IRR Is a Cash-Flow Equation, Not a Reinvestment Account
An internal rate of return is defined by the rate that makes the present value of a specified set of cash flows equal to zero. That definition does not describe what an investor actually does with money received before the final date. If an interim receipt is spent, held as cash, or invested at a different rate, the project's IRR equation is unchanged, but the investor's ending wealth can differ.
This distinction matters when interpreting an IRR. It is valid to say that the cash flows have an IRR of a particular rate. It is a further assumption to say that every interim receipt earns that same rate until the end of the project. Under that assumption, the IRR gives a convenient terminal-value identity. Without it, the identity need not describe realized wealth.
Unless stated otherwise, examples use annual periods, an effective annual rate, and cash flows at the ends of the stated periods. The cash flows and their timing are treated as certain for calculation purposes. The IRR is assumed to be the relevant root identified by the question; cash-flow patterns can have multiple roots or none, as discussed in the earlier material on multiple or missing IRRs.
Why Reinvesting at the IRR Gives a Terminal-Value Identity
Suppose the IRR equation is multiplied by \((1+i_{\mathrm{IRR}})^n\). Each cash flow at time \(t\) is then accumulated from time \(t\) to time \(n\), giving
This is an algebraic consequence of the NPV equation. For the usual project with an initial investment \(C_0=-I\), no further outflows, and positive interim receipts, rearranging gives
The left side is the terminal value of the receipts if each is reinvested at the IRR until time \(n\). The right side is the initial investment accumulated at that same rate for \(n\) periods. Thus, under this reinvestment assumption, the two terminal values are equal.
The timing exponent is important. A receipt arriving at the end of period \(t\) can earn interest only in periods \(t+1\) through \(n\), so it is accumulated for \(n-t\) periods. In particular, a receipt at the end of year 1 in a three-year project earns for years 2 and 3: its accumulation factor is \((1+i)^2\), not \((1+i)^3\). A receipt at the final date has exponent zero and earns no further interest within the project.
Worked Examples
Worked Example: Verifying the IRR and Its Terminal-Value Identity
A project requires an initial payment of $1,000 and pays $300 at the end of year 1 and $880 at the end of year 2. Verify that its IRR is 10% effective annually, then compare terminal values when the interim receipt is reinvested at 10%.
Method. Substitute the proposed rate into the NPV equation. Then accumulate the cash flows to the end of year 2. The calculation treats the year-1 receipt as arriving after year 1 has elapsed.
At 10%, the present value of the two receipts is $$ \frac{300}{1.10}+\frac{880}{(1.10)^2} =272.7273+727.2727 =1{,}000. $$
The net present value is therefore \(-1{,}000+1{,}000=0\), so 10% is an IRR. To find the terminal value of the receipts at the end of year 2, the year-1 receipt earns one year of interest, and the year-2 receipt earns none: $$ 300(1.10)^{2-1}+880(1.10)^{2-2} =330+880 =1{,}210. $$
The initial $1,000 accumulated for two years at 10% is $$ 1{,}000(1.10)^2=1{,}210. $$ The terminal values agree, as the identity predicts. Common mistake avoided: accumulating the year-1 receipt for two years would incorrectly treat it as if it arrived at the start of year 1.
Worked Example: A Different Reinvestment Rate Changes Realized Wealth
A project has cash flows of \(-\$1{,}000\) at time 0, \(+\$400\) at the end of year 1, and \(+\$770\) at the end of year 2. Its IRR is 10% effective annually. Suppose the year-1 receipt is reinvested for the second year at 5%, while the final receipt is retained at the project end. Find the terminal wealth and the annual compound rate implied by that wealth relative to the initial $1,000.
Method. First confirm the IRR from the discounted cash flows. Then accumulate the year-1 receipt at the actual reinvestment rate, not at the project's IRR. The final receipt arrives at the end of year 2 and is not accumulated.
At 10%, the present values are $$ \frac{400}{1.10}=363.6364,\qquad \frac{770}{(1.10)^2}=636.3636. $$ Their sum is $1,000, so the NPV is zero and the IRR is 10%.
The year-1 receipt earns 5% for the one remaining year: $$ 400(1.05)=420. $$ Adding the year-2 receipt gives terminal wealth of $$ 420+770=1{,}190. $$ For comparison, if the year-1 receipt had instead earned the IRR of 10%, terminal wealth would be $$ 400(1.10)+770=440+770=1{,}210, $$ which equals \(1{,}000(1.10)^2\). At the actual 5% reinvestment rate, the annual compound rate equivalent to growing $1,000 to $1,190 over two years is $$ \left(\frac{1{,}190}{1{,}000}\right)^{1/2}-1 =\sqrt{1.19}-1 \approx 0.09087. $$ Thus the equivalent annual compound rate is 9.087% to three decimal places. Common mistake avoided: reporting 10% as the realized compound rate merely because it solves the IRR equation. Here, the IRR remains 10%, while the terminal-wealth equivalent is lower because the interim receipt earns only 5%.
Worked Example: Comparing IRR Accumulation With Actual Terminal Wealth
A project requires $1,200 initially and pays $500 at the end of year 1 and $1,250 at the end of year 2. Verify its IRR, then calculate the terminal wealth if the year-1 receipt earns 10% for the remaining year. Also find the annual compound rate equivalent to that terminal wealth.
Method. Verify the IRR using the NPV equation, then keep the IRR calculation separate from the actual accumulation assumption.
At 25%, the present values are $$ \frac{500}{1.25}=400,\qquad \frac{1{,}250}{(1.25)^2} =\frac{1{,}250}{1.5625} =800. $$ Their sum is $1,200, so the NPV is zero and the IRR is 25%.
The year-1 receipt is reinvested for one year at 10%, yielding $$ 500(1.10)=550. $$ The terminal receipt of $1,250 arrives at the end of year 2, so it has no remaining accumulation period. Actual terminal wealth is therefore $$ 550+1{,}250=1{,}800. $$ If both receipts were instead accumulated at the IRR, their terminal value would be $$ 500(1.25)+1{,}250=625+1{,}250=1{,}875, $$ equal to the initial investment accumulated at 25% for two years: $$ 1{,}200(1.25)^2=1{,}200(1.5625)=1{,}875. $$ The annual compound rate equivalent to the actual terminal wealth is $$ \left(\frac{1{,}800}{1{,}200}\right)^{1/2}-1 =\sqrt{1.5}-1 \approx 0.2247. $$ This is 22.47% to two decimal places. Common mistake avoided: using the IRR to accumulate the year-1 receipt even though the problem specifies a 10% reinvestment rate.
Using the Identity Carefully
For a conventional investment with a single initial outflow and subsequent positive receipts, it is natural to describe the left side of the terminal-value identity as the accumulated receipts. If there are later outflows as well, the same algebraic identity still holds with signed cash flows. But a negative cash flow is not a receipt to reinvest: it may represent an additional funding requirement. In that case, the identity is a signed accumulation equation, and interpreting every term as investor wealth requires care.
The annual compound rate calculated from initial investment and terminal wealth is meaningful as an equivalent rate only under the stated timing and reinvestment assumptions. It summarizes the growth from the initial amount to the final amount as if there were no intervening cash flows. It does not change the original cash-flow IRR, and it should not be presented as a rate earned on each receipt unless the cash flows and reinvestment actually support that interpretation.
Use the dated, signed cash flows in the NPV equation. Do not infer a reinvestment strategy from the root alone.
If a receipt's actual reinvestment rate is specified, use it for the time remaining after the receipt arrives. If none is specified, distinguish the IRR identity from an assertion about actual wealth.
A receipt at the end of period \(t\) is multiplied only by factors for periods \(t+1\) through \(n\). A final-period receipt has no further accumulation.
Report the IRR as the rate solving the cash-flow equation. Report any terminal-wealth equivalent separately, with its reinvestment assumptions.
Common Exam Trap
A frequent error is to give a receipt one extra period of interest. Suppose a receipt of $200 arrives at the end of year 1 and the terminal date is the end of year 3. At a constant reinvestment rate \(j\), the incorrect calculation is
That exponent counts three periods after a receipt that arrives only after the first period has ended. The correct accumulation is for years 2 and 3, so it is
A related interpretation trap is to treat reinvestment at the IRR as an unavoidable feature of the IRR itself. It is not. The rate is determined by the original cash flows and dates; reinvestment assumptions enter only when translating interim receipts into a realized terminal value.
Check Your Understanding
For each question, distinguish the IRR equation from the assumed reinvestment of interim receipts. Amounts are in dollars, and all cash flows occur at the ends of annual periods unless indicated otherwise.
- A project costs $500 at time 0 and pays $200 at the end of year 1 and $385 at the end of year 2. Verify that 10% is its IRR. If the first receipt is reinvested at 10%, what is the terminal value at year 2? (Answer: NPV at 10% is zero; terminal value is $605.)
- A project costs $1,000 and pays $400 at the end of year 1 and $770 at the end of year 2. Its IRR is 10%. If the year-1 receipt earns 6% for the remaining year, find terminal wealth and the equivalent annual compound rate over two years. (Answer: Terminal wealth is $1,194; equivalent annual rate is approximately 9.27%.)
- A project costs $2,000 and pays $600 at the end of year 1 and $2,160 at the end of year 2. Verify the IRR of 20%. If the first receipt is reinvested at 12%, compare actual terminal wealth with the terminal value under IRR reinvestment. (Answer: Actual terminal wealth is $2,832; the IRR-based terminal value is $2,880.)
- A project pays $250, $400, and $500 at the ends of years 1, 2, and 3. The year-1 receipt earns 5% during year 2 and 8% during year 3; the year-2 receipt earns 8% during year 3. Find terminal wealth at the end of year 3. (Answer: $1,215.50.)