# Using Statistics to Understand Return Characteristics

### Investment statistics can be used in two ways:

1. To compare the performance histories of multiple investment managers.
2. To try to predict a range of future returns for an investment.

### A. Comparing Managers’ Track Records and Time Periods

When using statistics to predict an investment’s return, it is critical to note that the length of the investment’s track record and the time frame will dramatically affect the calculations. For example, the average annual return of the S&P 500 Index, over the 36 ½ -year period from January 1975 to June 2011, was 11.84%. However, as Figure 1 highlights, if we assess the same data using only 1- or 3-year rolling returns, they range between 61% and -43% on a 1-year rolling basis, and between 33% and -16% on a 3-year rolling basis.

Similarly, if we assess a 5-year rolling period, the returns range between -3% and 20%. As Figure 2 illustrates, only when we lengthen the period to 10 years do we begin to see a true reversion to the mean, or a narrowing of the spread of actual returns close to the long-term average or mean return. This means that if an investment has a 20% return one year, perhaps the best prediction for its return the next year is “I don’t know.” A one-year time period doesn’t provide sufficient information from which to draw conclusions. Therefore, investors should not rely exclusively on statistics that only cover 1-, 3- or even 5-year periods, since they may not be significant or meaningful over the long term.

### B. Understanding Investment Return Characteristics

In this section, we review some of the methods and statistics used to predict investment returns, including standard deviation, skewness and kurtosis, and Monte Carlo simulation.

### Using Standard Deviation to Predict Possible Return Ranges

Can we use historical returns to predict future investment returns? As you can see with Figure 2, despite all of our carefully analyzed averages of historical returns, the S&P 500 Index still experienced one of its worst returns ever in 2008.

To help us predict future returns, we can generate a range of probabilities for the expected returns using standard deviation as a mathematical measure of predictability, rather than using historical averages. Standard deviation enables us to generate a probable range of expected returns. To demonstrate this, we can assess the returns of the S&P 500 Index and develop a normal, bell-shaped distribution of returns for the Index. Figure 3 illustrates the distribution of monthly returns for the S&P 500 Index.

From Figure 3, we see that the mean monthly return for the S&P 500 Index is 1.04% for the period January 1975 to June 2011. If we try to predict next month’s return, based on this information alone, there is a 50% chance that the return will exceed 1.04%, and a 50% chance it will not achieve this return. As Figure 4 illustrates, there is a 75% chance that the next monthly return will be greater than -1.66%, according to the shaded area under the curve. While some might find this information beneficial, there are significant problems with relying too heavily upon standard deviation as a predictive statistic. Perhaps the biggest problem is that very few investments display a normal distribution.

Investors who are required to select and monitor investment managers should develop a basic understanding of investment statistics. Quantitative tools can provide you with good insight that you can use in your qualitative interviews with managers and when monitoring your investments.