Advocates of active management admit that only a minority of mutual funds will outperform their benchmarks, but they argue there is still a significant probability of success. According to the data Rick Ferri provides in his new book, The Power of Passive Investing (Wiley, 2010), the odds of an actively managed fund beating an index fund are as high as 42% in any given year, and about 30% over five years. Surely by zeroing in on funds with good track records and low costs you can give yourself a decent chance of building an index-beating portfolio. Right?

Ferri takes this argument apart with simple math. The key point here is that almost no one builds a portfolio with a single mutual fund. It’s far more common to select several funds in various asset classes. But here’s the rub: to determine the odds that the *entire portfolio* will beat its benchmark, you need to combine the probabilities that each individual fund will outperform.

Here’s a simplified example. Since the probability of flipping heads is one in two, you’ll flip two heads in a row one time in four (2 × 2), and three in a row just one time in eight (2 × 2 × 2). The same principle is at work when you select mutual funds in a diversified portfolio. What are the chances you will pick multiple winners?

### Not zero — but close

To test this idea, Ferri took the performance data of actual mutual funds and programmed a computer to create thousands of portfolios using three, five, and 10 active funds. Then he compared these with portfolios of index funds in the same asset classes.

He found that over five-year periods, the probability of a five-fund portfolio beating comparable index funds was 16%. Even then, only 5% of these outperformed the index-fund portfolio by more than 0.5%. By contrast, 63% of the portfolios underperformed by more than 0.5%.

When he went out 10 years, the odds were reduced even further. A trivial 0.8% of the portfolios outperformed the index funds by more than 0.5%, while 70% underperformed by at least that much.

Of course, most people have a horizon much longer than 10 years. And as Ferri’s model demonstrates, over a typical investor’s lifetime, the probability that an actively managed mutual fund portfolio will outperform index funds is vanishingly small.

So it’s simply not true to say that actively managed funds have no chance of earning higher returns than index funds over the long term. The probability is not zero — it’s just statistically indistinguishable from zero.

*Coming up next*: I’ll be posting excerpts from an interview I did with Rick Ferri about his book. At the end of the series, I’ll give readers an opportunity to win a copy of The Power of Passive Investing.

Interesting post. However, I believe that you have made an inadvertent error in your use of the multiplication rule of probability. What follows is an example demonstrating the error: consider two funds, F1 and F2, which have respective benchmarks of 5.0% and 5.5%.

Moreover, assume that fund F1 has a probability of 0.6 of returning 4%, and a probability of 0.4 of returning 7%; and fund F2 has a probability of 0.8 of returning 3%, and a probability of 0.2 of returning 7%. Hence F1 has a probability of 0.4 of beating its benchmark, and F2 has a probability of 0.2 of beating its benchmark.

The result of holding the funds F1 and F2 in equal proportions (50% held in one, and 50% held in the other) causes us to have a probability of 0.48 of returning 3.5%, a probability of 0.32 of returning 5%, a probability of 0.12 of returning 5.5%, and a probability of 0.08 of returning 7%. The comparable benchmark in this case is the average of F1’s benchmark and F2’s benchmark, i.e., 5.25%.

Your claim is that the result of holding funds F1 and F2 causes us to have a probability of 0.4 x 0.2 = 0.08 of beating the combined benchmark. This is not the case — the chance of exceeding this value is 0.12 + 0.08 = 0.2.

Note, however, that this example is not entirely realistic, as the expected return of fund F1 is higher than its benchmark — which would never be the case for any mutual fund that I know of, unless they were comparing their fund’s holdings to the wrong benchmark, which is something that we see happen all too often.

Raman is right in the sense that the computations are more involved than just straight multiplication of probabilities, but this distinction is mostly unimportant. The article’s conclusion is correct and the explanation is clearly intended to be accessible to a broad audience rather than pleasing to mathematicians.

The odds of outperforming indexes with actively-managed mutual funds over a lifetime is very small. But not as small as the odds of winning a lottery. However, buying one lottery ticket has a downside of only $2. The downside of losing with actively-managed mutual funds is likely to be hundreds of thousands of dollars.

Nice post.

What a great reminder why my wife and I got out of high-priced mutual funds. Even “the good funds” can only be good for so long.

@Michael – I still play the lottery. Small thrills I guess.

I look forward to an opportunity to win the book, sounds like a good read.

Cheers,

Mark

@Raman and MJ: Many thanks for the explanation. I defer to your superior math skills on this one.

The key issue I want readers to take away is that every time you add an actively managed fund to a portfolio, you decrease the likelihood that the portfolio as a whole will outperform. But of course, unlike a series of coin flips, it is impossible to determine that probability in advance: we cannot know any specific fund’s chances of beating a given benchmark. All we can do is look at the aggregate record of mutual funds in the past and estimate.

I should make it clear, however, that Ferri’s experiment used actual historical returns, not just probabilities. There’s a lengthy explanation of his methodology in the book.

Interesting. Does this similarly mean that splitting one’s portfolio between fewer indices is better? Or not, as the goal there is to meet, not beat, the indices in question?

@Maxwell: No, these findings don’t suggest that it’s futile to diversify your portfolio with several index funds. When we talk about “the market” in this sense, each individual asset class can be considered a different market. Also, the goal of diversifying across several equity asset classes is to reduce volatility, not necessarily to achieve higher returns.

I always wondered how well active funds would do if you were to not include the management fee. Seems to me that by definition, half should outperform. Would it be possible to find a fund that outperforms consistently without including the MER?

@Open source: Absolutely, many active funds would outperform before costs. Most funds that trail the indexes do so by an amount less than their MER. The point isn’t that active managers are idiots. It’s just that whatever skill they have is not sufficient to overcome the cost of executing their strategies.

Though they can offer greater returns in the long haul actively managed funds rarely beat index funds.. Try this screen for dividend-seeking funds with winning long-term records..