Introduction
An integrated disability income valuation requires more than multiplying an annual benefit by the probability of being disabled. The claim must satisfy the waiting-period condition, remain payable under the benefit limit, and occur at a payment date. Premiums may also depend on the insured’s state. This tutorial combines those features in one hypothetical contract and makes the timing conventions explicit.
The model is annual and discrete: transitions occur between anniversaries, and all payments occur at anniversaries. The calculation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. It is not an official exam question.
The hypothetical contract pays $1,000 at each eligible anniversary, for at most two payments per continuous disability spell. There is no benefit after death. The annual effective interest rate is \(i=5\%\), so \(v=1/(1+i)=1/1.05\). The transition probabilities are constant from year to year:
Each row sums to 1. From healthy, the insured remains healthy with probability 0.90, becomes disabled with probability 0.08, and dies with probability 0.02. From disabled, recovery occurs with probability 0.25, continued disability with probability 0.65, and death with probability 0.10. Death is absorbing.
Occupancy Probabilities and Payment Eligibility
Let \(p_H(t)\) and \(p_D(t)\) denote the probabilities of being healthy and disabled, respectively, at anniversary \(t\). Since the initial state is healthy, \(p_H(0)=1\) and \(p_D(0)=0\). The annual transition model gives the recursions
The second equation adds two distinct ways to be disabled at the next anniversary: transition from healthy during the year, or remain disabled throughout it. For a benefit at time \(t+1\), however, being disabled at \(t+1\) is not sufficient. The insured must also have been disabled at time \(t\) and then remain disabled for the year. Consequently, the probability of an eligible benefit at \(t+1\), before applying any remaining benefit limit, is \(0.65p_D(t)\).
The two-payment limit is also applied by spell, not by policy year in isolation. In the three-year horizon used below, an insured starts healthy and can receive a first payment at time 2 and a second at time 3 after becoming disabled at time 1. An insured first becoming disabled at time 2 can receive a first payment at time 3. No spell can have accumulated more than two payments by time 3.
Worked Example: Full Benefit and Premium Calculation
Worked Example: Full Benefit and Premium Calculation
For the contract defined above, calculate the expected present value of benefits, the expected present value factor for premiums, and the level annual premium \(P\) under the equivalence principle. The premium is paid at times 0, 1, and 2 only if the insured is healthy at that anniversary.
Method. First calculate state probabilities anniversary by anniversary. Then determine eligibility using persistence in the disabled state, discount benefits and premiums at their actual payment times, and equate their expected present values.
At time 1, the insured is healthy with probability 0.90 and disabled with probability 0.08:
At time 2:
The eligible benefit probability at time 2 is the probability of being disabled at time 1 and remaining disabled through the next year:
At time 3, a benefit is eligible for anyone disabled at time 2 who remains disabled through the year. This includes both continuing claims and claims that began at time 2:
With \(v=1/1.05\), \(v^2=0.9070294785\), and \(v^3=0.8638375985\), the expected present value of benefits is
Premiums are due at time 0 with probability 1, at time 1 with probability 0.90, and at time 2 with probability 0.83. The premium present value factor is therefore
The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits:
Check. Substituting the unrounded premium gives \(44.7478(2.6099773)\approx \$116.79\), matching the benefit value. This avoids treating every disabled insured at an anniversary as benefit-eligible or charging premiums after the insured has left the healthy state.
Why the Benefit Probability Uses Persistence
The eligibility formula follows directly from conditioning on the state at the prior anniversary. A time-\(t+1\) benefit requires disability at both \(t\) and \(t+1\). Given disability at \(t\), the probability of still being disabled at \(t+1\) is 0.65. Therefore
This is a direct application of conditional probability and the transition model. It does not assume that the state at \(t+1\) alone identifies the duration of the current disability. New entrants to \(D\) are included in \(p_D(t+1)\), but they have not yet satisfied the one-year waiting period for a payment at \(t+1\).
Use the transition probabilities to calculate the probability of each relevant state at each anniversary.
For a payment at the next anniversary, require disability at both endpoints of the year.
Count payments within each continuous disability spell; do not pay beyond the specified maximum.
Use the discount factor for the anniversary at which each benefit or premium is paid.
Divide the benefit EPV by the EPV factor for one unit of the state-contingent premium.
Worked Example: Separating the Time-3 Claim Cohorts
Worked Example: Separating the Time-3 Claim Cohorts
Decompose the time-3 eligible benefit probability into claims that began at time 1 and claims that began at time 2. Find the present value of the time-3 benefit attributable to the time-2 onset cohort.
Method. At time 2, the disabled probability \(0.124\) has two sources. The time-1 disabled cohort has probability \(0.08(0.65)=0.052\). The time-2 onset cohort has probability \(0.90(0.08)=0.072\), because the insured must be healthy at time 1 and then become disabled during the next year.
For the time-1 cohort to receive a time-3 benefit, it must remain disabled in both annual transitions from time 1 through time 3. Its probability is
For the time-2 onset cohort to receive a time-3 benefit, it must remain disabled for the year from time 2 to time 3. Its probability is
Adding the cohorts gives \(0.0338+0.0468=0.0806\), the same time-3 eligibility probability obtained from \(p_D(2)(0.65)=0.124(0.65)\). The time-2 onset cohort’s time-3 benefit has present value
The time-2 onset cohort is receiving its first payment at time 3. The time-1 onset cohort is receiving its second and final payment, so the two-payment limit is respected. The cohort decomposition is a useful check that the transition calculation includes both continuing and newly eligible claims.
Worked Example: Changing the Waiting Period
Worked Example: Changing the Waiting Period
Suppose the same contract instead has a two-year waiting period, with the first possible payment at time 3. Calculate the probability of an eligible payment at time 3 and its expected present value. Assume the two-payment maximum remains, although no second payment can occur within this three-year valuation horizon.
Method. The insured must be disabled at times 1, 2, and 3. Since the insured starts healthy at time 0, the required path is healthy to disabled at time 1, followed by two consecutive years of continued disability.
The only benefit in the horizon is paid at time 3. Its expected present value is
A time-2 onset does not qualify at time 3 under a two-year waiting period: it has persisted for only one year. That cohort would qualify at time 4 if it remained disabled, but time 4 is outside this valuation horizon. The common mistake is to count all insureds in state \(D\) at time 3; the correct calculation requires tracking how long the insured has been disabled.
Common Exam Trap: Omitting Newly Eligible Claims
A frequent error is to calculate the time-3 payment using only claims already disabled at time 1. That would give probability \(0.08(0.65)^2=0.0338\) and omit the insureds who first become disabled at time 2 and then persist through time 3. Those insureds have completed the one-year waiting period by time 3.
Correction: Include the time-2 onset cohort as well: \(0.08(0.65)^2+(0.90)(0.08)(0.65)=0.0338+0.0468=0.0806\). Equivalently, use \(p_D(2)(0.65)=0.124(0.65)=0.0806\). The omitted cohort adds \(1000(0.0468)v^3=\$40.43\) to the benefit EPV.
The timing convention determines which probability is appropriate. If benefits were paid continuously or monthly, the annual endpoint calculation would not by itself describe eligibility or payment timing; the contract and transition model would need to be specified at the relevant time scale. Likewise, if a policy covered partial years, had a different recurrence rule, or imposed a limit across multiple spells rather than per spell, the simple anniversary conditions used here would need to change.
Check Your Understanding
Use the transition model and timing conventions in this tutorial unless a question explicitly changes them. Answers are rounded to the nearest cent where money is requested.
- What is the probability that an eligible benefit is paid at time 2 under the one-year waiting period? (Answer: \(0.0520\))
- For a benefit at time 3, what is the probability contributed by claims first entering disability at time 2? (Answer: \(0.0468\))
- At 5% interest, what is the present value at time 0 of the time-2 onset cohort’s expected time-3 payment? (Answer: \(\$40.43\))
- If the waiting period is two years, what is the probability of an eligible time-3 payment and its expected present value? (Answer: Probability \(0.0338\); EPV \(\$29.20\))
- Under the original one-year waiting period, what is the premium EPV factor if premiums are paid only while healthy at times 0, 1, and 2? (Answer: \(2.60998\))