Sensitivity of NPV to Duration Profits

Why One Duration’s Profit Change Has a Specific NPV Effect

A profit signature describes profit by duration, but a dollar of profit at one duration does not necessarily add a dollar to the contract’s NPV. Its effect depends on the chance that the relevant policy or event reaches that duration and on how far in the future the profit emerges. This tutorial isolates the effect of changing one duration’s profit while holding the other assumptions fixed.

The calculation is useful for evaluating a proposed change to a benefit, expense, or other feature when its effect on profit at a particular duration is known. It also gives a direct check on a revised NPV: the new NPV must equal the old NPV plus the discounted, probability-weighted change.

Definition: Let $p_t$ be the profit at duration $t$, conditional on the policy or relevant event reaching that duration, and let $w_t$ be the probability weight applied to that profit. For an effective annual discount rate $i$, the NPV effect of changing only $p_t$ by $\Delta p_t$ is $$ \Delta\mathrm{NPV}_t = w_t\Delta p_t(1+i)^{-\tau_t}, $$ where $\tau_t$ is the time in years from issue to the profit’s emergence. For a year-end profit at duration $t$, $\tau_t=t$; for a midyear profit in duration $t$, $\tau_t=t-\tfrac12$. This formula assumes the weight, discount rate, timing convention, and all other profits are unchanged.

Here, $w_t$ is the probability weight appropriate to the profit being valued. For example, if the profit is earned only if a policy remains in force at the start of duration $t$, the weight can be the probability of being in force then. The exact definition depends on the profit signature’s convention. If $p_t$ is already an unconditional expected profit, it already incorporates the probability weight, so use $w_t=1$ rather than weighting it a second time.

A positive $\Delta p_t$ increases NPV, while a negative $\Delta p_t$ decreases it. A later profit change has a smaller absolute NPV effect than the same change at an earlier time when $i>0$, because its discount factor is smaller. Midyear emergence is earlier than year-end emergence at the same duration, so it has a larger discount factor.

Why the Sensitivity Formula Holds

For a finite signature with conditional profits $p_k$, weights $w_k$, and emergence times $\tau_k$, its NPV is the sum of the discounted weighted profits. Changing one profit changes just one term of that sum. Subtracting the original NPV from the revised NPV therefore leaves the change in that single term.

\begin{aligned} \mathrm{NPV} &= \sum_k w_k p_k(1+i)^{-\tau_k},\\ \mathrm{NPV}_{\mathrm{new}} &= \mathrm{NPV}_{\mathrm{old}} +w_t\bigl(p_t+\Delta p_t-p_t\bigr)(1+i)^{-\tau_t},\\ \Delta\mathrm{NPV}_t &= \mathrm{NPV}_{\mathrm{new}}-\mathrm{NPV}_{\mathrm{old}} =w_t\Delta p_t(1+i)^{-\tau_t}. \end{aligned}

This is an exact finite-change relationship, not an approximation, provided only that profit at duration $t$ changes and the other inputs stay fixed. Equivalently, the NPV’s rate of change with respect to $p_t$ is $w_t(1+i)^{-\tau_t}$. That rate says how much NPV changes for each additional dollar of conditional profit at that duration.

Worked Applications

Worked Example: A Midyear Profit Increase

A hypothetical insurer’s profit at duration 2 is conditional on the policy remaining in force to the beginning of that duration. The probability of being in force is $0.75$. A design change increases the conditional profit at that duration by $90 per policy, with the profit emerging at midyear. The effective annual discount rate is $8\%$. All other profits and assumptions are unchanged. Find the change in NPV per policy.

Method. Apply the one-duration sensitivity formula. Because the profit emerges halfway through duration 2, its time from issue is $\tau_2=2-\tfrac12=1.5$ years. The probability weight is $w_2=0.75$, and the profit change is $\Delta p_2=\$90$.

First calculate the probability-weighted profit change:

w_2\Delta p_2=0.75(90)=\$67.50.

Next calculate the discount factor at 1.5 years:

(1.08)^{-1.5} =\frac{1}{1.08^{1.5}} =\frac{1}{1.122368923\ldots}.

Thus,

\Delta\mathrm{NPV}_2 =\frac{67.50}{1.122368923\ldots} =60.14065\ldots \approx \$60.14.

The change increases NPV by $60.14 per policy. As a timing check, if this same weighted change emerged at the end of duration 2, its present value would be $67.50/(1.08)^2=67.50/1.1664=\$57.87$ to the nearest cent. The midyear value is larger because it is discounted for 1.5 rather than 2 years. The common mistake avoided here is using the year-end exponent for a midyear profit.

Worked Example: A Midyear Loss Increase Compared With Year-End

A hypothetical policy has a conditional profit decrease of $70 at duration 2. Assume the policy is certain to reach the relevant duration, so $w_2=1$. The effective annual discount rate is $6\%$. Calculate the NPV change if the decrease emerges at midyear, then compare it with the change if it emerges at year-end.

Method. Use $\Delta p_2=-\$70$, and discount at the time the change emerges. The negative sign must remain in the calculation because this is a reduction in profit.

For midyear emergence, $\tau_2=1.5$:

\Delta\mathrm{NPV}_{\mathrm{midyear}} =(-70)(1.06)^{-1.5} =\frac{-70}{1.06^{1.5}} =\frac{-70}{1.091336795\ldots} =-64.1415\ldots \approx -\$64.14.

For year-end emergence, $\tau_2=2$:

\Delta\mathrm{NPV}_{\mathrm{year-end}} =(-70)(1.06)^{-2} =\frac{-70}{1.1236} =-62.29975\ldots \approx -\$62.30.

Both changes reduce NPV. The midyear reduction has the greater absolute present value because the loss occurs earlier: $\$64.14>\$62.30$. A common sign error is to report the midyear value as positive because the discount factor is positive. Discounting preserves the sign of the profit change; it does not turn a loss into a gain.

Worked Example: Updating an Existing NPV

A hypothetical long-term contract has an existing NPV of $18.40 per policy. At duration 3, the probability of being in force at the start of the duration is $0.60$. A revised expense allowance increases the conditional profit at duration 3 by $120 per policy, emerging at year-end. The effective annual discount rate is $5\%$. Find the revised NPV, assuming every other duration’s profit is unchanged.

Method. First find the change in NPV for duration 3. Then add that change to the existing NPV. Since the profit is conditional on being in force, apply the $0.60$ weight once.

The weighted change in conditional profit is

w_3\Delta p_3=0.60(120)=\$72.

Year-end emergence at duration 3 means $\tau_3=3$. The discount factor is

(1.05)^{-3} =\frac{1}{1.05^3} =\frac{1}{1.157625}.

Therefore, the NPV change is

\Delta\mathrm{NPV}_3 =\frac{72}{1.157625} =62.1963\ldots \approx \$62.20.

The revised NPV is

\mathrm{NPV}_{\mathrm{new}} =18.40+62.1963\ldots =80.5963\ldots \approx \$80.60.

The answer is an increase of $62.20 and a revised NPV of $80.60 per policy. A common mistake is to use the full $120 as the weighted change, which would ignore the probability of reaching the duration. Another is to apply the weight twice if the $120 had already been stated as an unconditional expected profit.

Assumptions, Scope, and Exam Traps

The sensitivity calculation is deliberately local: it measures the NPV effect of a change in one duration’s profit, not the effect of a change to the entire product design. Its direct use requires that the probability weight, discount basis, and emergence time remain fixed. If a benefit redesign also changes persistency, the relevant weights may change too; the resulting total NPV change then includes both effects and cannot be found by changing $p_t$ alone.

The timing convention is equally important. A duration label identifies the profit period, but does not by itself determine the discount exponent. Read whether the profit is at the beginning, middle, or end of the duration and measure the time from issue. This tutorial uses year-end and midyear emergence; a different stated timing convention requires its corresponding $\tau_t$.

Omitting the probability weight: if the profit is conditional on reaching the duration, weight its change by the appropriate probability. Do not omit that factor.
Applying a probability weight twice: if the input is already an unconditional expected profit, do not multiply it by the same probability again.
Using the wrong exponent: a midyear profit at duration $t$ is discounted for $t-\tfrac12$ years, not $t$ years.
Dropping the sign: a profit decrease has $\Delta p_t<0$, so its NPV effect is negative.
Changing several inputs silently: the one-duration formula holds other profits, weights, timing, and the discount rate fixed. If they change, account for those changes separately.

A quick check is to compare the size and sign of the result with the inputs. For a positive rate and a weight between zero and one, the absolute NPV change cannot exceed the absolute conditional profit change. A negative profit change must give a negative NPV change. For the same weighted profit change, earlier emergence must have a larger absolute present value than later emergence.

Key takeaway: For an isolated change to a conditional profit at duration $t$, multiply the profit change by its probability weight and discount it from its actual emergence time. Add that signed amount to the original NPV to obtain the revised NPV.

Check Your Understanding

For each problem, assume the stated probability weights and other inputs remain fixed. Round dollar answers to the nearest cent.

A conditional profit at duration 1 increases by $50. The probability weight is $0.80$, and the profit emerges at year-end. The effective annual discount rate is $4\%$. Find the NPV change. (Answer: +$38.46)
A conditional profit at duration 2 decreases by $100. The probability weight is $0.90$, and the profit emerges at year-end. The effective annual discount rate is $5\%$. Find the NPV change. (Answer: -$81.63)
A conditional profit at duration 2 increases by $80 with probability weight $0.75$. It emerges at midyear, and the effective annual discount rate is $8\%$. Find the NPV change. (Answer: +$53.46)
A conditional profit at duration 3 increases by $40 with probability weight $0.50$. At an effective annual discount rate of $6\%$, compare its NPV change if it emerges at year-end with the change if it emerges at midyear. (Answer: year-end, +$16.79; midyear, +$17.29)
At an effective annual discount rate of $5\%$, a year-end conditional profit at duration 1 increases by $30 with probability weight $0.70$, while a year-end conditional profit at duration 2 decreases by $50 with probability weight $0.40$. Find the combined NPV change. (Answer: +$1.86)

What You'll Learn