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At 1:37 PM -0800 3/29/02, Lance Fisher wrote:
>Thanks for the informative reply. It's amazing how a seemingly simple
>calculation can become complex when it's examined closely.
>
>Last night after posting my original questions, I continued reading the
>following paper... http://www.stanford.edu/~wfsharpe/art/sr/sr.htm
>
>I'm sure you're familiar with this, but there are issues that further
>complicate things, such as time dependence. This basically states that
>an annual SR will differ from, say, a six month SR.
Actually, that is not an issue if you annualize the numbers. If you
are using monthly log returns, you multiply the returns by 12 and the
monthly standard deviations by SQRT(12). You will get the same value
for Sharpe Ratio if the distribution is "normal" which it is quite
closely.
The issue of normality is a religious issue with many - but a bogus
one. The returns in any single period tend to have fatter tails and
are slightly narrower than a normal distribution. (It is really a
stable Paretian distribution of which the normal distribution is a
special case.) But as you add the returns for each period, the
distribution of the sum (hence the annual average) become very nearly
"normal" after adding only a few monthly terms - as dictated by the
"Central Limit Theorem". Do not let anyone snow you with the
non-normal excuse.
>Sharpe also wrote about two versions of the SR, ex-ante & ex-post, the
>difference in which I don't fully understand at this point. Though it
>appears that the ex-ante version is the SR as you've described it in
>previous posts - Sharpe = (Total %Return - %Risk free return) / StdDev%.
>Whereas the ex-post appears to be - Sharpe = (AVERAGE of differential
>returns) / StdDev of differential returns.
No. Ex-post is calculated after the fact on realized results. Ex-ante
means calculating it on expected returns in the future, (which is
what is important for making money, of course). The problem is how to
get expected returns. The usual way is to extrapolate the past so you
tend to get into circular reasoning.
>So, to recap based on what has been discussed so far, I think I can
>assume the following things concerning the Sharpe Ratio.
>
>1) For the purposes of testing and comparing my own systems, it is safe
>to exclude the risk free rate of return from the SR calculation. This
>must be done with the realization that this will create a small error in
>the final result, and the size of this error will be inversely
>proportionate to the %Return in the equation.
If trading futures, it should be left out. Trading stocks there is an
error if you leave it out but you can estimate the error. I usually
use 5% when trading stocks although that seems high now.
>2) When a particular Sharpe Ratio is claimed for a given system, it is a
>virtually meaningless number if it is not accompanied by some additional
>information, such as - Assumed Risk Free Rate, Time Period over which it
>was calculated, & even Method of Calculation.
It is not as bad as you think. Lots of people calculate it wrong,
including Futures Truth. The reason I furnish software to do it is to
form a de-facto standard. I have communicated with Prof. Sharpe and
he said it looked as if I understood the issues and was calculating
it correctly. (He didn't look through all the cases in detail so I
don't mean to imply that he endorses my methods.) The software
automatically normalizes for time, etc. If you normalize correctly,
the results should be consistent. Errors of 20% are considered minor
- we are concerned whether it is 3 or 4 of 5, not 2.2 or 2.3... The
main problem is that Prof. Sharpe suggested the "measure" and many
text books got it wrong since he was not specific on a lot of things.
His web pages sort of finesses the problem of measuring returns.
>3) The method for calculating SR for a trading system is, Sharpe =
>(Total %Return(over x time) - %Risk free return(over x time)) / StdDev%
>of Individual Trade Returns (over x time).
Calculate periodic % return per my previous post depending upon
whether you are trading constant size or scaling trade size with a
growing account. That is a major source of error. Assume monthly as
an example (but it could be weekly samples, daily samples, etc.)
If you are trading a constant number of contracts of something like
the Nasdaq futures, you have a problem since the value of equity
exposure moved by a factor of three over the past few years. (The
slope of the ideal equity curve [Sharpe Ratio = infinity] in that
case is equal to the value of the cash NDX since that is the rate the
exposure varied over time.)
Then, for stocks, subtract the interest rate on T-Bills for each
period (convert yearly rate to monthly first). That gives you the
excess return for each period (month).
Then annualize the returns my multiplying the returns by the number
of sample periods in a year:
12 for monthly, 52 for weekly, 253 for daily, etc.
(Using the log returns in the scale-trading case automatically takes
care of compounding.)
Then, annualize the standard deviation of excess returns by
multiplying the monthly standard deviation by the the square root of
the above numbers based upon the period. (You are, in effect,
multiplying the "variance" by the above numbers since the standard
deviation is the square root of the variance.)
Now divide the annualized_excess_return by the
annualized_standard_deviation_of_excess_returns to get the Sharpe
Ratio.
Aren't you sorry you asked?
Bob
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