Choosing Return Conventions for Portfolio Performance and Risk Analysis
Summary
The document distinguishes return measures according to their use. To summarize a monthly portfolio series as an annual growth rate, it recommends compounding the simple monthly returns and converting that compounded growth into an annualized simple return. It illustrates why an arithmetic average of periodic percentage returns can misrepresent realized growth: a 100% gain followed by a 50% loss leaves the investment flat, despite a positive arithmetic average.
Log returns are described as convenient for analysis because they add across time and are often used in risk calculations such as volatility, correlation, and value at risk under common modeling assumptions. Simple percentage returns remain familiar for performance reporting, and the answer cites their use in investment performance standards. The appropriate measure depends on purpose: geometric compounding supports growth summaries across periods, while arithmetic or log-based statistics may suit different analytical tasks. The discussion does not offer one return convention for every application.
Key ideas
- Annualized growth can be computed by compounding simple monthly returns across the year.
- Arithmetic averages can misstate realized growth when gains and losses compound.
- Log returns add across time and are useful inputs for several risk analyses.
- Simple percentage returns are familiar for portfolio performance reporting.
- Choose the return convention according to whether the goal is growth reporting or statistical analysis.
Tags
Full text
# What returns to use?
# What returns to use?
I have monthly returns of my portfolio... I would like to summarize the performance over a longer period in one overall figure.
Should I use log returns per month then use geometric mean on the log returns to get the correct return?
or
Log returns per month then arithmetic mean? or simple returns then geometric or arithmetric mean?
What is most correct and why? :S
## Answer by nitin23 (score 4)
https://quant.stackexchange.com/a/38907
Since you're looking to summarize the performance of a monthly return series in a single number, it is best to compute the annualized return. This is the standard used in the investment management industry. You could also compare your portfolio returns with that of an industry benchmark like S&P 500 on an annualized basis.
Assuming your returns are in decimals, here's how you'd compute it in Excel:
- Compute compounded returns g = 1 + r (assuming monthly returns r are in decimals, not percentages)
- Annualized return = GEOMEAN(g)^12 - 1
Computing arithmetic mean doesn't make much sense. To illustrate this, take the following example: you have an asset (like present day cryptos), that goes up 100% in month1 and drops 50% in month2. In reality you made no money, so your geometric mean would be 0, but the arithmetic mean would be +25%...quite meaningless.
## Answer by David Addison (score 1)
https://quant.stackexchange.com/a/38896
The most correct answer depends on how the metrics will be used.
Logarithms are definitely the most convenient to work due to their additive properties. Logs are also the inputs which risk metrics usually expect. For example, if you assume that logarithmic returns are normally distributed, a-la Geometric Brownian Motion, then you would use these to calculate standard deviations, correlations, VaR, etc.
Most investors are most accustomed to simple percent returns. In fact, the CFA Institute’s GIPS advises asset managers and advisors to use simple percent returns for presenting performance.
The use cases for geometric means are mostly limited as a means to convert simple returns from different bases and timeframes into another simple return. You typically won’t want to do grunt work with or present geometric returns. For example, annualized growth rates are simple arithmetic, but can also be found from the geometric means of simple monthly returns.
My advice is to work in logs and present in percents since the conversion from logs to percents is facile. E.g., $(1+\mu_g)^{t} \equiv e^{\mu_{log} t}$.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.