# Measures of Return

Sharpe [1966], Treynor [1965] and Jensen [1968] agree that the profitability of an investment should be measured by the mean rate-of-return, but there is disagreement regarding the measure of risk.

For any investment, return can be defined as capital appreciation plus income. The rate-of-return over a period of time t can be formally defined as follows:1

where it is the rate-of-return in period t, Pt the price of the investment at the end of period t, and Dt the income received during period t.

For investment periods in excess of one period, the geometric mean formula is as follows:

where, n denotes the number of periods.

However, logarithmic returns are often used, as they facilitate mathematical compounding and statistical testing. With the geometric compound return, the cumulative active return cannot be calculated simply by taking the difference between the cumulative portfolio return and the cumulative benchmark return. This calculation can only be undertaken with logarithmic returns:

Although the above calculations appear straightforward, it is important to note that as estimated sample moments (e.g. average returns) have to be used instead of the true population moments (e.g. expected returns), the precision of mean return estimates depends solely on the length of the sample. In practice this requires a long historic sample, consistent in both quality and construction.2 For example, consider an asset with an expected return of 8% and an annual standard deviation of 20%. The standard error of the mean estimate equals:

where T is the number of years in the sample. For the mean to be significantly different from zero a t-statistic of at least 2 is normally required. Hence:

Therefore:

In contrast, estimation of volatility does not require the length of the sample to be long. It only requires the size of the sample to be large, which is often achieved by increasing the sampling from monthly to weekly or daily observations. However, with both unfulfilled by current real estate indices, the requirement cannot be met, making an accurate assessment of the 'true' return difficult. The issues associated with measuring returns on real estate are considered in section 2.4.2, while choice of the appropriate benchmark is discussed in section 4.5.6.

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1For a comprehensive discussion see Gujarati [1995]; for an application to finance, see Levy & Sarnat [1971a].

2It also assumes that the returns are `easy' to calculate. Appendix A on page 247 illustrates the difficulties involved in constructing one of the most important and widely used of market proxies, the S&P 500.