Linear Interpolation Between Trial Rates

Why Interpolate Between Trial Rates?

For a project with uneven cash flows, it is often possible to calculate net present value at a chosen rate but inconvenient to solve the IRR equation directly. If two trial rates produce NPVs with opposite signs, the IRR is bracketed between them, provided the NPV is continuous over that interval. Linear interpolation estimates where NPV would reach zero by treating the change between the two calculated values as a straight line.

This method is a useful estimate, not generally an exact solution. NPV is usually a curved function of the rate because each cash flow is discounted by a different power of $1+r$. The estimate is most useful when the trial rates are reasonably close and the NPV curve is not changing sharply across the interval.

Definition: For cash flows $C_0,C_1,\ldots,C_n$ at equally spaced period ends, the net present value at effective rate $r$ per period is $$ \operatorname{NPV}(r)=\sum_{t=0}^{n}\frac{C_t}{(1+r)^t}, \qquad r>-1. $$ An internal rate of return is a rate $r$ for which $\operatorname{NPV}(r)=0$. Linear interpolation estimates such a rate from two trial rates and their NPVs.

The periods and rates must match: if the cash flows are annual, use effective annual trial rates; if the cash flows are quarterly, use effective quarterly rates. The interpolation fraction is dimensionless, so it can be applied to rates expressed consistently as decimals or percentages. The estimated rate must use the same units as the two trial rates.

The Interpolation Formula

Let $r_L<r_H$ be two trial rates, and let $N_L=\operatorname{NPV}(r_L)$ and $N_H=\operatorname{NPV}(r_H)$. The rates bracket a zero when $N_L$ and $N_H$ have opposite signs. The straight line joining the points $(r_L,N_L)$ and $(r_H,N_H)$ gives the interpolation estimate:

\widehat{r} =r_L+\frac{N_L}{N_L-N_H}(r_H-r_L).

This expression works whether the lower-rate NPV is positive or negative, as long as the two NPVs have opposite signs. For example, if $N_L>0$ and $N_H<0$, then $N_L/(N_L-N_H)$ lies between 0 and 1. If $N_L<0$ and $N_H>0$, both numerator and denominator are negative, so the fraction still lies between 0 and 1.

Evaluate NPV at both trial rates.
Use the stated cash-flow timing and the correct rate per period. Retain enough digits in the calculations to avoid unnecessary rounding error.

Check the signs.
The NPVs must have opposite signs to bracket a zero. If they have the same sign, this pair of trial rates does not support an interpolation estimate of a root between them.

Calculate the interpolation fraction.
Use $N_L/(N_L-N_H)$, with the rates and NPVs paired in the same lower-to-higher order.

Move that fraction of the rate interval.
Add the fraction times $r_H-r_L$ to $r_L$, then report the result as an estimate.

Why the Formula Holds—and Why It Is Approximate

The formula follows by writing the equation of the straight line through the two trial points. At an intermediate rate $r$, the line's NPV value is

N_{\mathrm{line}}(r) =N_L+\frac{N_H-N_L}{r_H-r_L}(r-r_L).

Set this straight-line value equal to zero and solve for $r$:

0=N_L+\frac{N_H-N_L}{r_H-r_L}(r-r_L) \quad\Longrightarrow\quad r=r_L+\frac{N_L}{N_L-N_H}(r_H-r_L).

This derivation explains both the method and its limitation. The calculation finds the zero of the line joining the two actual NPV points. It finds the exact IRR only when the actual NPV function has that same straight-line behavior over the interval, or when the line happens to cross zero at the actual root. In general, $\operatorname{NPV}(r)$ is not linear in $r$, so interpolation and the true root can differ.

Worked Example: Interpolation Near a Known IRR

A hypothetical equipment project costs $900 now and returns $330, $363, and $399.30 at the ends of years 1, 2, and 3. Estimate its annual IRR by interpolating between 8% and 12%. Then compare the estimate with the exact IRR. Use effective annual rates. The method is linear interpolation; it avoids treating an estimate as an exact root.

At 8%, the three receipts have present values

\frac{330}{1.08}=305.5556,\qquad \frac{363}{1.08^2}=\frac{363}{1.1664}=311.2140,\qquad \frac{399.30}{1.08^3}=\frac{399.30}{1.259712}=316.9772.

Therefore,

\operatorname{NPV}(0.08) =-900+305.5556+311.2140+316.9772 \approx 33.7468.

At 12%, the discounted receipts are

\frac{330}{1.12}=294.6429,\qquad \frac{363}{1.12^2}=\frac{363}{1.2544}=289.3814,\qquad \frac{399.30}{1.12^3}=\frac{399.30}{1.404928}=284.2138.

Thus,

\operatorname{NPV}(0.12) =-900+294.6429+289.3814+284.2138 \approx -31.7619.

The NPVs have opposite signs, so the zero is bracketed. Using the unrounded NPV calculations in the interpolation formula,

\widehat{r} =0.08+\frac{33.7468}{33.7468-(-31.7619)}(0.12-0.08) \approx 0.100606.

The interpolated estimate is about 10.061%. To check the exact root, test 10%: the present value of each receipt is $300, since

\frac{330}{1.10}=300,\qquad \frac{363}{1.10^2}=300,\qquad \frac{399.30}{1.10^3}=300.

The three present values total $900, which offsets the initial $900 outlay. The exact IRR is therefore 10%. The interpolation estimate is about 0.061 percentage points higher. The small difference is due to the curvature of NPV between 8% and 12%, not a failure of the arithmetic.

How Curvature Can Affect the Estimate

A wider rate interval or more pronounced curvature can make the straight-line estimate less accurate. The next example compares interpolation with an exact solution for a project with two future receipts. This is still an annual cash-flow model, with effective annual trial rates. The method is interpolation followed by an algebraic check; the common mistake to avoid is assuming that opposite-sign NPVs make the interpolated result exact.

Worked Example: A Wider Difference from the Exact IRR

A hypothetical project requires $1,000 now and returns $600 at the end of each of the next two years. Estimate the annual IRR using trial rates of 10% and 20%, then solve for the exact IRR to assess the estimate.

At 10%,

\operatorname{NPV}(0.10) =-1{,}000+\frac{600}{1.10}+\frac{600}{1.10^2} =-1{,}000+545.4545+495.8678 \approx 41.3223.

At 20%,

\operatorname{NPV}(0.20) =-1{,}000+\frac{600}{1.20}+\frac{600}{1.20^2} =-1{,}000+500+416.6667 \approx -83.3333.

The NPVs bracket a zero. Interpolating between them gives

\widehat{r} =0.10+\frac{41.3223}{41.3223-(-83.3333)}(0.20-0.10) \approx 0.13315.

The estimate is about 13.315%. To find the exact IRR, let $x=1+r$. Setting NPV equal to zero and multiplying by $x^2$ gives

-1{,}000+\frac{600}{x}+\frac{600}{x^2}=0 \quad\Longrightarrow\quad 1{,}000x^2-600x-600=0.

The positive solution is

x=\frac{0.6+\sqrt{0.6^2+2.4}}{2} =\frac{0.6+\sqrt{2.76}}{2} \approx 1.1306624, \qquad r\approx 0.1306624.

The exact IRR is about 13.066%, so interpolation overestimates it by about 0.249 percentage points. The difference is larger than in the preceding example. The two NPVs define a chord, whereas the actual NPV function is curved. The interpolated value is useful as an estimate within the bracket, but it is not a substitute for solving the IRR equation when greater accuracy is required.

Using NPV Results Supplied in a Problem

Sometimes a problem supplies trial rates and their NPVs, rather than asking you to calculate the NPVs from the underlying cash flows. The same method applies. Confirm that the rate units match the cash-flow periods, check the signs, and then interpolate. Do not add assumptions about cash flows that are not given.

Worked Example: Interpolating from a Valuation Table

A project analyst has calculated annual NPVs of $62.40 at 7% and $-$37.60$ at 11%. Estimate the annual IRR. The figures are hypothetical. The method is direct interpolation from the reported values; it avoids giving the two endpoints equal weight when their NPV magnitudes are unequal.

The lower-rate NPV is positive and the higher-rate NPV is negative, so the trial rates bracket a zero. The NPV change from the first endpoint to the second is

$$ 62.40-(-37.60)=100.00. $$

The fraction of the four-percentage-point interval needed to move from the $62.40 NPV to zero is

\frac{62.40}{100.00}=0.624.

Therefore, the interpolated rate is

\widehat{r}=7\%+0.624(11\%-7\%) =7\%+2.496\% =\mathbf{9.496\%}.

This is an estimate based on the two table entries. The table alone does not establish that the exact IRR equals 9.496%; the actual NPV curve may not be linear between the trial rates.

Common Mistakes and Practical Checks

Interpolation is a short calculation, but errors often arise from misreading the signs, mixing units, or rounding too early. Use these checks before reporting an estimate:

Do not interpolate without a sign change. Two positive NPVs or two negative NPVs do not bracket a zero. A root might exist elsewhere, but this pair of trial rates does not locate it between the endpoints.
Keep each NPV attached to its rate. Write the lower and higher trial rates alongside their NPVs before substituting. Reversing one pairing changes the calculation.
Use the whole NPV change in the denominator. The denominator is $N_L-N_H$, which equals the total NPV difference when the signs are opposite. It is not just the magnitude of one endpoint's NPV.
Do not average the trial rates automatically. The zero is at the midpoint only when the two endpoint NPVs have equal magnitudes. For the valuation-table example, the midpoint is 9%, but the NPVs are not equal in magnitude, so the interpolated result is 9.496%.
Keep rate units consistent. If the rates are 7% and 11%, the interval is four percentage points. If using decimals, use 0.07 and 0.11 throughout.
Do not label the estimate as exact. The interpolation formula zeros the chord between two NPV points. The actual NPV function can be curved.

A useful reasonableness check is that the estimate should lie between the two trial rates when the NPVs have opposite signs. If it does not, check the signs, the endpoint order, and the denominator. Retain full calculator precision through the interpolation and round only the final rate; report the requested number of decimal places or significant figures.

Key takeaway: Linear interpolation estimates an IRR by finding the zero of the straight line between two opposite-sign NPV values. It is a quick, interpretable estimate inside the bracket, but the exact IRR may differ because NPV is generally nonlinear in the rate.

Check Your Understanding

For each problem, use linear interpolation and remember that the result is an estimate unless the exact root is established separately.

A hypothetical investment has NPV $+$48$ at 6% and $-$32$ at 10%. Estimate the IRR. (Answer: 8.4%)
A project has NPV $-$24$ at 9% and $+$56$ at 13%. Estimate the IRR, taking care to keep the rates in increasing order. (Answer: 10.2%)
An NPV table gives $+$90$ at 4% and $-$30$ at 7%. Estimate the rate at which NPV is zero. (Answer: 6.25%)
A project costs $1,000 now and returns $550 at the end of each of the next two years. Its effective annual NPVs are to be evaluated at 5% and 15%. Calculate both NPVs, then interpolate. (Answer: NPV at 5% is approximately $22.6757; NPV at 15% is approximately $-$105.8601$; interpolated IRR is approximately 6.764%)
A project has NPV $+$20$ at 8% and $-$20$ at 12%. What does interpolation estimate, and does that alone prove the exact IRR is 10%? (Answer: It estimates 10%; no, equal and opposite endpoint NPVs do not prove that the actual NPV curve is linear or that its exact zero is at the midpoint.)

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