## Wednesday, July 29, 2009

### Interpreting the IRR

I recently stumbled upon an article that I found quite interesting: "What Does an IRR (or Two) Mean?," by David Johnstone (Journal of Economic Education, Winter 2008). David is the National Australia Bank Professor of Finance at the University of Sydney School of Business. I found two things in particular quite insightful.

First, you may be aware that with the IRR we run the risk of having multiple solutions. And although there are techniques to help identify the number of potential solutions, the process is still fraught with challenges. David pointed out that multiple solutions will only occur "when the balance in the investment is at one or more times negative. That is, at some stage in its life, more is taken out than exists in the account." (page 79) I found a second article by Eschenbach, Baker & Whittaker ("Characterizing the Real Roots for P, A, and F with Applications to Environmental Remediation and Home Buying Problems," The Engineering Economist, 2007) which supported this claim. Clearly there are cases when the multiple solutions problem will be an issue, but how frequently will we find an investment portfolio go into the red? Dare I say virtually never?

The second insight I gained from David's piece was his "simple but intuitively meaningful interpretation of the notion of IRR." He uses the following example:
• at time = 0 (the starting point), we begin with\$1,200
• at time = 1 (end of year 1), the client withdrawals \$500
• at time = 2 (end of year 2), the client withdrawals \$850
• at time = 3 (end of year 3), we end with \$500.
The IRR is found to be 25 percent. So, what does this 25% represent? It's what we earn throughout the period. The following table should help convey this:

 t = 1 t = 2 t = 3 Balance at (t-1) 1,200 1,000 400 Period t interest (25.00%) 300 250 100 1,500 1,250 500 Cash flow at t -500 -850 Balance at t 1,000 400 500

We can see that with the IRR, we're able to reconcile our values throughout.

This, of course, is something you can't do with time-weighting. I expect to discuss this at greater length in the August issue of Performance Perspectives.

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