Annualizing Log Returns Without Confusing Growth Rates and Losses
Summary
The document explains how to interpret an annualized log return when a daily simple return is negative. It describes log returns as continuously compounded rates and notes their useful properties, including converting compounded price changes into sums. The apparent puzzle is that multiplying the logarithm of a daily loss by the number of trading days can produce a value below negative one, even though an asset cannot lose more than its entire value.
The answer resolves this by distinguishing the annualized log rate from the corresponding simple return: exponentiating the cumulative log rate gives the ending wealth as a fraction of its starting value. The example converts a one-percent daily loss into an ending value of roughly eight percent after a trading year. This assumes the same daily return compounds throughout the period; it illustrates the transformation rather than offering a general risk-modeling prescription.
Key ideas
- A cumulative log return is a continuously compounded rate, not a simple percentage return.
- Exponentiating the cumulative log return converts it to a wealth multiplier.
- A negative log rate below negative one does not imply that the asset lost more than its full value.
- The example assumes repeated compounding of the stated daily loss across trading days.
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Full text
# Annualzing the log of daily returns riddle
# 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)?
## Answer by Joshua Chance (score 10, accepted)
https://quant.stackexchange.com/a/1443
You're forgetting that -2.52 is still in natural logarithm terms. So the correct answer is 2.71828183 raised to the -2.52 power which equals 0.08. Your ending portfolio value is 8% of what it was a year ago.
## Answer by ensabahnur (score 0)
https://quant.stackexchange.com/a/39763
To get the final value, just get $e^{rt}=e^{-0.01 \cdot 252}=8.04 \%$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.