Morningstar® yet again paints a gloomy picture of the so-called “average investor returns.” The thesis is that these returns are generally lower than mutual fund returns because of investors’ poor timing: they make contributions before market downturns and withdrawals before rebounds, similarly to what they do with ETFs. Morningstar’s annual findings have been subsequently propagated by articles in The New York Times and MarketWatch.

To calculate the investor return, Morningstar takes into account the initial value of the fund’s assets, all inflows and outflows for the fund, and the end asset value, all obtained from the fund’s filings in a given period. This calculation is similar to that of the internal rate of return (IRR).

However, there is one major issue with this methodology: the IRR calculated from the fund’s aggregate cash flows is *not* the same as the average of IRRs realized by all investors in the fund. To determine the latter, Morningstar would have to take a representative sample of investors in the fund, calculate their individual IRRs, and then take an average of those IRRs.

Such a representative sample would need to have at least 20-30 investors, which, if multiplied by over 23,000 funds (all separate share classes), and applied on a regular basis, would be impractical. Hence the convenient shortcut of using only the overall cash flows of the fund and attributing the resulting IRR to a “typical investor.”

This approach not only results in inaccurate figures but also assumes that there exists such a hypothetical average investor whose cash flows into and out of the fund precisely mimicked (in proportion) the composite cash flows of the fund. As Alpholio™ stated in previous posts, it is highly unlikely that such an investor exists.

To illustrate the point, Alpholio™ constructed a simple Microsoft® Excel® simulation of a mutual fund (spreadsheet is available upon request). The simulation spans a period of 12 months and assumes that the fund had 30 investors; in this case, the sample size is the entire population. The simulation has six parameters governing its outcomes (all random variables have normal distributions):

- The amount and standard deviation of the initial investment in the fund. For example, $10,000 and 10%, meaning that 99.7% of the investors initially invested between $7,000 and $13,000 in the fund.
- The annualized return and standard deviation of return of the fund. For example, 10% and 15%, respectively, which models the typical attributes of the S&P 500® index.
- Inflow/outflow base amount and corresponding standard deviation. For example, $10,000 and 10%. This results in random contributions to and withdrawals from the fund each investor would make monthly. (The limit, of course, is that an investor cannot withdraw more than he/she has left in the fund.)

All initial investments are assumed to be made at the end of the month preceding the start of the simulation (e.g. December 31, 2012). All additional contributions to and withdrawals from the fund are assumed to be made on the first day of each month (e.g. from January to December 2013). The return of the fund randomly varies monthly. The spreadsheet can be recalculated by pressing the F9 key, upon which a new series of random scenarios is generated and charted.

The simulation calculates individual IRRs of all 30 investors based on each investor’s random cash flows. It also calculates the overall IRR based on the aggregate cash flows of the fund. The difference between the latter and the former is the IRR error.

It turns out that across many simulation runs, the the average IRR error is relatively small, i.e. around +0.1%. This means that, on average, the fund flow IRR *overestimates* the true average investor IRR by that small amount. However, the standard deviation of the error is relatively large, i.e. about 0.6%. This means that the error is typically distributed in the -1.7% to +1.9% range. This puts into question the accuracy of 10-year “return gaps” cited by Morningstar in the negative 1.66% to 3.14% range for various mutual fund categories.

Here is a sample distribution of the error in 1000 simulations:

While the interpolation line is somewhat jagged, a normal-like distribution shape clearly emerges.

In conclusion, while this simple simulation is by no means perfect or exhaustive, it does demonstrate that the calculation of a “typical investor return” based solely on the composite flows of the fund can be quite inaccurate. Due to the non-linear, iterative nature of the IRR calculation, the IRR of the aggregate cash flows is not the same as the average IRR of individual investor cash flows. Hence, both investors and the media should interpret the “average investor return” figures with caution.