Why Duration-Specific Cash Flows Matter
A prospective reserve depends on the cash flows still expected after the valuation time. For long-term contracts, those flows need not stay level: a premium may change after an introductory period, a benefit may increase at a specified duration, or expenses may differ in later policy years. In a multi-state model, the amount may also depend on the policyholder’s state or on the transition occurring during the period.
The calculation principle is unchanged: value the next period’s cash flow and the reserve at the next valuation time, conditional on the current state. The essential discipline is to use the cash-flow amount and transition probabilities that apply to that particular duration. The methods below use an annual discrete-time model; they do not address continuous payment timing.
The cash flow and reserve are conditional on the transition destination in this notation. A payment that depends only on the starting state can be represented by setting the same amount \(C_{k+1}(i,j)\) for every possible destination \(j\). A payment triggered by a transition can instead have different amounts for different destinations. If a premium is received at period-end, it enters \(C\) as a negative amount. This convention makes each period’s timing explicit; cash flows at other times require a different valuation convention.
The Backward Projection Method
A reserve projection usually begins with a known terminal reserve, often zero after a policy has terminated or after all contractual obligations have ended. Work backward one duration at a time. At each step, use the transition probabilities for the period being valued, add the period’s net outflow to the reserve at the destination state, take the conditional expectation, and discount one year.
Specify the reserve at the final valuation time in every possible state. A terminated state commonly has reserve zero when no further payments are due.
For the period from \(k\) to \(k+1\), identify its own transition probabilities and cash-flow amounts. Do not substitute values from another duration unless the model says they are unchanged.
For each possible destination \(j\), add \(C_{k+1}(i,j)\) to \(V_{k+1}(j)\), weight by \(p_{ij}^{(k)}\), and sum over destinations.
Multiply the expected value by \(v\) to obtain \(V_k(i)\). Repeat for each state at the preceding duration.
The recursion follows directly from the prospective definition. Conditional on being in state \(i\) at time \(k\), the next state is \(j\) with probability \(p_{ij}^{(k)}\). In that event, the insurer’s end-of-period outflow is \(C_{k+1}(i,j)\), and the value at time \(k+1\) of obligations remaining then is \(V_{k+1}(j)\). The total value at time \(k+1\) for that outcome is their sum. Its conditional expected value is therefore the probability-weighted sum over \(j\); discounting that expected end-of-period value by \(v\) gives the value at time \(k\). This also explains why the reserve at the destination state, not the starting state, appears inside the sum.
Worked Projections
Worked Example: Two Living States and Changing Benefits
A hypothetical two-year contract has two nonterminal states, Active (A) and Disabled (D). The annual effective interest rate is 5%, so \(v=1/1.05=20/21\). At the end of year 2, all obligations have ended, so \(V_2(A)=V_2(D)=0\). During year 1, the end-of-year net outflow is $100 from A and $300 from D, regardless of the destination state. The transition probabilities in year 1 are A to A 0.8, A to D 0.2, D to A 0.25, and D to D 0.75. During year 2, the corresponding net outflows are $120 from A and $360 from D; probabilities are A to A 0.7, A to D 0.3, D to A 0.2, and D to D 0.8. Find both reserves at time 0.
Method. Apply the recursion backward from the time-2 boundary. Since the year-2 cash flow depends on the starting state only, the destination probabilities sum to one in each row.
At time 1, in state A, the expected year-2 outflow is \(0.7(120)+0.3(120)=120\). Thus \(V_1(A)=v(120)=2400/21=\$114.2857\ldots\). In state D, it is \(0.2(360)+0.8(360)=360\), so \(V_1(D)=v(360)=7200/21=\$342.8571\ldots\).
At time 0 in state A, the expected year-1 outflow plus time-1 reserve is \(0.8(100+V_1(A))+0.2(100+V_1(D))\). Equivalently, because the year-1 outflow is $100 in either destination, this is \(100+0.8(2400/21)+0.2(7200/21)=100+3360/21=260\). Therefore \(V_0(A)=v(260)=5200/21=\$247.6190\ldots\).
In state D, the year-1 outflow is $300 in either destination. The expected amount before discounting is \(300+0.25(2400/21)+0.75(7200/21)=300+6000/21=1700/5? \) More directly, \(6000/21=285.714285\ldots\), so the amount is \(12300/21=585.714285\ldots\). Hence \(V_0(D)=(20/21)(12300/21)=246000/441=\$557.8231\ldots\).
Check. The year-2 outflows exceed the year-1 outflows, so the time-1 reserves should exceed the present value of year-1 outflows alone in both states. Both do. The method avoids treating the year-1 amounts as level over the whole contract.
Worked Example: Transition Payments and a Termination State
A hypothetical policy has Active (A), Disabled (D), and Terminated (T) states. T is absorbing and has no reserve after entry. The annual effective interest rate is 5%, so \(v=20/21\). For year 2, transitions from A are to A with probability 0.8 and to D with probability 0.2; transitions from D are to A with probability 0.25 and to D with probability 0.75. The year-2 end-of-year transition outflows are \(C_2(A,A)=\$100\), \(C_2(A,D)=\$300\), \(C_2(D,A)=\$150\), and \(C_2(D,D)=\$500\). Take \(V_2(A)=V_2(D)=V_2(T)=0\). For year 1, probabilities from A are 0.7 to A, 0.2 to D, and 0.1 to T; from D they are 0.2 to A, 0.7 to D, and 0.1 to T. Year-1 transition outflows are respectively $80, $280, and $50 from A, and $120, $400, and $200 from D. Find \(V_0(A)\) and \(V_0(D)\).
Method. First value year 2 by transition, then use those state-specific reserves in the year-1 recursion. The payment on entry to T is included in the transition outflow, while \(V(T)=0\).
At time 1 in A, the expected year-2 outflow is \(0.8(100)+0.2(300)=140\), so \(V_1(A)=140(20/21)=\$133.3333\ldots\). In D, it is \(0.25(150)+0.75(500)=37.5+375=412.5\), so \(V_1(D)=412.5(20/21)=\$392.8571\ldots\).
At time 0 in A, the expected year-1 amount is \(0.7(80+V_1(A))+0.2(280+V_1(D))+0.1(50+0)\). The three terms are \(0.7(213.3333\ldots)=149.3333\ldots\), \(0.2(672.8571\ldots)=134.5714\ldots\), and \(5\). Their sum is \(288.9047\ldots\), so \(V_0(A)=(20/21)(288.9047\ldots)=\$275.1473\ldots\).
For D, the expected year-1 amount is \(0.2(120+V_1(A))+0.7(400+V_1(D))+0.1(200+0)\). The terms are \(0.2(253.3333\ldots)=50.6667\ldots\), \(0.7(792.8571\ldots)=555\), and \(20\). The sum is \(625.6667\ldots\), giving \(V_0(D)=(20/21)(625.6667\ldots)=\$595.8730\ldots\).
Check. Each row of transition probabilities sums to one, and the termination payment is counted once, at the transition into T. The common error avoided here is to omit that payment merely because the terminated-state reserve is zero.
Worked Example: A Premium That Changes by Duration
A hypothetical three-year contract has Active (A), Disabled (D), and terminated states. Only A and D have future reserves. The annual effective interest rate is 10%, so \(v=10/11\). At time 3, \(V_3(A)=V_3(D)=0\). In year 3, net outflows at period-end are $110 in A and $230 in D. In year 2, they are $60 in A and $180 in D. In year 1, the gross benefit is $150 in A and $260 in D; a $100 premium is received only in A at year-end, so year-1 net outflows are $50 in A and $260 in D. The transitions in year 3 are A to A 0.75, A to D 0.25, D to A 0.2, and D to D 0.8. In year 2 they are A to A 0.75, A to D 0.25, D to A 0.2, and D to D 0.8. In year 1 they are A to A 0.8, A to D 0.2, D to A 0.1, and D to D 0.9. There are no further outflows on termination. Find the time-0 reserves.
Method. Combine the premium and benefit with the stated sign convention, then project backward. The premium changes the year-1 net outflow only; it is not deducted again in later years.
At time 2, only year-3 outflows remain. Since they depend on the starting state and the transition probabilities in each row sum to one, \(V_2(A)=(10/11)(110)=100\), and \(V_2(D)=(10/11)(230)=2300/11\).
At time 1 in A, the year-2 expectation is \(60+0.75(100)+0.25(2300/11)=60+75+575/11=2060/11\). Thus \(V_1(A)=(10/11)(2060/11)=20600/121=\$170.2479\ldots\).
At time 1 in D, it is \(180+0.2(100)+0.8(2300/11)=180+20+1840/11=4040/11\). Therefore \(V_1(D)=(10/11)(4040/11)=40400/121=\$333.8843\ldots\).
At time 0 in A, the year-1 net outflow is $50. The expected amount before discounting is \(50+0.8(20600/121)+0.2(40400/121)=50+24560/121=30610/121\). Hence \(V_0(A)=(10/11)(30610/121)=306100/1331=\$229.9775\ldots\).
At time 0 in D, the year-1 net outflow is $260. The expected amount is \(260+0.1(20600/121)+0.9(40400/121)=260+38420/121=69880/121\). Hence \(V_0(D)=(10/11)(69880/121)=698800/1331=\$525.0188\ldots\).
Check. The year-1 premium reduces the active-state outflow from $150 to $50, and appears exactly once. The method avoids the sign error of adding a premium as though it were an insurer outflow.
Exam Traps and Practical Checks
The preceding examples illustrate why a reserve calculation should be organized by duration and state, rather than by one formula applied with an unexamined set of constant inputs. In particular, when a question specifies changing benefits, premiums, expenses, or transition probabilities, each period must be valued with its own data. A useful working table has one row per duration and separate columns for each state’s reserve, the relevant transition probabilities, and the cash flows that apply during that period.
- Using the wrong duration’s amount. A benefit increase in year 2 does not change the year-1 benefit unless the contract says it does. Label every cash flow by the period in which it is payable.
- Ignoring the destination state. If the next reserve depends on the state reached, calculate a probability-weighted sum of cash flow plus destination reserve. Do not use an unweighted average of reserves.
- Reversing the premium sign. Under the stated convention, premiums received are negative outflows. If gross benefits and premiums are listed separately, form the net outflow consistently before applying the recursion.
- Misplacing a transition payment. A payment triggered by entering a state belongs to that transition’s cash flow. A zero reserve after termination does not erase a termination payment made at the transition.
- Using a boundary value too early. Apply the terminal reserve only at its stated valuation time. Earlier reserves include the intervening expected cash flows and discounted continuation values.
At each duration, check that transition probabilities from a given starting state sum to one, that all listed outflows use the same sign convention, and that the reserve has the stated conditioning state. As a final check, substitute the calculated reserves into the recursion. If a cash flow changes by duration, a calculation that uses the same amount in every period needs explicit justification from the model.
Check Your Understanding
Use end-of-year net outflows, positive for insurer payments and negative for premiums received. Unless stated otherwise, a terminated state has reserve zero. Round dollar answers to the nearest cent.
- A one-year contract has a 5% annual effective interest rate. From its current state, it continues with probability 0.9, producing a $50 end-of-year outflow and a time-1 reserve of $40; it terminates with probability 0.1 and pays $300 at termination. Calculate the current reserve. (Answer: $105.71)
- At a time with zero interest, a policy in state A transitions to A with probability 0.6 and to D with probability 0.4. The transition outflows are $10 and $40, respectively, and the next reserves are \(V_1(A)=\$100\) and \(V_1(D)=\$250\). Calculate \(V_0(A)\). (Answer: $155.60)
- At 4% annual effective interest, a one-period model has states A and D. From A, probabilities to A and D are 0.75 and 0.25; from D they are 0.2 and 0.8. The end-of-period outflow is $80 in A and $200 in D, independent of destination, and both next reserves are zero. Find both current reserves. (Answer: \(V_0(A)=\$76.92\), \(V_0(D)=\$192.31\))
- At 5% annual effective interest, a policy in state A has next-state probabilities 0.8 to A, 0.1 to D, and 0.1 to T. The respective transition outflows are $20, $100, and $300; next reserves are $150 in A, $400 in D, and zero in T. Calculate the current reserve in A. (Answer: $154.29)
- A two-year, one-state policy has 0% interest and no transitions. Its end-of-year net outflows are $40 in year 1 and $100 in year 2, with zero reserve at time 2. Calculate its time-0 reserve and identify the amount used for each year. (Answer: $140; use $40 in year 1 and $100 in year 2.)