Compounding and Annualizing Returns for Portfolio Analysis

Summary

The document clarifies the difference between compounding simple returns across periods and scaling returns for an annualized portfolio statistic. For a constant average daily simple return, compounding converts it to a yearly level return by applying the daily growth factor repeatedly. For log returns, aggregation across days is additive, so the average daily log return can be multiplied by the number of trading days in a year.

The answer notes that daily log returns and simple returns are close when daily moves are small, which helps explain why linear scaling of daily returns is often a usable approximation. This distinction is relevant when constructing annualized return inputs and interpreting a Sharpe ratio. The post also proposes scaling daily volatility by the square root of the annualization period, but its accepted answer focuses on return compounding and does not discuss assumptions behind volatility scaling, serial dependence, or whether the mean return used is arithmetic or geometric. Those details matter when applying the formulas to real portfolio data.

Key ideas

• Simple returns compound multiplicatively across periods to produce a level return.
• Log returns aggregate additively, so their average can be scaled by the number of periods.
• For small daily returns, simple and log returns are close, making linear scaling a useful approximation.
• The post distinguishes return annualization from compounding but does not examine assumptions behind volatility scaling.

Tags

# Compounding vs Annualizing Returns in a Portfolio Optimization Context


# Compounding vs Annualizing Returns in a Portfolio Optimization Context












This might be a rather basic question that might be closed... but I can't for the life of me understand why in many Google search results the annualization of daily returns is done like this:

r_yearly = (1+r_daily)^252 - 1

However, this is not the correct way to annualize returns in a portfolio optimization context. Meaning, if I were to annualize the daily returns and daily volatility of a stock to generate an annualized Sharpe ratio, it should be like:

r_yearly = r_daily * 252

vol_yearly = vol_daily * 252^0.5

Sharpe_yearly = r_yearly / vol_yearly

The reason why I am posting this is because I have wasted so much time searching about daily returns annualization and I have realized that there is a difference between annualizing and compounding. Most of the Google search results that stated annualization actually mean compounding. Can anyone confirm this and give some useful insights to understand/remember this well?

## Answer by phdstudent (score 4, accepted)

https://quant.stackexchange.com/a/76885

Let's get two things straight:

- Denote $R$ the average daily return of an asset. Then denote $r \equiv \log(1+R) $

- Assume time period is 1 day. Then if working with levels the yearly return is: $R_{year} = (1+R)^{252} - 1$. If working in logs: $r_{year} = 252*r$.

Now for daily returns: $\log(1+R) \approx R$ which implies: $252*R \approx 252*r$ and therefore it does not matter much.

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