Simple or Log Returns?

One thing that puzzles many in quantitative finance is when log return is appropriate and when simple return should be used.

Simple Return Formula:        
          

Log Return Formula:      

                   

While there are several benefits of using log returns like log-normality, time additiveness, approximate raw-log equality and mathematical ease, a common fallacy is to use is at places where it is not appropriate. The most important difference between simple and log returns is the features of time-additivity and asset-Additivity.

Simple returns are asset-additive: Portfolio return is the weighted average of the stocks in the portfolio. 

where weights are the proportion of an individual stock in the portfolio and the sum of the weights is 1. Therefore, if an investor puts an equal amount of money in each stock it will be called “equally-weighted” portfolio. The return of an equally-weighted portfolio is just the average return of all stocks in the portfolio. 

On the contrary, if investment in each stock is in proportion to its total outstanding market shares it is termed as “value-weighted” portfolio, where the weight of an asset is equal to the proportion of its value to the total value of all assets in the portfolio. A value-weighted portfolio is more realistic and easy to maintain as no frequent rebalancing is required.

Log returns are not asset-additive. The weighted average of log returns of individual stocks is not equal to the portfolio return. In fact, log returns are not a linear function of asset weights. In comparison, if simple returns are used than the portfolio return is the weighted average of assets in that portfolio. So one of the advantages of simple return is that it can be used where portfolios are formed and portfolio returns have to be calculated because of its asset-additive property.

Log returns are time-additive: The logarithmic return of an asset over a period of t to T is the sum of all logarithmic returns between the t and T. In other words, the log return over n periods is merely the difference in log between initial and final periods. This is an advantage because the sum of a normally distributed variable is also normally-distributed.

As for simple return, the product of returns over n periods is the return for a period.

This is a negative feature of simple returns as probability theory states that the product of normally distributed variables is not normally-distributed.

Takeaways:

1. When a cross-section of assets is being studied use simple returns

2. Use continuously compounded return when temporal behavior of return is the focus of interest
Reference: 
1. Campbell, J. Y., Lo, and A. W., MacKinlay (2007). The econometrics of financial markets. PrincetonUniversity Press, Princeton, New Jersey. 

2. William G Schwert Teaching Notes