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

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?

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.