What is, here, the relationship between "compound" and "arithmetic return" and "volatility"?

Question

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 1

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.