Worked Disability and Death Benefit Valuation

Valuing Benefits Triggered by Disability or Death

A member of an active pension plan may leave active service through disability or death. These are competing exits: the member can experience one exit cause first, but not both as an active member. A valuation must therefore account for both the probability of each cause and the time at which its benefit is paid.

Disability benefits often need an additional step. A pension may continue after the member leaves active service, so its value at disability depends on survival in the disabled state. That post-exit survival is a different assumption from the probability of remaining active before disability or death.

Definition: In a discrete annual competing-risk model, let $q_k^{(D)}$ be the probability that an active member exits through disability during year $k$, conditional on being active at the start of that year. Let $q_k^{(M)}$ be the corresponding probability of death in service, and let $p_k=1-q_k^{(D)}-q_k^{(M)}$ be the probability of remaining active throughout year $k$. The cause probabilities are mutually exclusive and conditional on being active at the start of the year.

These probabilities describe exits during a year, with benefits paid at the end of the year of exit unless the problem states otherwise. Define $A_k$ as the probability of being active at the start of year $k$. Then $A_1=1$, and for $k\geq 2$,

A_k=\prod_{j=1}^{k-1}p_j.

Thus the probability of disability exit during year $k$ is $A_kq_k^{(D)}$, and the probability of death in service during year $k$ is $A_kq_k^{(M)}$. If $i$ is the effective annual interest rate and $v=1/(1+i)$, a benefit paid at the end of year $k$ is discounted by $v^k$.

Value the Disability Pension at the Disability Date

Suppose a pension of $B$ is paid annually in arrears while the disabled member is alive. Let $r_j$ be the probability that a member who is alive at the start of the $j$th year after disability remains alive to receive the payment at the end of that year. Assume that disability does not recover, that death ends the pension, and that these survival probabilities apply conditional on entering the disabled state.

Result: The expected present value at the date of disability of a pension paying $B$ at the end of each post-disability year is $$ B\sum_{j=1}^{n}v^j\prod_{h=1}^{j}r_h $$ if payments are limited to $n$ years. If post-disability survival is a constant $r$ each year and the pension continues for life, the value is $B\,rv/(1-rv)$, provided $rv<1$.

The product $\prod_{h=1}^{j}r_h$ is the probability of surviving all $j$ post-disability years and receiving the $j$th payment. Each expected payment is discounted from its payment date to the disability date. In the constant-survival, lifelong case, the resulting series is geometric:

B\left(rv+(rv)^2+(rv)^3+\cdots\right) = B\frac{rv}{1-rv}.

The first payment is one year after disability, so the series begins with $rv$, not with 1. The formula does not apply unchanged if recovery can end the pension, benefits escalate, payments are made at different times, or the pension is guaranteed for a period regardless of survival; those conditions require their own payment and decrement terms.

From Exit Probabilities to Present Values

For a disability exit in year $k$, first calculate the value of the disability benefits at the end of that year. This may include a lump sum payable on disability plus the value of any future pension, measured at the disability date. Then multiply that value by the probability of disability exit in year $k$ and by $v^k$. Treat a death-in-service benefit in the same way, using its own amount and the death exit probability.

Identify the active-state exits.
Write down each annual cause probability and the probability of remaining active. Check that the cause probabilities plus the active probability sum to 1 in each year.

Find each cause-specific exit probability.
Multiply the probability of being active at the start of the year by the conditional probability of the relevant exit during that year.

Value benefits at the appropriate date.
For disability, include any post-disability pension value at the disability date. For death in service, use the benefit payable on that death.

Discount and sum.
Discount a year-$k$ exit benefit by $v^k$, then add the disability and death contributions over all covered years.

When the cause probabilities are constant, write $q^{(D)}$, $q^{(M)}$, and $p=1-q^{(D)}-q^{(M)}$. If benefits are covered for $n$ years, a constant value $C_D$ at each disability exit and a constant benefit $C_M$ at each death exit have present value

\sum_{k=1}^{n}p^{k-1}v^k \left(q^{(D)}C_D+q^{(M)}C_M\right).

Here $C_D$ includes any pension value measured at the disability date. This expression assumes that both benefits are paid at the end of the year of exit and that the stated exit probabilities apply in each covered year. If the cause probabilities or benefits vary by year, calculate each year’s contribution separately instead.

Why the Constant-Rate Exit Weight Has This Form

For a constant annual cause probability $q$, a member who remains active with probability $p$ each year, and a benefit of 1 paid at the end of the exit year, the present value $W$ satisfies a one-year recursion. An exit in the first year contributes $qv$. If the member remains active, which has probability $p$, the same valuation problem begins again one year later and is discounted by $v$. Therefore,

$$ W=qv+pvW. $$

Solving gives

W=\frac{qv}{1-pv}.

This is the discounted exit weight for an unlimited active period under constant assumptions. It is not the probability of eventual exit: the probability weights are also discounted. If coverage ends after a finite number of years, use the finite sum $\sum_{k=1}^{n}p^{k-1}qv^k$, not the unlimited-period expression.

Worked Example: Two-Year Disability and Death Benefits

A hypothetical plan covers an active member for two years. In each year, the conditional probability of disability exit is $0.04$, the probability of death in service is $0.01$, and the probability of remaining active is $0.95$. The effective annual interest rate is $5\%$. A disability exit pays $ \$12{,}000$ at the end of the exit year; a death exit pays $ \$40{,}000$ at the end of that year. There are no covered exits after year 2. Find the present value of the expected benefits.

Method. Calculate the discounted exit weight for each cause over the two covered years, then multiply by the corresponding benefit. The discount factor is $v=1/1.05=20/21$.

The disability weight is

0.04v+0.95(0.04)v^2 =0.04\left(\frac{20}{21}+0.95\frac{400}{441}\right) =0.04\left(\frac{800}{441}\right) =\frac{32}{441} \approx 0.0725624.

The death-in-service weight has the same active-state probabilities, with cause probability $0.01$:

0.01v+0.95(0.01)v^2 =0.01\left(\frac{800}{441}\right) =\frac{8}{441} \approx 0.0181406.

Therefore the expected present value is

\$12{,}000\left(\frac{32}{441}\right) +\$40{,}000\left(\frac{8}{441}\right) =\frac{\$704{,}000}{441} \approx \$1{,}596.37.

Check. The undiscounted probabilities of disability exit and death exit over the two years are $0.04+0.95(0.04)=0.078$ and $0.01+0.95(0.01)=0.0195$, respectively. Together with the probability $0.95^2=0.9025$ of remaining active through year 2, they sum to 1. The discounted weights are smaller than the corresponding exit probabilities, as expected. The result is rounded to the nearest cent.

Common mistake avoided. Using $0.04$ and $0.01$ as the total two-year exit probabilities omits year-2 exits among members who remain active after year 1. The second-year terms must include the year-1 active probability $0.95$.

Worked Example: Disability Pension With a Two-Year Active Coverage Period

A hypothetical plan covers active exits for two years. In each year, the disability probability is $0.02$, the death-in-service probability is $0.01$, and the probability of remaining active is $0.97$. The effective annual interest rate is $5\%$. Disability pays a lump sum of $ \$20{,}000$ at exit and a pension of $ \$5{,}000$ at the end of each future year while the disabled member is alive. Post-disability survival is $0.90$ each year, with no recovery. Death in service pays $ \$30{,}000$ at the end of the year of death. Find the expected present value.

Method. First value the pension at the disability date, then combine that value with the disability lump sum. Next calculate the discounted exit weights for the two covered active years. The discount factor is $v=20/21$.

The pension is lifelong, payable in arrears, and contingent on survival. Its value at disability is

\$5{,}000\frac{0.90v}{1-0.90v} = \$5{,}000\frac{0.90/1.05}{1-0.90/1.05} = \$5{,}000(6) = \$30{,}000.

Adding the disability lump sum gives a total disability-date value of $ \$20{,}000+\$30{,}000=\$50{,}000$.

The discounted disability exit weight over the two active years is

0.02v+0.97(0.02)v^2 = \frac{0.02}{1.05}+\frac{0.97(0.02)}{1.05^2} = \frac{404}{11025}.

The death-in-service weight is half as large because its annual cause probability is half the disability probability:

0.01v+0.97(0.01)v^2 = \frac{202}{11025}.

Thus the total expected present value is

\$50{,}000\left(\frac{404}{11025}\right) +\$30{,}000\left(\frac{202}{11025}\right) = \$1{,}832.20+\$549.66 = \$2{,}381.86.

Check. The two-year disability exit probability is $0.02+0.97(0.02)=0.0394$, and the death exit probability is $0.01+0.97(0.01)=0.0197$. Their sum, $0.0591$, plus the probability $0.97^2=0.9409$ of remaining active through year 2 is 1. The present value is rounded to the nearest cent.

Common mistake avoided. Counting only year-1 exits would give $0.02(\$50{,}000)+0.01(\$30{,}000)=\$1{,}300$, omitting year-2 exits and discounting. The two-year valuation must include the year-2 exit weights and the end-of-year payment timing.

Worked Example: A Disability Pension Limited to Three Payments

A hypothetical plan covers active exits for two years. In each active year, disability has probability $0.04$, death in service has probability $0.01$, and remaining active has probability $0.95$. The effective annual interest rate is $25\%$. Disability pays $ \$10{,}000$ at the exit date and a $ \$1{,}000$ pension at the end of each of the next three years, provided the member survives to that payment. The post-disability survival probabilities for those three years are $0.75$, $0.75$, and $0.75$. Death in service pays $ \$20{,}000$ at the end of the exit year. Find the expected present value.

Method. Value the three post-disability pension payments at the disability date, add the disability lump sum, and apply the two-year discounted exit weights. Here $v=1/1.25=0.8$.

The probability of receiving the first, second, and third pension payments is $0.75$, $0.75^2=0.5625$, and $0.75^3=0.421875$, respectively. Including discounting, the pension value at disability is

\$1{,}000\left(0.75(0.8)+0.75^2(0.8)^2+0.75^3(0.8)^3\right) = \$1{,}000(0.600+0.360+0.216) = \$1{,}176.

The total disability-date value is $ \$10{,}000+\$1{,}176=\$11{,}176$. The two-year discounted disability weight is

$$ 0.04(0.8)+0.95(0.04)(0.8)^2 =0.032+0.02432 =0.05632. $$

The death-in-service weight is

$$ 0.01(0.8)+0.95(0.01)(0.8)^2 =0.008+0.00608 =0.01408. $$

Combining both causes gives

\$11{,}176(0.05632)+\$20{,}000(0.01408) = \$629.43232+\$281.60 = \$911.03.

Check. The year-1 disability and death probabilities sum to $0.05$. The year-2 exit probability is $0.95(0.05)=0.0475$, so the probability of any covered exit is $0.0975$; the probability of remaining active through year 2 is $0.95^2=0.9025$. These sum to 1. The present value is rounded to the nearest cent.

Common mistake avoided. Using $0.75$ as the probability of receiving every future payment ignores survival through earlier post-disability years. The third payment, for example, requires survival through all three years, with probability $0.75^3$.

Checks and Common Traps

A competing-risk valuation is easiest to audit when the active-state probabilities, exit weights, benefit values, and discount factors are kept separate until the final sum. In particular, distinguish these two quantities:

Quantity	What it measures	Typical use
Cause-specific exit probability	Probability of that exit, including the probability of being active beforehand	Multiply by a benefit to find an undiscounted expected payment
Discounted exit weight	Cause-specific exit probability multiplied by the discount factor for its payment time	Multiply by a benefit to find its present-value contribution

A frequent error is to use the probability of remaining active after an exit has occurred to value a disability pension. Active survival determines whether a disability exit can occur in a later year; post-disability survival determines whether a pension payment is received after that exit. These are separate phases and may have different assumptions.

Also check the payment date. A benefit payable at the end of year $k$ is discounted by $v^k$. A pension valued at the disability date starts with the first payment one year later if it is payable annually in arrears. Changing either timing convention changes the factors, even if the benefit amounts and probabilities are unchanged.

Key takeaway: For each exit cause, multiply the probability of being active at the start of the year by its conditional exit probability, value the resulting benefit at the exit date, and discount from the stated payment dates. For a disability pension, use post-disability survival separately from active-state survival.

Check Your Understanding

For each problem, state the cause-specific contributions before combining them. Unless otherwise stated, benefits are paid at the end of the year of exit.

A plan covers two years. Each year, disability and death-in-service probabilities are $0.03$ and $0.02$, respectively, and the member remains active with probability $0.95$. At $i=4\%$, the disability benefit is $ \$8{,}000$ and the death benefit is $ \$25{,}000$. Find the expected present value. (Answer: $ \$1{,}361.50$)
Active members face constant annual disability and death-in-service probabilities of $0.015$ and $0.005$, with no other exit. The assumptions continue indefinitely and $i=4\%$. Disability pays $ \$12{,}000$ at exit and death in service pays $ \$30{,}000$ at exit. Find the expected present value. (Answer: $ \$5{,}500.00$)
At $i=5\%$, a disability pension pays $ \$2{,}000$ at the end of each of the next two years, provided the disabled member survives. Post-disability survival probabilities are $0.80$ for the first year and $0.50$ for the second. A $ \$5{,}000$ lump sum is paid at disability. Find the value at the disability date. (Answer: $ \$7{,}249.43$)
A plan covers two years at $i=5\%$. In year 1, disability and death probabilities are $0.02$ and $0.01$, leaving $0.97$ active. In year 2, conditional on being active, they are $0.04$ and $0.02$. Disability pays $ \$10{,}000$ at exit; death in service pays $ \$25{,}000$. Find the expected present value. (Answer: $ \$1{,}220.41$)

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