Arithmetic vs. Logarithmic Rates of Return

Say you hold a stock as it increases from $100 to $105. Usually, this is reported as a return of 5%. The formula for this return (which we’ll call arithmetic) is as follows:

r_\alpha = \frac{FV}{PV} - 1 = \frac{FV - PV}{PV} = \frac{105}{100} - 1 = 5 \%

This simple definition of return serves us well for most uses, but there are some quirks that make arithmetic returns difficult to use in some academic and valuation settings. For example, continuously compounded arithmetic returns are not symmetric. If a position appreciates 15% and then depreciates 15%, the total change is -2.25%.

FV = PV(1 + r_\alpha)

FV = PV(1+.15)(1 - .15) = PV(0.9775)

\frac{FV}{PV} - 1 = 0.9775 - 1 = -2.25 \%

To avoid this quirk, practitioners sometimes use log returns, which are defined as follows:

r_\ell = ln(\frac{FV}{PV})

The function  ln(x) answers the question “e to what power is equal to x?” The constant e will be defined in a future post. In excel, this is the function =LN().

Rearranging the above formula allows one to solve for a Future Value given a certain rate of return and initial Present Value.

FV = PVe^{r_\ell}

The equivalence between these returns is given by the following formulas:

r_\ell = ln(r_\alpha + 1)

r_\alpha = e^{r_\ell} - 1

For a stock growing from $100 to $105, this yields a log return of 4.88%. For small returns, arithmetic and logarithmic returns will be similar, but, as returns get further away from zero, these two formulations will produce increasingly different answers.

If one is modeling the stock market, it is common to assume that returns are normally distributed. In this context, log returns are far superior to arithmetic returns since the sum of repeated samples from a normal distribution is normally distributed. However, the product of repeated samples from a normal distribution is not normally distributed.

However, that’s not to say arithmetic returns aren’t without their benefits. Arithmetic returns aggregate well across portfolios. The arithmetic return for a portfolio is simply equal to the weighted average of each constituent’s arithmetic return.

But, more importantly for forensic settings, arithmetic returns are widely understood by the public. Confidently explaining complex topics to laymen separates good and great expert witnesses. For that reason alone, we normally rely on arithmetic growth rates when it will not produce an error.