From the textbook, sharpe ratio is (return-riskfree rate)/risk
However I wonder if I can use (return-index return)/risk, where the index acts as the benchmark, to calculate the sharpe ratio?
I am quite confused about the difference between these two.. Thanks
Indeed, the Sharpe ratio utilises the risk-free rate. When you're using another benchmark then the risk-free rate, say the market, the ratio is often referred to as the Information Ratio.
In addition, the denominator becomes the standard deviation of the difference between the market return with your portfolio returns instead of standard deviation of the difference between the risk-free rate and your portfolio return
Sure you could use this formula so-called Ex-post Sharpe ratio in which you consider in the numeratorof the formula the differential between the realized asset return and its benchmark. Using this approach always disclose for transparency purposes the benchmark under consideration, in particular if you're doing some comparison analyses. You'll consequently avoid some biases than resorting to the problematic selection of an adequate risk-free rate ( flat yields on T-bills or adhoc rates on money market instruments). Hope it helps.
