Flat-Rate IRR Versus Spot-Curve NPV

Why IRR and Spot-Curve NPV Can Differ

A project’s internal rate of return compresses its dated cash flows into one rate. A spot curve instead assigns a discount rate to each maturity. These approaches answer different questions: IRR identifies a rate that makes the project’s flat-rate NPV equal to zero, while spot-curve NPV measures value using the rates applicable to the individual cash-flow dates.

The distinction matters when the term structure is not flat. If a project has cash flows at several dates, discounting all of them at its IRR is guaranteed to produce zero NPV by definition. That result does not establish that the project is worth zero at rates that vary by maturity. For valuation, compare the project’s dated cash flows with a discount curve appropriate to those dates.

Definition: Let $C_t$ be a project cash flow at time $t$, with $C_0$ the initial cash flow. The flat-rate IRR $r$ is a rate that solves $ \sum_{t=0}^{n} C_t(1+r)^{-t}=0 $. If the cash flows have one initial outflow followed by positive inflows, there is typically one economically relevant IRR. For a nonflat curve, let $s_t$ be the effective annual spot rate for maturity $t$ years. The project’s spot-curve NPV is $ \sum_{t=0}^{n} C_t(1+s_t)^{-t} $, with the time-zero cash flow included without discounting.

These formulas assume annual cash-flow dates and annual effective rates, so a cash flow at time $t$ is discounted by the factor $(1+s_t)^{-t}$. For a different payment frequency or noninteger date, the rate and time convention must be adjusted consistently. A spot rate is maturity-specific: the one-year spot rate discounts a cash flow at year 1, and the two-year spot rate discounts a cash flow at year 2. It is not appropriate to use one of those rates for every maturity unless the curve is flat.

The Identity Behind the Difference

Let $r$ be an IRR for the project. Its defining equation is $ \sum_{t=0}^{n} C_t(1+r)^{-t}=0 $. Subtracting this zero from the spot-curve NPV gives an exact expression for the difference between the two valuations:

\begin{aligned} \operatorname{NPV}_{\text{spot}} &= \sum_{t=0}^{n} C_t(1+s_t)^{-t} \\ &= \sum_{t=0}^{n} C_t(1+r)^{-t} + \sum_{t=1}^{n} C_t\left[(1+s_t)^{-t}-(1+r)^{-t}\right] \\ &= \sum_{t=1}^{n} C_t\left[(1+s_t)^{-t}-(1+r)^{-t}\right]. \end{aligned}

The time-zero terms cancel in the difference because neither discounting method changes $C_0$. If every spot rate equals the IRR, each bracketed term is zero and spot-curve NPV is zero as well. If rates differ by maturity, the terms generally do not cancel. The formula also explains why there is no universal direction to the difference: the size and sign of each contribution depend both on the cash flow and on how its spot rate compares with the IRR.

For a conventional project with positive future cash flows, if every relevant spot rate is below the IRR, each spot discount factor exceeds the corresponding IRR discount factor. The spot-curve NPV is then positive. If every relevant spot rate is above the IRR, the spot-curve NPV is negative. This sign conclusion relies on positive future cash flows; with cash flows of mixed signs, individual terms can offset one another.

Worked Comparisons

Worked Example: A Zero-IRR NPV but a Negative Spot-Curve NPV

A hypothetical project requires an initial outlay of $1,000 and returns $480 at the end of year 1 and $864 at the end of year 2. Find its IRR, then value it using a one-year spot rate of 10% and a two-year spot rate of 30%. All rates are annual effective.

Method. First solve the IRR equation using one rate for both maturities. Then discount each cash flow using the spot rate for its own maturity.

Step 1: Solve for the IRR. At 20%, the year-1 present value is $480/1.20=400$, and the year-2 present value is $864/(1.20)^2=864/1.44=600$. Their sum is $1,000, equal to the initial outlay. Therefore $r=20\%$ is the IRR:

-1000+\frac{480}{1.20}+\frac{864}{(1.20)^2} =-1000+400+600=0.

Step 2: Use the spot curve. The year-1 payment is discounted at 10%, and the year-2 payment at 30%:

\begin{aligned} \operatorname{NPV}_{\text{spot}} &=-1000+\frac{480}{1.10}+\frac{864}{(1.30)^2}\\ &=-1000+436.363636\ldots+511.242604\ldots\\ &=-52.393760\ldots \end{aligned}

The spot-curve NPV is $-\$52.39$ to the nearest cent. The project has zero NPV when every future payment is discounted at its 20% IRR, but a negative NPV under the stated maturity-specific rates. The common mistake is to treat the zero at the IRR as if it were the project’s value under every discount basis.

Worked Example: IRR Ranking Reverses the Spot-NPV Comparison

Two hypothetical projects each require $1,000 now. Project A pays $1,200 at the end of year 1. Project B pays $1,600 at the end of year 2. The one-year spot rate is 10%, and the two-year spot rate is 30%. Calculate both IRRs and both spot-curve NPVs, then compare the rankings.

Method. Solve each project’s IRR from its own cash-flow equation. For the spot valuation, use the appropriate rate for the date of each project’s payment.

Project A IRR. Its equation is $-1000+1200/(1+r_A)=0$. Thus $1+r_A=1.2$, so $r_A=20\%$.

Project B IRR. Its equation is $-1000+1600/(1+r_B)^2=0$. Therefore $(1+r_B)^2=1.6$, and $r_B=\sqrt{1.6}-1=0.264911\ldots$, or 26.4911% to four decimal places.

Project A spot-curve NPV. Its payment occurs at year 1, so use the one-year spot rate:

\operatorname{NPV}_{A,\text{spot}} =-1000+\frac{1200}{1.10} =-1000+1090.909091\ldots =\$90.91.

Project B spot-curve NPV. Its payment occurs at year 2, so use the two-year spot rate:

\operatorname{NPV}_{B,\text{spot}} =-1000+\frac{1600}{(1.30)^2} =-1000+\frac{1600}{1.69} =-\$53.25.

Project B has the higher IRR, but Project A has the higher spot-curve NPV. In this comparison, the IRRs rank each project’s implied rate of return; they do not rank value under the stated spot curve. Treating the higher IRR as sufficient evidence that B is preferable would ignore the different maturity-specific discounting and would select the project with negative spot-curve NPV.

Worked Example: Interpreting a Positive Spot-Curve NPV

A hypothetical project has cash flows of $-\$450$ now, $+\$240$ at year 1, and $+\$360$ at year 2. Its one-year and two-year spot rates are 5% and 12%, respectively. Find its IRR and spot-curve NPV, and explain why the NPV is positive.

Method. Verify the IRR by discounting both future payments at one common rate. Then use each maturity-specific spot rate for the curve valuation.

Step 1: Verify the IRR. At 20%, the year-1 payment has present value $240/1.20=200$, and the year-2 payment has present value $360/(1.20)^2=360/1.44=250$. Together these equal the $450 outlay, so the IRR is 20%.

Step 2: Calculate spot-curve NPV. The one-year payment is discounted at 5%; the two-year payment is discounted at 12%, whose two-year accumulation factor is $1.12^2=1.2544$:

\begin{aligned} \operatorname{NPV}_{\text{spot}} &=-450+\frac{240}{1.05}+\frac{360}{1.2544}\\ &=-450+228.571429\ldots+286.989796\ldots\\ &=\$65.56. \end{aligned}

Both spot rates are below the 20% IRR, and both future cash flows are positive. Each payment therefore has a larger present value under the spot curve than under 20% discounting: the increases are $228.571429\ldots-200=\$28.571429\ldots$ at year 1 and $286.989796\ldots-250=\$36.989796\ldots$ at year 2. Together they account for the positive spot-curve NPV of $65.56. The conclusion depends on these positive future flows; with mixed-sign flows, rate comparisons alone would not determine the NPV’s sign.

Common Exam Trap: Using the IRR as the Valuation Curve

A frequent error is to find an IRR and then report zero as the project’s NPV, even when the question specifies a nonflat spot curve. For the first worked example, discounting at its 20% IRR gives $-1000+480/1.20+864/(1.20)^2=0$. That calculation is correct as an IRR check, but it is not the requested spot-curve valuation. With the given 10% and 30% spot rates, the correct NPV is $-\$52.39$.

A related error is to apply the one-year spot rate to all future payments, or the two-year rate to all of them. The rate must match the cash-flow maturity. Before calculating, list each dated cash flow beside its corresponding spot rate and discount factor.

Keep the questions distinct.
Use one flat rate to find the IRR; use maturity-specific spot rates to find spot-curve NPV.

Match each rate to its date.
For a cash flow at time $t$, use the discount factor $(1+s_t)^{-t}$.

Interpret the results separately.
An IRR is a break-even flat discount rate. Spot-curve NPV is the value using the stated maturity-specific rates.

This distinction is useful whenever an investment’s cash flows span maturities with different rates. IRR is a compact summary of a project’s cash-flow pattern, but it does not preserve the spot curve’s maturity-by-maturity valuation. For decisions based on the curve, calculate NPV directly from the dated cash flows and their corresponding spot rates. The two measures agree at zero when the relevant spot rates are all equal to the project’s IRR; otherwise, any equality or difference in value must be established by calculation.

Check Your Understanding

For each problem, use annual effective rates and match each spot rate to the payment’s maturity.

A project pays $-\$900$ now, $+\$480$ at year 1, and $+\$720$ at year 2. Verify its IRR and calculate its spot-curve NPV when $s_1=5\%$ and $s_2=15\%$. (Answer: IRR $=20\%$; spot-curve NPV $=\$101.57$.)
A project costs $1,200 now and pays $1,800 at year 2. Find its IRR and its spot-curve NPV if $s_2=25\%$. (Answer: IRR $=\sqrt{1.5}-1=22.4745\%$ to four decimal places; spot-curve NPV $=-\$48.00$.)
A project pays $-\$1,000$ now, $+\$480$ at year 1, and $+\$864$ at year 2. Verify its IRR and calculate its spot-curve NPV when $s_1=5\%$ and $s_2=30\%$. (Answer: IRR $=20\%$; spot-curve NPV $=-\$31.61$.)
Project A costs $1,000 and pays $1,100 at year 1. Project B costs $1,000 and pays $1,400 at year 2. Find their IRRs and spot-curve NPVs when $s_1=5\%$ and $s_2=20\%$. Which project has the higher IRR, and which has the higher spot-curve NPV? (Answer: A: IRR $=10\%$, NPV $=\$47.62$; B: IRR $=\sqrt{1.4}-1=18.3216\%$ to four decimal places, NPV $=-\$27.78$. B has the higher IRR; A has the higher spot-curve NPV.)
A conventional project has positive future cash flows at years 1 and 2, an IRR of 15%, and spot rates $s_1=8\%$ and $s_2=12\%$. Without knowing the cash-flow amounts, determine whether its spot-curve NPV is positive, zero, or negative, and justify your conclusion. (Answer: Positive, because both spot rates are below the IRR, so each positive future cash flow has a larger present value at its spot rate than at 15%.)

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