Logarithmic Returns

We will usually be thinking of logarithmic returns, when we explore what is happening with more complicated situations such as actual stock market returns.  We will also often use logarithms of prices; with the compound interest situation, with log prices, the curve becomes linear:

By changing to a log scale on the y-axis we create a linear plot.  The slope of the plot is the logarithmic return.

When we deal with a series of market prices, we will find it convenient to deal with log prices and log returns.  If we take set of sequential prices:

The set of their sequential decimal returns is;

All the denominators are different so this method has a problem that the returns cannot be easily added or subtracted.  And the returns are not symmetric, i.e. if one starts with a fixed amount and one day has a return of  -0.01 and the next day have a return of +0.01, the initial value of the investment has not been regained.  Logarithmic returns can be simply added.  From here on log will be considered the natural logarithm to the base e.  

Here is the set of log returns for the same series:

or

The sum of the series of returns now is the same as the return over the whole interval since all but the first and last log prices cancel out:

The average return over all the prices becomes:

Note that the mean log return of a time series of prices is determined by only three numbers, the first price, the last price and the number of time intervals between the prices.  This is true even if the series consisted of thousands of returns.  This "expected" return thus gives no information about what happened over the sub intervals.  On a log price plot this is the same as determining the rate by fitting the price series by drawing a line through the first and last prices.  When you are quoted a historical return you usually are given this result converted to an annual decimal return.

© Copyright 2007 mathestate    Fri 14 Dec 2007