Why Arithmetic Return and Volatility Do Not Determine Compound Return
Summary
The note distinguishes three common performance measures: compound return over a sequence of periods, arithmetic return as the average of period returns, and annualized volatility as a scaled estimate of return variability. Compound return reflects the cumulative growth of an investment across the observed periods, while arithmetic return summarizes the series with its mean. Annualized volatility is derived from volatility measured at a shorter interval; the example uses a square-root-of-time scaling based on the assumed number of trading days in a year.
The key conclusion is that these quantities cannot in general be converted into one another from only two values. The arithmetic mean and compound outcome depend on the full pattern of returns, and volatility alone does not supply that missing sequence or distributional information. The note gives definitions and an illustrative annualization convention, but the source data’s precise conventions remain uncertain; annualization assumptions can vary, and the stated product formula omits a subtraction of one if expressed as a net compound return.
Key ideas
- Arithmetic return is the mean of the period returns in a series.
- Compound return describes cumulative investment growth across the periods.
- Annualized volatility can be estimated by scaling shorter-period volatility by the square root of the assumed periods per year.
- Two of arithmetic return, compound return, and volatility do not generally determine the third.
Tags
Full text
# What is, here, the relationship between "compound" and "arithmetic return" and "volatility"?
# What is, here, the relationship between "compound" and "arithmetic return" and "volatility"?
I'm trying to find the exact (ie, not an approximate) relation between the "Compound Return", "Arithmetic Return", and the "Annualised Volatility" as given the assumptions below, and from there the precise meanings/definitions that should underly these numbers as presented here.
These numbers are from a bank's long-term assumptions. I can't quite figure it out, also not by trying several possibilities. I assume the "arithmetic return" is calculated from the other two. There seem quite a few possibilities (eg, continuously compounded vs annually compounded, ln-transformations, arithmetic or geometric volatility etc.) The source does not provide any mathematically precise definitions.
Can you figure out which definitions are used here?
## Answer by skoestlmeier (score 4)
https://quant.stackexchange.com/a/42885
There is no possibility to convert any two of your mentioned variables into the remaining one. For the compound and arithmetic return you can derive an inequality, but that's the best you can do.
The definitions for your statements are:
$$r_{\mathrm{compound}}= \prod_{t=0}^{n}{\left( 1+r_t \right)}$$
$$r_{\mathrm{arithmetic}}=\frac{1}{n} \sum_{t=0}^n{r_t}$$
$$\sigma_{\mathrm{year}} = \sigma_{\mathrm{day}}\cdot \sqrt{252}$$
where $t$ and $n$ denotes the beginning and ending period of time and $\sigma_t$ the volatility measured in time period $t$.
In fact, the arithmetic return is just the mean of a given return series. The compound return gives an indication of how much money would have been made by an investor, who invested one dollar in the corresponding asset/portfolio. The annualized volatility is derived from a volatility (i.e. standard deviation of returns) measured in a shorter period of time. The factor $\sqrt{252}$ is used because one can assume to be 252 daily trading days within one year. For an extended description on this factor you may look here.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.