# 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

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:

1. Compute compounded returns g = 1 + r (assuming monthly returns r are in decimals, not percentages)

2. 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.

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., $(1+\mu_g)^{t} \equiv e^{\mu_{log} t}$.