mathematics

Modified internal rate of return

Modified internal rate of return is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Modified internal rate of return rather than just read about it. In short: The modified internal rate of return (MIRR) is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of unequal size.

Key takeaways

Modified internal rate of return belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Modified internal rate of return to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Modified internal rate of return from memory before moving on to harder problems.

Reference excerpt

The modified internal rate of return (MIRR) is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of unequal size. As the name implies, MIRR is a modification of the internal rate of return (IRR) and as such aims to resolve some problems with the IRR.

Problems associated with the IRR While there are several problems with the IRR, MIRR resolves two of them. Firstly, IRR is sometimes misapplied, under an assumption that interim positive cash flows are reinvested elsewhere in a different project at the same rate of return offered by the project that generated them. This is usually an unrealistic scenario and a more likely situation is that the funds will be reinvested at a rate closer to the firm's cost of capital. The IRR therefore often gives an unduly optimistic picture of the projects under study. Generally for comparing projects more fairly, the weighted average cost of capital should be used for reinvesting the interim cash flows. Secondly, more than one IRR can be found for projects with alternating positive and negative cash flows, which leads to confusion and ambiguity. MIRR finds only one value.

Calculation MIRR is calculated as follows:

MIRR = F V ( positive cash flows, reinvestment rate ) − P V ( negative cash flows, finance rate ) n − 1 {\displaystyle {\text{MIRR}}={\sqrt[{n}]{\frac {FV({\text{positive cash flows, reinvestment rate}})}{-PV({\text{negative cash flows, finance rate}})}}}-1} , where n is the number of equal periods at the end of which the cash flows occur (not the number of cash flows), PV is present value (at the beginning of the first period), FV is future value (at the end of the last period). The formula adds up the negative cash flows after discounting them to time zero using the external cost of capital, adds up the positive cash flows including the proceeds of reinvestment at the external reinvestment rate to the final period, and then works out what rate of return would cause the magnitude of the discounted negative cash flows at time zero to be equivalent to the future value of the positive cash flows at the final time period. Spreadsheet applications, such as Microsoft Excel, have inbuilt functions to calculate the MIRR. In Microsoft Excel this function is =MIRR(...).

Example If an investment project is described by the sequence of cash flows:

then the IRR r is given by

NPV = − 1000 + − 4000 ( 1 + r ) 1 + 5000 ( 1 + r ) 2 + 2000 ( 1 + r ) 3 = 0 {\displaystyle {\text{NPV}}=-1000+{\frac {-4000}{(1+r)^{1}}}+{\frac {5000}{(1+r)^{2}}}+{\frac {2000}{(1+r)^{3}}}=0} . In this case, the answer is 25.48% (with this conventional pattern of cash flows, the project has a unique IRR). To calculate the MIRR, we will assume a finance rate of 10% and a reinvestment rate of 12%. First, we calculate the present value of the negative cash flows (discounted at the finance rate):

P V ( negative cash flows, finance rate ) = − 1000 + − 4000 ( 1 + 10 % ) 1 = − 4636.36 {\displaystyle PV({\text{negative cash flows, finance rate}})=-1000+{\frac {-4000}{(1+10\%)^{1}}}=-4636.36} . Second, we calculate the future value of the positive cash flows (reinvested at the reinvestment rate):

F V ( positive cash flows, reinvestment rate ) = 5000 ⋅ ( 1 + 12 % ) 1 + 2000 = 7600 {\displaystyle FV({\text{positive cash flows, reinvestment rate}})=5000\cdot (1+12\%)^{1}+2000=7600} . Third, we find the MIRR:

MIRR = 7600 4636.36 3 − 1 = 17.91 % {\displaystyle {\text{MIRR}}={\sqrt[{3}]{\frac {7600}{4636.36}}}-1=17.91\%} . The calculated MIRR (17.91%) is significantly different from the IRR (25.48%).

Comparing projects of different sizes Like the internal rate of return, the modified internal rate of return is not valid for ranking projects of different sizes, because a larger project with a smaller modified internal rate of return may have a higher net present value. However, there exist variants of the modified internal rate of return which can be used for such comparisons.

References

Worked examples

Example 1 — a first encounter with Modified internal rate of return

Start with the simplest possible case. Write down what Modified internal rate of return claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Modified internal rate of return before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Modified internal rate of return ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Modified internal rate of return

In research: Modified internal rate of return appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Modified internal rate of return in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Modified internal rate of return is common in secondary-school and first-year university syllabi. It links to neighbouring topics Capital budgeting, Investment, Mathematical finance, so understanding it makes those chapters shorter.
In everyday life: Look for Modified internal rate of return outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Modified internal rate of return in 20 minutes

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Frequently asked questions

What is Modified internal rate of return in simple terms?

The modified internal rate of return (MIRR) is a financial measure of an investment's attractiveness. It is used in capital budgeting to rank alternative investments of unequal size.

Why does Modified internal rate of return matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Modified internal rate of return?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Modified internal rate of return.