I am trying to understand why the Sharpe ratio changes (increases) when I simulate leveraging my portfolio by multiplying all the time series of daily returns by a leverage factor (e.g. 5).
I understand that the Sharpe ratio should not change when a portfolio is leveraged (other things being equal).
However I find that the annualized Sharpe ratio (calculated geometrically with formula:
return = (product of 1+ daily returns ^ (262/number of returns))-1,
stdev = stdev(returns)*(sqrt(262)) deos increase (e.g. from 3.1 to 4.3).
However, the daily Sharpe ratio (calculated as the arithmetic average of returns divided by standard deviation) remains identical (mathematically identical).
I am assuming a risk free rate of zero, so the Sharpe is simply return divided by
I'm sure it's something obvious, but can anyone explain why?