Comparing NPV Across Discount Bases

Why the Discount Basis Must Be Comparable

A profit signature records profit amounts by duration. Its net present value (NPV) depends not only on those amounts but also on the discount basis used to weight them. Comparing two NPVs is meaningful only when the rates are interpreted consistently: a quoted annual effective rate is not the same thing as a nominal annual rate convertible monthly, even if the quoted percentages match.

This tutorial holds the profit signature fixed and changes the discount basis. The central distinction is between equivalent ways to express the same accumulation and different assumptions about accumulation. Equivalent rates must produce the same NPV when applied to the same cash flows at the same times. Different rates can produce different NPVs, but that difference reflects a changed basis rather than a changed profit signature.

Definition: Let $\Pi_t$ be the profit at time $t$, measured in years, and let the fixed profit signature consist of the amounts $\Pi_t$ and their payment times. Under an annual effective interest rate $i$, the NPV of profits at integer year-ends $t=0,1,\ldots,n$ is $$ \operatorname{NPV}(i)=\sum_{t=0}^{n}\Pi_t v^t =\sum_{t=0}^{n}\frac{\Pi_t}{(1+i)^t}, \qquad v=\frac{1}{1+i}. $$ A profit at time zero is undiscounted. Profits received later are discounted according to their timing.

The definition requires the rate and timing convention to be specified. For year-end profits, an annual effective rate directly supplies the one-year accumulation factor $1+i$. For profits at fractional times, an annual effective rate by itself does not specify how accumulation is interpolated within the year. A continuous-force convention is one standard way to define that interpolation; it must be stated rather than silently assumed.

Converting Common Discount Bases

The notation distinguishes an effective annual interest rate $i$, a force of interest $\delta$, a nominal annual rate $i^{(m)}$ convertible $m$ times per year, and an effective annual discount rate $d$. If these rates are equivalent for a year, their accumulation and discount factors agree:

1+i=e^\delta =\left(1+\frac{i^{(m)}}{m}\right)^m =\frac{1}{1-d}, \qquad v=\frac{1}{1+i}=e^{-\delta}=1-d.

Thus, the conversions are

\delta=\ln(1+i),\qquad i^{(m)}=m\left((1+i)^{1/m}-1\right),\qquad d=\frac{i}{1+i}.

The nominal rate $i^{(m)}$ is not itself the effective annual rate. Its periodic rate is $i^{(m)}/m$, and its effective annual accumulation is $\left(1+i^{(m)}/m\right)^m$. Similarly, the effective discount rate satisfies $v=1-d$; it is not generally equal to the effective interest rate $i$.

Worked Example: Revaluing One Signature Under Equivalent Bases

A hypothetical product has profits of $\Pi_0=-\$100$, $\Pi_1=\$60$, and $\Pi_2=\$60$, with the latter two amounts paid at the ends of years 1 and 2. Revalue this fixed signature using an annual effective rate of 10%, its equivalent force of interest, its equivalent nominal rate convertible quarterly, and its equivalent effective discount rate. Report NPVs to the nearest cent.

Method. First convert every quoted basis to the same annual accumulation factor. Then discount the profits at their unchanged times.

The effective annual rate gives $1+i=1.10$, so $v=1/1.10=0.9090909$. The NPV is

\operatorname{NPV} =-100+\frac{60}{1.10}+\frac{60}{1.10^2} =-100+54.5454545+49.5867769 =4.1322314.

The force of interest equivalent to 10% is $\delta=\ln(1.10)=0.0953102$. Since $e^\delta=1.10$, its year-end discount factors are $e^{-\delta}=1/1.10$ and $e^{-2\delta}=1/1.10^2$. The NPV is therefore again $-100+54.5454545+49.5867769=\$4.13$.

For quarterly conversion, the equivalent nominal annual rate is $i^{(4)}=4((1.10)^{1/4}-1)$, approximately $0.096455$, or $9.6455\%$. Its quarterly rate is approximately $0.0241137$, and its annual accumulation is $(1+i^{(4)}/4)^4=1.10$. It therefore gives the same NPV, $\$4.13$.

Finally, the equivalent effective discount rate is $d=0.10/1.10=0.0909091$, or $9.0909\%$. Since $1-d=0.9090909=v$, it too gives the same NPV. The calculations reconcile because all four descriptions represent the same annual accumulation and the same profit timing.

Common mistake avoided: treating $i^{(4)}$ as though it were an effective annual rate would use $1.096455$, not $1.10$, as the annual accumulation. The nominal rate must first be converted through its quarterly periods.

Why Equivalent Bases Give the Same NPV

The equality of NPVs is a direct consequence of the definition of an equivalent rate. A year-end profit at time $t$ is multiplied by the discount factor for $t$ years. If two bases produce the same one-year accumulation, their discount factors agree at every integer year.

Start with the annual accumulation.
If the effective annual rate is $i$, one year of accumulation multiplies a value by $1+i$.

Express another basis as the same accumulation.
For a force of interest, $e^\delta=1+i$. For a nominal rate convertible $m$ times, $(1+i^{(m)}/m)^m=1+i$.

Invert to obtain discounting.
The discount factor for one year is $v=1/(1+i)$. Under equivalent bases it is also $e^{-\delta}$, $(1+i^{(m)}/m)^{-m}$, and $1-d$.

Apply the factor at each profit time.
At integer time $t$, each equivalent basis gives $v^t$. Multiplying each unchanged $\Pi_t$ by that same factor and summing gives the same NPV.

This argument concerns equivalent rates and matching cash-flow times. If the quoted rates imply different annual accumulations, the discount factors will not agree. If the cash-flow timing changes, the comparison no longer holds even when the annual rates are equivalent.

Worked Example: Same Quoted Percentage, Different Rate Convention

A hypothetical contract pays a single profit of $\$112$ at the end of year 1. Compare its NPV under (a) 12% effective annual interest and (b) 12% nominal annual interest convertible monthly. Report each NPV to the nearest cent.

Method. Interpret each rate according to its stated convention, find the one-year accumulation, and discount the $\$112$ payment. The nominal rate in part (b) has a monthly rate of $0.12/12=0.01$.

Under 12% effective annual interest, the year-end discount factor is $1/1.12$. Therefore,

\operatorname{NPV}_{\mathrm{effective}} =\frac{112}{1.12} =\$100.00.

Under 12% nominal interest convertible monthly, the annual accumulation is $(1.01)^{12}=1.12682503$, not $1.12$. Thus,

\operatorname{NPV}_{\mathrm{nominal\ monthly}} =\frac{112}{(1.01)^{12}} =\frac{112}{1.12682503} \approx \$99.39.

The nominal monthly basis implies an effective annual rate of approximately $12.6825\%$, so it discounts the payment slightly more heavily. The NPV difference is approximately $\$100.00-\$99.39=\$0.61$. This is a genuine difference in discount assumptions, not a failure of an equivalent-rate conversion.

Common mistake avoided: substituting 12% directly as an effective annual rate in part (b) would give $\$100.00$. That calculation ignores monthly conversion and incorrectly equates two different annual accumulations.

Preserving Timing, Including Fractional Years

For fractional payment times, use a convention that specifies the accumulation between year-ends. Under a constant force of interest $\delta$, the accumulation over $t$ years is $e^{\delta t}$, so the discount factor is $e^{-\delta t}$. If $\delta=\ln(1+i)$, this can also be written $(1+i)^{-t}$. The latter fractional power is appropriate here because the continuous-force interpolation has been specified.

Timing rule: Under a force of interest $\delta$, a profit $\Pi_t$ paid at time $t$ has present value $\Pi_t e^{-\delta t}$. Replacing $t$ by a whole year or by a different duration changes the cash-flow timing and therefore changes the NPV.

Worked Example: Equivalent Bases with Half-Year Profits

A hypothetical product produces profits of $\$110$ at time $0.5$ years and $\$133.10$ at time $1.5$ years. Use a 21% effective annual rate with continuous-force interpolation, and then its equivalent force of interest. Compare with the incorrect calculation that treats the profits as occurring at the ends of years 1 and 2. Report NPVs to the nearest cent.

Method. Keep each payment at its stated time. The equivalent force is $\delta=\ln(1.21)$. Since $1.21=1.1^2$, the half-year accumulation factor is $\sqrt{1.21}=1.1$.

Using the effective rate with fractional powers, the first present value is $110/(1.21)^{0.5}=110/1.1=\$100$. The second is $133.10/(1.21)^{1.5}=133.10/(1.1^3)=133.10/1.331=\$100$. Hence the NPV is $\$100+\$100=\$200.00$.

Using the force of interest gives the same result. For the first payment, $110e^{-0.5\ln(1.21)}=110/1.1=\$100$; for the second, $133.10e^{-1.5\ln(1.21)}=133.10/1.331=\$100$. The total remains $\$200.00$, as required for equivalent bases at the same times.

If the payments are incorrectly moved to year 1 and year 2, the calculation becomes $110/1.21=90.9091$ and $133.10/1.21^2=90.9091$, for an erroneous total of $\$181.82$. The difference arises from changing both payment times, not from changing between equivalent discount bases.

Common mistake avoided: a duration label such as “first year” does not automatically mean a profit occurs at year-end. Use the time of payment specified in the signature.

Practical Checks for an NPV Comparison

When comparing NPVs under alternative bases, make the comparison auditable. Record the profit amount and its time, identify the rate convention, convert it to an annual accumulation or discount factor, and then apply the factor at that time. A useful reconciliation is to calculate the annual accumulation implied by every rate. Equivalent bases should produce the same factor before any profits are discounted.

Quoted basis	Annual accumulation	Equivalent discount factor for one year
Effective annual interest $i$	$1+i$	$1/(1+i)$
Force of interest $\delta$	$e^\delta$	$e^{-\delta}$
Nominal annual interest $i^{(m)}$, convertible $m$ times	$(1+i^{(m)}/m)^m$	$(1+i^{(m)}/m)^{-m}$
Effective annual discount $d$	$1/(1-d)$	$1-d$

The discount basis matters because NPV gives different weights to profits at different times. A higher effective annual accumulation gives smaller present values for future positive profits, while it makes a future negative profit less negative in present-value terms. The net effect on a multi-year signature depends on its pattern of positive and negative profits; the conversion rules themselves do not determine whether the overall NPV rises or falls.

A concise exam-style presentation states whether profits are at time zero, year-ends, or fractional times; writes the rate conversion; shows at least the discount factors used; and reports the NPV with an appropriate rounding convention. These steps are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Key takeaway: To compare a fixed profit signature across discount bases, hold every profit amount and payment time fixed. Convert each rate according to its convention; equivalent annual accumulations must produce identical NPVs under the same timing convention.

Check Your Understanding

For each problem, state the discount convention you use and preserve the profit timing given.

A hypothetical signature has profits of $\$50$ at year 1 and $\$60$ at year 2. Find its NPV at 10% effective annual interest. (Answer: $\$95.04$)
Convert 5% effective annual interest to an equivalent force of interest, nominal annual rate convertible quarterly, and effective annual discount rate. Give rates to four significant figures. (Answer: $\delta=0.04879$, $i^{(4)}\approx0.04909$ or $4.909\%$, and $d\approx0.04762$ or $4.762\%$)
A hypothetical contract pays $\$112$ at year 1. Compare NPV at 12% effective annual interest and 12% nominal annual interest convertible semiannually. Report to the nearest cent. (Answer: $\$100.00$ and $\$99.68$, respectively)
A hypothetical profit of $\$121$ is paid at time $0.5$ years. Under 21% effective annual interest with continuous-force interpolation, find its present value. (Answer: $\$110.00$)
A hypothetical fixed signature has profits $-\$200$ at time 0, $\$80$ at year 1, and $\$150$ at year 2. Find its NPV using 10% effective annual interest, and state the equivalent force of interest that would give the same NPV. (Answer: $-\$3.31$; equivalent force $\delta=\ln(1.10)\approx0.09531$)

Quoted basis	Annual accumulation	Equivalent discount factor for one year
Effective annual interest \(i\)	\(1+i\)	\(1/(1+i)\)
Force of interest \(\delta\)	\(e^\delta\)	\(e^{-\delta}\)
Nominal annual interest \(i^{(m)}\), convertible \(m\) times	\((1+i^{(m)}/m)^m\)	\((1+i^{(m)}/m)^{-m}\)
Effective annual discount \(d\)	\(1/(1-d)\)	\(1-d\)

What You'll Learn