## Tuesday, January 28, 2014

### A Portfolio with 3 Different Returns

The Canadian Securities Administrators (CSA) are making changes to reporting rules on client portfolio statements. Among the changes is the requirement to report clients’ yearly returns in dollar-weighted terms (also called the internal rate of return (IRR)). This gives clients a better idea of how their portfolios have performed, but opens the door to the amusing possibility of multiple return values.

For an accessible explanation of the difference between time-weighted and dollar-weighted returns, see Neil Jensen’s excellent article. More mathematically-inclined readers can see Wikipedia’s internal rate of return (IRR) page.

Dollar-weighted returns are a great idea because they give more weight to the returns you get when you have more money invested. After all, the return you get on a few dollars is much less important than the return you get on millions. However, dollar-weighted returns can have some mathematical quirks in rare circumstances. One of those rare quirks is that there can be more than one correct return value.

Here is the most plausible scenario that I could concoct that gives multiple returns:

Start of year portfolio value: \$200,000.

Mar. 1: Portfolio jumps to \$303,000. Client withdraws \$300,000.

May 1: Client withdraws \$2000.

July 1: Client withdraws \$1000.

Nov. 1: Client deposits \$301,000.

End of year: A poor final two months leaves a portfolio value of \$198,000.

If you add up the cash flows into and out of the account, you’ll see that a 0% return fits the data. However, a return of 10% every 2 months also fits the data. Even stranger is that a return of -10% every two months fits the data as well!

Compounding these returns to a full year, the client statement could show -46.9%, 0%, or +77.2%. No one of these returns is any more correct than the others.

On a practical level, this isn’t a big problem. When the return is ambiguous, it means that the cash flows were so unusual that the return isn’t very meaningful anyway. And we could just make up a simple rule about which return to show, or when to show no return at all.