Risk-adjusted performance measurement: Log returns vs. simple returns and geometric vs. arithmetic mean return

Question

I have just simulated 49 weeks of correlated returns on 5 different stocks, assuming returns being lognormally distributed. Next, I am supposed to assume that the simulated 49 weeks of returns represents the actual performance of the 5 different stocks the last 49 weeks, and thereby measure the performance of each stock using any performance measures I find suitable.

My first question relates to whether I should use (1) simple returns or (2) log-returns when evaluating the performance of each stock using performance measures based on volatility (e.g. Sharpe ratio), extreme risk (e.g. reward-to-VaR) and lower partial moments (e.g. Sortino ratio)?

Also, depending upon the correct answer to the first question, when calculating the average weekly return (i.e. mean return) on each stock, should I calculate a arithmetic or geometric mean of the simple return/log-return?

Answer 1

In theory, stock prices are lognormally distributed.

People usually prove lognormality by referring to positivity and right skewness of stock prices. Mathematically (or philosophically if you wish), lognormality follows from the following equation $\frac{S}{dS}={\mu}dt+{\sigma}dW$, which you may see a lot in quantitative finance ("random walk") or in physics ("brownian motion" or diffusion). If you solve this equation, you'll see that the price $S$ is lognormal indeed.

However, if you're not dealing with continuously compounding returns, especially with longer periods like weeks or years, you may not have a chance to observe continuous exponential compounding. Such a process can be very well described (or approximated if you wish) by geometric compounding.

Finally, back to your question. My advice for calculating risk adjusted measures in your case would be: unless you can prove, graphically or by statistical testing, that your weekly returns are lognormal, go with geometric averaging of simple percent changes. This will allow for (1) compounding and (2) better interpretability. Arithmetic averaging will give you wrong results when compounding. Lognormality will be an unnecessary complication in your case.

Answer 2

Simulations are commonly based on a geometrical brownian motion. So in this case using lognormal returns approach is appropriate. This also bounds you to calculate a geometrical mean.