Benefit Design Comparisons Using NPV

Why Benefit Design Comparisons Need a Profit Signature

Two hypothetical contracts can have the same premium pattern and still produce different profit emergence because their benefits differ in amount, trigger, or payment date. Comparing only the total undiscounted benefit is not enough: a benefit payable sooner has a different present value, and a benefit payable only on a particular event must be weighted by the probability of that event.

A disciplined comparison traces the effect of each design through the duration profits first, then discounts those profits to time zero. This keeps three questions separate: what cash flow changes, in which duration does it emerge, and what probability weight applies? The resulting NPV comparison is an expected-value comparison under the stated assumptions, not a prediction of the profit that will occur on any one policy.

Definition: For a discrete annual model, let $c_t$ be the expected profit during duration $t$, conditional on the policy being in force at the start of that duration. Let $w_t$ be the probability that it is in force at that time. The unconditional expected profit in duration $t$, or profit-signature entry, is $\pi_t=w_t c_t$. If profits emerge at the end of each duration and the effective annual discount rate is $i$, then $v=1/(1+i)$ and the net present value is $\operatorname{NPV}=\sum_t v^t\pi_t$.

This tutorial uses end-of-duration profit timing throughout. A different timing convention would require different discount exponents, so it must not be mixed into the same comparison. The figures $c_t$ and $\pi_t$ are expected profits: they include expected benefit outgo, not the payment that would occur in every individual policy outcome.

Translate the Benefit Change into Duration Profits

Start by specifying the benefit trigger and payment duration. For example, suppose an additional death benefit of amount $\Delta B$ is paid at the end of duration $t$, if death occurs during that duration. If the conditional probability of death during the duration is $q_t$, the change in conditional expected profit is $-q_t\Delta B$. The minus sign records that additional benefit outgo reduces profit. The corresponding unconditional signature change is $-w_tq_t\Delta B$.

For an additional survival benefit paid at the end of duration $t$ only if the policyholder survives that duration, let $p_t=1-q_t$. Its change in conditional expected profit is $-p_t\Delta B$, and its unconditional signature change is $-w_tp_t\Delta B$. If survival to a later duration is needed, the exposure probability $w_t$ must already reflect the probability of remaining in force through the preceding durations.

\begin{aligned} \Delta c_t&=-q_t\Delta B &&\text{(additional death benefit payable in duration }t\text{)},\\ \Delta \pi_t&=w_t\Delta c_t,\\ \Delta\operatorname{NPV}&=\sum_t v^t\Delta\pi_t. \end{aligned}

These expressions isolate the benefit effect. If other contract cash flows, assumptions, or expenses also change, their effects must be included separately in the relevant duration profits. A benefit payable at the end of a year is not automatically a certain outgo in that year: the event trigger and its probability determine its expected cost.

Why Discounted Signature Differences Give the NPV Difference

The key justification is the linearity of present value. Suppose designs A and B have unconditional duration profits $\pi_t^A$ and $\pi_t^B$, with identical timing and discount rate. Their NPVs are sums of their discounted duration profits. Subtracting the two sums duration by duration gives:

\begin{aligned} \operatorname{NPV}_B-\operatorname{NPV}_A &=\sum_t v^t\pi_t^B-\sum_t v^t\pi_t^A\\ &=\sum_t v^t(\pi_t^B-\pi_t^A)\\ &=\sum_t v^t\Delta\pi_t. \end{aligned}

Thus, a benefit comparison can be made directly from the difference signature, without recomputing cash flows common to both designs. This works because the same discount factors apply to both designs and the profit entries use the same timing convention. If the designs use different discount assumptions, calculate each NPV on its own basis before comparing them.

Worked Example: Increasing a Two-Year Death Benefit

A hypothetical two-year contract has conditional expected profits of $500 at the end of duration 1 and $700 at the end of duration 2 under design A. The probability of death during duration 1, conditional on being in force at its start, is $q_1=0.10$. Conditional on reaching duration 2, it is $q_2=0.20$. There are no lapses, so $w_1=1$ and $w_2=0.90$. Design B increases the death benefit by $2,000 in either duration. Use an effective annual discount rate of $i=5\%$. Compare the designs' NPVs.

Method. Calculate the added expected benefit outgo in each duration, apply the in-force probability to get the unconditional signature, and discount each signature entry. The common mistake avoided here is applying the duration-2 death probability to all original policies instead of only those in force at the start of duration 2.

Under design A, the unconditional profits are $\pi_1^A=1(500)=500$ and $\pi_2^A=0.90(700)=630$. Therefore:

\operatorname{NPV}_A=\frac{500}{1.05}+\frac{630}{1.05^2} =476.1905+571.4286=1{,}047.6191.

The increase in conditional expected benefit outgo is $0.10(2{,}000)=200$ in duration 1 and $0.20(2{,}000)=400$ in duration 2. Hence design B has conditional profits $500-200=300$ and $700-400=300$. Its unconditional signature is $\pi_1^B=1(300)=300$, $\pi_2^B=0.90(300)=270$, giving:

\operatorname{NPV}_B=\frac{300}{1.05}+\frac{270}{1.05^2} =285.7143+244.8980=530.6123.

The difference is $\operatorname{NPV}_B-\operatorname{NPV}_A=530.6123-1{,}047.6191=-517.0068$. Thus the benefit increase lowers NPV by $517.01, rounded to the nearest cent. As a check, the difference signature is $(-200,-360)$, and its discounted value is $-200/1.05-360/1.05^2=-517.0068$, consistent with the direct comparison.

Payment Timing Can Change NPV Even When Total Outgo Is Unchanged

A timing change moves profit between durations. If a benefit payment is brought forward, the earlier duration has greater expected outgo and lower profit; the later duration has less expected outgo and higher profit. The undiscounted sum of the changes may be zero, but their discounted sum generally is not. With a positive interest rate, moving an otherwise identical outgo earlier usually reduces NPV.

Worked Example: Moving Part of a Benefit Payment Earlier

A hypothetical guaranteed contract has end-of-duration profits of $400 and $600 in durations 1 and 2 under design A. Design B moves $500 of benefit outgo from the end of duration 2 to the end of duration 1. The payment is certain under both designs, so no event-probability adjustment is needed. Use an effective annual rate of $i=4\%$.

Method. Record the moved outgo as a $500 reduction to duration-1 profit and a $500 increase to duration-2 profit relative to design A. The common mistake avoided is treating the same undiscounted total outgo as proof that the NPVs are equal.

Design A has NPV:

\operatorname{NPV}_A=\frac{400}{1.04}+\frac{600}{1.04^2} =384.6154+554.7337=939.3491.

Design B's duration profits are $400-500=-100$ and $600+500=1{,}100$. Therefore:

\operatorname{NPV}_B=\frac{-100}{1.04}+\frac{1{,}100}{1.04^2} =-96.1538+1{,}017.0118=920.8580.

The change is $920.8580-939.3491=-18.4911$. The payment is moved earlier, so NPV falls by $18.49, rounded to the nearest cent. The difference-signature check gives $-500/1.04+500/1.04^2=-18.4911$. Although the undiscounted changes sum to zero, the earlier loss receives the larger discount factor.

Compare Different Triggers on a Common Basis

A benefit amount alone does not determine its expected cost. A death benefit and a survival benefit of the same size can have different expected costs because their triggering probabilities differ, and payments in different durations receive different discount factors. The comparison should therefore state the conditional event probabilities, the exposure probability at each duration, and the payment date.

Worked Example: Death Benefit in Year 1 or Survival Benefit in Year 2

A hypothetical contract has $q_1=0.10$, so $p_1=0.90$. Conditional on surviving duration 1, the duration-2 death probability is $q_2=0.20$, so $p_2=0.80$. Compare two additions to an otherwise identical contract: design A adds a $2,000 death benefit payable at the end of duration 1; design B adds a $2,000 survival benefit payable at the end of duration 2 if the policyholder survives both durations. Assume no lapses and $i=5\%$. Find each design's NPV change relative to the common contract and identify which produces the lower NPV.

Method. Use the event probability for the payment trigger, then the probability of being exposed to that duration. The common mistake avoided is using the duration-2 survival probability alone without also requiring survival through duration 1.

For design A, the additional benefit is payable with probability $q_1=0.10$. Its duration-1 unconditional profit change is $-0.10(2{,}000)=-200$, so:

\Delta\operatorname{NPV}_A=\frac{-200}{1.05}=-190.4762.

For design B, the probability of surviving both durations is $p_1p_2=0.90(0.80)=0.72$. The expected additional outgo per original policy is $0.72(2{,}000)=1{,}440$ at the end of duration 2. Equivalently, the conditional profit change among policies entering duration 2 is $-0.80(2{,}000)=-1{,}600$, weighted by $w_2=0.90$ to give an unconditional change of $-1{,}440$. Thus:

\Delta\operatorname{NPV}_B=\frac{-1{,}440}{1.05^2}=-1{,}306.1224.

Design B produces the lower NPV. Its change relative to design A is $-1{,}306.1224-(-190.4762)=-1{,}115.6462$, or $1,115.65 lower, rounded to the nearest cent. The result reflects both the different trigger probabilities and the payment durations; it is not a comparison of the two $2,000 face amounts alone.

Common Mistakes and a Practical Comparison Sequence

When comparing benefit designs, a plausible-looking calculation can still be wrong if the probability weighting or duration is misplaced. In particular, conditional profit changes are not automatically unconditional signature changes. If a conditional duration-2 profit loss is $-$400$ and only 90% of original policies reach duration 2, the unconditional signature change is $0.90(-400)=-$360$, not $-$400$.

Another frequent error is discounting a year-2 payment by $v$ rather than $v^2$. Under the stated end-of-year convention, duration 1 is discounted one year and duration 2 is discounted two years. Using the wrong exponent changes the value even if the profit amounts and probabilities are correct.

A reliable workflow is:

Specify what changes.
State the benefit amount, trigger, and payment duration for each design. Keep all shared assumptions and cash flows common where possible.

Calculate the conditional effect.
Multiply the changed benefit by the conditional probability of the event that triggers payment, with a negative sign for additional outgo.

Form the unconditional signature change.
Multiply the conditional change by the probability of being in force at the start of that duration.

Discount and compare.
Apply the discount factor for the payment duration and sum the entries. Check that a direct NPV difference agrees with the discounted difference signature.

The signature also explains when a design changes profit, not just the total change in present value. A design may create a larger loss in an early duration and a smaller loss later, or shift losses across durations while leaving undiscounted totals unchanged. Those distinctions matter when interpreting profit emergence as well as when selecting the higher-NPV design.

Check Your Understanding

Assume end-of-duration profit timing for each question. Answers are rounded to the nearest cent where they are monetary amounts.

A one-year benefit increase of $3,000 is payable on death. The conditional death probability is 0.08, and $i=4\%$. Find the NPV change. (Answer: $-$230.77$)
A design change produces unconditional profit-signature changes of $-$120$ in duration 1 and $+$250$ in duration 2. If $i=5\%$, find the NPV change. (Answer: $+$112.47$)
A death benefit increases by $1,500 in each of two durations. The conditional death probabilities are $q_1=0.10$ and $q_2=0.25$; there are no lapses, and $i=4\%$. Find the NPV change per original policy. (Answer: $-$456.27$)
A certain $800 benefit outgo is moved from the end of duration 3 to the end of duration 1. If $i=5\%$, find the NPV change in profit. (Answer: $-$70.83$)
Compare a $4,000 death-benefit increase payable in duration 1, where $q_1=0.12$, with a $3,000 survival-benefit increase payable at the end of duration 2, where $p_1=0.88$ and $p_2=0.75$. There are no lapses and $i=4\%$. How much lower is the survival-benefit design's NPV change than the death-benefit design's? (Answer: $1,369.08 lower)

What You'll Learn