Analyzing Investment Returns

The arithmetic mean and standard deviation are the first and most simple of the basic statistical concepts used in investing. The mean and standard deviation of investment returns provide the basic profile of any security with respect to risk and return.

The mean. in statistics, is merely the arithmetic average. For example, if you have average returns for 10 years, you simply add them together and divide by 10 to find the mean. (See the general form of the calculation and an example below.) The term mean usually refers to the arithmetic mean but there’s also a geometric mean. which will be discussed in the section on comparing mutual funds with key statistics (Compare Funds on the menu bar).

The mean is used as an estimate of expected future returns of mutual funds because it’s the best estimate we can make solely from past data. Analysts have many resources at their disposal to help them project the future returns of securities and they have the time required to put those resources to work. However, most of us don’t have the time and resources to do detailed securities analysis and, as the focus of this site is mutual fund investing, it’s really not relevant to us. For all intents and purposes, predicting future mutual fund performance from historical data is appropriate as long as any trends are noted and taken into consideration. So we use the mean to project the future and always deliver the obligatory caveat that past returns are not necessarily indicative of future returns.

The standard deviation is a measure of variability which is used as the standard measure of the total risk of individual assets and portfolios of assets. There are two variants of standard deviation: population and sample. The sample standard deviation is used when working with historical returns, as they are deemed to be samples unless 100% of the data points are used in the calculation. The population standard deviation is only used when working with 100% of the data points. Daily NAVs from a fund’s inception through the most recent trading day would be considered to be a population. Monthly returns for the past ten years is a sample.

In plain English, the standard deviation is the absolute value of the average deviation of the data points from the mean. In mathematical terms the it is the square root of the sample variance and the sample variance is the sum of the squared deviations divided by the number of data points less one, (n — 1). To compute the population variance, you would simply divide by n instead of (n — 1) and the population standard deviation would be the square root of the population variance.

The arithmetic mean is computed as follows:

n = the number of data points and i = 1 through n .

When r _Avg is used as an estimate of future returns it is

referred to as the expected value of r. E(r) .

The population standard deviation is computed as follows:

Population Standard Deviation = S = V 1/2

Where V 1/2 is equivalent to the square root of V .

The sample standard deviation is computed as follows:

Sample Standard Deviation = S = V 1/2

Where V 1/2 is equivalent to the square root of V .

Here’s an example calculation of the mean and standard deviation of of investment returns for a sample:

In the Interest Rates subsection I introduced the risk-free rate of return, r*, as being the basis from which securities are priced. As the T-Bill rate is used as a proxy for r*, the basis can vary quite a bit over time. Indeed, the T-Bill rate was above 17% in 1981 and as low as 0.01% in 2009. Therefore, the proper way to evaluate returns is with excess returns. which is the actual return less the average T-Bill rate for the period over which the returns were computed. In the example above, the period was yearly, so the average T-Bill rate for each year should have been deducted from each year’s return. However, you won’t find this in most published data, so you should do it yourself. (Historic T-Bill rates can be found on the U.S. Treasury’s web site.) If the T-Bill rate was relatively stable over the period being analyzed, then it’s not terribly important, as you’ll be comparing everything on a relative basis. But if you compare raw returns for 2009 to 1981, you’ll not be comparing apples to apples. If your only means of projecting future returns is by using historic returns, then you definitely should work with excess returns.

Security returns have been found to be approximately normally distributed, which means it’s relatively safe to use the normal distribution to make some general inferences of what can be expected if the past is repeated in the future. This is a pretty good assumption for the variability of returns but not the returns themselves. But, as mentioned above, that’s all most of us have so that’s what we use, thus the standard disclaimer that past returns are not necessarily indicative of future returns.

Standard Deviation Rules of Thumb

Here are some rules of thumb regarding the standard normal distribution. (The probabilities have been rounded, as these are just rules of thumb.): 68% of the probability lies within one standard deviation of the mean, and as the distribution is symmetric, that 68% can be interpreted as being centered on the mean. What’s that mean, you say? It means that the actual return for any given year could have been expected to be within one standard deviation of the mean 68% of the time. In the example above, there’s a 68% probability that the return in any year selected at random from the 10-year sample would be between -10.5% (13.4 — 23.9) and +37.3% (13.4 + 23.9). As the distribution is symmetric, this can be refined to a 34% probability that the return would be between 13.4% and -10.5% and a 34% probability that it would be between 13.4% and 37.3%. But that only covers 68% of the probability.

There’s nearly a 96% probability that the return for any given year was within two standard deviations of the mean and nearly a 100% probability that it was within three. Again, this is centered on the mean, so it can be interpreted as +/- 2 SD and +/- 3 SD respectively.

Standard Deviation Rules of Thumb