Why Transition Timing Matters
A lump sum may be payable precisely when a person moves from one state to another: for example, on disability onset, retirement, or entry into a specified care level. Its value depends not only on the amount but also on the probability of that transition and when the payment is made. In a model with competing exits, a transition can occur only while the person remains in its originating state.
The central valuation task is to combine the model’s transition probabilities with the cash-flow rule. A transition payment is different from a payment made continuously while a person occupies a benefit state; the latter is a separate type of cash flow. The methods here apply whether the transition probabilities come from a continuous-time state model or from a discrete-time Markov model. The material is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Continuous-Time Transition Payments
Suppose a person starts in state \(i\) at time 0. Let \(p_{oi}(t)\) denote the probability of being in state \(i\) at time \(t\), given the initial state \(o\), and let \(\mu_{ij}(t)\) be the transition intensity from \(i\) to \(j\). Under the usual continuous-time Markov model, conditional on being in state \(i\) at time \(t\), the probability of a transition to \(j\) during a short interval of length \(dt\) is \(\mu_{ij}(t)dt+o(dt)\).
If a benefit \(b_{ij}(t)\) is paid immediately on that transition, its expected present value over a time horizon \(T\), at constant force of interest \(\delta\), is:
This expression counts only transitions out of state \(i\), and it discounts each payment from its actual transition time. If several transition types pay benefits, add their expected present values. A transition intensity is not itself a transition probability over a finite period: the probability also depends on remaining in the originating state and on competing transitions.
Worked Example: Disability-Onset Benefit with a Competing Exit
A hypothetical policyholder starts active. While active, disability occurs with constant intensity \(\alpha=0.20\) per year and death occurs with constant intensity \(\beta=0.10\) per year. Both exits are absorbing, and a benefit of $10,000 is paid immediately on disability onset. The force of interest is \(\delta=0.10\), and the valuation horizon is \(T=2\) years. Find the expected present value.
Method. While active, the probability of remaining active to time \(t\) is \(e^{-(\alpha+\beta)t}=e^{-0.30t}\). The disability transition density at time \(t\) is this active-state probability multiplied by its disability intensity, \(0.20e^{-0.30t}\). Thus:
The transition intensity is multiplied by the probability of still being active; death competes with disability by reducing that probability. The payment is discounted at the force of interest from the transition time. Check: the probability of disability within two years is \(\frac{0.20}{0.30}(1-e^{-0.60})\), approximately \(0.3008\); the undiscounted expected payment is therefore about $3,008, greater than the discounted value of $2,753.36. A common error is to use \(0.20\) as the probability of disability in each year or to ignore death when calculating the transition density.
Discrete-Time Transition Benefits
Some models specify transition probabilities over fixed periods rather than transition intensities. Let \(p_{ij}^{(k)}\) be the probability of moving from state \(i\) to state \(j\) during period \(k\), conditional on being in state \(i\) at the start of that period. If \(n_i^{(k)}\) is the probability of occupying state \(i\) at that start, then the probability of an \(i\)-to-\(j\) transition during period \(k\) is \(n_i^{(k)}p_{ij}^{(k)}\). If its benefit \(b_{ij}^{(k)}\) is paid at the end of the period, the period’s expected payment is that probability times the benefit.
For an annual model with effective annual interest rate \(i\), a benefit paid at the end of year \(k\) is discounted by \(v^k\), where \(v=1/(1+i)\). State probabilities for later years are found by carrying forward the probabilities from the previous year using the transition matrix. This ensures the calculation accounts for all paths that can lead to each state.
Worked Example: Retirement, Disability, and Recovery Payments
A hypothetical two-year annual model has states active (A), disabled (D), retired (R), and dead (X). The initial state is A. In each year, the transition probabilities are: from A, 0.70 to A, 0.20 to D, and 0.10 to R; from D, 0.10 to A, 0.70 to D, and 0.20 to X. R and X are absorbing. At the end of a year, the policy pays $5,000 on A-to-D, $12,000 on A-to-R, and $800 on D-to-A transitions. The effective annual interest rate is \(i=0.05\). Find the expected present value of these benefits over two years.
Method. In year 1 the policyholder starts in A. The expected payment at year-end is:
At the start of year 2, the state probabilities are \(P(A)=0.70\), \(P(D)=0.20\), and \(P(R)=0.10\). From A, the expected transition payment is still $2,200. From D, only the D-to-A transition in the specified benefit rules pays, with expected payment \(0.10(800)=\$80\). Therefore the year 2 expected payment is:
The discount factor is \(v=1/1.05=0.95238095\). The expected present value is:
The answer is rounded to the nearest cent. Check: the undiscounted expected total is $3,756, greater than the present value. A common mistake is to value year 2 as though the policyholder must still be active: 20% are disabled at the start of year 2, and the D-to-A benefit must be included.
Piecewise Intensities and Delayed Payments
Intensities can change with time. In that case, the transition density at each time still has the form \(p_{oi}(t)\mu_{ij}(t)\), but the state probability must reflect all intensity changes that occurred earlier. A fixed delay between transition and payment also changes the discount factor. If payment is guaranteed \(c\) years after the transition, the discount factor for a transition at time \(t\) is \(e^{-\delta(t+c)}\). This adjustment assumes the payment is still made on that date under the stated contract rule; a benefit conditional on survival until payment requires an additional survival factor.
Worked Example: Piecewise Onset Intensity and a Payment Delay
A hypothetical insured starts active and may become disabled or die. During year 1, the disability intensity is 0.10 and the death intensity is 0.10 per year. During year 2, the disability intensity is 0.20 and the death intensity is 0.10 per year. These intensities are constant within each year. A $10,000 benefit is paid exactly 0.25 years after disability onset, regardless of subsequent state. The force of interest is \(\delta=0.05\), and disability onsets are counted through time 2. Find the expected present value.
Method. For a transition in year 1, the total exit intensity is 0.20. For a transition in year 2, it is 0.30. The year 1 present-value contribution per dollar of benefit is:
At time 1, the probability of remaining active is \(e^{-0.20}=0.8187308\). For a disability transition at time \(1+u\), where \(0\leq u\leq1\), survival in the active state through the first year and through the fraction \(u\) of year 2 gives probability \(e^{-0.20}e^{-0.30u}\). Discounting to the transition time gives the year 2 contribution per dollar:
The immediate-payment value is therefore \(10{,}000(0.0884797+0.1314224)=\$2{,}199.02\). The payment delay adds a further discount factor \(e^{-0.05(0.25)}=e^{-0.0125}=0.9875778\), so:
The answer is rounded to the nearest cent. Check: the delay reduces the immediate-payment value, as the factor \(0.9875778\) is less than 1. A common mistake is to discount only from time 0 to onset; here the cash is not paid until 0.25 years after onset.
Valuation Checks and Common Traps
A reliable transition-benefit calculation makes the event, the payment date, and the conditioning state explicit. In a continuous-time model, the occupancy probability and transition intensity together give the transition density. In a discrete-time model, multiply the probability of occupying the originating state at the start of the period by the conditional transition probability. Then apply the payment rule and discount to the actual payment date.
| Situation | Probability or density used | Discount to use |
|---|---|---|
| Continuous transition, paid immediately at time \(t\) | \(p_{oi}(t)\mu_{ij}(t)\,dt\) | \(e^{-\delta t}\) |
| Annual transition during year \(k\), paid at year-end | \(n_i^{(k)}p_{ij}^{(k)}\) | \(v^k\) |
| Continuous transition at time \(t\), paid after fixed delay \(c\) | \(p_{oi}(t)\mu_{ij}(t)\,dt\) | \(e^{-\delta(t+c)}\) |
Watch for four frequent errors:
- Using an intensity as a finite-period probability. An intensity is a rate; competing transitions and time in the originating state determine the probability of a transition over an interval.
- Forgetting the originating-state probability. A transition cannot occur from state \(i\) when the person is not in \(i\).
- Discounting to the wrong date. A payment at year-end is discounted by \(v^k\), not by \(v^{k-1}\); a delayed payment is discounted beyond its transition time.
- Counting a transition benefit as an occupancy benefit. A lump sum triggered at entry is not paid merely because the person remains in the destination state.
These calculations are building blocks for integrated valuations: a state model supplies the probabilities, while the contract specifies which transitions trigger payments and when those payments occur. A final check should compare discounted and undiscounted values, verify that the included transitions are contractually payable, and confirm that all cash flows lie within the stated horizon or payment rule.
Check Your Understanding
For each problem, identify the transition event and payment timing before calculating.
- A one-year annual model starts in active status. The probability of disability during the year is 0.20, and a $4,000 benefit is paid at year-end if disability occurs. The effective annual interest rate is 4%. Find the expected present value. (Answer: $769.23)
- An insured starts active. Disability and death have constant intensities 0.15 and 0.05 per year, respectively. A $5,000 benefit is paid immediately on disability, with force of interest 0.04, over a one-year horizon. Find the expected present value. (Answer: $666.79)
- A two-year annual model starts active. From active, the probabilities of remaining active, becoming disabled, and dying are 0.60, 0.30, and 0.10. From disabled, the probabilities of remaining disabled and dying are 0.70 and 0.30. A $3,000 benefit is paid on active-to-disabled transitions and $1,000 on disabled-to-death transitions, at year-end. The effective annual interest rate is 5%. Find the two-year expected present value. (Answer: $1,428.57)
- In a one-year annual model, an employed member retires with probability 0.12, dies with probability 0.03, and remains employed with probability 0.85. Retirement pays $20,000 and death pays $5,000 at year-end. The effective annual interest rate is 5%. Find the expected present value. (Answer: $2,428.57)
- A transition with constant intensity 0.10 competes with another exit of intensity 0.05. A $12,000 benefit is guaranteed 0.5 years after the transition, and the force of interest is 0.03. The transition is possible indefinitely. Find its expected present value. (Answer: $6,567.41)