under Competing Risks

Why Disability Pension Values Need Two Stages

A disability pension depends on two separate events. First, a member must leave active service because of disability rather than retirement, withdrawal, or death. Then, after disability, the member must survive to receive each pension installment. The active-service probabilities determine how likely the pension is to begin; the post-disability survival and payment assumptions determine its value once it begins.

This distinction matters because mortality after disability need not be the same as mortality while actively employed. A sound valuation identifies each basis and the timing of every payment before applying a formula. The methods below use a discrete annual model, consistent with annual benefit payments and annual multiple-decrement probabilities.

Definition: Suppose each year an active member can exit for disability $I$, retirement $R$, withdrawal $W$, or death $D$. Let $q_{I,k}^A$, $q_{R,k}^A$, $q_{W,k}^A$, and $q_{D,k}^A$ be the probabilities of each cause of exit during year $k$, conditional on being active at the start of that year. The probability of remaining active through year $k$ is $p_k^A=1-q_{I,k}^A-q_{R,k}^A-q_{W,k}^A-q_{D,k}^A$. The probability of being active at the start of year $k$ is ${}_{k-1}p^A=\prod_{r=1}^{k-1}p_r^A$, with ${}_0p^A=1$.

The exit probabilities for a year are mutually exclusive and exhaustive under this model: an active member either exits for exactly one of the listed causes or remains active. Thus each probability is between zero and one, and their sum cannot exceed one. If the assumptions are constant from year to year, the annual active survival probability is $p^A$, so ${}_{k-1}p^A=(p^A)^{k-1}$.

After disability, assume for now that the pension stops at death, there is no recovery to active service, and no survivor benefit is included. Write $q_j^{D}$ for the conditional probability that a disabled pensioner dies during the $j$th year after disability, and ${}_jp^{D}$ for the probability of surviving $j$ years as a disabled pensioner. This is a separate basis from the active-member death probability $q_{D,k}^A$.

Value at the Date of Disability

Assume an annual pension of $B$ dollars is paid at each year-end after disability, provided the pensioner is then alive. If the pension has at most $m$ installments, its value at the date of disability is

V_D^{(m)}=B\sum_{j=1}^{m}v^j\,{}_jp^D, \qquad v=\frac{1}{1+i}.

Here $i$ is the effective annual interest rate. In the infinite-payment case, replace the finite sum by $\sum_{j=1}^{\infty}v^j\,{}_jp^D$, but only when that series converges. For a constant post-disability death probability $q^D$, ${}_jp^D=(1-q^D)^j$, so the series has ratio $r=(1-q^D)/(1+i)$. Its value is finite exactly when $|r|<1$, assuming $i>-1$. Under the usual assumption $i\geq 0$, $q^D>0$ is sufficient for convergence, but the ratio condition is the general test.

Timing condition: The formula above assumes the first pension installment is paid one year after disability, and later installments are annual. If the pension instead pays immediately at the disability date, the payment at that date must be added; the value is not the same annuity-immediate value.

Combining Active-Service Risks and Pension Payments

If disability occurs at the end of year $k$, its probability is ${}_{k-1}p^A q_{I,k}^A$. The pension value at that time is discounted back $k$ years. If disability can trigger the pension only during years $1$ through $n$, the expected present value at the start of active service is

\operatorname{EPV} =\sum_{k=1}^{n}v^k\,{}_{k-1}p^A q_{I,k}^A\,V_{D,k}.

Here $V_{D,k}$ is the value, at the end of year $k$, of the pension triggered by disability in that year. It may vary by year if the pension amount, payment term, or post-disability assumptions vary. Retirement, withdrawal, and active-member death affect the EPV through the probability of remaining active and then exiting for disability. They do not also reduce the disabled pension annuity unless the pension terms explicitly make it contingent on those active-service events.

Find the active-service probability.
For each possible disability year, multiply the probability of surviving active through all preceding years by that year's disability exit probability.

Value the pension at disability.
Use the post-disability survival probabilities, interest rate, payment term, and stated payment timing.

Discount and combine.
Discount the disability-date pension value from the end of the disability year to the valuation date, weight it by the probability of disability in that year, and sum over eligible years.

The formula follows by conditioning on the year and cause of the first active-service exit. A pension is paid only in the disability branch. Conditional on that branch in year $k$, its value at that date is $V_{D,k}$; the probability of reaching the branch is ${}_{k-1}p^A q_{I,k}^A$. Discounting that conditional value by $v^k$ gives the $k$th term. The disability-year events are mutually exclusive, so adding their expected present values gives the total.

Worked Example: Lifetime Pension After Disability

A hypothetical plan pays an annual disability pension of $12,000 at each year-end after disability, while the pensioner is alive. Disability can occur at the end of either of the next two active-service years. In each year, the active-member exit probabilities are $q_I^A=0.10$, $q_R^A=0.15$, $q_W^A=0.05$, and $q_D^A=0.10$. The effective annual interest rate is $i=0.05$. After disability, the annual death probability is constant at $q^D=0.20$. Find the EPV at the start of active service. Assume the pension is lifelong and the first installment is one year after disability.

Method. First calculate the value of the pension at the date of disability using disabled-member mortality. Then weight the possible disability dates by their active-service probabilities and discount to the start of service.

The annual active-service probability is $p^A=1-0.10-0.15-0.05-0.10=0.70$. Since the four exits sum to 0.40, the correct remaining active probability is $0.60$. Thus ${}_0p^A=1$, and ${}_1p^A=0.60$. (The exit probabilities sum to 0.40, so $p^A=0.60$.)

For the post-disability pension, $v=1/1.05=20/21$ and disabled survival for $j$ years is ${}_jp^D=0.80^j$. The ratio of the infinite series is $v(0.80)=16/21$, which is less than one, so the series converges. Its value per dollar of annual pension is

\sum_{j=1}^{\infty}\left(\frac{16}{21}\right)^j =\frac{16/21}{1-16/21} =\frac{16}{5} =3.2.

The pension value at disability is therefore $V_D=12{,}000(3.2)=\$38{,}400$. The probability-weighted discount factor for disability in year 1 or year 2 is

0.10v+0.60(0.10)v^2 =0.10\left(\frac{20}{21}\right)+0.06\left(\frac{400}{441}\right) =\frac{22}{147}.

Hence the EPV is $38{,}400(22/147)=\$5{,}746.94$, rounded to the nearest cent. As a check, the disability probabilities are 0.10 in year 1 and 0.06 in year 2; each is discounted to the valuation date and multiplied by the same value at disability.

Common mistake avoided: Using the active death probability of 0.10 to value the pension would give an annuity factor of $0.9v/(1-0.9v)=6$, not 3.2. That substitutes active-member mortality for the stated disabled-member mortality and overstates this EPV. The post-disability basis must be applied after the disability event.

Worked Example: A Finite Pension With Year-Varying Disability Incidence

A hypothetical plan pays $9,000 annually for at most four year-end installments after disability, conditional on the pensioner being alive. The effective annual interest rate is 5%, and the disabled-member death probability is 20% each year. Disability benefits can be triggered during the first three active-service years. In those years the disability probabilities are 0.08, 0.12, and 0.10. The probabilities of remaining active to the next year are 0.70 after year 1 and 0.70 after year 2. Find the EPV at the start of service.

Method. Compute one four-payment annuity value at disability, then use the year-specific active survival and disability probabilities. The inputs already give the active survival probabilities, so do not infer them by subtracting only the disability probabilities.

The post-disability discount-survival ratio is $r=(1/1.05)(0.80)=16/21$. The four-payment annuity factor is

\sum_{j=1}^{4}r^j =\frac{16}{21}+\frac{256}{441}+\frac{4096}{9261}+\frac{65536}{194481} =\frac{412624}{194481} \approx 2.121668.

Thus the pension value at disability is $V_D=9{,}000(2.121668)=\$19{,}095.01$, rounded to cents. The probability of disability in year 1 is 0.08. The probability of disability in year 2 is $0.70(0.12)=0.084$, and in year 3 it is $0.70(0.70)(0.10)=0.049$.

With $v=1/1.05=20/21$, the discounted probability weight is

0.08v+0.084v^2+0.049v^3 =0.08\left(\frac{20}{21}\right) +0.084\left(\frac{400}{441}\right) +0.049\left(\frac{8000}{9261}\right) \approx 0.194709.

The EPV is $19{,}095.01(0.194709)=\$3{,}717.97$, rounded to cents. The three disability probabilities sum to 0.213, which is no greater than one; the smaller discounted weight is expected because later disability is both less likely and more heavily discounted.

Common mistake avoided: Applying the four-installment term to the active-service period would confuse the period during which disability can begin with the period over which a pension is paid. The four installments are counted after each disability event.

Worked Example: Payment Timing and the Disability-Year Weight

A hypothetical plan pays $6,000 annually for life after disability. Disability may occur at the end of either of two active-service years. In each year, the disability probability is 0.20; the probability of remaining active through the first year is 0.50. The effective annual interest rate is 25%, and the post-disability annual death probability is 20%. The first pension installment is paid one year after disability. Find the EPV, and quantify the error from treating the pension as payable immediately at disability.

Method. Use an annuity-immediate value at the disability date and then discount each possible disability-date value to the start of service.

The post-disability discount-survival ratio is $r=(1/1.25)(0.80)=0.64$. The lifelong annuity-immediate factor is $r/(1-r)=0.64/0.36=16/9$, so $V_D=6{,}000(16/9)=\$10{,}666.67$.

The year 1 disability probability is 0.20. The year 2 probability is $0.50(0.20)=0.10$. With $v=0.80$, the discounted probability weight is $0.20(0.80)+0.10(0.80^2)=0.16+0.064=0.224$. Therefore the EPV is $10{,}666.67(0.224)=\$2{,}389.33$, rounded to cents.

If the pension were payable immediately at disability, its value would include an installment at that date. The corresponding annuity-due factor is $1/(1-r)=25/9$, giving an EPV of $6{,}000(25/9)(0.224)=\$3{,}733.33$. This is $1,344.00 higher. The difference is the immediate payment, which has value $6,000 at disability and contributes $6{,}000(0.224)=\$1{,}344$ at the valuation date.

Common mistake avoided: Counting a payment at the disability date when the terms specify the first payment one year later changes an annuity-immediate into an annuity-due. The payment date must follow the plan terms, not an assumed convention.

Checks, Limits, and Exam Traps

Before finalizing a valuation, check that active exit probabilities are conditional on being active at the start of the relevant year, and that their sum leaves a valid probability of remaining active. Then check that disabled survival is conditional on disability having occurred, not on continued active service. Finally, confirm whether the pension is finite or lifelong and whether the first installment is immediate or one year after disability.

An infinite pension formula also requires a convergence check. With constant post-disability mortality and effective annual interest rate $i>-1$, the payment-series ratio is $r=(1-q^D)/(1+i)$. The geometric series is finite only if $|r|<1$. For example, $q^D=0.20$ and $i=-0.50$ give $r=0.80/0.50=1.6$, so the infinite series diverges despite positive mortality. Under the usual nonnegative-interest assumption, a positive death probability makes the ratio less than one; stating the ratio condition avoids relying on that narrower assumption.

A further trap is to treat all exits from active service as though they trigger the disability pension. Retirement, withdrawal, and death can preclude disability, but they are not disability claims. Their effect belongs in the active survival probability that precedes a disability event. Benefits contingent on those other exits require their own cause-specific valuation.

Key takeaway: A disability pension EPV is the sum, over eligible disability years, of the discounted probability of disability in that year multiplied by the pension value at the disability date. The active-service decrement basis and the post-disability payment basis serve different purposes and should remain separate.

Check Your Understanding

For each problem, use the stated payment timing and distinguish active-service survival from post-disability survival.

A member can become disabled only at the end of the next active-service year, with probability 0.10. If disability occurs, a $1,000 annual pension is paid for at most four year-ends after disability. The effective annual interest rate is 0%, and annual disabled-member mortality is 25%. Find the EPV at the start of service. (Answer: $305.08)
For each of two active-service years, the disability probability is 0.20 and the probability of remaining active through the year is 0.50. The effective annual interest rate is 0%. After disability, the annual death probability is 50%, and a $2,000 pension is paid at each year-end while alive. Find the EPV. (Answer: $600.00)
Disability probabilities in active-service years 1 and 2 are 0.10 and 0.20. The probability of remaining active after year 1 is 0.60. The effective annual interest rate is 25%. After disability, annual death probability is 20%, and the pension is $900 annually for life, starting one year after disability. Find the EPV. (Answer: $250.88)
A disabled pensioner receives $1,200 at each year-end while alive. The effective annual interest rate is $-20\%$, and the constant annual post-disability death probability is 50%. Find the value at disability and state whether the infinite series converges. (Answer: $2,000.00; it converges because the series ratio is 0.625.)

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