The Modified Internal Rate of Return, often just called the MIRR, is a powerful and frequently used investment performance indicator. Yet, it’s commonly misunderstood by many finance and commercial real estate professionals. In this post we’ll take a deep dive into the concept of the MIRR. We’ll define the MIRR, look at the logic and intuition behind the MIRR, dispel some common mistakes and misconceptions, and finally we’ll tie it all together with a relevant example.

## What is MIRR?

First of all, what is the definition of MIRR? The **Modified Internal Rate of Return (MIRR)** is a variation of the traditional Internal Rate of Return (IRR) calculation in that it computes IRR with explicit reinvestment rate and finance rate assumptions. The MIRR accounts for the reinvestment of any positive interim cash flows by using a reinvestment rate, and it also accounts for any negative cash flows by using a finance rate (also known as a safe rate).

The reason why these two rates are used is because it allows for any positive cash flows thrown off by an investment over the holding period to be reinvested at the “reinvestment rate”. It also allows any negative cash flows to be discounted back to the present time at the “finance rate” to determine how much needs to be set aside today in order to fund the future cash outflows.

By using this approach, the MIRR boils a set of cash flows down to just two numbers: 1) a single initial investment amount at the present time and 2) a total accumulated capital amount at the end of the holding period. Then, a single rate of return can be calculated using only these two numbers, which results in what’s known as the MIRR.

## MIRR Example

Let’s take an example of the modified internal rate of return to see how this works. Suppose we have the following set of cash flows:

We invest $100,000 today and in return we receive $18,000 per year for 5 years, plus at the end of year 5 we sell the asset and get back $100,000. If we use the traditional Internal Rate of Return (IRR) calculation, we get an IRR of 18%.

As you may recall, one of the problems with the traditional IRR calculation is that it doesn’t account for the reinvestment of interim cash flows. So, how can we use the Modified Internal Rate of Return to eliminate this problem?

First, let’s explicitly define a reinvestment rate for all of the $18,000 interim cash flows. In order to account for the yield we can earn on these interim cash flows, let’s assume we can reinvest them at a rate of 10%. Note that this rate is lower than the above calculated IRR. This could be for a variety of reasons. For example, it could be the case that we can’t find any other investments that yield higher than 10%.

As shown above, we simply take each of our interim cash flows of $18,000 and then compound them forward at a rate of 10% to the end of year 5. When we add up all of our cash flows at the end of year 5 we get a total of $209,892. By doing this we have transformed our initial set of cash flows into a different time value of money problem, which takes into account the yield we earn on interim cash flows that are reinvested elsewhere. Now we can simply take our new set of cash flows and solve for the IRR, which in this case is actually the MIRR since it’s based on our modified set of cash flows.

As you can see, the MIRR when using a 10% reinvestment rate is 15.98%. This is less than the 18% IRR we initially calculated above. Intuitively, it’s lower than our original IRR because we are reinvesting the interim cash flows at a rate lower than 18%. Also take note again that the MIRR calculation here is simply the IRR calculation. The only difference is that now we’ve transformed our initial set of cash flows into a new, modified, set of cash flows. That means that when we now calculate the IRR it’s a *modified* IRR.

## MIRR Example With Negative Cash Flows

Using the reinvestment rate on positive interim cash flows like we did above is how MIRR is commonly used, but sometimes there is more than one negative cash outflow during the holding period. Consider the following set of cash flows:

We have the same initial $100,000 upfront investment, but now we also have to come out of pocket $50,000 in year 2 for a capital expenditure. However, once this improvement is realized, our cash flows increases from $18,000 to $25,000 and now we can also sell the property at the end of year 5 for a higher price. This results in a higher IRR of 19.33%, but what does it do to our MIRR?

Let’s first tackle the positive interim cash flows by compounding them forward to the end of year 5.

This is the same process we followed in our first MIRR example, but now we simply ignore the negative cash outflow in year 2. This leaves us with a -$100,000 initial investment, a -$50,000 cash outflow in year 2, and a $309,104 cash inflow at the end of year 5. Next, let’s discount our -$50,000 outflow back to the present time at our finance rate or safe rate.

This simply tells us that if we want to have $50,000 available to spend in 2 years, then we need to set aside $45,351 today in an account earning 5% annually. So, now we’ve transformed our original set of cash flows into a new modified set of cash flows that has just two figures: a $145,351 initial investment and a $309,104 accumulated capital amount at the end of the holding period.

Now we can simply calculate an IRR on the above modified set of cash flows to get a Modified Internal Rate of Return of 16.29%. This modified internal rate of return now accounts for the funds we need to set aside today at a safe rate in order to fund future capital outlays. It also accounts for the reinvestment of all interim cash flows at our expected reinvestment rate.

## How MIRR Solves the Multiple IRR Problem

You may recall that one of the problems with the traditional IRR calculation is that there are as many solutions to IRR as there are sign changes in a set of cash flows. Let’s take a look at an example set of cash flows:

When you run an IRR calculation on the above set of cash flows you indeed get multiple solutions. For the above set of cash flows we get 3 different IRR solutions: 0%, 100%, and 200%. So, which one is correct? The answer is that all of them are correct! Why is this? The short answer is that the IRR formula is not a linear equation but instead it’s a polynomial which can generate multiple solutions. This is also the reason why the IRR function in Excel asks for a “guess” as an input. This is used to help Excel determine which solution is correct in case there are multiple solutions.

The good news is that the MIRR eliminates this well-known problem with IRR. To see how, let’s run the MIRR on the above set of cash flows using the same procedure we followed above. We’ll skip the interim steps of discounting negative cash flows at the safe rate and compounding interim positive cash flows at the reinvestment rate. However, this process is exactly the same as we followed above and it leaves us with the following modified set of cash flows:

And now when we calculate an IRR on this modified set of cash flows we get 6.50%. Using the modified internal rate of return eliminates the multiple IRR problem because we are explicitly defining our safe rate and reinvestment rate. This boils the set of cash flows down to just two figures, resulting in a single MIRR figure.

## Conclusion

In this article we discussed the logic and intuition behind the modified internal rate of return, or simply the MIRR. The MIRR is a powerful investment metric that is gaining in popularity since it eliminates the problems with the traditional IRR calculation and also provides a more realistic measure of return. In this article we broke down the MIRR calculation step by step to make understanding the mechanics of MIRR easy to understand.

This is a nice summary of the logic issues with complex cash flows and how MIRR can solve them. It can also be calculated very easily on a financial calculator.

Thanks Tom

This is the best definition of MIRR ever , Thank a lot

Similar to your IRR example, can you show the “underneath the hood” math for the MIRR calculation? In particular, I was hoping you could break this down into the “Return on Capital” and “Return of Capital” format you used earlier in your IRR example.

Hi, The ROC and ROIC will not be recovered from the investment cash flow (NCF) if MIRR is used. Only at IRR both ROC an ROIC can be recovered. MIRR produce boundless solution and not based on NCF but external income (not generated by the investment)! MIRR is a dubious exercise.

Rob:

First, reinvestment assumption is invalid and therefore MIRR is redundant when there is no reinvestment.

Second, MIRR is dubious. See your example of multiple IRR cash flow. The cumulative NCF is 0 i.e. the total income minus total expenditure without discounting is zero, no investor will waste their time to estimate the IRR!!

As I suggested earlier, I have discussed these issues in my papers:

1. “A New Method to Estimate NPV from the Capital Amortization Schedule and an Insight into Why NPV is Not the Appropriate Criterion for Capital Investment Decision”: http://ssrn.com/abstract=2899648

2. IRR Performs Better than NPV: A Critical Analysis of Cases of Multiple IRR and Mutually Exclusive and Independent Investment Projects: https://ssrn.com/abstract=2913905

3. The Controversial Reinvestment Assumption in IRR and NPV Estimates: New Evidence Against Reinvestment Assumption (February 16, 2017): https://ssrn.com/abstract=2918744

Dr Kannapiran Arjunan

You argument about MIRR being redundant rests on the premise that you don’t use the components of MIRR that make it useful in the first place. If you remove the assumptions that make the MIRR useful (reinvestment rate and safe rate), then yes, it’s redundant. But the purpose of using the MIRR is so you can actually make assumptions about reinvestment and safe rates.

Your point about the sum of the cash flows being 0 is interesting. But how do you take into account the time value of money? A simple summation will not do this. If you have a real safe rate and a real reinvestment rate, then the TIMING of the cash flows must be taken into account, as well as the returns that could be earned over that time. MIRR accomplishes this.

As an example, when using the MIRR on the set of cash flows (that admittedly has a net sum of 0), we still end up with a 2.3% return, as shown in the above article. Why? Because the timing matters. In other words, even if the sum of cash flows is 0, you can still end up with a positive return due to your assumptions about the reinvestment rate and safe rate.

Thanks Rob:

Really appreciate your simple way of explaining the concept that on the downside easily expose the weakeness.

1. In your case MIRR is 16.3% and so also IRR. MIRR collapsed to IRR because there is no intermediate income to be reinvested. There is only end of project income (yr 5).

Pls refer to the Excel MIRR function help sample. Here is the results:

MIRR 11.8% 12.6% 13.1% 13.9% 16.1% 30.1%

IRR 13.1% 13.1% 13.1% 13.1% 13.1% 13.1%

4,087 1,409 0 (2,478) (8,615) (37,522)

Logical Error: When MIRR is higher than IRR, NPV is negative (Red) above IRR rate . Only at IRR (13.1), NPV = 0. If you increase the reinvestment rate the MIRR is increasing limitlessly as if NCF will support that MIRR. This details are in my forthcoming paper.

2. In your multiple IRR case, the PV of cash flow should be less than PV of the Capital cost. Then where from the MIRR is 2.3% ?? Its dubious also this happened because by discounting the 2 yr neg CF the total negative flow is 117833 instead of 120000!

Let us keep talking. Cheers Kannan

Excellent

Hi Rob:

Are you sure the MIRR is 2.3% for the last NNCF data? When I use financing rate of 5% and reinvesrment rate 10%, the MIRR is 6.5% and when used 10% and 10%, the MIRR is 9.5%.

I have included your data then ends with a MIRR 2.3% in my recent paper entitled:

A Resolution to the Problem of Multiple IRR: A Modified Capital Amortization Schedule (MCAS) Method for Non-Normal Cash flow (NNCF) to Obtain a Unique IRR (July 11, 2017). Available at this link: SSRN: https://ssrn.com/abstract=

The comments that relates to the MIRR example discussed here is furnished below that formed part of my paper.

“Many authors suggested to use MIRR to overcome the problem of multiple IRR. Using the same NNCF data, both Kyd (2017) and Schmidt (2015), showed that the data led to multiple IRRs (0%, 100% and 200%). They concluded that all the three IRRs are right but this conclusion needs solid evidence. According to these authors, the problem of multiple IRR could be eliminated by MIRR. Kyd used a finance rate (FR) of 8% and a reinvestment rate (RR) of 3% and estimated a MIRR of 5.8%. For the same data, Schmidt used FR of 10% and RR of 5% and estimated a MIRR of 2.3% (the correct MIRR must be 7.7%).

For the given NNCF, there are multiple IRRs (three IRRs: 0%, 100% and 200%) under the CAS and DCF methods. For all the three IRRs, the NPV is zero and therefore the result is consistent with the mathematical relationships. However, the NPV at 10 % (assumed hurdle rate) is negative (-1284) and that being the case the IRR 100% and 200% are spurious or dubious. With 0% or 100% or 200% IRRs, the ROIC is zero as per CAS. There are positive ROICs of $40000 and $60000 and matching negative ROICs at 100% and 200%, respectively. This is obviously caused by the opening balances are positives that leads to interest to be paid by the investor to achieve that false IRR of 100% and 200%.

The net benefit stream or income is not adequate to support each one of the MIRRs and therefore the MIRRs are all over estimate. The ROIC at that MIRRs are just left unrecovered as closing balance (see last column of Table 3). The claim by Kyd (2017) and Schmidt (2015) that the MIRR solves the problem of multiple IRR is doubtful. MIRRs are unique return but not the real return or feasible return.

The three different IRR solutions: 0%, 100%, and 200% may be the roots of the polynomial (NPV equation). Then the zero return is the real return and others are not real. The MCAS estimated IRR is 0% which is the real return and is consistent with NCF and NPV. The real return must be zero as indicated by the ROIC component for all other returns (IRRs and MIRRS) are not recovered and remain as closing balance. For example, at MIRR 5.8% the ROIC of $4119 is not recoverable from the income stream and remains as closing balance. The MCAS eliminated the reinvestments and that is reason for the NPV at 10% with MCAS is less than the NPV at 10% with CAS and DCF method. MIRR estimate, by including a reinvestment rate, is not consistent with NCF.

In summary, the real IRR is 0% estimated by the MCAS method. All other IRRs and MIRRs are not real returns. MIRR could never eliminate the real problem of multiple IRR. Next, the discussion will be about the AIRR and GIRR that are proposed based on strong mathematical analysis of the properties of the polynomials (NPV equation).

Which metric is better if you didn’t reinvest the returns? Say if you we’re calculate a real estate investment with uneven cash flows. Do you just use 0 both for the reinvestment rate and finance rate for MIRR? As you can tell, I’m not a finance person. Any help is greatly appreciated!

why the discount rate on how to find mirr on a negative value changes to 5%

anyone to help ASAP, i have an exam today