# Portfolio risk versus returns: the statistics

Portfolio returns are expected to follow a normal distribution. Deviations from this pattern are a form of investment risk, described below.

## Intro

Fund returns are constantly changing. Instead of showing these changes on a performance chart (over time), count the number of times a return falls on each value. Create a plot with returns on the horizontal scale (X-axis), min to max; the vertical scale (Y-axis) with the number of times the return fell on that value. This format is called a distribution plot, as it shows how the values are distributed across the range of returns.

Statistics is the science of the collection, organization, and interpretation of data. Statistical methods use this plot to analyze past performance and to predict future returns. For example; since most returns fall near plot center, it seems likely that future returns will also be near the plot center.

Based on usage, this plot is known as a statistical distribution. For many types of fund returns, the outline of the plot has the shape of a bell (like the shape of the Liberty Bell) and is called a normal distribution.

- insert graphs here - Plot over time, then show distribution

Identify: normal distribution, left side, right side, fat tail, long tail.

The shape of the curve is described using 4 different terms (you need all 4)...

## Rough outline

• Fat tails
• Long tails
• Left tale risk
• Right tale opportunity

The first four moments of a distribution:

Investors define volatility to be the standard deviation. For example, William F. Sharpe defines the Return per Unit of Risk (Sharpe Ratio) as some Return divided by the Standard Deviation.

### Statistics discussions

• Median of a distribution and how skewness affects the relationship between the mean and the median
• Correlations and confidence intervals

#### Log-normal distributions

• Why for annual returns the log normal distribution, rather than the normal distribution, should be used

The proper distribution for a normal model would be a log-normal distribution... The reason for the log-normal distribution is that changes are relative to other changes. The probability that the market drops by 20% in the second half of the year is relatively independent of what happened in the first half of the year, but it is a larger point loss if the first half of the year was a bull market.