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Annualizing a Sharpe vs. Sortino Ratio

Below are functions that (I hope) calculate Sharpe and Sortino ratios.
I have made their interfaces identical.
From here, "Commonly, Sharpe Ratios on a daily, weekly or monthly basis are annualized by multiplying by the square root of the higher frequency time period. This is because the effective return is proportional to time. Assuming a Weiner process governs stock prices, variance is proportional to time. Hence standard deviation is proportional to the square root of time. So you would scale a Sharpe Ratio by multiplying by t/√t = √t, where t is the frequency you are annualizing from."
So why doesn't this logic apply to a Sortino Ratio, i.e. why don't we have an equivalent np.sqrt(N) term in the last line?
Thanks

def annualised_Sharpe(daily_ret, yearly_benchmark_rate=0.05,N=252):
MAR = yearly_benchmark_rate/N
excess_daily_ret = daily_ret - MAR
return np.sqrt(N) * excess_daily_ret.mean() / excess_daily_ret.std()

def annualised_Sortino(daily_ret, yearly_benchmark_rate=0.05, N=252):
MAR = yearly_benchmark_rate/N
excess_daily_ret = daily_ret - MAR
target_downside_deviation = np.sqrt(np.mean(minimum(excess_daily_ret,0.0)**2))
sortino = excess_daily_ret.mean() / target_downside_deviation
# Add Period correction??
#return np.sqrt(N) * sortino
return sortino

2 responses

Thank you for posting those.
And with Bill's ok, this is an effort to calculate Beta, for both portfolio and individual securities. Ballpark. Needs improvement from others.

Clone Algorithm
20
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Backtest from to with initial capital
Total Returns
--
Alpha
--
Beta
--
Sharpe
--
Sortino
--
Max Drawdown
--
Benchmark Returns
--
Volatility
--
Returns 1 Month 3 Month 6 Month 12 Month
Alpha 1 Month 3 Month 6 Month 12 Month
Beta 1 Month 3 Month 6 Month 12 Month
Sharpe 1 Month 3 Month 6 Month 12 Month
Sortino 1 Month 3 Month 6 Month 12 Month
Volatility 1 Month 3 Month 6 Month 12 Month
Max Drawdown 1 Month 3 Month 6 Month 12 Month
# Backtest ID: 561e97ee6a74db10d3de64b8
There was a runtime error.

Hi Bill,
Your Sortino ratio calculation is correct. As for annualizing the Sortino, in my opinion, it shouldn't matter as long as you are consistent with whatever you do. The annualizing factor is just a scalar multiple of the calculation result, so it can't change the outcome of a ranking system using the Sortino as long as the same scalar is used in every case. Annualizing is common practice in industry, but algorithms don't care if you do it or not. I'm not 100% what the industry standard is for the Sortino, but a sqrt(252) multiple seems reasonable to me.