internal rate of return

[David G. Hough at validgh]

The recent discussion of how to compute the internal rate of return overlooked
the most important point about it, namely whether it's worth computing.

Computing internal rate of return is a fascinating mathematical problem,
amounting to solving a polynomial that can be quite difficult numerically.
For instance, recent
HP business calculators go to a lot of trouble to provide IRR solutions.
The problem fascinates the mathematically inclined even when none of the
real solutions make much sense financially.

When comparing two investments involving more than one payment in or more
than one payment out, the question that's interesting is how much
money do you have to have up front, and how much you'll have at the end.
You may not have to pay in all the money up front, but you have to keep it
in a safe investment until it's needed. So an interesting parameter is
what rate of interest is available in a safe investment until all the payments
in are accomplished.

And if there is more than one payment out, then the question is what interest
rate can you get on reinvesting that payment until the final payment is made.
That rate might be higher than the safe rate mentioned above.

IRR is based on the supposition that that the safe investment rate and the
reinvestment rate are the same as the internal rate of return of the
investment, which is ridiculous, once you think about it. A simpler
analysis would be:

1) Discount all the payments in back to the first payment using the safe
investment rate.

2) Discount all the payments out forward to the last payment using the
reinvestment rate.

3) Calculate the rate of return as a simple compound interest problem from
the one net discounted payment in and the one net discounted payment out.
For this simple case, rate of return is easy to compute and easy to understand.

This rate of return based on one payment in and one payment out
is easier to compute than the usual IRR and focuses the analysis on the real
hard part, financially speaking, which is predicting the safe interest rate
and the reinvestment interest rate over the lifetime of the investment, not
to mention the uncertainties in those payments out. For complicated series
of payments in and out, IRR puts the attention elsewhere -
an old trick of magicians. Interval arithmetic could be used to bound
the effect of uncertainties in the payments out, but with most complicated
investments, any realistic bound on the certainty of the payments out would
disclose potentially negative rates of return.

If Roy Martin were here, he'd explain this better, as he explained it to me
long ago.

--

David G Hough validdgh@validghdotcom
Consultant on system correctness, performance evaluation, and
IEEE 754 binary floating-point arithmetic --- http://wwwdotvalidghdotcom

[bsh...@efn.org]

> The recent discussion of how to compute the internal rate of return overlooked
> the most important point about it, namely whether it's worth computing.

[snip]

> When comparing two investments involving more than one payment in or more
> than one payment out, the question that's interesting is how much
> money do you have to have up front, and how much you'll have at the end.
> You may not have to pay in all the money up front, but you have to keep it
> in a safe investment until it's needed. So an interesting parameter is
> what rate of interest is available in a safe investment until all the payments
> in are accomplished.
>
> And if there is more than one payment out, then the question is what interest
> rate can you get on reinvesting that payment until the final payment is made.
> That rate might be higher than the safe rate mentioned above.
>
> IRR is based on the supposition that that the safe investment rate and the
> reinvestment rate are the same as the internal rate of return of the
> investment, which is ridiculous, once you think about it. A simpler
> analysis would be:
>
> 1) Discount all the payments in back to the first payment using the safe
> investment rate.

So...

first paymentIn + SUM{ (paymentIn)/(1+sr)^t = A

Where sr = safe rate, and t is time from first payment to the one
being discounted? This is the same as finding the sum of the
present values of the payments when t=0, right?

> 2) Discount all the payments out forward to the last payment using the
> reinvestment rate.

So...

last paymentOut + SUM{ paymentOut*(1+rr)^t = B

Where rr = reinvestment rate, and t is time from any given
payment out to final payment out. Rr is the same as the yield,
for a stock, right? But, what if the rr varies? This is a sum of
the future values of the payments, right?

> 3) Calculate the rate of return as a simple compound interest problem from
> the one net discounted payment in and the one net discounted payment out.
> For this simple case, rate of return is easy to compute and easy to understand.

So...

A(1 + r)^t = B

Defines r, the rate of return is this scheme. t is the time from
the first payment in to the last payment out. Does r have to be
solved by guessing?

>
> This rate of return based on one payment in and one payment out
> is easier to compute than the usual IRR and focuses the analysis on the real
> hard part, financially speaking, which is predicting the safe interest rate
> and the reinvestment interest rate over the lifetime of the investment, not
> to mention the uncertainties in those payments out. For complicated series
> of payments in and out, IRR puts the attention elsewhere -
> an old trick of magicians. Interval arithmetic could be used to bound
> the effect of uncertainties in the payments out, but with most complicated
> investments, any realistic bound on the certainty of the payments out would
> disclose potentially negative rates of return.
>
> If Roy Martin were here, he'd explain this better, as he explained it to me
> long ago.

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