Discounting Profit Signatures to Obtain NPV

Why Discounting a Profit Signature Matters

A profit signature records profit by time, but profit amounts at different times are not directly comparable. A dollar of profit received later has a smaller present value than a dollar received now when the annual effective interest rate is positive. Discounting converts the entries to a common valuation time so that they can be added and compared.

This tutorial uses a profit signature as already determined from annual results. Its concern is the next step: applying the correct discount factor to each entry. The essential discipline is to identify the time at which each profit occurs before using a formula. A payment at the end of year 2 is discounted for two years; a payment at the beginning of year 2 is discounted for one year.

Definition: A profit signature is a sequence of profit amounts indexed by their payment or recognition times. If $\Pi_t$ is the profit amount at time $t$, its time-zero net present value (NPV), at annual effective interest rate $i$, is the sum of each amount multiplied by its discount factor $v^t$, where $v=1/(1+i)$. The amounts may be signed: a loss or outflow is negative, and a gain or inflow is positive.

For this calculation, the timing and basis of the entries must be specified. An entry might be the expected profit per policy issued, the expected aggregate profit for a block, or a conditional profit for a policy still in force. These are not interchangeable. If a signature is already stated as expected profit per policy originally issued, its entries already incorporate the probabilities used to form that expectation. Do not multiply by survival probabilities again. If entries are conditional on being in force, first establish the appropriate expected amounts for the basis being valued.

The NPV Formula and Its Timing

Suppose the profit amounts $\Pi_0,\Pi_1,\ldots,\Pi_n$ occur at times $0,1,\ldots,n$, respectively. With a constant annual effective interest rate $i$, the time-zero NPV is

\operatorname{NPV}_0=\sum_{t=0}^{n}\Pi_t v^t =\sum_{t=0}^{n}\frac{\Pi_t}{(1+i)^t}.

The term at time zero is undiscounted because $v^0=1$. If the signature instead contains profits at the ends of years 1 through $n$, the formula is $\sum_{t=1}^{n}\Pi_t v^t$. If it contains profits at the beginnings of years 1 through $n$, those times are 0 through $n-1$, so the formula is $\sum_{t=0}^{n-1}\Pi_{t+1}v^t$. The label “year 1 profit” alone does not determine the exponent; the stated payment timing does.

Justification: At annual effective rate $i$, one dollar at time zero accumulates to $(1+i)^t$ dollars at time $t$. Therefore, the time-zero amount equivalent to $\Pi_t$ at time $t$ is $\Pi_t/(1+i)^t=\Pi_t v^t$. Adding the time-zero equivalents of all signed profit amounts gives the NPV. If each $\Pi_t$ is an expected profit, the same result follows by linearity of expectation: the expected present value is $\sum_t v^t E[\Pi_t]$, provided the discount rate is fixed and the expectations and timing use the same basis.

This is a valuation of the profit signature at the specified interest rate. It is not, by itself, a measure of the variability of profit or a full assessment of risk. A positive NPV means that the discounted positive amounts exceed the discounted negative amounts on the stated basis; it does not guarantee that every policy or every outcome is profitable.

Worked Example: Discounting an End-of-Year Profit Signature

A hypothetical contract has an initial profit of $-\$100$ at time 0, followed by profits of $\$70,\$80,\$60$ at the ends of years 1, 2, and 3. The annual effective interest rate is $5\%$. Calculate the time-zero NPV to the nearest cent.

Method. The initial amount is at time 0 and is not discounted. The remaining entries occur at times 1, 2, and 3. Here $i=0.05$ and $v=1/1.05$.

\operatorname{NPV}_0 =-100+\frac{70}{1.05}+\frac{80}{1.05^2}+\frac{60}{1.05^3}.

The individual present values are approximately $\$66.6666667$, $\$72.5623583$, and $\$51.8302559$. Retaining the unrounded values, the NPV is approximately

$$ -100+66.6666666667+72.5623582766+51.8302559136 =91.0592808569. $$

Rounded to the nearest cent, the time-zero NPV is $\$91.06$. The check is that the initial loss remains $-\$100$, while each later positive profit is discounted for its actual number of years.

Common mistake avoided: Summing displayed rounded present values and calling that the unrounded NPV can introduce a small error. The calculation above uses the exact discount expressions before rounding the final result.

Beginning-of-Year and End-of-Year Profits

Annual labels can conceal a one-period timing difference. A profit at the beginning of year 1 is at time 0; a profit at the end of year 1 is at time 1. More generally, a beginning-of-year amount for year $k$ is paid at time $k-1$, while an end-of-year amount for year $k$ is paid at time $k$.

Description	Time of payment	Discount factor
Beginning of year 1	0	$v^0=1$
End of year 1	1	$v^1$
Beginning of year 2	1	$v^1$
End of year 2	2	$v^2$

The following comparison makes the timing effect explicit. The data and interest rate are identical; only the stated payment times differ.

Worked Example: Re-indexing Beginning-of-Year Profits

A hypothetical design produces profits of $\$8,\$12,\$16$, each paid at the beginning of years 1, 2, and 3, respectively. The annual effective interest rate is $5\%$. Find the NPV. Then calculate the result that would apply if the same amounts were paid at the ends of those years.

Beginning-of-year case. The payments occur at times 0, 1, and 2. Thus

\operatorname{NPV}_{\text{begin}} =8+\frac{12}{1.05}+\frac{16}{1.05^2} =8+11.4285714286+14.5124716553 =33.9410430839.

The beginning-of-year NPV is $\$33.94$ to the nearest cent.

End-of-year case. If the same amounts instead occur at times 1, 2, and 3, then

\operatorname{NPV}_{\text{end}} =\frac{8}{1.05}+\frac{12}{1.05^2}+\frac{16}{1.05^3} \approx 32.323169.

The end-of-year NPV is $\$32.32$ to the nearest cent. It is lower because each amount is received one year later than in the beginning-of-year case.

Common mistake avoided: Treating “beginning of year 2” as time 2 discounts that amount for one year too many. Its payment time is 1, so its factor is $v^1$, not $v^2$.

Comparing Hypothetical Premium Designs

NPV can compare alternative designs when their profit signatures use the same valuation basis, valuation time, interest rate, and timing convention. A design with larger early profits can have a higher NPV than one with larger later profits, even if the latter has a larger undiscounted total. The comparison is a comparison of the supplied signatures; it does not explain how premiums, expenses, claims, or other annual results were determined.

Worked Example: Comparing Two Premium Designs

An insurer considers two hypothetical premium designs for the same type of contract. Each signature is per policy issued, includes all relevant expected profit amounts on that basis, and records profits at time 0 and at the ends of years 1, 2, and 3. At an annual effective interest rate of $10\%$, the signatures are:

Design A: $(-20,40,40,40)$. Design B: $(-5,25,45,60)$. Amounts are in dollars. Calculate both time-zero NPVs, determine which design has the higher NPV, and find the difference to the nearest cent.

Design A. The time-zero loss is not discounted. The year-end profits are discounted by $1.1$, $1.1^2$, and $1.1^3$:

\begin{aligned} \operatorname{NPV}_A &=-20+\frac{40}{1.1}+\frac{40}{1.1^2}+\frac{40}{1.1^3}\\ &=-20+36.3636363636+33.0578512397+30.0525920350\\ &=79.4740796383. \end{aligned}

Thus Design A has an NPV of $\$79.47$ to the nearest cent.

Design B. Using the same time indexing and rate,

\begin{aligned} \operatorname{NPV}_B &=-5+\frac{25}{1.1}+\frac{45}{1.1^2}+\frac{60}{1.1^3}\\ &=-5+22.7272727273+37.1900826446+45.0788888054\\ &=99.9962441773. \end{aligned}

Thus Design B has an NPV of $\$100.00$ to the nearest cent. Using unrounded NPVs, the advantage of Design B is

$$ 99.9962441773-79.4740796383=20.5221645390, $$

or $\$20.52$ to the nearest cent. Design B has the higher NPV on the stated basis and at the stated rate.

Common mistake avoided: Comparing only the undiscounted totals gives Design A a total of $\$100$ and Design B a total of $\$125$, but those totals do not account for timing. The NPV comparison discounts each year-end amount consistently, including the initial time-zero entries.

Common NPV Traps

A reliable calculation begins by writing the time index beside every entry. The most frequent errors are not difficult algebra; they are inconsistencies about what an entry means.

Discounting a time-zero entry. A time-zero amount is multiplied by $v^0=1$, not by $v$.
Using the year label as the exponent without checking timing. The end of year $k$ is time $k$, but the beginning of year $k$ is time $k-1$.
Changing the basis partway through the calculation. Do not combine amounts per policy issued with conditional amounts per policy in force unless they have been converted to a common basis.
Applying probabilities twice. If a signature already contains expected profits per policy issued, discount those entries; do not apply the same transition or survival probabilities again.
Rounding intermediate values too soon. Keep full precision in the discount factors and present values, then round the final NPV or difference as requested.
Ignoring the sign convention. An expense or loss must reduce NPV. Changing its sign can materially reverse the comparison.

Set the valuation basis.
Check whether each entry is expected per policy issued, conditional on being in force, or an aggregate amount.

Assign payment times.
Translate beginning- or end-of-year wording into time points such as 0, 1, and 2.

Apply discount factors.
For an amount at time $t$, multiply by $v^t=(1+i)^{-t}$, preserving its sign.

Sum and check.
Add the unrounded present values, confirm that earlier positive profits receive less discounting, and round only at the end.

Key takeaway: NPV is the sum of signed profit amounts brought to a common time-zero basis. Correct timing and a consistent profit basis matter as much as the discount arithmetic.

Check Your Understanding

For each problem, use the stated annual effective interest rate and timing. Round money to the nearest cent only after calculating with unrounded values.

A hypothetical contract has profits of $\$50$ and $\$60$ at the ends of years 1 and 2. At $4\%$, calculate its time-zero NPV. (Answer: $\$103.55$)
A signature has profits of $\$6,\$8,\$10$ at the beginnings of years 1, 2, and 3. At $5\%$, calculate its time-zero NPV. (Answer: $\$22.69$)
A hypothetical contract has a time-zero loss of $\$25$, followed by profits of $\$20$ at the ends of years 1, 2, and 3. At $4\%$, calculate its NPV. (Answer: $\$30.50$)
At $5\%$, Design A has profits of $\$15$ and $\$25$ at the ends of years 1 and 2. Design B has profits of $\$10$ and $\$34$ at those same times. Which has the larger NPV, and by how much to the nearest cent? (Answer: Design B, by $\$3.40$)
A hypothetical signature has four profits of $\$3$, each paid at the beginning of years 1 through 4. At $6\%$, calculate its time-zero NPV. (Answer: $\$11.02$)

Description	Time of payment	Discount factor
Beginning of year 1	0	\(v^0=1\)
End of year 1	1	\(v^1\)
Beginning of year 2	1	\(v^1\)
End of year 2	2	\(v^2\)

What You'll Learn