# Annualzing the log of daily returns riddle

Two popular ways to measure returns are Arithmetic returns and Log returns. Let's define arithmetic (simple period) returns as: P(t) - P(t-1) / P(t-1). Let's define log return as Ln( P(t)/P(t-1) ) or equivalently Ln(1 + arithmetic return).

The log returns have nice properties -- logarithmic returns prevents security prices from becoming negative; we can interpret log returns as continuously compounded returns; they are approximately equal to simple returns at small values; if the security price follows Brownian motion then the log returns are normally distributed; logs convert products into sums. Jorion (2001) has a brief description of the properties and application of the log description (pg. 94): Jorion 2001

Let's say a security lost 1% (simple return) today. We would like to annualize this daily return (as it is an input to some risk model whose outputs we would like on an annualized basis). So: Ln(1 -.01) * 251 trading days = -2.52 annualized return. However, a security cannot lose more than 100% of its value. How do we explain this outcome (i.e. what is the right operation such that the lower bound on the log return is > -1)?

To get the final value, just get $$e^{rt}=e^{-0.01 \cdot 252}=8.04 \%$$.