In the CFAI book it says that Sharpe ratio is time dependent- overall Sharpe ratio increases proportionally with the square root of time, so an annual Sharpe will be square root of 12 times bigger than a monthly Sharpe ratio.
However in the 2017 past exam paper it says that st dev increases when you increase the reporting from let’s say monthly to quarterly. Wouldn’t this decrease the Sharpe ratio and doesn’t this contradict the first sentence above?
Thanks in advance.
you have a monthly standard deviation of 1% . .
now think is your annual standard deviation (on average) going to be closer to 12% or is it going to be < 1% ?
sharpe ratio is a product of return/risk
assume rf is 1% monthly or 3% quarterly (in reality;the quarterly rate is 1.01^3 -1 or 3.03%)
assume your portfolio performs at 2% monthly or 6% quarterly (in reality, the quarterly rate is 1.02^3-1 or 6.12%)
assume monthly std dev of portfolio is 1%, the quarterly std dev is going to be 3^0.5*1 or roughly 1.7%
now let’s count the monthly and quarterly sharpe ratio:
monthly: 2-1/1 = 1
quarterly: 6.12-3.03/1.7 = 1.8
even using the simple compounded return numbers: 6-3/1.7 = 1.76
look at the calculations, as time frame increases, std deviation increases by a factor of square root while returns increases by an exponential factor!
yes, std deviation (the denominator) increases as time frame increases, but return (the numerator) also increases almost always higher than std deviation does in the same time frame choice.