Rates of Return

Holding Period Return

0A holding period return, R, is the return earned from holding an asset for a single specified period of time.

where the subscript indicates the time of the price or income; (t=0) is the beginning of the period; and (t=1) is the end of the period.

A holding period return can be computed for a period longer than one year. For example, an analyst may need to compute a three-year holding period return from three annual returns. In that case, the three-year holding period return is computed by compounding the three annual returns:

where R₁, R₂, and R₃ are the three annual returns.

Arithmetic or Mean Return

The arithmetic or mean return is denoted by $\bar{R_i}$ and given by the following equation for asset i, where $R_{it}$ is the return in period t and T is the total number of periods:

Geometric Mean Return

A geometric mean return provides a more accurate representation of the growth in portfolio value over a given time period than the arithmetic mean return.

The geometric mean return is denoted by $\bar{R}_{Gi}$ and given by the following equation for asset i:

where $R_{it}$ is the return in period t and T is the total number of periods.

In general, the difference between the arithmetic and geometric means increases with the variability within the sample; the more disperse the observations, the greater the difference between the arithmetic and geometric means.

The Harmonic Mean

The harmonic mean, $\bar{X}_H$, is appropriate in cases in which the variable is a rate or a ratio.

Harmonic Mean Formula.

The harmonic mean of a set of observations $X_1, X_2,\dots,X_n$ is:

with $X_i > 0$ for $i = 1,2,\dots,n$.

The harmonic mean is the value obtained by summing the reciprocals of the observations,

the terms of the form $\frac{1}{X_i}$, and then averaging their sum by dividing it by the number of observations, n, and, then finally, taking the reciprocal of that average,

The trimmed mean removes a small defined percentage of the largest and smallest values from a dataset containing our observation before calculating the mean by averaging the remaining observations.

The winsorised mean is calculated after replacing extreme values at both ends with the values of their nearest observations, and then calculating the mean by averaging the remaining observations.