A financial asset's total return consists of two components: an income yield consisting of cash dividends or interest payments, and a return reflecting the capital gain or loss resulting from changes in the price of the financial asset.

Holding Period Return

It is the percentage by which the value of an investment has grown for a particular period. It is the sum of investment income and capital gains divided by the initial value.

It has two important characteristics:

• It has an element of time attached to it: monthly, quarterly or annual returns. HPR can be computed for any time period.
• It has no currency unit attached to it; the result holds regardless of the currency in which prices are denominated.

Example

A stock is currently worth $60. If you purchased the stock exactly one year ago for $50 and received a $2 dividend over the course of the year, what is your holding period return?

R_t = ($60 - $50 + $2)/$50 = 0.24 or 24%

Arithmetic or Mean Return

The arithmetic mean is what is commonly called the average. It is the most widely used measure of central tendency. When the word "mean" is used without a modifier, it can be assumed to refer to the arithmetic mean. The mean is the sum of all scores divided by the number of scores.

• All interval and ratio data sets (e.g., incomes, ages, rates of return) have an arithmetic mean.
• All data values are considered and included in the arithmetic mean computation.
• A data set has only one arithmetic mean. This indicates that the mean is unique.
• The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. Deviation from the arithmetic mean is the distance between the mean and an observation in the data set.

The arithmetic mean has the following disadvantages:

• The mean can be affected by extremes, that is, unusually large or small values (outliers). In such cases we can either do nothing, or delete all the outliers (trimmed mean), or replace the outliers with another value (winsorized mean).
• The mean cannot be determined for an open-ended data set (i.e., n is unknown).

Geometric Mean Return

The geometric mean has three important properties:

• It exists only if all the observations are greater than or equal to zero. In other words, it cannot be determined if any value of the data set is zero or negative.
• If values in the data set are all equal, both the arithmetic and geometric means will be equal to that value.
• It is always less than the arithmetic mean if values in the data set are not equal.

It is typically used when calculating returns over multiple periods. It is a better measure of the compound growth rate of an investment. When returns are variable by period, the geometric mean will always be less than the arithmetic mean. The more dispersed the rates of returns, the greater the difference between the two. This measurement is not as highly influenced by extreme values as the arithmetic mean.

The Harmonic Mean

The harmonic mean of n numbers x_i (where i = 1, 2, ..., n) is:

The special cases of n = 2 and n = 3 are given by:

and so on.

For n = 2, the harmonic mean is related to arithmetic mean A and geometric mean G by:

The harmonic mean may be viewed as a special type of weighted mean in which an observation's weight is inversely proportional to its magnitude. It is used most often when the data consists of rates and ratios, such as P/Es.

Which Mean to Use

It depends on lots of factors: outliers, symmetric distribution? compounding?

• Include all values, including outliners? Arithmetic mean.
• Compounding required? Geometric mean.
• Minimize the impact of extreme outliners? Harmonic mean, trimmed mean, winsorized mean.

In general:

Harmonic Mean <= Geometric Mean <= Arithmetic Mean