Reading an NPV Profile

What an NPV Profile Shows

Net present value summarizes a project at one discount rate. An NPV profile extends that calculation: it shows the project’s NPV over a range of discount rates. This makes it possible to see how sensitive a project’s value is to the rate used and where its NPV changes sign.

The profile’s horizontal axis is the discount rate $r$; its vertical axis is the resulting NPV, measured in dollars. Each point on the curve is calculated using the same project cash flows and a different rate. The profile therefore does not describe changing forecasts. It describes how the present value of fixed forecasts changes when the discount rate changes.

Definition: For project cash flows $C_0,C_1,\ldots,C_n$, where $C_t$ is the cash flow at the end of period $t$, the NPV profile is the function $$ \operatorname{NPV}(r)=\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}. $$ Here, $r$ is the effective discount rate per period and $r>-1$. Cash flows and the discount rate must use matching time periods. The time-0 cash flow $C_0$ is not discounted.

The formula assumes cash flows occur at the stated dates and remain fixed as $r$ varies. For annual end-of-year cash flows, $r$ must be an annual effective rate. If the cash flows are quarterly, the discount rate must instead be appropriate to a quarter. The profile’s units are currency on the vertical axis and rate on the horizontal axis.

The Zero Crossing and the IRR

A point where the profile crosses the horizontal axis has NPV equal to zero. The discount rate at that point satisfies the internal rate of return equation, because IRR is a rate that makes the project’s NPV zero. For a conventional project—an initial outflow followed only by inflows—the NPV profile decreases as the rate increases. If it crosses zero, increasing the rate beyond that crossing makes NPV negative; decreasing the rate below it makes NPV positive.

A profile need not cross zero over the range being displayed. Also, if cash flows change sign more than once, its shape and zero crossings need not follow the simple decreasing pattern of a conventional project. In that case, do not infer a unique IRR merely from the profile’s appearance over a limited range.

When two calculated NPVs of opposite signs bracket a zero crossing, straight-line interpolation gives an estimate. If $r_L$ and $r_H$ are the lower and higher rates, with corresponding NPVs $N_L>0$ and $N_H<0$, the estimated crossing is

r_{\text{est}}=r_L+\frac{N_L}{N_L-N_H}(r_H-r_L).

This method treats the curve between the two points as approximately a straight line. An NPV profile is generally curved, so interpolation estimates rather than solves the exact zero. The estimate is more useful when the rates are close and the bracket is reasonably narrow. If both NPVs have the same sign, these points do not bracket a crossing, so this interpolation formula should not be used.

Worked Example: Estimating a Zero Crossing

A hypothetical project costs $1,000 immediately and produces receipts of $600 at the end of each of the next two years. Its annual discount rate is tested at 10% and 15%. Calculate both NPVs, estimate the zero crossing by interpolation, and check the estimate against the IRR equation.

Method. Calculate NPV at both rates using annual end-of-year cash flows, verify that the values bracket zero, and then interpolate. This method avoids treating a single NPV as the IRR.

At 10%, the present values of the two receipts are

\frac{600}{1.10}=\$545.4545,\qquad \frac{600}{1.10^2}=\frac{600}{1.21}=\$495.8678.

Thus,

\operatorname{NPV}(10\%)=-1{,}000+545.4545+495.8678 =\$41.3223.

At 15%, the present values are

\frac{600}{1.15}=\$521.7391,\qquad \frac{600}{1.15^2}=\frac{600}{1.3225}=\$453.6863,

\operatorname{NPV}(15\%)=-1{,}000+521.7391+453.6863 =-\$24.5746.

The NPVs have opposite signs, so the rates bracket a zero. Interpolating,

r_{\text{est}}=10\%+ \frac{41.3223}{41.3223-(-24.5746)}(15\%-10\%) \approx 13.14\%.

As a check, the exact zero satisfies $-1{,}000+600/(1+r)+600/(1+r)^2=0$. Setting $x=1+r$ and multiplying by $x^2$ gives $1{,}000x^2-600x-600=0$. Its positive solution is $x=(3+\sqrt{69})/10$, giving $r\approx 13.0662\%$. The interpolated estimate is close but not exact, as expected for a curved profile. The common mistake is to report the interpolation estimate as an exact IRR.

Why the Profile Slopes Down

The slope tells us how quickly NPV changes as the discount rate changes. To find it, differentiate the NPV formula with respect to $r$. The time-0 amount is constant, so its derivative is zero. For a cash flow at time $t$, differentiating its discount factor gives

\frac{d}{dr}\left[C_t(1+r)^{-t}\right] =-tC_t(1+r)^{-(t+1)}.

Adding these terms gives the slope of the NPV profile:

\operatorname{NPV}'(r) =-\sum_{t=1}^{n}\frac{tC_t}{(1+r)^{t+1}}.

The factor $t$ is important: a later cash flow contributes more to rate sensitivity because its present value is affected by discounting over more periods. For a conventional project, all $C_t$ after the initial outflow are positive. Each term in the derivative is then negative, so the profile slopes downward. If the discount rate is expressed as a decimal, the derivative is dollars per one-unit change in that decimal rate. A one-percentage-point increase is a change of $0.01$, not $1$.

A second derivative helps explain why a straight line only approximates the profile. For positive future cash flows,

\operatorname{NPV}''(r) =\sum_{t=1}^{n}\frac{t(t+1)C_t}{(1+r)^{t+2}}>0.

The profile is therefore convex for a conventional project: as the rate increases, its negative slope becomes less steep. This curvature is one reason that a local slope estimate and a straight-line interpolation need not match the exact change.

Worked Example: Interpreting a Profile’s Slope

A hypothetical project has cash flows of $-\$500$ at time 0, followed by receipts of $100, $200, and $400 at the ends of years 1, 2, and 3. Calculate NPV at 10% and 11%, calculate the slope at 10%, and use that slope to estimate the effect of a one-percentage-point increase.

Method. Calculate the discounted cash flows at each rate, then differentiate the NPV formula for the local slope. This method avoids omitting the time multiplier from later cash flows.

At 10%, the present values of the receipts are

\frac{100}{1.10}=\$90.9091,\qquad \frac{200}{1.10^2}=\$165.2893,\qquad \frac{400}{1.10^3}=\$300.5259.

Therefore,

\operatorname{NPV}(10\%)=-500+90.9091+165.2893+300.5259 =\$56.7243.

At 11%, the corresponding present values are

\frac{100}{1.11}=\$90.0901,\qquad \frac{200}{1.11^2}=\$162.3245,\qquad \frac{400}{1.11^3}=\$292.4764.

Thus,

\operatorname{NPV}(11\%)=-500+90.0901+162.3245+292.4764 =\$44.8910.

At 10%, the derivative is

\begin{aligned} \operatorname{NPV}'(0.10) &=-\left( \frac{1(100)}{1.10^2} +\frac{2(200)}{1.10^3} +\frac{3(400)}{1.10^4} \right)\\ &=-(82.6446+300.5259+819.6164)\\ &\approx-\$1{,}202.79\text{ per unit change in }r. \end{aligned}

A one-percentage-point increase is $\Delta r=0.01$. The local estimate of the NPV change is therefore $-1{,}202.79(0.01)=-\$12.03$, giving an estimated NPV of $56.7243-12.03\approx \$44.70$. The exact NPV at 11% is $44.89, a change of about $-\$11.83$. The estimate is close, but the slope is local and the profile is curved. The common mistake is to multiply the year-2 term by 1 instead of 2, or to treat a one-percentage-point change as a full unit change.

Reading a Profile in Practice

A useful profile is not just a plotted curve. It should be based on correctly timed cash flows, labeled rates, and NPVs calculated consistently. A small table of rates and NPVs can serve as a numerical profile before a graph is drawn. The graph then makes the direction, approximate steepness, and any zero crossing easier to see.

Fix the cash-flow timeline.
List each amount at its actual period. Do not discount the time-0 amount.

Choose rates in matching periods.
For annual cash flows, use annual effective rates; keep the same cash flows at every rate.

Calculate and plot NPV.
Compute the NPV at each selected rate and plot rate horizontally and NPV vertically.

Interpret the result.
For a conventional project, a zero crossing is its IRR if it exists; the slope indicates sensitivity near the rate being examined.

The profile can also help check calculations. For a conventional project, increasing the discount rate should not increase NPV. If a table shows NPV rising as the rate rises, check for errors such as discounting the initial cost, using the wrong period in an exponent, or mismatching the cash-flow and rate units. A downward pattern alone does not verify every number, but it is a useful diagnostic.

Slope should be interpreted locally. If the slope is $-\$1{,}202.79$ per unit change in $r$, that does not mean a one-percentage-point rate increase reduces NPV by $1,202.79. The first-order estimate for a change of $0.01$ is approximately $-\$12.03$. For larger changes, recalculate NPV at the new rate or use a profile table rather than relying on the local estimate.

Common Errors to Avoid

Using a rate that does not match the periods. An annual rate cannot be applied as though each quarterly cash flow were a year away. Match the discount period to the cash-flow timing.
Discounting the initial investment. A time-0 cash flow is already at the valuation date, so its discount factor is $1$.
Using the wrong sign or reading the wrong axis. A negative NPV is below zero on the vertical axis; the rate is on the horizontal axis.
Interpolating without a bracket. Opposite signs at two rates are needed to estimate a zero crossing between them. Two positive NPVs do not bracket a crossing.
Omitting the period multiplier in a slope. Differentiation contributes a factor of $t$ to the term for a cash flow at time $t$.
Confusing rate units. A one-percentage-point increase is $0.01$ when rates are decimals. Multiply the slope by $0.01$ to estimate that change.
Calling an estimate exact. Interpolation and slope-based changes are approximations; recalculate the NPV at the target rate when an exact value is needed.

Key takeaway: An NPV profile maps discount rates to NPVs for fixed cash flows. Its zero crossing satisfies the IRR equation, while its slope measures local rate sensitivity. For a conventional project, NPV decreases as the discount rate increases, but interpolation and slope calculations remain estimates unless the full NPV equation is solved or evaluated directly.

Reading profiles accurately connects NPV calculations with rate sensitivity and IRR interpretation. This skill is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, use the stated end-of-year timing and annual effective rate. Answers are rounded to the nearest cent or, for rates, to two decimal places.

A hypothetical project costs $400 at time 0 and pays $250 at the end of each of the next two years. Calculate its NPV at 5%. (Answer: $64.85.)
A project has cash flows $-\$800$ at time 0, $300 at the end of year 1, and $600 at the end of year 2. Calculate its NPV at 10%. (Answer: $-\$31.40$.)
A project costs $500 immediately and pays $200 at each year-end for three years. Calculate its NPV at 8%. (Answer: $15.42.)
A project receives $200 at the end of year 1 and $300 at the end of year 2, with no time-0 cash flow. Calculate the slope of its NPV profile at 5%, in dollars per unit change in the annual rate. (Answer: approximately $-\$699.71$ per unit change in the rate.)
A project costs $1,000 immediately and receives $400 at the end of year 1 and $800 at the end of year 2. Its NPV is approximately $24.79 at 10% and $-\$47.26$ at 15%. Estimate the zero crossing by linear interpolation. (Answer: approximately 11.72%.)

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