Reinvestment Assumptions Behind IRR and Terminal Wealth

IRR and the Value of Interim Proceeds

An investment can return cash before its final date. The internal rate of return (IRR) summarizes the project’s dated cash flows as a single annual rate, but it does not tell us what an investor will actually earn on those interim receipts. That depends on what happens to the cash after it is received. To compare the two ideas, bring every project receipt to the same terminal date using an explicitly stated reinvestment rate.

Throughout this tutorial, assume cash flows occur at integer year-ends. A conventional project has one initial outflow $C_0>0$ at time 0 and nonnegative receipts $C_1,\ldots,C_n$ at the ends of years 1 through $n$, with at least one receipt positive. Rates are effective annual rates. Under these assumptions the project has a unique nonnegative IRR when its net present value at rate 0 is positive. The calculations below concern the project cash flows themselves, not additional deposits or withdrawals by an investor.

Definition: The internal rate of return $i$ is the rate that makes the net present value of the project’s cash flows equal to zero: $$ -C_0+\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}=0. $$ For a stated effective annual reinvestment rate $j>-1$, the terminal wealth at time $n$ is $$ W_n(j)=\sum_{t=1}^{n}C_t(1+j)^{n-t}. $$ A receipt at time $t$ earns interest for $n-t$ years, while a receipt at time $n$ earns none. The annualized return on terminal wealth, relative to the initial outlay, is $$ R_j=\left(\frac{W_n(j)}{C_0}\right)^{1/n}-1, $$ provided $W_n(j)>0$.

The IRR equation determines $i$ from the project cash flows. The reinvestment rate $j$ is a separate assumption used to accumulate receipts. In particular, the IRR calculation does not require the investor actually to reinvest proceeds at $i$. But if we want to interpret $i$ as the annualized growth rate from the initial outlay to the terminal value of all receipts, that interpretation corresponds to accumulating interim receipts at $i$.

Why Reinvestment at the IRR Reproduces the IRR

The connection follows directly from the present-value equation. Multiply the IRR equation by $(1+i)^n$. Each receipt at time $t$ is then multiplied by $(1+i)^{n-t}$, exactly the factor that accumulates it from year $t$ to year $n$ at rate $i$.

-C_0+\sum_{t=1}^{n}\frac{C_t}{(1+i)^t}=0 \quad\Longrightarrow\quad C_0(1+i)^n=\sum_{t=1}^{n}C_t(1+i)^{n-t}=W_n(i).

Dividing by $C_0$ and taking the $n$th root gives $R_i=i$. This equality relies on using the IRR as the rate for accumulating every interim receipt, with the same annual periods and terminal date as in the IRR equation. If the actual reinvestment rate is $j\ne i$, terminal wealth generally differs from $C_0(1+i)^n$, so its annualized return generally differs from $i$ as well.

Key result: For a conventional project with one initial outflow and nonnegative subsequent receipts, reinvestment of all interim receipts at the project’s IRR makes the annualized terminal-wealth return equal to the IRR. At another reinvestment rate, calculate terminal wealth using that rate; do not assume the IRR remains the realized terminal-wealth return.

Worked Calculations

For a conventional project, increasing $j$ increases the terminal value of any positive receipt made before year $n$, because its accumulation factor $(1+j)^{n-t}$ increases. The size of the effect depends on how early the receipts arrive and how long they can earn interest. The examples show how to keep the IRR calculation separate from the reinvestment calculation.

Worked Example: Comparing IRR With a Lower Reinvestment Rate

A project requires $C_0=\$1{,}000$ at time 0, pays $\$300$ at the end of year 1, and pays $\$880$ at the end of year 2. Find its IRR. Then find the terminal wealth and annualized terminal-wealth return if the year-1 receipt is reinvested at 5% effective annually.

Method and IRR. Set the present value of the two receipts equal to the initial outlay. Test $i=10\%$:

\frac{300}{1.10}+\frac{880}{1.10^2} =272.7273+727.2727 =1{,}000.

Thus $i=10\%$ is the IRR. The displayed present values are rounded for readability; the exact fractions sum to $\$1{,}000$.

Terminal wealth at $j=5\%$. The year-1 receipt earns interest for one year, and the year-2 receipt earns none:

W_2(0.05)=300(1.05)^{2-1}+880(1.05)^{2-2} =315+880 =\$1{,}195.

Annualized return. The project runs for two years, so $$ R_{0.05}=\left(\frac{1{,}195}{1{,}000}\right)^{1/2}-1 =\sqrt{1.195}-1 \approx 0.09316. $$ The annualized terminal-wealth return is 9.316% (rounded to three decimal places), below the 10% IRR because the interim receipt earns only 5% while being accumulated.

Check and common mistake avoided. At 10%, terminal wealth would be $300(1.10)+880=\$1{,}210$, which equals $1{,}000(1.10)^2$, confirming the IRR identity. A common mistake is to compound the year-1 receipt for two years, calculating $300(1.05)^2$. It is received after one year and can earn interest for only the remaining one year; the correct factor is $1.05^{2-1}$.

Worked Example: Comparing Several Reinvestment Scenarios

A project costs $\$1{,}000$ initially and returns $\$200$, $\$300$, and $\$759$ at the ends of years 1, 2, and 3. Find its IRR, then calculate terminal wealth and the annualized terminal-wealth return under reinvestment rates of 4% and 12%.

Method and IRR. Check 10% by accumulating each receipt to year 3:

200(1.10)^2+300(1.10)+759 =242+330+759 =1{,}331 =1{,}000(1.10)^3.

Equivalently, discounting the receipts at 10% gives $200/1.10+300/1.10^2+759/1.10^3=\$1{,}000$. Thus the IRR is 10%.

At a 4% reinvestment rate. Accumulate each receipt for the years remaining until year 3:

W_3(0.04)=200(1.04)^2+300(1.04)+759 =216.32+312+759 =\$1{,}287.32.

The annualized terminal-wealth return is $$ R_{0.04}=\left(\frac{1{,}287.32}{1{,}000}\right)^{1/3}-1 \approx 0.08782, $$ or 8.782% to three decimal places.

At a 12% reinvestment rate.

W_3(0.12)=200(1.12)^2+300(1.12)+759 =250.88+336+759 =\$1{,}345.88.

Therefore, $$ R_{0.12}=\left(\frac{1{,}345.88}{1{,}000}\right)^{1/3}-1 \approx 0.10408, $$ or 10.408% to three decimal places. The terminal wealth and its annualized return rise as the reinvestment rate rises because the positive year-1 and year-2 receipts earn more. The IRR remains 10%: it is a property of the project cash flows, not a new IRR for each reinvestment scenario.

Check and common mistake avoided. At $j=10\%$, terminal wealth is $242+330+759=\$1{,}331$, and its annualized return is exactly 10%. Do not change the project’s IRR merely because a different reinvestment rate is specified; recalculate terminal wealth and its annualized return instead.

Worked Example: Finding the Reinvestment Rate for a Target Wealth

A project costs $\$2{,}000$ and pays $\$400$ at the end of year 1, $\$600$ at the end of year 2, and $\$1{,}700$ at the end of year 3. What constant effective annual reinvestment rate $j$ makes terminal wealth equal to $\$2{,}800$? What annualized terminal-wealth return does that target imply?

Method. Let $x=1+j$. The year-1 receipt earns for two years, and the year-2 receipt earns for one year. The year-3 receipt earns for zero years. Therefore:

400x^2+600x+1{,}700=2{,}800, $$ $$ 4x^2+6x-11=0.

Using the quadratic formula, the roots are $\bigl(-6\pm\sqrt{36+176}\bigr)/8$. Since $x=1+j$ must be positive, choose $$ x=\frac{-6+\sqrt{212}}{8} =\frac{-3+\sqrt{53}}{4} \approx 1.0700275. $$ Thus $j\approx 0.0700275$, or 7.003% to three decimal places. The other root is negative and cannot equal $1+j$ for a rate $j>-1$.

Annualized return and check. The target wealth is 1.4 times the initial outlay over three years, so $$ R_j=\left(\frac{2{,}800}{2{,}000}\right)^{1/3}-1 =1.4^{1/3}-1 \approx 0.11869. $$ The annualized terminal-wealth return is 11.869% to three decimal places. This is not the reinvestment rate: $j$ is the rate earned by interim receipts, whereas $R_j$ measures the growth from the initial outlay to the specified terminal wealth. Substituting $x\approx1.0700275$ into $400x^2+600x+1{,}700$ gives $\$2{,}800$, as required.

Interpretation and Exam Traps

An IRR can be useful as a rate that summarizes a project’s cash flows, but it is not, by itself, a prediction of the return an investor will realize. The interpretation as an annualized growth rate from the initial outlay to terminal wealth depends on the assumed rate earned by interim proceeds. If receipts can be reinvested at a lower rate than the IRR, the annualized terminal-wealth return will generally be lower; if they can be reinvested at a higher rate, it will generally be higher. This comparison uses the conventional-project assumptions above and a common terminal date.

IRR is not the reinvestment rate observed in the market. The IRR is obtained from the project cash flows. A stated reinvestment rate is a separate scenario assumption.
Use the time remaining, not the time elapsed. A receipt at year $t$ is multiplied by $(1+j)^{n-t}$ to reach year $n$.
Do not confuse an annualized terminal-wealth return with $j$. The former compares total terminal wealth with the initial outlay; $j$ accumulates interim receipts.
Keep the assumptions in view. With later negative cash flows, multiple initial outlays, or nonconventional cash-flow patterns, the simple interpretation as reinvested positive proceeds needs additional care. The terminal accumulation equation can still be written, but a negative project cash flow is not a receipt available for reinvestment.

Key takeaway: First solve the IRR equation from the dated project cash flows. Then, for any stated reinvestment rate, accumulate each receipt for exactly the time remaining to the common terminal date and calculate the annualized return from terminal wealth. The two rates coincide when receipts are accumulated at the IRR.

Check Your Understanding

For each problem, use year-end cash flows and effective annual rates. Round percentage returns to two decimal places unless otherwise stated.

A project costs $\$1{,}500$, pays $\$400$ at year 1, and $\$1{,}375$ at year 2. Verify that its IRR is 10%. If the year-1 receipt earns no interest before year 2, find terminal wealth and the annualized terminal-wealth return. (Answer: IRR $=10\%$; terminal wealth $=\$1{,}775$; annualized return $\approx 8.78\%$.)
A project costs $\$2{,}500$ and pays $\$600$, $\$600$, and $\$1{,}600$ at years 1, 2, and 3. At a 5% reinvestment rate, find terminal wealth and its annualized return. (Answer: terminal wealth $=\$2{,}891.50$; annualized return $\approx 4.96\%$.)
A project costs $\$3{,}000$, pays $\$500$ at year 1 and $\$1{,}000$ at year 2, and has a final receipt $X$ at year 3. Find $X$ so that the project’s IRR is 8%. (Answer: $X=\$2{,}115.936$.)
A project costs $\$1{,}000$ and pays $\$200$, $\$500$, and $\$900$ at years 1, 2, and 3. If interim receipts are reinvested at 6%, find terminal wealth and the annualized terminal-wealth return. (Answer: terminal wealth $=\$1{,}654.72$; annualized return $\approx 18.28\%$.)
For a conventional project with at least one positive receipt before the terminal date, how does raising the constant reinvestment rate affect terminal wealth? When does the annualized terminal-wealth return equal the IRR? (Answer: raising the reinvestment rate increases terminal wealth; the annualized terminal-wealth return equals the IRR when receipts are accumulated at the IRR.)

What You'll Learn