Integrated Reserve and Sensitivity Workshop

Why Integrate Reserves and Sensitivity?

A change in mortality, lapse, expenses or interest can affect more than one part of a profit test. It may change the expected cash flows, the number of contracts remaining in force and the reserve held for each surviving contract. The resulting profit movement is therefore not necessarily the direct cost of the changed assumption: it can also reflect when a reserve is established or released.

This workshop uses a hypothetical two-year term-insurance portfolio to connect these effects. The model is deliberately discrete and single-state: contracts are either in force at the start of a year or have terminated. It is not a multi-state reserve model. All monetary outputs are reported to cents; intermediate calculations retain full precision.

Model and timing: At the beginning of year $t$, each in-force contract has reserve $V_t$, receives premium $P_t$, and incurs expense $e_t$. The net amount is invested for one year at effective annual rate $i$. At year-end, death occurs with probability $q_t$, lapse with probability $l_t$, and survival in force has probability $p_t=1-q_t-l_t$. Death and lapse benefits are paid at year-end, and $V_{t+1}$ is held for each contract surviving in force. The probabilities are conditional on being in force at the start of the year, and must satisfy $q_t\geq 0$, $l_t\geq 0$, and $q_t+l_t\leq 1$.

Reserve-Adjusted Profit and Net Present Value

Let $B$ be the death benefit, $S_t$ the lapse benefit, and $n_t$ the expected number of contracts in force at the start of year $t$. Thus $n_{t+1}=n_t p_t$. With the timing above, the expected profit at the end of year $t$, per contract in force at the start of that year, is the accumulated starting reserve and net premium, less expected termination benefits and the reserve for survivors.

\pi_t=(V_t+P_t-e_t)(1+i)-q_tB-l_tS_t-p_tV_{t+1}, \qquad \Pi_t=n_t\pi_t.

Here $\pi_t$ is per starting in-force contract and $\Pi_t$ is the aggregate profit for the cohort, expressed per initial portfolio unit if $n_0=1$. A zero lapse benefit means $S_t=0$, not that the lapse probability is zero. For a term contract reaching scheduled expiry, the terminal reserve is commonly set to $V_T=0$ if no benefit or liability remains after expiry.

For a reserve calculated on the same assumptions as the projection, the one-year prospective reserve relation is

V_t=v\bigl(q_tB+l_tS_t+p_tV_{t+1}\bigr)-(P_t-e_t), \qquad v=\frac{1}{1+i}.

This relation is a convenient reserve basis for the examples, not a statement that every regulatory or reporting reserve must use the profit-test assumptions. If the valuation basis differs, the projected profit need not be zero even when experience matches the projection assumptions.

The net present value (NPV) of the profit signature discounts each year-end profit to time zero. It is important to discount at the same rate and from the same profit timing used in the projection.

\operatorname{NPV}=\sum_{t=0}^{T-1}v^{t+1}\Pi_t.

A Short Justification of the Profit Formula

At the start of a year, a contract brings the reserve $V_t$ and receives net cash $P_t-e_t$. Together these funds earn interest for the year, giving $(V_t+P_t-e_t)(1+i)$. At year-end, the expected death and lapse payments per starting contract are $q_tB$ and $l_tS_t$. The expected reserve needed for continuing contracts is $p_tV_{t+1}$. Subtracting these three amounts from the accumulated funds leaves the expected year-end surplus, which is $\pi_t$. Multiplying by $n_t$ gives the cohort profit $\Pi_t$.

The reserve relation follows by setting this one-year expected profit to zero on the reserve basis. Solving $(V_t+P_t-e_t)(1+i)=q_tB+l_tS_t+p_tV_{t+1}$ for $V_t$ gives the displayed formula. This also explains why a reserve change can shift profit between years: profit is the residual after the required closing reserve, not simply the period’s premium less claims and expenses.

Worked Workshop

Worked Example: Establishing the Baseline Profit Signature

A hypothetical insurer projects 1,000 identical two-year term contracts. Each contract pays a death benefit of $B=\$1{,}000$, has no lapse benefit, and receives an annual premium of $P=\$20$ while in force. The initial expense is $e_0=\$8$; the renewal expense is $e_1=\$4$. The effective annual interest rate is $i=5\%$. In year 1, $q_0=0.01$ and $l_0=0.05$; in year 2, $q_1=0.02$ and $l_1=0.04$. The terminal reserve is $V_2=0$. Calculate the reserve at the start of year 2, both years’ expected profits, and the NPV per initial contract.

Method. Use the year-2 assumptions to calculate $V_1$, then use the reserve-adjusted profit equation and discount the year-end profit signature. The in-force probabilities are $p_0=1-0.01-0.05=0.94$ and $p_1=1-0.02-0.04=0.94$. Since the lapse benefit and terminal reserve are zero,

V_1=\frac{0.02(1000)+0.04(0)+0.94(0)}{1.05}-(20-4) =\frac{20}{1.05}-16 =3.047619\ldots

Thus $V_1=\$3.05$ per contract in force at the start of year 2. In year 1, the accumulated funds per initial contract are $(0+20-8)(1.05)=12(1.05)=\$12.60$. Expected death claims are $0.01(1000)=\$10$, and the expected closing reserve per initial contract is $0.94(3.047619\ldots)=\$2.864762\ldots$. Therefore,

\Pi_0=1\bigl(12.60-10-2.864762\ldots\bigr) =-0.264762\ldots

There are $1(0.94)=0.94$ expected contracts in force at the start of year 2 per initial contract. For each such contract, accumulated funds are $(3.047619\ldots+20-4)(1.05)=19.047619\ldots(1.05)=\$20$. Expected death claims are $0.02(1000)=\$20$, and $V_2=0$. Year-2 profit is therefore zero, both per contract and in aggregate. The NPV per initial contract is

\operatorname{NPV} =\frac{-0.264762\ldots}{1.05}+\frac{0}{1.05^2} =-0.252154\ldots

Check. The expected cash flows give the same NPV. The initial net cash flow is $20-8=\$12$; at time 1, expected renewal net cash is $0.94(20-4)=\$15.04$ and year-1 claims are $\$10$; at time 2, expected claims are $0.94(0.02)(1000)=\$18.80$. Hence $12+(15.04-10)/1.05-18.80/1.05^2=-0.252154\ldots$. The answer is $\operatorname{NPV}=-\$0.25$ per initial contract, rounded to cents. A common mistake is to discount the year-2 claim by one year rather than two; it is paid at the end of year 2.

Worked Example: Recalculating Reserves Under a Combined Shock

Use the same hypothetical product, but suppose the shocked projection has $i=4\%$, initial expense $e_0=\$9$, renewal expense $e_1=\$5$, and probabilities $(q_0,l_0)=(0.015,0.04)$ in year 1 and $(q_1,l_1)=(0.03,0.03)$ in year 2. Premiums, death benefit, lapse benefit and terminal reserve are unchanged. Recalculate the year-2 starting reserve, the year-1 profit, the expected reserve held for survivors, and the NPV. Compare the results with the baseline.

Method. Recalculate the reserve on the shocked assumptions before projecting profit. The shocked in-force probabilities are $p_0=1-0.015-0.04=0.945$ and $p_1=1-0.03-0.03=0.94$. The year-2 reserve is

V_1^{\mathrm{shock}} =\frac{0.03(1000)}{1.04}-(20-5) =\frac{30}{1.04}-15 =13.846154\ldots

The per-contract reserve movement is $13.846154\ldots-3.047619\ldots=\$10.798535\ldots$, an increase of $\$10.80$ per in-force contract, rounded to cents. The expected year-1 closing reserve per initial contract is $0.945(13.846154\ldots)=\$13.084615\ldots$, compared with $\$2.864762\ldots$ at baseline. The cohort reserve exposure has increased by $\$10.219853\ldots$ per initial contract.

In year 1, the accumulated funds are $(20-9)(1.04)=\$11.44$. Expected death claims are $0.015(1000)=\$15$, and the expected reserve for survivors is $\$13.084615\ldots$. Thus the year-1 profit per initial contract is

\Pi_0^{\mathrm{shock}} =11.44-15-13.084615\ldots =-16.644615\ldots

By construction, the year-2 profit per in-force contract is zero: $(13.846154\ldots+20-5)(1.04)=\$30$, exactly equal to the expected year-2 claim $0.03(1000)=\$30$. Therefore, the shocked NPV is $-16.644615\ldots/1.04=-16.004438\ldots$, or $-\$16.00$ per initial contract to cents. Relative to the baseline NPV of $-\$0.25$, the NPV change is $-\$15.75$ per initial contract to cents.

Check. The shocked expected cash flows give $11+(0.945(20-5)-15)/1.04-0.945(0.03)(1000)/1.04^2=-16.004438\ldots$, which agrees with the profit-signature NPV. This comparison changes mortality, lapse, expenses and interest together; it does not identify the effect of any one assumption by itself.

Worked Example: Isolating an Interest-Rate Sensitivity

Return to the baseline mortality, lapse and expense assumptions, but reduce the effective annual interest rate from $5\%$ to $4\%$. Recalculate $V_1$, the year-1 profit and the NPV. Then measure the NPV change from baseline.

Method. Hold all assumptions except interest fixed and recalculate the reserve. This is an interest-only sensitivity, not a combined shock. The year-2 reserve is

V_1^{(4\%)} =\frac{0.02(1000)}{1.04}-(20-4) =\frac{20}{1.04}-16 =3.230769\ldots

The reserve increases by $3.230769\ldots-3.047619\ldots=\$0.183150\ldots$ per in-force contract, or $\$0.18$ to cents. Expected year-1 closing reserves are $0.94(3.230769\ldots)=\$3.036923\ldots$ per initial contract. The year-1 accumulated funds are $(20-8)(1.04)=\$12.48$. Subtracting expected claims and the survivor reserve gives

\Pi_0^{(4\%)} =12.48-0.01(1000)-0.94(3.230769\ldots) =-0.556923\ldots

Year-2 profit is again zero because $(3.230769\ldots+20-4)(1.04)=\$20$, matching the expected year-2 claim per in-force contract. The NPV is $-0.556923\ldots/1.04=-0.535503\ldots$, or $-\$0.54$ per initial contract to cents. The change from the baseline NPV is $-0.535503\ldots-(-0.252154\ldots)=-\$0.28$ per initial contract to cents.

Check. The issue-date cash-flow NPV is $12+(0.94(20-4)-10)/1.04-0.94(0.02)(1000)/1.04^2=-0.535503\ldots$. It agrees with the profit-signature result. The common mistake is to change the discount rate for NPV but leave the reserve unchanged; that mixes bases and can misstate both profit emergence and the sensitivity.

Interpreting the Movements

The combined shock increased the reserve per surviving contract and also changed the expected number of survivors. These are separate effects. In the example, the per-contract reserve rose by $\$10.80$, while the expected reserve held per initial contract rose by about $\$10.22$, because the in-force probability also changed. Reporting only the per-contract reserve movement would not describe the total cohort exposure.

A reserve increase is not, by itself, an equal and opposite change in NPV. The reserve affects the timing and amount of projected profits; the NPV aggregates those profits using a stated discount convention. When assumptions are changed, recalculate the in-force counts, cash flows and reserves consistently, then discount the resulting profit signature. For attribution, change one assumption at a time or use a clearly specified two-way comparison; a combined shock alone cannot identify individual contributions.

The reconciliation to expected cash flows is a useful check when the profit-test and NPV conventions match. It can fail if premiums, expenses, claims, reserves or profits are placed at inconsistent times, if the discount basis is changed in only one part of the calculation, or if a reserve floor is applied in one projection but not the other. The calculations here assume no reserve floor, no surrender benefit and no policy-level variation. They are illustrative hypothetical projections, not statements about actual insurers.

Key takeaway: State the timing and assumptions first. Recalculate reserves and in-force counts under the chosen scenario, construct the reserve-adjusted profit signature, and discount profits from their payment dates. Reconcile to expected cash flows before interpreting the sensitivity.

Check Your Understanding

For each problem, use the annual timing and reserve-adjusted profit conventions stated above. Report monetary answers to cents.

A one-year reserve calculation has $B=\$800$, $q=0.025$, $l=0.05$, lapse benefit $S=\$40$, $i=4\%$, $P=\$18$, $e=\$3$, and $V_1=0$. Calculate the starting reserve. (Answer: $\$6.15$)
At the start of a year, $V_t=0$, $P=\$30$, $e=\$10$, and $i=3\%$. At year-end, $q=0.02$, $l=0.08$, $B=\$900$, $S=\$5$, and $V_{t+1}=\$12$. Calculate profit per starting in-force contract. (Answer: $-\$8.60$)
A two-year profit signature per initial contract has year-end profits $\Pi_0=\$4$ and $\Pi_1=-\$3$. If $i=5\%$, calculate its time-zero NPV. (Answer: $\$1.09$)
A term contract has $B=\$1{,}200$, $P=\$25$, no lapse benefit and $V_2=0$. Under the baseline year-2 assumptions, $q_1=0.015$, $l_1=0.035$, $e_1=\$5$, and $i=5\%$. Under a shocked basis, $q_1=0.025$, $l_1=0.025$, $e_1=\$7$, and $i=4\%$. Calculate the shocked-minus-baseline movement in $V_1$. (Answer: $+\$13.70$ per in-force contract)
For a one-year projection, $V_t=0$, $P=\$24$, $e_t=\$6$, $i=4\%$, $q_t=0.01$, $l_t=0.09$, $B=\$1{,}500$, lapse benefit $S_t=\$20$, and $V_{t+1}=\$8$. Calculate the year-end profit and its time-zero NPV. (Answer: profit $-\$5.28$; NPV $-\$5.08$)

What You'll Learn