Why Present Value Matters for Investment
A project’s receipts and costs may occur at different dates, so simply adding them does not measure their value in today’s terms. Funds available today can earn a return, while funds received later cannot be used until they arrive. Net present value (NPV) puts project cash flows on a common time basis by discounting future amounts to the present.
For an investment decision expressed in real terms, both the project cash flows and the discount rate must be real. A real cash flow is stated in purchasing power at a chosen base date; the real discount rate measures the return after allowing for inflation. The earlier material on real interest rates distinguishes this from a nominal rate, which includes expected inflation.
This calculation assumes that cash flows are assigned to the stated dates, that the rate is an annual effective real rate, and that the discount rate appropriately reflects the opportunity cost and risk of the project’s cash flows. The examples below treat the stated cash flows and rate as given. For a project with a time-zero outlay, do not discount \(I_0\): it is already measured at the present date.
Discounting Real Cash Flows
A cash flow \(C_t\) received at the end of year \(t\) has present value \(C_t/(1+r)^t\). The exponent is the number of annual periods between the present and the cash-flow date. Thus a year-two amount is divided by \((1+r)^2\), not merely by \(1+r\).
The reason is the opportunity-cost comparison. If a present amount \(P\) earns the annual effective real rate \(r\) for \(t\) years, it grows to \(P(1+r)^t\). To have exactly \(C_t\) at year \(t\), the amount needed now therefore satisfies \(P(1+r)^t=C_t\), giving \(P=C_t/(1+r)^t\). Adding the present values of all dated net cash flows and subtracting the initial outlay gives NPV.
Worked Example: A Project Just Below the Break-Even Value
A hypothetical equipment upgrade costs \(\$700\) now and produces real net cash flows of \(\$400\) at the end of year 1 and \(\$400\) at the end of year 2. The annual effective real discount rate is \(10\%\). Calculate NPV and state the decision.
Method. Discount each future real flow by the appropriate power of \(1.10\), then subtract the time-zero cost.
The year-one present value is \(400/1.10=\$363.6364\). The year-two present value is \(400/(1.10)^2=400/1.21=\$330.5785\). Thus $$ \operatorname{NPV}=-700+363.6364+330.5785=-5.7851. $$
Rounded to the nearest cent, NPV is \(-\$5.79\). Since it is negative, reject the project at a \(10\%\) real rate. The common mistake is to report \(-\$7.44\); the discounted amounts above sum to \(\$694.2149\), so subtracting \(\$700\) gives \(-\$5.7851\), not \(-\$7.44\). This total provides a direct check.
The sign of NPV is the decision signal, but its magnitude also matters: it measures the project’s net value in present-dollar terms at the specified discount rate. NPV is not simply the undiscounted sum of cash flows, nor does a positive total of future receipts alone guarantee acceptance.
Worked Example: Three Annual Real Inflows
A hypothetical project costs \(\$1{,}200\) now and produces a real net inflow of \(\$500\) at each of the next three year-ends. The annual effective real rate is \(6\%\). Calculate NPV and give the decision.
Method. Discount the three equal inflows separately, using exponents 1, 2, and 3.
The present values are \(500/1.06=\$471.6981\), \(500/(1.06)^2=500/1.1236=\$444.9982\), and \(500/(1.06)^3=500/1.191016=\$419.8096\). Their sum is \(\$1{,}336.5060\), so $$ \operatorname{NPV}=1{,}336.5060-1{,}200=136.5060. $$
Rounded to the nearest cent, NPV is \(\$136.51\), so accept the project. The common timing error is to discount every payment by one year; doing so would overstate the value of the year-two and year-three receipts. The three distinct denominators confirm that each payment has been discounted for its own waiting period.
How a Change in the Real Rate Affects NPV
When every future net cash flow is nonnegative, a higher real discount rate lowers or leaves unchanged the present value of each flow. For any \(C_t\geq 0\), \(t\geq1\), and \(r>-1\), increasing \(r\) increases \((1+r)^t\), so \(C_t/(1+r)^t\) cannot increase. If at least one future flow is strictly positive, its present value falls strictly as \(r\) rises; consequently, total NPV falls strictly.
The same conclusion follows by differentiating NPV with respect to \(r\), when the dated cash flows are fixed: $$ \frac{d\,\operatorname{NPV}}{dr} =-\sum_{t=1}^{T}\frac{tC_t}{(1+r)^{t+1}}. $$
If every \(C_t\) is nonnegative, every term in the sum is nonnegative, so the derivative is nonpositive. Any strictly positive future flow makes at least one term positive and the derivative negative. This justifies the usual conclusion that a higher real rate reduces NPV for a project with future net inflows. It does not, by itself, say that the acceptance decision must change: NPV may remain positive at both rates or negative at both rates.
Worked Example: Does the Decision Change at a Higher Rate?
A hypothetical asset costs \(\$800\) now and pays real net inflows of \(\$500\) at the end of each of the next two years. Compare NPV at annual effective real rates of \(8\%\) and \(14\%\), and determine whether the decision changes.
Method. Recalculate the present value of the same dated cash flows at each rate; do not change the cash flows when the discount rate changes.
At \(8\%\), the year-one value is \(500/1.08=\$462.9630\), and the year-two value is \(500/(1.08)^2=500/1.1664=\$428.6694\). Therefore NPV is \(462.9630+428.6694-800=\$91.6324\), or \(\$91.63\) to the nearest cent. Accept at \(8\%\).
At \(14\%\), the year-one value is \(500/1.14=\$438.5965\), and the year-two value is \(500/(1.14)^2=500/1.2996=\$384.7338\). Therefore NPV is \(438.5965+384.7338-800=\$23.3303\), or \(\$23.33\) to the nearest cent. Accept at \(14\%\) as well.
NPV decreases as the rate rises, but remains positive at both rates, so the decision does not change. The common mistake is to infer that any increase in the discount rate forces rejection; the decision depends on whether NPV actually crosses zero.
For a project with nonnegative future net flows, the break-even discount rate, if one exists, is the rate at which NPV equals zero. Comparing NPV at two rates can show whether the decision changes without requiring an exact break-even calculation. If a project has both future inflows and future outflows, however, the simple monotonicity conclusion may fail.
Worked Example: Converting Consistently Between Real and Nominal Values
A hypothetical project costs \(\$1{,}000\) today and has real net cash flows of \(\$300\), \(\$400\), and \(\$500\) at the ends of years 1, 2, and 3. Expected annual inflation is \(2\%\), and the annual effective real rate is \(5\%\). Find the equivalent nominal rate and nominal cash flows, then calculate NPV.
Method. Use the exact Fisher relation for annual effective rates with the same annual inflation assumption: \(1+j=(1+r)(1+\pi)\), where \(j\) is the nominal rate, \(r\) the real rate, and \(\pi\) inflation. Convert each real flow to nominal dollars for its payment date, or discount the real flows directly.
The nominal rate is \(j=(1.05)(1.02)-1=0.071\), or \(7.10\%\). The nominal flows are \(\$300(1.02)=\$306.00\), \(\$400(1.02)^2=\$416.16\), and \(\$500(1.02)^3=\$530.604\).
Discounting in real terms gives present values \(300/1.05=\$285.7143\), \(400/(1.05)^2=\$362.8118\), and \(500/(1.05)^3=\$431.9188\). Their sum is \(\$1{,}080.4449\), so NPV is \(\$80.4449\), or \(\$80.44\) to the nearest cent. Discounting the corresponding nominal flows at \(7.10\%\) gives the same present values, since \(306/1.071=300/1.05\), with the same equality holding at years 2 and 3. Mixing real cash flows with a nominal rate, or nominal cash flows with a real rate, would be inconsistent.
Future Outflows and a Common Rate Trap
The usual “higher rates lower NPV” rule requires nonnegative future net cash flows. If a project has a future net outflow, that outflow has a negative present value. A higher discount rate makes the outflow’s present value less negative, which can offset some or all of the decline in the present value of future inflows. In that case, the derivative formula still applies, but its terms can have different signs; the direction of the total change must be calculated.
Worked Example: A Later Net Outflow
A hypothetical project requires \(\$1{,}000\) now, produces a real net inflow of \(\$1{,}200\) at the end of year 1, and requires a real net outflow of \(\$300\) at the end of year 2. The annual effective real rate is \(10\%\). Calculate NPV and explain the rate-sensitivity issue.
Method. Keep the year-two cash flow negative and discount it for two years: $$ \operatorname{NPV}=-1{,}000+\frac{1{,}200}{1.10}-\frac{300}{(1.10)^2}. $$
The year-one inflow has present value \(\$1{,}200/1.10=\$1{,}090.9091\). The year-two outflow has present value \(-\$300/1.21=-\$247.9339\). Thus NPV is \(-1{,}000+1{,}090.9091-247.9339=-\$157.0248\), or \(-\$157.02\) to the nearest cent. Reject at \(10\%\).
The outflow contributes a negative present value whose magnitude falls as the rate rises, opposing the decrease in the discounted inflow. The common mistake is to treat every future amount as a positive benefit or to assume that the presence of an inflow alone guarantees that a higher rate lowers NPV. Here the year-two amount is a cost, so the rate effect must be assessed using the signed cash flows.
Check Your Understanding
For each problem, use annual effective real rates and round dollar answers to the nearest cent.
- A hypothetical project costs \(\$900\) now and pays real net inflows of \(\$500\) at the ends of years 1 and 2. At a \(5\%\) real rate, find NPV and state the decision. (Answer: NPV \(=\$29.71\); accept.)
- A project costs \(\$1{,}500\) and returns real net inflows of \(\$600\) at each of the next three year-ends. Calculate NPV at a \(8\%\) annual effective real rate and state the decision. (Answer: NPV \(=\$46.26\); accept.)
- A project costs \(\$2{,}400\) now and returns real net inflows of \(\$1{,}000\), \(\$900\), and \(\$800\) at the ends of years 1, 2, and 3. At a \(4\%\) real rate, calculate NPV. (Answer: NPV \(=\$104.84\); accept.)
- A project costs \(\$1{,}000\) and pays \(\$400\) in real terms at the end of each of the next three years. Compare NPV at \(4\%\) and \(10\%\) real rates, and say whether the decision changes. (Answer: NPV \(=\$110.04\) at \(4\%\), accept; NPV \(=-\$5.26\) at \(10\%\), reject; yes, the decision changes.)
- A project costs \(\$1{,}000\) now, receives a real net inflow of \(\$700\) at year 1, and pays a real net outflow of \(\$200\) at year 2. At a \(10\%\) real rate, calculate NPV, retaining the sign of each flow. (Answer: NPV \(=-\$528.93\); reject.)