Numerical Valuation of Multi-Period Contracts

Why Value a Contract Period by Period?

A disability income contract can make different payments depending on whether the insured is active, disabled, or dead at a payment date. Recovery can return a disabled insured to active status, while a new disability can begin from the active state. When transition assumptions or benefits vary by period, a numerical valuation is a direct way to combine these features without forcing them into a closed-form expression.

The essential procedure is to work backward from the end of the contract. At each period, condition on the state reached at the next payment date, add the cash flow due in that state to the value of any remaining contract, weight by the transition probability, and discount. This tutorial uses a discrete-time model with annual transitions and end-of-year payments. Its methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let the possible states be active $A$, disabled $D$, and dead $Z$. At time $t$, let $p_{ij}(t)$ be the probability that the insured is in state $j$ at time $t+1$, given that the insured is in state $i$ at time $t$. For each starting state $i$, the transition probabilities satisfy $\sum_j p_{ij}(t)=1$. Let $c_{t+1}(j)$ be the insurer’s net cash flow at the end of period $t+1$ if the insured is then in state $j$, with benefits treated as positive outflows and premiums as negative outflows. Let $V_t(i)$ be the expected present value, at time $t$, of future net outflows if the insured is in state $i$ then.

For a contract with $N$ annual periods and effective annual interest rate $i$, the discount factor for one year is $v=1/(1+i)$. The model assumes that the transition probabilities and cash flows are specified for each period, the next state is sufficient for determining the stated cash flow, and each cash flow occurs at the end of its period. If payments occur at the beginning of a period or within a period, their timing must be represented differently.

The Backward Recursion and the Boundary Condition

At the end of the final period, no future contract value remains after the final payment has been made. Thus the terminal boundary condition is $V_N(j)=0$ for every state $j$. The final payment itself belongs in $c_N(j)$; it is not part of $V_N(j)$. For each earlier period, the value follows from the next period’s values:

V_t(i)=v\sum_j p_{ij}(t)\left[c_{t+1}(j)+V_{t+1}(j)\right], \qquad t=N-1,\ldots,0.

The recurrence follows by conditioning on the state at the end of the next period. If that state is $j$, the insurer pays the net amount $c_{t+1}(j)$ then and holds a contract with value $V_{t+1}(j)$ immediately afterward. Their sum is a time-$t+1$ amount. Multiplying by $p_{ij}(t)$ weights that amount by the probability of reaching $j$, summing covers all possible next states, and multiplying by $v$ discounts the expected amount back one year. The terminal condition prevents any value from being carried beyond the contract’s final date.

Specify the states and transitions.
For each period and starting state, list the probabilities of moving to each possible next state. Check that each row sums to 1.

Specify end-of-period cash flows.
Assign benefits and premiums by the state at the payment date. Use a consistent sign convention, such as positive outflows for benefits and negative outflows for premiums.

Set the terminal values.
After the final payment, set the remaining contract value in every state to zero.

Recurse backward.
For each earlier period and starting state, use the transition-weighted cash flows plus next-period values, discounted one period.

Interpret the result.
A positive value is an expected present value of net outflows; a negative value means expected premiums exceed expected benefits under the specified assumptions.

Worked Valuations

The examples use hypothetical contracts. Unless stated otherwise, death is an absorbing state with no further payments. The value is calculated from the insurer’s perspective, so a premium received reduces net outflows.

Worked Example: Two-Year Benefit and Active-State Premium

A hypothetical two-year contract pays $1,000 at each year-end when the insured is disabled at that date and receives a $300 premium when the insured is active. There is no payment on death. The annual effective interest rate is 5%. In each year, the transition probabilities are: from active, 0.90 to active, 0.08 to disabled, and 0.02 to dead; from disabled, 0.20 to active, 0.75 to disabled, and 0.05 to dead. Calculate the value at time 0 for an insured who is initially active and for one who is initially disabled.

Method. Use backward recursion with $v=1/1.05$. The end-of-year net cash flows are $-300$ in active, $1{,}000$ in disabled, and $0$ in dead. After the second year’s payment, the contract has ended, so $V_2(A)=V_2(D)=V_2(Z)=0$.

First calculate the values at time 1. If active at time 1, the expected year-2 cash flow is

$$ 0.90(-300)+0.08(1{,}000)+0.02(0)=-270+80=-190. $$

Therefore $V_1(A)=-190/1.05=-180.952381$. If disabled at time 1, the expected year-2 cash flow is

$$ 0.20(-300)+0.75(1{,}000)+0.05(0)=-60+750=690, $$

so $V_1(D)=690/1.05=657.142857$. The value in the dead state is zero.

Now start in the active state at time 0. The next-period amount is the year-1 cash flow plus the time-1 value in the reached state:

\begin{aligned} V_0(A) &=\frac{0.90[-300+(-180.952381)]+0.08[1{,}000+657.142857]+0.02(0)}{1.05}\\ &=\frac{-432.857143+132.571429}{1.05} =\frac{-300.285714}{1.05} =-285.986395. \end{aligned}

For an insured initially disabled,

\begin{aligned} V_0(D) &=\frac{0.20[-300+(-180.952381)]+0.75[1{,}000+657.142857]+0.05(0)}{1.05}\\ &=\frac{-96.190476+1{,}242.857143}{1.05} =\frac{1{,}146.666667}{1.05} =1{,}092.063492. \end{aligned}

Rounded to the nearest cent, the values are $-$285.99$ from active and $1,092.06 from disabled. The negative active-state value is consistent with the specified assumptions: when initially active, expected premium income is greater than expected benefit outgo in present-value terms.

Common mistake avoided: Treating a year-1 disability as if it guarantees the year-2 benefit. The year-2 value depends on the state transition from disabled, including the possibility of recovery or death.

Worked Example: Period-Varying Assumptions and Starting Disabled

A hypothetical two-year policy pays $800 if disabled at the end of year 1 and $1,100 if disabled at the end of year 2. It receives a $250 premium at either year-end if the insured is active, with no payment on death. The annual effective interest rate is 4%. In year 1, the transition probabilities from active are 0.85 to active, 0.12 to disabled, and 0.03 to dead; from disabled they are 0.25 to active, 0.70 to disabled, and 0.05 to dead. In year 2, the corresponding probabilities are 0.88, 0.09, and 0.03 from active, and 0.30, 0.65, and 0.05 from disabled. Find the value at time 0 if the insured is disabled initially.

Method. Work backward using year-specific transitions and cash flows. Set all time-2 continuation values to zero. For the final year, the cash flows are $-250$ in active, $1{,}100$ in disabled, and zero in dead. Thus, from active at time 1,

V_1(A)=\frac{0.88(-250)+0.09(1{,}100)+0.03(0)}{1.04} =\frac{-220+99}{1.04} =-116.346154.

From disabled at time 1,

V_1(D)=\frac{0.30(-250)+0.65(1{,}100)+0.05(0)}{1.04} =\frac{-75+715}{1.04} =615.384615.

The value from an initially disabled state uses the year-1 transitions. A transition to active gives a year-1 premium of $-250$ and continuation value $V_1(A)$; a transition to disabled gives the year-1 benefit of $800$ and continuation value $V_1(D)$. Therefore,

\begin{aligned} V_0(D) &=\frac{0.25[-250+(-116.346154)]+0.70[800+615.384615]+0.05(0)}{1.04}\\ &=\frac{-91.586538+990.769231}{1.04} =\frac{899.182693}{1.04} =864.598743. \end{aligned}

The value is $864.60 to the nearest cent. The transition probabilities must be matched to the period: using year-1 probabilities in the final-year calculation would change the result.

Common mistake avoided: Applying the year-2 benefit amount in year 1. The cash flow is determined by both the payment period and the state at that period’s end.

Worked Example: A Final-Year Settlement and the Boundary Condition

A hypothetical two-year contract pays a $1,000 disability benefit at the end of year 1 if the insured is disabled then. It also receives a $200 premium at that date if the insured is active. At the end of year 2, it pays a final settlement of $500 if active or $1,200 if disabled, and receives a $200 premium if active. There is no payment on death. The annual effective interest rate is 5%. In year 1, the transition probabilities from active are 0.80 to active, 0.15 to disabled, and 0.05 to dead. In year 2, they are 0.85, 0.10, and 0.05. From disabled, the year-1 probabilities are 0.30 to active, 0.65 to disabled, and 0.05 to dead; the year-2 probabilities are 0.25, 0.70, and 0.05. Find the value at time 0 for an initially active insured.

Method. In year 2, combine the final settlement and premium into one net cash flow: active has $500-200=300$, disabled has $1{,}200$, and dead has zero. The contract ends after that payment, so $V_2=0$ in every state. At time 1,

\begin{aligned} V_1(A)&=\frac{0.85(300)+0.10(1{,}200)}{1.05} =\frac{255+120}{1.05}=357.142857,\\ V_1(D)&=\frac{0.25(300)+0.70(1{,}200)}{1.05} =\frac{75+840}{1.05}=871.428571. \end{aligned}

For the first year, the cash flows are $-200$ in active and $1{,}000$ in disabled. Starting active, the time-1 amounts after a transition to active and disabled are $-200+357.142857=157.142857$ and $1{,}000+871.428571=1{,}871.428571$, respectively. Hence,

\begin{aligned} V_0(A) &=\frac{0.80(157.142857)+0.15(1{,}871.428571)+0.05(0)}{1.05}\\ &=\frac{125.714286+280.714286}{1.05} =\frac{406.428572}{1.05} =387.074830. \end{aligned}

The value is $387.07 to the nearest cent. The final-year settlement is included in the final-year cash flow, while the value after that cash flow is zero.

Common mistake shown and corrected: A mistaken setup might assign $V_2(A)=500$ and $V_2(D)=1{,}200$, then include those amounts again as year-2 cash flows. That counts the final settlement twice. The correct boundary condition is $V_2(A)=V_2(D)=V_2(Z)=0$, with the settlement included once in $c_2(j)$.

Checks and Interpretation

Numerical recursion is systematic, but a well-organized calculation still depends on correct timing and state assignment. Before accepting a result, check that every transition row sums to 1, that each payment is attached to the specified state and date, and that the number of discount factors matches the number of years between valuation and payment. If the contract has a benefit limit or another termination rule, transitions and cash flows after termination must not create additional payments.

A useful audit is to inspect the last period first: its cash flow should contain every payment due at the contract boundary, and its continuation value afterward should be zero. Then verify each earlier step by checking that its expected amount is a probability-weighted sum over all next states. A negative value is not automatically an error; it can result when expected premium income exceeds expected benefits.

Key takeaway: For a finite-period disability contract with end-of-period payments, value backward from a zero post-contract boundary. At each step, combine the cash flow due in the next state with that state’s continuation value, average using the appropriate transition probabilities, and discount one period.

Check Your Understanding

For each question, use the stated insurer-outflow sign convention and round money to the nearest cent.

A one-year contract has an annual effective interest rate of 4%. An initially active insured transitions with probabilities 0.80 to active, 0.15 to disabled, and 0.05 to dead. The year-end premium is $100 if active and the benefit is $500 if disabled. Find the value at time 0. (Answer: $-$4.81$)
A two-year contract has an annual effective interest rate of 10%. In each year, the transition probabilities from active are 0.80 to active, 0.15 to disabled, and 0.05 to dead; from disabled they are 0.25 to active, 0.70 to disabled, and 0.05 to dead. At each year-end, the premium is $100 if active and the benefit is $600 if disabled. Find the time-0 value for an initially active insured. (Answer: $64.67)
For a one-year contract at 5% annual effective interest, an initially disabled insured has probabilities 0.40 of being active, 0.55 of remaining disabled, and 0.05 of dying at year-end. The premium is $150 if active and the benefit is $700 if disabled. Find the value at time 0. (Answer: $309.52)
A two-year contract has an annual effective interest rate of 5%. In year 1, from active the probabilities are 0.70 to active, 0.20 to disabled, and 0.10 to dead; from disabled they are 0.30, 0.60, and 0.10. In year 2, from active they are 0.80, 0.10, and 0.10; from disabled they are 0.20, 0.70, and 0.10. The year-1 cash flow is a $150 premium if active or a $700 benefit if disabled. In year 2, the final settlement is $400 if active or $900 if disabled, and a $150 premium is received if active; there is no payment on death. Find the time-0 value for an initially active insured, applying the terminal boundary correctly. (Answer: $340.82)

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