Calculating MIRR From Project Cash Flows

Why Modified IRR Separates Two Rates

The internal rate of return (IRR) is one summary of a project’s cash flows, but its interpretation depends on assumptions about how cash flows are valued between their dates. The modified internal rate of return, or MIRR, makes two rates explicit: a finance rate for project outflows and a reinvestment rate for project inflows. The result is one annualized rate that links the present value of outflows to the terminal value of inflows.

MIRR is useful when an analyst wants an annual return measure while using rates that reflect the assumed cost of financing and return on reinvested receipts. It is not a replacement for specifying those assumptions: a different finance or reinvestment rate can produce a different MIRR for the same cash flows. The method below assumes equally spaced annual periods and constant effective annual rates.

Definition: For project cash flows over periods $0,1,\ldots,n$, the finance rate $f$ is the rate used to discount negative cash flows to time 0. The reinvestment rate $r$ is the rate used to accumulate positive cash flows to time $n$. The MIRR is the annual rate that equates the present value of outflows, accumulated to time $n$ at the MIRR, to the terminal value of inflows.

Let $X_t$ be the signed cash flow at time $t$. Separate its positive and negative parts as $X_t^+=\max(X_t,0)$ and $X_t^-=\max(-X_t,0)$, so that $X_t=X_t^+-X_t^-$. The two values needed for the calculation are

PV_-=\sum_{t=0}^{n}\frac{X_t^-}{(1+f)^t}, \qquad FV_+=\sum_{t=0}^{n}X_t^+(1+r)^{n-t}.

Here $PV_-$ is the present value of outflows at time 0, and $FV_+$ is the terminal value of inflows at time $n$. The convention includes cash flows at both endpoints: a time-0 outflow is not discounted, and a time-$n$ inflow is not compounded.

The MIRR Formula and Its Justification

Once the two values have been calculated, MIRR is the rate that grows $PV_-$ from time 0 to time $n$ until it equals $FV_+$. Thus,

PV_-(1+\operatorname{MIRR})^n=FV_+, \qquad \operatorname{MIRR} =\left(\frac{FV_+}{PV_-}\right)^{1/n}-1.

This formula requires $n>0$, $PV_->0$, and $FV_+>0$. It also requires rates greater than $-100\%$ so that the accumulation and discount factors are defined and positive. If a project has no outflows or no inflows, this standard MIRR calculation is not defined. The rate periods must match the cash-flow periods: with annual cash flows, $f$, $r$, and MIRR are effective annual rates.

The justification is the meaning of an equivalent annual return. Discounting each outflow at $f$ expresses the cost of the outflow in time-0 dollars. Compounding each receipt at $r$ expresses the accumulated value of receipts at time $n$. If the present-value cost earned a constant rate $M$ over $n$ years, its terminal value would be $PV_-(1+M)^n$. Setting that equal to the accumulated receipts gives the MIRR equation. Taking the positive $n$th root and rearranging yields the formula.

Set the timeline.
Identify time 0, the final period $n$, and the date of every cash flow. Do not omit a period with no cash flow when identifying the horizon.

Value the outflows.
Discount each negative cash flow’s positive magnitude to time 0 at the finance rate $f$, then add the present values to obtain $PV_-$.

Value the inflows.
Accumulate each positive cash flow to time $n$ at the reinvestment rate $r$, then add the terminal values to obtain $FV_+$.

Solve and check.
Use $(FV_+/PV_-)^{1/n}-1$. Check that raising $1+\operatorname{MIRR}$ to the $n$th power approximately reproduces $FV_+/PV_-$.

Worked Calculations

Worked Example: A Single Initial Outlay

A hypothetical project requires $1,000 at time 0 and produces receipts of $300, $400, and $500 at the ends of years 1, 2, and 3. The finance rate is 8% effective annually and the reinvestment rate is 10% effective annually. Calculate MIRR, rounded to two decimal places.

Method. The project lasts $n=3$ years. The only outflow is at time 0, so $PV_-=\$1{,}000$. Accumulate each receipt to the end of year 3 at the 10% reinvestment rate:

FV_+=300(1.10)^2+400(1.10)+500 =363+440+500 =\$1{,}303.

Now take the three-year equivalent annual rate:

\operatorname{MIRR} =\left(\frac{1{,}303}{1{,}000}\right)^{1/3}-1 =(1.303)^{1/3}-1 \approx 0.0922 =9.22\%.

Check. $1.0922^3$ is approximately $1.3029$, close to the ratio $1.303$; the small difference is due to rounding. The finance rate does not change this example’s present value of outflows because the sole outflow occurs at time 0. This is not a reason to omit the finance rate from the method: later outflows would be discounted using it.

Worked Example: An Outlay After Time 0

A hypothetical project has cash flows of $-\$600$ at time 0, $-\$300$ at the end of year 1, $+\$500$ at the end of year 2, and $+\$600$ at the end of year 3. Use a finance rate of 10% and a reinvestment rate of 8%, both effective annually. Calculate MIRR to two decimal places.

Method. The project horizon is $n=3$. Discount the year-1 outlay at the finance rate; the time-0 outlay is already at present value:

PV_-=600+\frac{300}{1.10} =600+272.7273 =\$872.7273.

Compound the year-2 receipt for one year and the year-3 receipt for zero years, both at the reinvestment rate:

FV_+=500(1.08)+600 =540+600 =\$1{,}140.

Therefore,

\operatorname{MIRR} =\left(\frac{1{,}140}{872.7273}\right)^{1/3}-1 \approx (1.30625)^{1/3}-1 \approx 0.0931 =9.31\%.

Check. $1.0931^3$ is approximately $1.3061$, consistent with $FV_+/PV_-$ after rounding. The method accounts for the extra outlay by discounting it to time 0 before solving for the equivalent return.

Worked Example: A Gap Between Receipt Dates

A hypothetical four-year project requires $2,000 at time 0 and produces receipts of $700 at the end of year 1, $900 at the end of year 2, and $1,000 at the end of year 4. There is no cash flow at the end of year 3. The finance rate is 6% and the reinvestment rate is 9%, effective annually. Calculate MIRR to two decimal places.

Method. The empty year 3 remains part of the four-year horizon, so $n=4$. The initial outlay is the entire present value of outflows: $PV_-=\$2{,}000$. Accumulate each receipt to the end of year 4:

FV_+ =700(1.09)^3+900(1.09)^2+1{,}000 =906.5203+1{,}069.29+1{,}000 =\$2{,}975.8103.

Then,

\operatorname{MIRR} =\left(\frac{2{,}975.8103}{2{,}000}\right)^{1/4}-1 =(1.48790515)^{1/4}-1 \approx 0.1044 =10.44\%.

Check. $1.1044^4$ is approximately $1.4877$, close to $1.48790515$. The calculation uses three years of accumulation for the year-1 receipt, two years for the year-2 receipt, and none for the year-4 receipt.

Common Rate and Timing Traps

A common mistake is to swap the two specified rates: to discount outflows at the reinvestment rate and accumulate inflows at the finance rate. In the second example, that incorrect approach would give

PV_-^{\text{wrong}}=600+\frac{300}{1.08}=\$877.7778, \qquad FV_+^{\text{wrong}}=500(1.10)+600=\$1{,}150,

and an incorrect result of

\left(\frac{1{,}150}{877.7778}\right)^{1/3}-1 \approx 9.42\%.

The correct rate assignments give 9.31%, not 9.42%. The finance rate discounts negative cash flows; the reinvestment rate accumulates positive cash flows. Another frequent error is using $n-t$ as the exponent for a negative cash flow or $t$ as the exponent for a positive cash flow. Those exponents belong to different operations: outflows are discounted from $t$ to 0, while inflows are accumulated from $t$ to $n$.

The project horizon also matters. A period with no cash flow still counts when it lies between the project’s start and end dates. If the final receipt occurs at year 4, use $n=4$, even if year 3 has no receipt. Finally, retain unrounded present and terminal values until the last step; early rounding can change the reported MIRR.

Key takeaway: MIRR has three distinct ingredients: the present value of negative cash flows at the finance rate, the terminal value of positive cash flows at the reinvestment rate, and the number of periods connecting those two values. Label each cash flow’s date before calculating.

Check Your Understanding

For each hypothetical project, use the stated effective annual rates and round MIRR to two decimal places.

A project costs $1,500 at time 0 and returns $600, $700, and $800 at the ends of years 1, 2, and 3. The finance rate is 7% and the reinvestment rate is 5%. Calculate the terminal value of receipts and MIRR. (Answer: Terminal value = $2,196.50; MIRR = 13.56%.)
A project costs $1,000 at time 0 and returns $500 at the end of year 1 and $700 at the end of year 2. The finance rate is 8% and the reinvestment rate is 12%. Find $PV_-$, $FV_+$, and MIRR. (Answer: $PV_-=\$1{,}000$; $FV_+=\$1{,}260$; MIRR = 12.25%.)
A project has outflows of $400 at time 0 and $220 at the end of year 1, followed by receipts of $400 at the end of year 2 and $500 at the end of year 3. Use a 10% finance rate and a 6% reinvestment rate. Find both valued cash-flow totals and MIRR. (Answer: $PV_-=\$600$; $FV_+=\$924$; MIRR = 15.48%.)
A project costs $1,200 at time 0 and returns $400, $500, and $600 at the ends of years 1, 2, and 3. The finance rate is 6% and the reinvestment rate is 10%. Accumulate the receipts to year 3 and calculate MIRR. (Answer: $FV_+=\$1{,}634$; MIRR = 10.84%.)
A project costs $1,000 at time 0, returns $500 at the end of year 1, has no cash flow at the end of year 2, and returns $700 at the end of year 3. The finance rate is 6% and the reinvestment rate is 8%. What horizon should be used, what is the terminal value of receipts, and what is MIRR? (Answer: Use $n=3$; $FV_+=\$1{,}283.20$; MIRR = 8.67%.)

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