## Statistics, and the Long-Term Return on Your Investments

- retailba_mike
- July 20, 2017

There is this dismal applied science that is called "statistics". It's usually packaged together with "probability". So you take a course called "Probability and Statistics 101". And one of the very first things that you learn is that a statistic is a number whose value is derived from and is characteristic of a distribution of possible outcomes. The distribution defines the probabilities.

Think of the annual return ratios of some fund over the last 10 years. A return ratio for any given year is simply the price of a share of the fund at the end of the year divided by the price at the start of the year (and we might have to add dividends to the numerator to get the total return ratio). The 10 ratios are a distribution of past outcomes and we have a natural inclination to assume that future outcomes will be similarly distributed (to which idea we resort if we lack anything better).

For example, we can define a simple statistic right away: the **median** would be the return ratio that is exceeded by half of the values in the distribution. So we might think that there's a 50% chance that next year's return ratio will be better than the median that we just calculated from the history of the last 10 years. I'd have to say that if all that we know is those return-ratio results of the past 10 years then that is roughly correct.

But, if you buy that fund, what is it *most likely* to be worth in *10* years? That's probably what you *really* want to know. Well you can't get that from the median of the annual return ratios over the past 10 years. You couldn't even get the answer to that particular question from the median of the next 10 annual return ratios if somehow you knew those in advance. Actually, unless the historical data were distributed in a particular way the historical median isn't even a statistic of interest. But others are and we'll get to them. Do read on. Today's entry is about the statistics on fund performance that could be made available to you and how you can use them.

#### SEC Form N-1A

At Item 26(b)1, Calculation of Performance Data, on this Securities and Exchange Commission form we find the instruction that we are to compute the average annual total return T of a fund according to the formula (1+T)^{n}=P_{n}/P_{0} where P_{0} is the starting investment amount and P_{n} is the value of the investment after n years. If the subject is not finance we are inclined to call the 1 + T that is computed that way the **geometric mean**. Here it's the geometric mean of the annual return ratios.

That makes sense if you simply realize that P_{n}/P_{0} = (P_{n}/P_{n-1})(P_{n-1}/P_{n-2})...(P_{2}/P_{1})(P_{1}/P_{0}). The equality here happens because numerators on the right cancel the denominators of the prior term. So the geometric mean, 1 + T , 1 plus the average annual total return, is the n^{th} root of the product of the annual ratios— the geometric mean rather than the arithmetic mean.

The distinction between the geometric and arithmetic means is demonstrated in a tutorial on compounding by Nobel laureate William F. Sharpe. We would get the **arithmetic mean** simply by averaging the annual return ratios, and then subtracting 1. Often the arithmetic mean is simply called the **mean** as it's *the* rule for many people engaged in diverse crafts, with the geometric version being the exception.

OK. So now we have *three* statistics: the median, and the geometric and arithmetic means. But thus far we've only talked about computing them, not about what they mean for the long term. Professor Sharpe does a lot of that in his tutorial, and we'll take our own look at the long-term situation a bit further below... albeit a somewhat breezier one.

#### Other SEC Advisements

Pay close attention to the fund’s 5- and 10-year returns. If the fund’s returns were stellar in the past year but unimpressive in the past five or 10 years (or over the life of the fund, if shorter), it is possible that the past year’s outperformance will not last. On the other hand, if the fund experienced steady returns in the past five or 10 years, but suffered a sharp loss in the past year, it may be unclear as to whether the recent loss signals the beginning of a trend or is an isolated occurrence. Just because a fund had one good year does not mean that positive investment returns will continue.Remember, past performance is not necessarily an indication of future results.

That's some advice to investors from the SEC. And the double emphasis of the last sentence is theirs, not mine, but I couldn't agree more. That's from an online document that has the title "Investor Bulletin: How to Read a Mutual Fund Shareholder Report".

My problem is with the prior sentences. No you can't just find funds that did well more than half the time in the last 10 years and suppose that the likelihood of them doing better than their benchmark in the coming few years is substantially better than that of funds that did worse in prior years. ** The truth is that the likelihoods will be much the same if the average returns of recent years are all that you have to go on.** Yes, picking funds or advisors based on past performance is tricker than it looks. See my related entry that is listed below for further discussion of this.

#### The Form of the Long-Term Distribution

Now we get into some profundities. If you really want to know why we kinda, sorta might know what the probability distribution for 10-year returns on our investment should look like, you have to do some reading. I can recommend the anonymously-authored Wikipedia article on the central limit theorem (and the classical version of it will do). The proof comes later on that page and I also really like the proof.

So, the result is what is called a "lognormal" distribution. Fellow mathematical types can review the theorem, but here we'll just show what the distribution looks like with charts. None of the programs of Retail Backtest rely upon the use of the theorem. But a bit further below we'll talk about what aspects of it can be relied upon.

Here's something mathematical that the theorem tells us: *the median and the geometrical mean of the lognormal distribution turn out to be the same.* That rule doesn't apply to the distribution of historical returns, but if the future annual returns closely resemble random samples of historical returns then the computed median and geometric mean will be approximately equal. So, accepting that, we might as well call the geometric mean the median. We are left with only the median and the mean to consider— with "mean" being the short-form expression for the remaining arithmetic mean. And we'll add to those two the **mode** which is simply the most likely outcome, the one having the highest probability. So we do finish with three statistics: the mode, the median and the (arithmetic) mean. That is the correct order, as we will now see!

#### One-Year

So this is the lognormal distribution for one-year returns, and it's ginned up so as to be roughly consistent with what we expect from the S&P500, with regard to the risk and return levels.

Of particular interest is the spread of the mode, median and mean values for the return ratio. They are respectively about 1.06, 1.08 and 1.09— meaning that the corresponding profits would be 6%, 8% and 9%— as indicated by the red, green and magenta dots. Already the mode, the most likely value, the value at which the probability distribution peaks, is not quite as good looking as the median.

And those X marks on the axis are the 16%-84% return ratio values, meaning that there is a 16% chance that the return ratio will be found to be outside of those marks— higher or lower. Another way to put it is that those are the 16^{th} and 84^{th} percentiles. Those odd percentages— really it's 15.86552539314571... and 100 minus that— correspond to the median return ratio ± one standard deviation from it (when the standard deviation and median are computed using the logarithms of the return ratios). So we see that for any given year the odds are 1so 6 in a hundred that we will have a loss of more than about 5%. (I should perhaps have assumed greater volatility as there are usually substantially worse down years than that in any decade... but this is just an example.)

#### One- and Ten-Year

So now we inquire as to the implications of the lognormal distribution as time goes by. We see that at 10 years the peak probability is at a substantially higher return ratio, but that the distribution has greatly broadened out.

#### Ten-Year

This is the very same lognormal distribution for the 10-year return ratio as in the figure above, but with the horizontal axis stretched out as in the first figure above. Note especially that the spread in the mode, median and mean values now seems much more pronounced, and that's true also of the 16^{th} and 84^{th} percentile values. Whereas the mode was down only 1.6% from the median on the 1-year chart, on this 10-year chart it's down 14.8%.

The mode, median and mean are now respectively 1.80, 2.11 and 2.29 . And the 16^{th} percentile is 33% down from the median on this 10-year chart whereas on the 1-year chart it is only 12% down from the median. Note especially that the median is also a bit less likely to occur than the mode. But there is the saving grace of the 16^{th} percentile being well above 1.0, so that losses are substantially less likely.

#### The Moral of the Story

Well, the first thing to understand is that while it is the case that stock market returns have been found to be approximately lognormally distributed, the lognormal distribution of future returns is a conception that might be viewed as being divorced from reality: If we make an investment in a fund then ten years from now that investment will have only one outcome for us; there will not be a distribution of outcomes. We have to very nearly imagine thousands of parallel universes in which copies of us did better or worse than we did, in order for the very concept of a projected distribution of future outcomes to make some kind of conceptual sense. Here's a numerical "Monte Carlo" approach that produces the same lognormal result as the central limit theorem: if you have ten years of historical data then pick, ten times at random, from a hat, one of the annual return ratios of those ten years (putting it back into the hat each time before randomly picking the next annual return); then compound the ten sampled returns to get your 10-year outcome; then repeat the process many, many times so as to compile a distribution of 10-year outcomes. "Whacky", you may say.

Well, the lognormal distribution of future returns is formally referred to as "ex ante", meaning "before the event". The distribution of actual historical returns is the "ex post" distribution, compiled "after the event". Certainly we should take the lognormal distribution of future returns with a grain of salt. But if we know nothing else but the historical data, nothing about current events at any time, then we could say that the lognormal distribution that is derived from historical return ratios is the best that we can do. The question then is what good is it? Well, the way in which the various statistics, the ones that are computed and discussed above, increasingly spread out with time within the lognormal distribution is probably realistic. We should have that expectation of increasingly wider ranges of outcomes with time, and to that degree.

In particular, recalling now that the SEC requirement is that fund returns are to be stated based on the geometric mean of annual return ratios, which corresponds to our lognormal median, it's significant that the most probable outcome (the *mode* of our lognormal distribution, not the median) lags substantially below the median return.