At the end of Part 1 of this article we had just reached what appeared to be an epiphany of sorts. We looked first at Discounted Cash Flow, Net Present Value and Profitability Index, and found them almost but not quite up to our needs as real estate investors. We then turned Discounted Cash Flow on its head, solving for the rate rather than the Present Value. That rate – the Internal Rate of Return – looked like it provided a good measure of investment return and an excellent way to compare alternative investments because it was sensitive to the interplay between the timing and the magnitude of our investment’s cash flows.

That’s when I issued a “Not so fast” admonition, with the unabashed purpose of luring you back for the second part of this discourse.

All right – what are the problems with IRR and how do we deal with them?

The Problems with IRR

Internal Rate of Return works well in many situations, but in not quite all. In the typical investment, you expect to have a single negative cash flow on day one (your cash outlay to acquire the investment) followed by a series of periodic positive cash flows. The last of these will be the proceeds of sale when you finally dispose of the investment. In such a scenario, IRR usually works pretty well.

The wheels can start to come off, however, when you use IRR to help you choose among alternative investments. Particularly vexing is a situation where your investment timeline expects to encounter some negative cash flows. Perhaps you’re projecting a significant increase in the interest rate on your financing; or you expect to have some major (but unfunded) repairs; or you want to play “what if?” to see what will happen if you lose an important tenant and seek to replace that tenant quickly. Any of these possibilities could throw your projected cash flow for a future year into the negative.

That’s where the arcane math behind IRR throws you a curve. In general, if you have more than one change of sign in the series of cash flows (and you must include the initial investment as one of the cash flows), then you may encounter “non-unique” results. That’s a polite way of saying the same of facts can give you more than one answer, which clearly is not helpful.

Consider this example from the classic text, Mastering Investment Real Estate (Messner, Schreiber, Lyon and Ward):

In this series of cash flow, the sign changes three times; therefore, there could be as many as three different internal rates of return, i.e., rates at which you could discount these cash flows so that their NPV would equal zero. Indeed, there are three such rates: 0%, 100% and 200%, and they’re all mathematically correct.

The IRR is of little value if it presents you with multiple solutions for the same set of data and invites you to pick one of those solutions.

If IRR’s relationship with negative cash flows is occasionally dysfunctional, it doesn’t get along as well as it should with positive cash flows either. Conventional wisdom has long held that IRR assumes that positive cash flows can be reinvested, until the end of the holding period, at the same rate as the IRR itself. There are also those who assert that IRR actually makes no assumption at all as to the rate of reinvestment of positive cash flows.

For our purposes the distinction may be literally academic because in either case the IRR does not attend to how positive cash flows are handled in the real world. You will reinvest positive cash flows at the best rate you can reasonable obtain, and that rate is likely to be closely tied to the size of the cash flow. If your cash flow is large, you may be able to reinvest it in another piece of real estate. If it is small, then passbook savings may be your only option.

FMRR

You can use a modified version of IRR to deal with the problems of non-unique results and the reinvestment of positive cash flows. Back in the ‘70s (the 1970s, that is), when I was a young investor in bell bottoms, the technique was called Financial Management Rate of Return (FMRR). This technique eliminated negatives by first discounting them back at the safe rate to the nearest previous positive cash flow, adding that discounted negative amount to the positive cash. If there were any negatives left, those would be discounted back to day one, also at the safe rate, and added to the initial investment. The procedure would then compound the remaining positive cash flows forward to the end of the holding period at a rate that was realistic for those cash flows. It was up to you, the analyst, of course to specify the safe and reinvestment rates.

This process would leave you with a string of cash flows on which you could perform a proper IRR, a series that included one initial outflow – a negative amount – followed by all positive or zero cash flows.

For example, say that you found these among your series of cash flows;

If your safe rate were 4%, you would discount the (20,000) Year 4 cash flow back one year at that rate. The result would be (19,231). Now in Year 3 you can combine the positive 30,000 with the negative (19,231) and at the same time eliminate the negative cash flow in Year 4.

MIRR

A some point, perhaps in the mid-‘80s, I observed that most investors and brokers were using a variation of this variation on IRR called Modified Internal Rate of Return, or MIRR. I can only speculate as to what caused this shift, but I have a theory. Microsoft published its Excel spreadsheet software with MIRR as a built-in function. MIRR is perhaps slightly less precise than FMRR, but I suspect that it demands less computing power to calculate. The difference with MIRR is that it discounts all negative cash flows to day one rather than trying to mix and match individual negatives with offsetting positives. The difference between it and FMRR is typically inconsequential.

Consider these cash flows, once again based on an example in the text by Messner et al.:

If you use a safe rate of 5% to discount negative cash flows, a reinvestment rate of 10% for positive cash flows, and perform the admittedly tedious task of figuring the FMRR, you will find that your FMRR equals 19.4%. Use Excel’s MIRR function with the same safe/reinvestment choices, and the result is 18.0%. If you believe this difference justifies the additional time and effort to calculate the FMRR, you may want to try switching to decaf.

It’s worth noting however that if you were to use Excel’s IRR function on these cash flows, using a “guess” rate of 20% to narrow the field of possible answers, you would get an IRR of 25.2% for these same cash flows. Clearly, the difference in this example between IRR and MIRR is quite meaningful. The MIRR yields a more conservative and probably more realistic measurement.

While MIRR addresses the chief deficiencies of IRR as a measure of return, it still comes up a bit short when you want to compare mutually exclusive alternative investments. The problem here is in accounting for both the duration and scale of your investment. Using MIRR to compare opportunities that require the same initial investment and will be held for the same length of time seems reasonable enough. What if the alternatives require different amounts of cash, or presume different holding periods?

Let’s say that you want to decide between two properties, one requiring a cash investment of $100,000, the other $60,000. Clearly, you must really have $100,000 in hand if you’re considering both options. To make an “apples-to-apples” comparison, you should invest $100,000 no matter which property you choose. When considering the property that requires only $60k, your analysis should involve both the return you expect to receive from the property plus the return you expect from the $40k cash that you were free to invest elsewhere.

Likewise, if you project that you will hold one property for 4 years but the other for 5, you should look at the after-tax proceeds from the 4-year property and take into account the return you could earn with those proceeds if you invested them elsewhere for one more year.

CpA

A Capital Accumulation (CpA) comparison is one way you might address these issues. This method allows you to compare investment alternatives not with a rate-of-return measurement but rather in terms of accumulated dollars, even if those dollars don’t remain in the property investment.

Consider two mutually exclusive investment opportunities, Property A and Property B, with the following cash flows:

To perform a Capital Accumulation comparison, you begin by eliminating the negative cash flows as you saw in the MIRR example above. This time you choose a safe rate of 4%.

Next, you’ll compound all positive cash flows to the end of your holding period, Year 5. You decide on a reinvestment rate of 9%.

You have one last consideration to deal with: Clearly, you must be starting out with $100,000 cash in your pocket or you wouldn’t be considering the purchase of Property A. If you decide to buy Property B, then you’ll have $40,000 left over to work with. To make a true comparison of these alternatives, you need to commit the same amount with either choice. You do that by assuming that you’ll invest the remaining $40k for the 5-year holding period at your reinvestment rate of 9%.

Your final series of cash flows, after discount negatives and compounding positives, looks like this:

Let’s have an instant replay of what you did here, first using just Property A. If some of this is starting to sound familiar, you’ve probably noticed that this process of computing CpA looks a great deal like what you saw for FMRR, above, except instead of looking for a rate of return you’re looking for a total accumulation of cash.

In the example you first needed to eliminate the negative cash flow of $15,000 that occurred in year 3. You did that by discounting it back at 4% per year until its discounted amount could be absorbed by an earlier positive cash flow. In this case you had to go back only one year. Negative 15,000 discounted at 4% for one year is negative 14,423, which could be absorbed by year 2’s positive 15,000, leaving a net for year 3 of the difference, 577.

(If you had some amount of negative cash flow that you could not absorb into positive cash flows, you would discount the negative cash back to Year 0 and add it to initial investment amount, again as described in FMRR.)

Next you compounded each of Property A’s positive cash flow forward to year 5 at the reinvestment rate of 9%. Why five years, when you expect to hold the property for just four? Because you’re comparing Property A to an alternative (and mutually exclusive) investment that you would hold for five years. So to keep the alternatives in balance you needed to keep your EOY 4 money in play (at the reinvestment rate) for the same amount of time as in Property B.

When you were done, your entire accumulation was a combined negative 100,000 and positive 281,287, or 181,287.

Property B worked the same way, but required one additional step: Equalizing your cash commitment by investing elsewhere the $40,000 that was not required to purchase Property B. By investing that $40k, you were able to compare two investment scenarios, each requiring $100k of cash.

As you can see, the total capital accumulation for Property A ($181,287) is greater than that for Property B ($163,815), so Property A is your preferred choice. Interestingly enough, if you were to do an IRR on the original cash flows, you would find that Property B, with 33.7% is greater than A’s 30.0%, while the B’s MIRR of 25.2% wins by a nose over A’s 24.8%.

Why might that be so? The answer should lie in one or both of the “equalizing” factors: holding period or initial investment. Perhaps in this example $40,000 is working harder for you invested in Property A, than it is invested independently if you chose Property B.

As promised in Part 1 of this article, we’ve really stirred the alphabet soup – NPV, IRR, FMRR, MIRR and CpA. Is CpA a silver bullet? Indeed is any one of these the only measure you’ll ever need?

Probably not. They all require judgments on your part – about cash flows, resale proceeds, and sometimes discount rate, safe rate or reinvestment rate. In regard to any specific deal you may be more confident about some of these judgments than you are about others, and that confidence – or lack of it – can influence how much you want to rely on any particular metric. The smart investor will understand them all, consider them all, and use the one (or several) that seems best suited to the actual investment decision at hand. After all, you wouldn’t keep just one kind of screwdriver or one size wrench in your basement toolbox. Do no less with your investments.

Need to calculate Capital Accumulation Comparison (CpA)? Use the Comparison Add-on for Real Estate Investment Analysis, Standard Edition.

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