Worked Valuation of a Multi-State Cash-Flow Contract

Valuing a Contract Across States and Payment Dates

A multi-state contract can generate cash flows for several different reasons. A premium might be due only when the insured is in a particular state; a recurring benefit might depend on the state at a payment date; and a separate amount might be paid only when a particular transition occurs. Expenses can also depend on the state. A reliable valuation keeps these payment rules distinct, identifies when each amount is paid, and discounts it from that date.

The method here applies to a finite set of payment dates and known transition probabilities. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The examples use generic states, so the cash-flow method does not depend on a particular product interpretation.

Definition: Let $X_t$ denote the state occupied at time $t$, where $t=0,1,\ldots,n$. Write $\pi_i(t)=P(X_t=i)$ for the probability of being in state $i$ at time $t$, and $p_{ij}(t)=P(X_{t+1}=j\mid X_t=i)$ for the probability of a transition from state $i$ to state $j$ over the period ending at $t+1$. Let $c_i(t)$ be the net outgo paid at time $t$ if the contract is in state $i$, and let $g_{ij}(t)$ be the net outgo paid at time $t$ if the transition over the preceding period was from $i$ to $j$. Outgo is positive; premiums received are negative.

A state payment is based on the state at its payment date. A transition payment is based on the pair of states at the two ends of the preceding period. For example, a payment triggered by an $A\to B$ transition at time $t$ uses the probability that the contract was in $A$ at $t-1$ and moved to $B$ at $t$. It is not generally determined by the probability of being in $B$ at time $t$.

Assume an effective annual interest rate $i$, with $v=1/(1+i)$, and payments at integer times. State-based payments may occur at times $0,\ldots,n$. Transition payments occur at times $1,\ldots,n$, since there is no preceding transition within the contract at time 0. The expected present value (EPV) is calculated from the expected net outgo at each payment date.

\pi_j(t+1)=\sum_i \pi_i(t)p_{ij}(t)

E[C_0]=\sum_i \pi_i(0)c_i(0), \qquad E[C_t]=\sum_i \pi_i(t)c_i(t) +\sum_i\sum_j \pi_i(t-1)p_{ij}(t-1)g_{ij}(t),\quad t=1,\ldots,n

\operatorname{EPV}=\sum_{t=0}^{n}v^tE[C_t]

The first expression propagates the state probabilities forward by one period. In the cash-flow expressions, state-based and transition-based amounts are separate terms: the former uses the probability of occupying a state at the payment date, while the latter uses the probability of the relevant transition. A cash flow that does not apply in a state or on a transition has amount zero there.

Why the Expected Cash-Flow Formula Works

The valuation formula follows from writing each payment as an amount multiplied by an indicator for the event that triggers it. Let $I_i(t)$ equal 1 if $X_t=i$ and 0 otherwise. For a state payment at time $t$, the amount is $c_i(t)I_i(t)$; taking expectations gives $c_i(t)P(X_t=i)=c_i(t)\pi_i(t)$. Summing over states gives the expected state-based cash flow.

Similarly, let $I_{ij}(t)$ equal 1 if the transition from $i$ at $t-1$ to $j$ at $t$ occurs, and 0 otherwise. The expected transition payment is $g_{ij}(t)P(X_{t-1}=i,X_t=j)$. By conditioning on the state at $t-1$, $P(X_{t-1}=i,X_t=j)=\pi_i(t-1)p_{ij}(t-1)$. Adding the state and transition payments is justified by linearity of expectation; the payment events need not be independent. Discounting each expected payment by $v^t$ then gives its present value at time 0.

This derivation assumes the listed cash flows are fully specified by the state at the payment date, by the immediately preceding transition, or by both as separate amounts. If a payment rule depends on additional history, such as the number of previous claim spells, the current state probabilities alone may not contain enough information to value it. The contract would need to represent that relevant history in the model or calculate its effect separately.

Worked Contract Valuations

Worked Example: State-Dependent Premiums and a Transition Benefit

A hypothetical two-year contract has states $A$, $B$, and $T$, starts in $A$, and has the following transition probabilities in each year: $A\to(A,B,T)=(0.6,0.3,0.1)$ and $B\to(A,B,T)=(0.2,0.6,0.2)$. State $T$ is absorbing. The effective annual interest rate is $5\%$. At time 0, an expense of $30 is paid and a premium of $100 is received in state $A$. At times 1 and 2, premiums of $100 in $A$ and $40 in $B$ are received. At those same times, a benefit of $200 is paid in $B$, an expense of $10 is paid in $A$, an expense of $20 is paid in $B$, and $300 is paid on each $A\to B$ transition. Find the EPV of net outgo.

Method. First find the state probabilities at each payment date and the probability of each $A\to B$ transition. Then calculate expected net outgo date by date and discount each amount. This avoids treating a transition benefit as if it were paid on every stay in state $B$.

At time 0, $\pi_A(0)=1$, so net outgo is $30−$100 = −$70. At time 1, the probabilities are $\pi_A(1)=0.6$, $\pi_B(1)=0.3$, and $\pi_T(1)=0.1$. The probability of an $A\to B$ transition is $1(0.3)=0.3$.

Expected premiums at time 1 are $0.6(100)+0.3(40)=72$. Expected recurring benefits are $0.3(200)=60$, expected expenses are $0.6(10)+0.3(20)=12$, and expected transition benefits are $0.3(300)=90$. Thus expected net outgo at time 1 is $60+12+90-72=90$.

At time 2, $\pi_A(2)=0.6(0.6)+0.3(0.2)=0.42$, and $\pi_B(2)=0.6(0.3)+0.3(0.6)=0.36$. The remaining probability, $0.22$, is in $T$. The time-2 $A\to B$ transition probability is $0.42(0.3)=0.126$.

Expected premiums at time 2 are $0.42(100)+0.36(40)=56.40$. Expected recurring benefits are $0.36(200)=72$, expenses are $0.42(10)+0.36(20)=11.40$, and transition benefits are $0.126(300)=37.80$. Therefore expected net outgo at time 2 is $72+11.40+37.80-56.40=64.80$.

Using $v=1/1.05$, the EPV is $-70+90/1.05+64.80/(1.05)^2=-70+85.714286+58.775510=74.489796$. Rounded to the nearest cent, the EPV of net outgo is $74.49. The check is that the three discounted date amounts, −$70.00, $85.71, and $58.78, sum to $74.49 after rounding.

Worked Example: Solving for a Level Premium

A hypothetical two-year contract starts in $A$, has states $A,B,T$, and has time-invariant transition probabilities $A\to(A,B,T)=(0.5,0.4,0.1)$ and $B\to(A,B,T)=(0.25,0.5,0.25)$, with $T$ absorbing. At an effective annual interest rate of $4\%$, a level premium $x$ is received at times 0, 1, and 2 whenever the contract is in $A$. A benefit of $160 is paid in $B$ at times 1 and 2, and $240 is paid for each $A\to B$ transition at those times. An expense of $20 is paid at time 0 and an expense of $10 is paid in $B$ at times 1 and 2. Find $x$ such that the EPV of net outgo is zero.

Method. Value benefits and expenses as positive outgo, and premiums as negative outgo. The premium EPV is $x$ times the discounted probability of being in $A$ at each premium date.

The state probabilities are $\pi_A(0)=1$; at time 1, $\pi_A(1)=0.5$ and $\pi_B(1)=0.4$. At time 2, $\pi_A(2)=0.5(0.5)+0.4(0.25)=0.35$ and $\pi_B(2)=0.5(0.4)+0.4(0.5)=0.40$. The probability of an $A\to B$ transition is $0.40$ at time 1 and $0.35(0.4)=0.14$ at time 2.

Expected benefits are $160(0.40)+240(0.40)=160$ at time 1 and $160(0.40)+240(0.14)=97.60$ at time 2. Expected expenses are $20 at time 0 and \(10(0.40)=$4 at each of times 1 and 2.

With $v=1/1.04=25/26$, the EPV of benefits and expenses is $20+4v+4v^2+160v+97.60v^2$. The expense part is $20+3.846154+3.698225=27.544379$; the benefit part is $153.846154+90.236686=244.082840$. Their total is $271.627219.

The premium EPV coefficient is $1+0.50v+0.35v^2=1+0.480769+0.323595=1.804364$. Setting net outgo to zero gives $271.627219-1.804364x=0$, so $x=271.627219/1.804364=150.539045$. The level premium is therefore $150.54 to the nearest cent.

As a check, using the unrounded premium gives a premium EPV of $271.627219, equal to the EPV of benefits and expenses. Using the rounded premium of $150.54 changes the net EPV by less than two-tenths of a cent. The method avoids discounting all three premiums as if they were certain: the time-1 and time-2 premiums are conditional on being in $A$.

Worked Example: Initial State Uncertainty and a Transition Trigger

A hypothetical one-year contract begins with probability 0.3 in $A$ and 0.7 in $B$. Its transition rows are $A\to(A,B,T)=(0.4,0.4,0.2)$ and $B\to(A,B,T)=(0.2,0.5,0.3)$, with $T$ absorbing. The effective annual interest rate is $3\%$. At time 0, premiums of $90 in $A$ and $40 in $B$ are received, while expenses of $12 in $A$ and $5 in $B$ are paid. At time 1, a benefit of $180 is paid in $B$, a benefit of $350 is paid on an $A\to B$ transition, and expenses of $8 in $A$ and $15 in $B$ are paid. Find the EPV of net outgo.

Method. Use the initial state distribution to calculate time-0 net outgo, then calculate both destination-state probabilities and the probability of the specific transition for time 1. This distinguishes the transition benefit from the recurring benefit in $B$.

At time 0, expected premiums are $0.3(90)+0.7(40)=27+28=55$, and expected expenses are $0.3(12)+0.7(5)=3.60+3.50=7.10$. Net outgo at time 0 is $7.10-55=-47.90$.

At time 1, $\pi_A(1)=0.3(0.4)+0.7(0.2)=0.12+0.14=0.26$, and $\pi_B(1)=0.3(0.4)+0.7(0.5)=0.12+0.35=0.47$. The $A\to B$ transition probability is $0.3(0.4)=0.12$. Expected benefits are $0.47(180)=$84.60$ and $0.12(350)=$42.00$. Expected expenses are $0.26(8)+0.47(15)=2.08+7.05=$9.13$. Thus time-1 expected net outgo is $84.60+42.00+9.13=$135.73$.

The EPV is $-47.90+135.73/1.03=-47.90+131.776699=83.876699$, or $83.88 to the nearest cent. The discounted time-1 amount is positive because it is an outgo; the time-0 premium makes the time-0 net amount negative.

Common mistake and correction. Multiplying the $350 transition benefit by $P(X_1=B)=0.47$ gives $164.50, but that counts every route ending in $B$, including $B\to B$. The correct trigger probability is $P(X_0=A,X_1=B)=0.12$, so the benefit is $0.12(350)=$42.00$. The mistaken calculation overstates expected time-1 outgo by $164.50-42.00=$122.50$, before discounting.

Checks That Prevent Valuation Errors

A useful calculation table has one row per payment date and separate columns for state-based premiums, state-based benefits, state-based expenses, and transition payments. It makes the payment trigger and timing visible before any amounts are combined. In particular, it helps prevent a transition probability from being replaced by a destination-state probability.

Before finalizing a valuation, check the following:

Probability accounting: state probabilities at each date are nonnegative and sum to 1, including absorbing states.
Transition accounting: the probability of a specified transition is the probability in its starting state multiplied by the relevant conditional transition probability.
Payment date: each cash flow is discounted to the date it is actually paid. A time-0 amount is not discounted; a time-$t$ amount is multiplied by $v^t$.
Sign: premiums received reduce net outgo, while benefits and expenses increase it.
Level premium: only include a premium at dates and in states where the contract requires it.

For calculation efficiency, state probabilities can be propagated date by date and reused for every state-based cash flow. Transition probabilities can then be calculated only for transitions with nonzero payments. Alternatively, the EPV can be calculated separately for premiums, benefits, and expenses and combined at the end, provided every component uses the same payment dates, probability rules, and sign convention.

These calculations produce an expected present value, not a description of any one contract’s realized outcome. A particular path through the states may involve more or fewer payments than the expected amounts. The EPV is appropriate for valuing expected cash flows under the stated model; it does not imply that every policy follows an average path.

Key takeaway: For each payment, identify its trigger, calculate the probability of that trigger, assign the amount to its actual payment date, and discount from that date. State-contingent payments use state probabilities; transition-contingent payments use transition probabilities.

Check Your Understanding

For each problem, track state and transition probabilities separately, apply the stated payment timing, and use the given effective annual interest rate.

A one-year contract starts in $A$, with transitions $A\to(A,B,T)=(0.7,0.2,0.1)$. At time 1 it pays $400 in $B$ and $100 on an $A\to B$ transition. At $i=5\%$, find the EPV of benefits. (Answer: $95.24)
A two-year contract starts in $A$ and has transitions $A\to(A,B,T)=(0.5,0.3,0.2)$ and $B\to(A,B,T)=(0.2,0.6,0.2)$. It pays $100 in $B$ and $200 on $A\to B$ at the end of each year. At $i=5\%$, find the EPV of benefits. (Answer: $132.52)
A one-year contract starts with probabilities 0.25 in $A$ and 0.75 in $B$. Transitions are $A\to(A,B,T)=(0.6,0.2,0.2)$ and $B\to(A,B,T)=(0.2,0.5,0.3)$. At time 1, it pays $120 in $B$, $400 on $A\to B$, and expenses of $6 in $A$ and $12 in $B$. At $i=4\%$, find the EPV of benefits and expenses. (Answer: $74.90)
A two-year contract starts in $A$, with transitions $A\to(A,B,T)=(0.6,0.25,0.15)$ and $B\to(A,B,T)=(0.2,0.6,0.2)$. A level premium $x$ is received in $A$ at times 0 and 1. Benefits of $100 and $120 are paid in $B$ at times 1 and 2, respectively. At $i=5\%$, find $x$ so that the EPV of benefits less premiums is zero. (Answer: $35.93)

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