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Since Bob doesn't seem to be around, I'll take a stab at this...
> If your average monthly return is M, your average annual return is
> 12*M.
You have to be careful how you express that.
If you return, say, 2% per month, and you withdraw all profits each
month, then your winnings don't compound, and your annual results are
indeed 2*12%. This is the constant-betsize approach that
Tradestation usually uses. (BTW if you use a constant betsize and
you DON'T withdraw your profits, then your effective monthly return
DECREASES as your account size grows.)
If you leave your winnings in the account to compound, and bet a
constant fraction of the account, you will NOT have 12*2% = 24%
growth at the end of the year. You'll have 2%^12 = 26.82%.
If you compound your profits, you can take the Log() of your monthly
profits. This allows you sum up your monthly log(return) to get the
annual return (which is equivalent to multiplying the monthly
returns), to multiply monthly returns times 12 to annualize them,
etc. In reality the difference between Sharpe values calculated with
returns and log(returns) is fairly small, but log(returns) is more
correct.
So log(1.02) = 0.0198, 12*0.0198 = 0.2376, and 12*e^0.2376 = 1.2682
or 26.82%, same as 2%^12.
In a constant-betsize system, a perfect system would produce a linear
equity curve. If you compound your winnings, a perfect system would
produce an exponential equity curve. A system should produce the
SAME Sharpe ratio regardless of whether it's fixed or compounding,
but you have to do your Sharpe calculations appropriately for the
betsize algorithm you're using. In particular you have to be careful
how you calculate your monthly returns: was the month's growth a
percentage of a fixed account (fixed betsize, linear equity curve) or
a percentage of an exponentially-growing account (fixed fraction,
exponential equity curve). Once you calculate the monthly returns
properly, your Sharpe values should be OK.
Gary
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