This document was presented during the 2016 NCREIF Winter Conference.
The author(s) take full responsibility for all content. This posting is for informational purposes only; neither NCREIF nor its Board express any opinion of the content presented herein.
Money-Weighted Rates of Return That Are Better/More Flexible Than IRR
Dean Altshuler
1. I LOVE IRR! It is mathematically fascinating, but…
2. IRR is a MWR that is NOT an average rate of return, but one that is artificially forced to be a constant rate
of return every cash flow period.
3. It is easy to show that, in forcing it to be constant, it becomes a ROR on a denominator that is therefore
mathematically concocted – to represent the sum of valuations along the way, without regard to what
the market does each period, i.e., what the valuations of the project might have been along the way.
4. By the way, it is not at all infrequent that one or more of those valuations implied by IRR are negative, but
the IRR user is blind to that!
a. That said, a scholar named Hazen proved that only the sign of the sum of the interim valuations
matters. If the sum is negative, then it looks like a shorted investment and a lower IRR is preferable to a
higher IRR!
i. Cash flows with multiple IRR solutions typically have summed valuations that are both
negative and positive. Hazen proved that each results in the same value added compared to
any chosen cost of capital.
This document was presented during the 2016 NCREIF Winter Conference.
The author(s) take full responsibility for all content. This posting is for informational purposes only; neither NCREIF nor its Board express any opinion of the content presented herein.
5. IRR is always wrong, because it is arbitrary. It ignores the valuation information TWR embraces.
6. However, if you truly have no good guess at interim valuations, then IRR is a useful metric.
7. Risk analysis is out with IRR! MWR attribution doesn’t really work.
a. Do we care about early period rates of return forgone, e.g., a great asset that later turned south,
resulting in a mediocre IRR? IRR masks that.
8. Like other MWR (and TWR) metrics, IRR has a numerator and denominator. Its denominator is the sum of
the BOP valuations that it concocts. Its numerator is merely the “whole dollar profit”, which is just the sum
of the cash flows - see Exhibit 1.
9. New MWR metrics allow you to use known (stock portfolio) or carefully(?)- estimated values of the
investment, instead of the concocted denominator, so as to compute a true “average rate of return”.
a. These metrics are simple “closed-form” algebraic expressions that don’t require a computer to
iterate and always produce a single solution!
10. Imagine a new MWR where the numerator, like IRR, is still the whole dollar profit, but the denominator can be
the sum of the actual BOP valuations you believe (eg real estate) or know (eg publicly traded stocks) to be good.
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This document was presented during the 2016 NCREIF Winter Conference.
The author(s) take full responsibility for all content. This posting is for informational purposes only; neither NCREIF nor its Board express any opinion of the content presented herein.
11. Actually, one of Professor Carlo Alberto Magni’s metrics, AROI, has a flexible denominator that allows you
to use whatever denominator you think is meaningful, not necessarily limited to just BOP values.
a. Analogous to the composites we use in TWR, one can show that adding together both period
numerators and denominators is equivalent to computing a weighted average rate of return where
the denominators are the weights. So AROI is a weighted-average rate of return.
b. What MWR would you get if, instead of using BOP values, you used the Modified Dietz denominators
in AROI to arguably better reflect the average denominators the numerator was earned on? TMWR
(shout out to Joe D)!
12. See Exhibits 1 and 2.
13. IRR is what mathematicians call a “special case” of the AROI in that if, somehow, the rates of return of
some investment, someday, truly were constant, AROI would produce the same result as IRR. 1
1
IRR has other problems, too, such as that it retroactively changes past period valuations as you add more periods to it, so as to
iterate to a new constant rate of return. There are other issues too, which are outside the scope of this presentation. One that
private equity folks often notice is that, when IRR starts off really high, you find that the rates of return earned in later periods don’t
move the needle nearly as much as you would expect. So, even though the longer track record may seem more statistically
significant (for a manager with control of cash flows, or an investment), IRR does a poor job with longer time frames. "Once a good
manager, always a good manager". if initial rates of return are negative, late rates of return move the needle excessively.
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This document was presented during the 2016 NCREIF Winter Conference.
The author(s) take full responsibility for all content. This posting is for informational purposes only; neither NCREIF nor its Board express any opinion of the content presented herein.
Exhibit 1. Implied Invested Capital That The IRR Is Earned On
End of
Year
Cash Flows
($30.00)
($30.00)
$20.00
$20.00
$45.00
Year
2011
2012
2013
2014
2015
Whole Dollar
Profit
$25.00
IRR as a NPV = 0 Solution:
IRR's Implied
Beginning Of
Year ("BOY")
Project Value
Rate of
Return
IRR's Implied
End of
Year ("EOY")
Project Value
$30.00
$64.03
$52.62
$39.68
13.42%
13.42%
13.42%
13.42%
$34.03
$72.62
$59.68
$45.00
Year End
IRR's Implied
Amount Added to
End of Year
(subtracted from)
Project Value
Project
- Net of Cash Flows
$30.00
$30.00
$30.00
$64.03
($20.00)
$52.62
($20.00)
$39.68
($45.00)
$0.00
Total Implied
Capital (BOY)
$186.32
13.42%
IRR as a ratio = Whole Dollar Profit/Total Implied Capital (BOY) = $25.00 / $186.32 = 13.42%
Exhibit 2. Actual Invested Capital That The AROI Is Earned On
Year
2011
2012
2013
2014
2015
End of
Year
Cash Flows
($30.00)
($30.00)
$20.00
$20.00
$45.00
Whole Dollar
Profit
$25.00
True
Beginning Of
Year ("BOY")
Project Value
Rate of
Return
True
End of
Year ("EOY")
Project Value
$30.00
$61.50
$45.00
$32.00
5.00%
5.69%
15.56%
40.63%
$31.50
$65.00
$52.00
$45.00
Year End
True
Amount Added to
End of Year
(subtracted from)
Project Value
Project
- Net of Cash Flows
$30.00
$30.00
$30.00
$61.50
($20.00)
$45.00
($20.00)
$32.00
($45.00)
$0.00
Total Invested
Capital (BOY)
$168.50
AROI = Whole Dollar Profit/Total True Capital (BOY) = $25.00 / $168.50 = 14.84%