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At 1:31 PM +0000 12/5/01, Michael Stewart wrote:
>If your futures system trades a variable number of contracts how can you
>fairly estimate the rate of return - would/could it be a) P&L / Total
>account size for the period or b) the sum of P&L's for each trade / account
>size for each trade or c) P&L / Average account size for the period
>
>Whilst on the subject of 'return' - are there any seasoned traders who maybe
>happy to share their own performance objectives/results - return for an
>'average' month for example as a return on initial margin (as return on
>account size can be so misleading).
A periodic return is usually the account value at the end of the
period, less the account value at the beginning of the period,
divided by the investment required to achieve the return. If you are
increasing your trade size as your account increases, then your ideal
account size would grow exponentially and you would calculate the
returns on the logarithm of the account value. If you are NOT
increasing your trade size as your account increases, then your ideal
account size would grow linearly and you would calculate the returns
on the account values directly.
At 1:56 PM +0000 12/5/01, Michael Stewart wrote:
>And if you traded twice - say one trade of 10 lots and another trade of 5
>lots within the same day?
>
>To work out the return you need to ascribe a value to the margin required
>for each lot - so if you look at the return based on the max account size
>your understating the return whereas if you take the min your overstating?
This is a very complicated question. Presumably, there is some reason
you changed the size. Perhaps the risk per contract was 2X higher in
the second case and you wanted the total risk of both trades to be
the same. In this case you could use the gain in the day divided by
the risk level to normalize the calculation to "percent return on
risk".
Some people use the account size to normalize the calculation to
"percent return on account".
But you are illustrating the problem of using return alone as the
measure. "Return" is a useless number without also considering the
risk. You can double the return by trading twice as many contracts.
You need to use a "risk-adjusted" return, such as the Sharpe Ratio to
properly evaluate any system.
How often do you see someone buy into this year's high-return mutual
fund only to see it tank the following year?
Periodic (logarithmic) returns have a probability distribution that
is approximately "normal" (Gaussian), as predicted by the Central
Limit Theorem. Looking at the return for any period gives you no
relevant information on what the return for a future period is likely
to be. But the standard deviation of the periodic returns can give
you a decent idea of future returns. About two thirds of the returns
will be within one standard deviation of the mean (average of all
periods) and about 95% will be within two standard deviations of the
mean. (The tails of the distribution tend to be fatter than "normal"
so this latter value is less accurate.)
For example, if you measure a return of 30% for this year, is that
unusual? If you measure the average return for many years as 20% with
a standard deviation of 10%, then a 30% return is equal one standard
deviation away from the mean (20% + 10% = 30%). Since about two
thirds of the occurrences are expected to be within one standard
deviation, this means that about one-sixth of the occurrences will be
in each tail of the distribution. So we can expect the returns to
exceed 30% in only one out of every six years. Similarly, we would
expect returns to be less than 10% in one out of every six years
(= 20% - 10%).
The Sharpe Ratio in this example is
Annualized_average_return / annualized_standard_deviation_of returns
= 20% / 10% = 2.0 (assuming you are futures)
Now if you double the number of contracts traded, your return would
be 60% in that year, your average return for many years would be 40%
with a standard deviation of 20%. The 60% return for this year is
still one standard deviation away from the mean so you would still
expect to see the 60% return only one year out of every six.
The Sharpe Ratio is still 40% / 20% = 2.0 and we see it is
independent of the number of contracts you traded.
Back to your original question, note that you can calculate the
dollar returns in a period and the dollar standard deviation of
returns without even converting to a percentage and their ratio (the
Sharpe Ratio) remains the same. So as long as you use the same
convention for normalizing the dollar values to percentages, the
Sharpe Ratio is independent of what you used. (Prof. Sharpe calls
this some "notional value").
Also, since the Sharpe Ratio calculation uses typically 30 to 200
different periodic samples of the equity curve for the calculation,
it almost impossible to "curve-fit" a good Sharpe Ratio. We all know
it is easy to get fooled by curve-fitting a single value of return,
net-profit, or other single numbers.
Bob Fulks
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