Duration Contribution Analysis

Why Decompose NPV by Duration?

A product’s total net present value (NPV) gives one summary of projected profitability, but it does not show which durations create that value. A duration contribution analysis makes the source of the total explicit: each period’s projected profit is discounted to the valuation date, and the resulting amounts are added. This makes it possible to identify the largest positive contributions, the periods that offset them, and whether the total depends heavily on one duration.

The analysis starts with the profit signature: the sequence of projected profits by duration. The key is to use the signature and its timing consistently. A profit already expressed per policy issued and including the probability of remaining in force must not be weighted by survival again. Conversely, if profits are stated conditional on being in force or alive, the relevant probability may still need to be applied.

Definition: Let $\Pi_k$ be the expected net profit at the end of duration $k$, measured per policy issued at time 0. With a constant effective annual interest rate $i$, set $v=1/(1+i)$. The duration-$k$ contribution to NPV is $C_k=v^k\Pi_k$, and the total NPV over $n$ durations is $\sum_{k=1}^{n}C_k$. If profit for duration $k$ is instead recognized at time $k-\tfrac12$, its contribution is $C_k=v^{k-1/2}\Pi_k$.

A contribution is a signed amount. A positive $C_k$ increases NPV, while a negative $C_k$ decreases it. For ranking, be clear about the question being asked: ranking by signed value identifies the largest additions to NPV, whereas ranking by absolute value identifies the durations with the greatest magnitude of influence, whether favorable or unfavorable.

Why the Contributions Add to NPV

Suppose $Z_k$ is the net profit amount associated with duration $k$, and profits are discounted at a fixed rate. For end-year payments, the present value of the full profit stream is the sum of each duration’s discounted amount. Taking expectations preserves that sum. The proof follows from the additivity of expectation:

\begin{aligned} \operatorname{NPV} &= E\left[\sum_{k=1}^{n}v^k Z_k\right]\\ &= \sum_{k=1}^{n}v^k E[Z_k]\\ &= \sum_{k=1}^{n} C_k, \end{aligned}

Here, $E[Z_k]=\Pi_k$, so $C_k=v^k\Pi_k$. The same argument applies to profits paid at specified midyear times, replacing $v^k$ with the discount factor for the actual payment time. No independence assumption between profits in different durations is needed for this additivity result. The essential conditions are that the discount factors are applied consistently, the duration profits use the same valuation basis, and the expected amounts are defined per the same unit of business.

This decomposition does not make an inaccurate profit signature accurate. It organizes the signature that has been supplied or calculated. If a profit amount excludes a benefit, expense, or decrement that belongs in the projection, every contribution and the total NPV may be misstated.

Worked Examples

Worked Example: Ranking End-Year Contributions

A hypothetical four-year contract has deterministic net profits per policy issued of $-\$100$, $\$80$, $\$100$, and $\$50$, paid at the ends of durations 1 through 4. The effective annual interest rate is $25\%$. Assume no mortality, lapse, or other decrement (annual mortality probability $q=0$); the stated profit amounts are deterministic, or equivalently have point-mass distributions at those amounts. Find each duration contribution, total NPV, and the duration with the greatest absolute contribution.

Method. Use the end-year factor $v^k$ for each profit. No survival weighting is required because the profits are already given per policy issued, and the example has no decrements.

First calculate the discount factor:

v=\frac{1}{1.25}=0.8.

The duration contributions are:

\begin{aligned} C_1&=(-100)(0.8)^1=-80,\\ C_2&=(80)(0.8)^2=(80)(0.64)=51.20,\\ C_3&=(100)(0.8)^3=(100)(0.512)=51.20,\\ C_4&=(50)(0.8)^4=(50)(0.4096)=20.48. \end{aligned}

Therefore,

\operatorname{NPV}=-80+51.20+51.20+20.48=\$42.88.

Duration 1 has the greatest absolute contribution, $\$80.00$, and it is negative. Durations 2 and 3 tie at $\$51.20$ each; duration 4 contributes $\$20.48$. Check: the positive contributions total $\$122.88$, and $\$122.88-\$80.00=\$42.88$, matching the NPV. The method avoids ranking durations by undiscounted profit alone: duration 3 has the largest undiscounted profit, but its discounted contribution is smaller in magnitude than duration 1’s.

Worked Example: Applying Survival Weighting Once

A hypothetical three-year benefit contract pays a net profit of $\$100$ at each year-end if the policyholder is alive at that date. Annual mortality is constant: the probability of death during each year, conditional on being alive at its start, is $q=0.20$. Assume mortality is the only decrement and the annual mortality probabilities are the same in each year. The effective annual interest rate is $25\%$. Determine the expected profit per policy issued at each duration, the duration contributions, and the total NPV.

Method. First calculate the probability of surviving to each payment date. Multiply the conditional profit by that probability to obtain the expected duration profit per policy issued. Then discount that expected amount to time 0.

The annual survival probability is $1-q=0.80$, so the probabilities of being alive at the ends of durations 1, 2, and 3 are $0.80$, $0.80^2=0.64$, and $0.80^3=0.512$. Thus the expected duration profits per policy issued are:

\begin{aligned} \Pi_1&=(100)(0.80)=\$80.00,\\ \Pi_2&=(100)(0.64)=\$64.00,\\ \Pi_3&=(100)(0.512)=\$51.20. \end{aligned}

At $i=0.25$, $v=0.8$. Discount each expected duration profit:

\begin{aligned} C_1&=(80)(0.8)=\$64.00,\\ C_2&=(64)(0.8)^2=(64)(0.64)=\$40.96,\\ C_3&=(51.20)(0.8)^3=(51.20)(0.512)=\$26.2144. \end{aligned}

The total is $64+40.96+26.2144=\$131.1744$, or $\$131.17$ to the nearest cent. Check: the direct contribution formula for duration $k$ is $100(0.80)^k(0.8)^k$, since the survival and discount factors are both $0.8^k$. This gives $64$, $40.96$, and $26.2144$, exactly as above. The method avoids multiplying by survival twice: the $\Pi_k$ values already include survival, so only discounting remains.

Worked Example: Discounting Midyear Profits

A hypothetical three-duration product has deterministic net profits of $-\$120$, $\$100$, and $\$80$, each recognized halfway through its duration. The effective annual interest rate is $44\%$. Assume no mortality or other decrement ($q=0$); the amounts have point-mass distributions at their stated values. Calculate the midyear contributions and NPV, then identify the largest contribution by absolute value.

Method. A profit in duration $k$ occurs at time $k-\tfrac12$, not at time $k$. Discount each amount using $v^{k-1/2}$.

Here $v=1/1.44$, and $\sqrt{1.44}=1.2$. The discount factors for the three midyear times are therefore $1/1.2$, $1/1.2^3$, and $1/1.2^5$. The contributions are:

\begin{aligned} C_1&=\frac{-120}{1.2}=-\$100.00,\\ C_2&=\frac{100}{1.2^3} =\frac{100}{1.728} =\$57.8704,\\ C_3&=\frac{80}{1.2^5} =\frac{80}{2.48832} =\$32.1502. \end{aligned}

Adding the unrounded contributions gives $-100+57.870370\ldots+32.150205\ldots=-\$9.979424\ldots$, or $-\$9.98$ to the nearest cent. Duration 1 has the greatest absolute contribution, $\$100.00$, and is a loss. Check: the two positive contributions total about $\$90.02$, which is $\$9.98$ less than the duration-1 loss. This confirms the negative total. The calculation avoids treating a midyear profit as though it were paid at year-end.

Interpreting Rankings and Avoiding Traps

A useful contribution table records, for each duration, the profit amount, the time of payment, the discount factor, and the resulting contribution. It also shows whether a profit is already an expected amount per policy issued or is conditional on survival or being in force. This makes the calculation auditable and reduces the risk of applying exposure weights twice or not at all.

Ranking question	Quantity to compare	Interpretation
Which duration adds most to NPV?	Signed $C_k$, from largest to smallest	The largest positive contribution
Which duration reduces NPV most?	Signed $C_k$, from smallest to largest	The most negative contribution
Which duration has the greatest magnitude of influence?	$\lvert C_k\rvert$	The largest contribution in absolute dollars, regardless of sign

Do not confuse a large undiscounted profit with a large NPV contribution. A profit paid later is discounted more heavily than an equal profit paid earlier. Also do not interpret the duration with the largest absolute contribution as necessarily the main source of positive value: its sign matters. A large negative contribution may be the strongest drag on NPV.

Common mistake—adding undiscounted profits. In the first worked example, adding $-100+80+100+50$ gives $\$130$. That is not the NPV, because it treats profits at different dates as if they were all paid at time 0. The correction is to discount each end-year amount separately: $-100(0.8)+80(0.8)^2+100(0.8)^3+50(0.8)^4=\$42.88$. The same timing discipline applies when a profit is recognized midyear: use its actual time exponent rather than the end-year exponent.

Duration contribution analysis is useful when a total NPV conceals offsetting gains and losses. It can reveal whether early losses are more than recovered later, whether one duration dominates the result, and which periods merit closer review of their projected assumptions. The decomposition is a diagnostic view of NPV, not a substitute for checking the underlying profit signature and its timing basis.

Key takeaway: Calculate each duration’s contribution using its expected profit per policy issued and the discount factor for its actual payment time. Sum the signed contributions to reconcile to NPV, and use signed or absolute rankings according to the question being answered.

Check Your Understanding

For each problem, use the stated timing and distinguish any conditional profit from expected profit per policy issued.

A hypothetical two-year contract has deterministic end-year profits of $\$50$ and $-\$20$, per policy issued. The effective annual interest rate is $10\%$, and there are no decrements. Calculate both contributions and total NPV. (Answer: $C_1=\$45.45$, $C_2=-\$16.53$, NPV $=\$28.93$.)
At an effective annual interest rate of $25\%$, a hypothetical three-year contract has deterministic end-year profits per policy issued of $-\$40$, $\$60$, and $\$30$, with no decrements. Find the duration contributions, NPV, and duration with the greatest absolute contribution. (Answer: contributions $-\$32.00,\ \$38.40,\ \$15.36$; NPV $=\$21.76$; duration 2 has the greatest absolute contribution.)
A hypothetical two-year contract pays $\$90$ at the end of year 1 and $\$120$ at the end of year 2 if the policyholder is alive. Annual mortality is $q=0.10$, constant in both years; interest is $i=0\%$, and mortality is the only decrement. Find the expected profits per policy issued and NPV. (Answer: expected profits are $\$81.00$ and $\$97.20$; NPV $=\$178.20$.)
A hypothetical product has profits of $\$121$ at the midpoint of duration 1 and $-\$110$ at the midpoint of duration 2. Profits are deterministic, there are no decrements, and $i=21\%$. Calculate the midyear contributions and NPV to the nearest cent. (Answer: $C_1=\$110.00$, $C_2=-\$82.64$, NPV $=\$27.36$.)
At $i=0\%$, a hypothetical three-duration product has deterministic end-year profits per policy issued of $\$100$, $-\$90$, and $\$20$, with no decrements. Find NPV and identify the greatest positive contribution and the greatest absolute contribution. (Answer: NPV $=\$30.00$; the greatest positive contribution is duration 1 at $\$100.00$, and the greatest absolute contribution is also duration 1 at $\$100.00$.)

Ranking question	Quantity to compare	Interpretation
Which duration adds most to NPV?	Signed \(C_k\), from largest to smallest	The largest positive contribution
Which duration reduces NPV most?	Signed \(C_k\), from smallest to largest	The most negative contribution
Which duration has the greatest magnitude of influence?	\(\lvert C_k\rvert\)	The largest contribution in absolute dollars, regardless of sign

What You'll Learn