Discounting State-Contingent Cash Flows

Discounting and Eligibility Are Separate

A state-contingent benefit depends on both when money is paid and whether the insured is in an eligible state at that time. These are separate parts of the valuation. The discount factor reflects the time value of money; the probability of eligibility reflects the state model. Combining them correctly is essential whether a contract pays at specified dates or continuously while a condition holds.

This tutorial uses a deterministic force of interest and state probabilities supplied by the model. All product descriptions and numerical data below are hypothetical. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $\delta(t)$ be a deterministic force of interest at time $t$, measured per year. The accumulated force from time 0 to time $t$ is $\int_0^t \delta(s)\,ds$, and the discount factor for a payment at time $t$ is $$ D(0,t)=\exp\left(-\int_0^t \delta(s)\,ds\right). $$ If the force is constant, $\delta(t)=\delta$, this becomes $D(0,t)=e^{-\delta t}$.

The integral, rather than the force at the payment date multiplied by the payment time, is the accumulated force when $\delta$ varies. A deterministic discount factor can be taken outside an expectation. The state-contingent part remains inside: for example, a payment is received only if an eligibility indicator equals 1.

Key result: If a deterministic amount $b_k$ is paid at time $t_k$ only when an event $E_k$ occurs, its expected present value is $b_kD(0,t_k)P(E_k)$. For a continuous payment rate $b(t)$ paid only while the insured is in an eligible state $j$, its expected present value over $[0,n]$ is $$ \int_0^n b(t)D(0,t)P(X_t=j)\,dt, $$ provided the expected value and integral exist. The state probability is not part of the discount factor.

Fixed-Date Payments

For payments at finitely many dates, value each possible payment at its own date. If a benefit is payable at $t_k$ only when the insured is eligible then, use the probability of eligibility at $t_k$, not the probability of eligibility at some other date. A finite sum of expected discounted payments gives the expected present value of the stream by linearity of expectation; independence between eligibility events at different dates is not required.

Worked Example: Disability Benefits at Three Annual Dates

A hypothetical policy pays $12,000 at each of years 1, 2, and 3 if the insured is in a specified disability state on the payment date. The probabilities of being in that state at those dates are 0.20, 0.30, and 0.25, respectively. The force of interest is constant at 0.04 per year. Find the expected present value.

Method. At each date, multiply the benefit by the eligibility probability and by the discount factor for that date. The force is constant, so $D(0,t)=e^{-0.04t}$.

At year 1, the discount factor is $e^{-0.04}=0.960789439$. The expected present value of that payment is $12{,}000(0.20)(0.960789439)=2{,}305.894654$.

At year 2, the discount factor is $e^{-0.08}=0.923116346$. The expected present value is $12{,}000(0.30)(0.923116346)=3{,}323.218846$.

At year 3, the discount factor is $e^{-0.12}=0.886920437$. The expected present value is $12{,}000(0.25)(0.886920437)=2{,}660.761311$.

Adding the three terms gives $2{,}305.894654+3{,}323.218846+2{,}660.761311=8{,}289.874811$. The expected present value is $8,289.87, rounded to the nearest cent. The calculation avoids the mistake of discounting the full benefit stream first and then applying one eligibility probability to all dates.

Time-Varying Forces of Interest

When the force changes over time, the discount factor depends on its accumulated value over the full interval from issue to payment. For example, if $\delta(t)=a+bt$, then $\int_0^t\delta(s)\,ds=at+\frac{1}{2}bt^2$, so $D(0,t)=e^{-at-\frac{1}{2}bt^2}$. This is not generally equal to $e^{-t\delta(t)}$.

Worked Example: A Benefit Under a Changing Force

A hypothetical contract pays $5,000 at the end of year 1 and another $5,000 at the end of year 2 if the insured is in an eligible care state at the respective payment date. The probabilities of eligibility are 0.40 at year 1 and 0.30 at year 2. The force of interest is $\delta(t)=0.02+0.01t$ per year. Find the expected present value.

Method. First integrate the force to each payment date. Then multiply the resulting discount factor by the expected amount payable at that date.

For year 1, the accumulated force is $\int_0^1(0.02+0.01s)\,ds=0.02+0.005=0.025$. Thus $D(0,1)=e^{-0.025}=0.975309912$. The expected present value at year 1 is $5{,}000(0.40)(0.975309912)=1{,}950.619824$.

For year 2, the accumulated force is $\int_0^2(0.02+0.01s)\,ds=0.04+0.02=0.06$. Thus $D(0,2)=e^{-0.06}=0.941764534$. The expected present value at year 2 is $5{,}000(0.30)(0.941764534)=1{,}412.646801$.

The total is $1{,}950.619824+1{,}412.646801=3{,}363.266625$, or $3,363.27 to the nearest cent.

Common mistake and correction. At year 2, using the force at the payment date times two gives $2\delta(2)=2(0.04)=0.08$, and an incorrect discount factor of $e^{-0.08}$. That would produce $5{,}000(0.30)e^{-0.08}=1{,}384.67$ for the second payment. The correct accumulated force is the integral, $0.06$, so the correct second term is $1,412.65.

Continuous Benefits While Eligible

For a benefit paid continuously at rate $b(t)$ while the insured occupies an eligible state, the payment over a short interval of length $dt$ is approximately $b(t)\,dt$ if the state is occupied. Its expected discounted amount is therefore approximately $b(t)D(0,t)P(X_t=j)\,dt$. Summing these contributions over time and taking the limit gives the expected present value integral.

Justification: Let $I_j(t)$ equal 1 when the insured is in eligible state $j$ at time $t$, and 0 otherwise. The present value of a continuous benefit stream is $\int_0^n b(t)D(0,t)I_j(t)\,dt$. Taking expectations and interchanging expectation and integration when the integral is integrable gives $$ E\left[\int_0^n b(t)D(0,t)I_j(t)\,dt\right] =\int_0^n b(t)D(0,t)E[I_j(t)]\,dt =\int_0^n b(t)D(0,t)P(X_t=j)\,dt. $$ The last equality follows because an indicator has expectation equal to the probability of its event. This derivation shows why discounting and eligibility enter as separate factors.

Worked Example: Continuous Payments in an Eligible State

A hypothetical long-term care contract pays continuously at $600 per year while the insured remains in an eligible state, for at most two years. The probability of still being in that state at time $t$ is $P(X_t=j)=1/(1+t)$. The force of interest is $\delta(t)=1/(1+t)$ per year. Find the expected present value.

Method. Integrate the benefit rate multiplied by the eligibility probability and the discount factor over the two-year term.

The accumulated force is $\int_0^t 1/(1+s)\,ds=\ln(1+t)$. Therefore the discount factor is $D(0,t)=e^{-\ln(1+t)}=1/(1+t)$.

At each time $t$, the expected discounted payment rate is consequently $600\left(1/(1+t)\right)\left(1/(1+t)\right)=600/(1+t)^2$. The expected present value is $\int_0^2 600/(1+t)^2\,dt$.

Since $\int(1+t)^{-2}dt=-(1+t)^{-1}$, the value is $600[-1/(1+t)]_0^2=600(1-1/3)=600(2/3)=400$. The expected present value is $400.00. The probability and discount factor happen to have the same functional form here, but they represent different quantities and must each be included.

Checks Before Valuing a State-Contingent Stream

A sound calculation identifies the payment rule, the relevant state probability, and the accumulated force separately. For each payment date, check that the probability refers to eligibility at that date. For continuous benefits, check that the state probability is integrated across the full payment period rather than replaced by a probability at the end of the term.

Identify the payment basis.
Decide whether the contract pays at specified dates or at a continuous rate, and state which event or state makes payment eligible.

Calculate discounting.
Use $e^{-\int_0^t\delta(s)\,ds}$ for a deterministic force, integrating from time 0 to the payment time.

Apply eligibility.
Multiply a fixed-date payment by its eligibility probability, or multiply the continuous rate by the probability of occupying the eligible state at each time.

Aggregate and check.
Sum fixed-date expected present values or integrate continuous expected discounted payments. Confirm that later payments are discounted over longer periods.

The most common errors are using a state probability as though it were an interest adjustment, discounting with the instantaneous force at the payment date instead of its integral, and using the same eligibility probability for every payment date without support from the model. The correct structure keeps each element visible: amount, probability of payment eligibility, and discount factor.

Check Your Understanding

For each problem, separate the discount factor from the state eligibility probability.

A hypothetical policy pays $4,000 at year 1 if the insured is eligible with probability 0.25, and $6,000 at year 2 if eligible with probability 0.40. The constant force of interest is 0.03 per year. Find the expected present value. (Answer: $3,230.68)
A $3,000 benefit is paid at year 2 if the insured is in an eligible state, with probability 0.50. The force is $\delta(t)=0.01+0.02t$. Find the expected present value. (Answer: $1,412.65)
A hypothetical policy pays continuously at $1,000 per year while eligible for three years. The eligible-state probability is $e^{-0.10t}$, and the force of interest is constant at 0.04. Find the expected present value. (Answer: $2,449.67)
A $2,000 payment is made at each of years 1 and 2 if eligible, with probabilities 0.80 and 0.60, respectively. The force is $\delta(t)=0.02t$. Find the expected present value. (Answer: $2,737.03)
A hypothetical benefit is paid continuously at $900 per year while an insured remains eligible for one year. The probability of eligibility at time $t$ is $1/(1+t)^2$, and $\delta(t)=1/(1+t)$. Find the expected present value. (Answer: $337.50)

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