Log Returns
If $100 grows to $120, what is the single period return?
Discretely... +20% because 120/100-1 = 20%
Continuously... log return is given by LN(P1/P0) or ~18.2%. (If the asset paid dividends, they are included in the numerator).
Period log returns are typically used in quantitative finance
Benefits of log return
* Time additive: Note that the two-period log return is identical to the sum of the each period’s log return. To get the n-period log return, we can simply add the consecutive single period log returns. Conversely, notice the simple return is not time additive.
* Mathematically convenient: logs and exponents are easier to manipulate with calculus. Theoretical models tend to assume, unrealistically but conveniently, continuously compounded rates of return. For example, if LogReturn = LN(P1/P0), then EXP[LogReturn] = P1/P0. If f(y) = EXP[LogReturn] then the first derivative, f’(y) is quite wonderfully also EXP[LogReturn]. In short, d/dx EXP[x] = EXP[x].
* Approximately good: for short periods (e.g., daily), the log return approximates the discrete return anyway
Drawbacks of log return
* Not "linear" in portfolio return: We would like to be able to say that portfolio return is a weighted sum of components (assets). However, we cannot say this under log returns: the log return is not linearly additive across portfolio components. But, the discrete return is linearly additive.
* Unrealistic: Markets tend to quote discrete returns
