Applying the Required Return in NPV

What the Required Return Does in an NPV Calculation

A project forecast gives cash flows and the dates on which they occur. To calculate net present value (NPV), those dated amounts must be expressed at a common valuation date. The required return supplies the rate used to make that conversion: it is the return required for the project’s risk and the period specified by the problem. It is not an additional project cost to subtract from each forecast cash flow.

The arithmetic depends on matching the rate to the timeline. An annual effective rate applies to a one-year interval; a quarterly effective rate applies to a quarter. If cash flows are dated at nonannual intervals, convert the supplied rate or use an equivalent expression that discounts to those exact dates. Do not divide an annual effective rate by the number of periods unless the stated rate convention supports that conversion.

Definition: Given a project’s cash flow $CF_t$ at time $t$, NPV is the sum of the cash flows expressed at time zero. The initial investment is included as a negative cash flow. With annual effective required return $r$ and time $t$ measured in years, $$ \operatorname{NPV}=\sum_{t} \frac{CF_t}{(1+r)^t}. $$ For cash flows at integer period $k$, if $j$ is the effective required return per period, $$ \operatorname{NPV}=\sum_{k} \frac{CF_k}{(1+j)^k}. $$

These expressions assume the stated required return is appropriate for discounting the project cash flows, its compounding convention is known, and the timing of each cash flow is measured consistently with that convention. A cash flow at time zero is not discounted. A date with no cash flow contributes zero; it does not change the period number of later cash flows.

Why Discounting Uses the Required Return

Discounting follows from the relationship between a present amount and its value after earning the required return. If $PV_t$ is the amount at time zero that grows to a cash flow $CF_t$ after $t$ years at annual effective rate $r$, then accumulation gives $CF_t=PV_t(1+r)^t$. Solving this equality for the present amount gives the discounting formula.

CF_t=PV_t(1+r)^t \quad\Longrightarrow\quad PV_t=\frac{CF_t}{(1+r)^t}

The required return therefore determines the conversion factor between a dated cash flow and its time-zero value. It does not change the forecasted cash flow itself. For example, a forecast receipt of $10,000 remains a $10,000 receipt at its stated date; the calculation then discounts that amount using the applicable return and time interval.

Set the timeline.
Record the initial investment at time zero and each later cash flow at its stated date. Use a negative sign for an outflow and a positive sign for an inflow.

Identify the return convention.
Determine whether the supplied rate is annual effective, effective per subannual period, or another stated convention. Do not infer a period rate by dividing unless that is valid for the convention given.

Discount each dated amount.
Use the number of periods from time zero to that cash flow, not the number of nonzero cash flows that have appeared so far.

Sum present values.
Add the discounted cash flows, including the time-zero amount. Keep full precision during intermediate calculations and round only the final result as requested.

Applying an Annual Effective Return

When a project has annual cash-flow dates and the required return is an annual effective rate, the period count is the number of years from time zero. Each cash flow is discounted separately, even if several amounts happen to be equal. This makes the date of each amount explicit and helps prevent applying the wrong exponent.

Worked Example: Annual Cash Flows at a Supplied Required Return

A hypothetical insurer is evaluating a software project that requires $100,000 immediately. It forecasts receipts of $33,000 at the end of year 1, $48,400 at the end of year 2, $39,930 at the end of year 3, and $29,282 at the end of year 4. The required return is 10% effective annually. Calculate NPV to the nearest cent.

Method. Place the initial investment at time zero and discount each receipt by the annual effective rate raised to its year number. The initial outlay is not discounted because it occurs at time zero.

\begin{aligned} \operatorname{NPV} &=-100{,}000+\frac{33{,}000}{1.10} +\frac{48{,}400}{1.10^2} +\frac{39{,}930}{1.10^3} +\frac{29{,}282}{1.10^4}\\ &=-100{,}000+30{,}000+40{,}000+30{,}000+20{,}000\\ &=\$20{,}000.00. \end{aligned}

The intermediate present values are $33,000/1.10 = $30,000; $48,400/1.21 = $40,000; $39,930/1.331 = $30,000; and $29,282/1.4641 = $20,000. They sum to $120,000. Including the $100,000 outflow gives the stated NPV of $20,000. The check is that the four discounted receipts total $120,000, exactly $20,000 more than the initial outlay.

Common mistake avoided: The 10% is not subtracted from each receipt as a dollar amount. It is used to discount each receipt according to its date.

Matching Subannual Cash Flows to the Rate

For cash flows within a year, the stated rate convention matters. If the required return is an annual effective rate $r$, the equivalent effective rate for one quarter is $j=(1+r)^{1/4}-1$, because four quarterly accumulation periods must multiply to $1+r$. More generally, for $m$ equal periods per year, the equivalent period rate is $(1+r)^{1/m}-1$. In contrast, if a problem supplies a nominal annual rate convertible $m$-thly, its period rate is the nominal rate divided by $m$. These conventions are not interchangeable.

A convenient check is to accumulate one unit over a full year using the proposed period rate. For a quarterly rate $j$, the one-year accumulation is $(1+j)^4$. It must agree with the annual accumulation implied by the supplied return. If it does not, the rate conversion is inconsistent.

Worked Example: Quarterly Cash Flows with an Annual Effective Return

A hypothetical project costs $35,000 now and pays $10,000 at the end of each of the next four quarters. Its required return is 21.550625% effective annually. Calculate NPV to the nearest cent.

Method. Convert the annual effective return to an effective quarterly rate. The given annual rate equals $(1.05)^4-1$, so the quarterly rate is 5%. This is a conversion from an annual effective rate, not a division of 21.550625% by four.

\begin{aligned} j&=(1+0.21550625)^{1/4}-1=0.05,\\ \operatorname{NPV} &=-35{,}000+\frac{10{,}000}{1.05} +\frac{10{,}000}{1.05^2} +\frac{10{,}000}{1.05^3} +\frac{10{,}000}{1.05^4}. \end{aligned}

The four present values, before rounding, are $9,523.8095, $9,070.2948, $8,638.3759, and $8,227.0248. Their sum is $35,459.5050. Subtracting the $35,000 time-zero cost gives NPV of $459.5050, or $459.51 to the nearest cent. As a check, the rate conversion gives $(1.05)^4=1.21550625$, which matches one plus the stated annual effective rate.

Common mistake avoided: Dividing the annual effective rate by four would give a quarterly rate of about 5.3877%, which does not produce the stated annual accumulation. The correct period rate is determined by equivalent accumulation.

Fractional-Year Dates

An annual effective rate can also be applied to dates that are a fraction of a year away. If the return is $r$ effective annually and time is measured in years, a cash flow at $t$ years is discounted by $(1+r)^t$. If the timeline instead uses equal half-year periods, converting the annual effective rate to an equivalent half-year rate and counting half-year periods gives the same result.

Worked Example: Half-Year Dates and Equivalent Period Rates

A hypothetical equipment project costs $20,000 now, produces a $10,000 receipt after six months, and produces a $12,000 receipt after eighteen months. The required return is 10.25% effective annually. Calculate NPV to the nearest cent.

Method. Since $1.1025=(1.05)^2$, the equivalent effective half-year rate is 5%. The first receipt is one half-year after time zero; the second is three half-year periods after time zero. The gap between the receipts is not the discounting period for the second receipt—the exponent counts from the valuation date.

\begin{aligned} \operatorname{NPV} &=-20{,}000+\frac{10{,}000}{1.05} +\frac{12{,}000}{1.05^3}\\ &=-20{,}000+9{,}523.8095+10{,}366.0512\\ &=-\$110.1393\\ &\approx -\$110.14. \end{aligned}

The discounted receipts sum to $19,889.8607, which is $110.1393 less than the initial outflow. The result is therefore a negative NPV of $110.14 when rounded to the nearest cent. As a check on the equivalent rate, two half-year periods accumulate by $1.05^2=1.1025$, the same as the stated annual effective rate.

Common mistake avoided: The eighteen-month receipt is discounted for three half-year periods, not one period and not two periods merely because it is the second receipt.

Common Exam Traps

Treating the required return as a cash flow. The return is the discount rate. It is not a percentage charge to subtract from each forecast receipt.
Mixing rate and time units. An annual rate cannot be paired with a count of quarters as though it were a quarterly rate. Convert the rate or measure time in years under the annual-effective convention.
Using the wrong exponent. The exponent is the time from valuation date to the cash-flow date. A missing cash flow in year 3 does not make a year-4 receipt a year-3 receipt.
Rounding too early. Retain precision in discount factors and present values; round only the final NPV unless the problem specifies otherwise.

Key takeaway: Apply the supplied required return by discounting each cash flow from its stated date to time zero. Preserve the cash-flow amounts and signs, use a rate convention consistent with the timeline, and include the initial outlay as a time-zero cash flow.

Check Your Understanding

For each problem, identify the correct period count, discount the dated amounts, and include the initial outlay. Answers are rounded as stated.

A project costs $25,000 now and pays $15,000 at the end of year 1 and $14,000 at the end of year 2. The required return is 10% effective annually. Calculate NPV to the nearest cent. (Answer: $206.61)
A project costs $18,000 and pays $10,000 at the end of each of years 1, 2, and 3. The required return is 10% effective annually. Calculate NPV to the nearest cent. (Answer: $6,868.52)
A project costs $30,000 now and pays $16,000 after six months and $17,000 after eighteen months. The required return is 10.25% effective annually. Calculate NPV to the nearest cent. (Answer: -$76.67)
A project costs $50,000 now and pays $12,000 at the end of quarter 1, $15,000 at the end of quarter 2, and $20,000 at the end of quarter 4. The required return is 21.550625% effective annually. Calculate NPV to the nearest cent. (Answer: -$8,511.94)
A project costs $9,000 now and pays $4,000 at the end of month 6 and $6,000 at the end of month 12. The required return is a nominal annual rate of 12% convertible monthly. Calculate NPV to the nearest cent. (Answer: $92.88)

What You'll Learn