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> I'm not sure I buy that hypothesis or the whole notion that Sharpe
> "punishes" upward volatility. There are two terms in the Sharpe
> formula: profit and stddev of profit. Downward volatility decreases
> profit; upward volatility increases profit. Upward volatility gives a
> higher Sharpe ratio every time.
Bingo.
I constructed a spreadsheet that calculates Sharpe ratio on
monthly returns. Then I fed it a 1% monthly return and a 0%
Tbill rate. Result: infinite Sharpe, since there is zero
volatility.
Then I added volatility to the results: Each 4th month
alternated between 1% + 2% = 3% and 1% - 2%. Other months stayed
at 1%. Equal volatility upward and downward. Result: 12-month
sharpe alternated between 3.93 (after the 3% months) and 2.80
(after the -1% months).
Next I changed each 4th month to be 1% - 2% = -1% -- only
negative volatility. Result: 12 month Sharpe = 1.90.
Next I changed each 4th month to be 1% + 2% = 3% -- only positive
volatility. Result: 12 month Sharpe = 5.80.
Finally I experimented with somewhat more realistic volatility.
Every 4th month flipped between -2 / +3 (pos & neg volatility
with a negative bias) and -1 / +4 (pos & neg volatility with a
positive bias). Result: negative bias Sharpe = 1.66/3.01,
positive bias Sharpe = 2.56/3.39.
So clearly positive volatility is NOT punished, at least not as
compared to negative or mixed volatility. Increased volatility
IS punished, so ideally the Sharpe seeks out zero volatility.
But in real-life system performance, you're not going to see zero
volatility. Given similar levels of volatility, a system with
positive volatility will get a higher Sharpe score than a system
with mixed or negative volatility. And that high-Sharpe system
is exactly the one you want to trade.
Gary
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