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Continued from part I
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The formula for Sharpe Ratio is:
Sharpe Ratio = Excess Annualized Return /
Annualized Standard Deviation of Returns
The Excess Annualized Return is the return you get for assuming risk.
If you trade stocks, the Excess Annualized Return is the Annualized
Return on your beginning account equity less the return you could
have gotten by investing in a risk-free investment (usually taken as
T-Bills). I usually use 5% for the T-Bill rate but it does vary a
little.
If you trade futures the Excess Annualized Return is the rate of
return on the amount of margin you have on deposit with your broker.
It also depends upon whether you are getting interest on the margin
on deposit with your broker. If your broker lets you invest the
margin in T-Bills, the Excess Annualized Return would be the excess
you get over the T-Bill rate. If your broker pays you no interest on
your margin deposit, the Excess Annualized Return is the actual
annualized return you achieve by trading.
If you trade a fixed number of contracts and withdraw the profits,
you get no compounding of returns. If you scale up the size of your
trades as your profits accumulate, you will get compounding and in
that case you would sample the logarithm of your account value
(instead of the actual account value) since you would ideally expect
the account value to grow exponentially.
A Sharpe Ratio of 1.0 is considered "pretty good". So with a return
of 20% per year trading stocks and a T-Bill rate of 5%, if the
Annualized Standard Deviation = 15%, the Sharpe Ratio will be:
(20% - 5%) / 15% = 1.0
Remembering that with a normal distribution, in about two thirds of
the years, your rate of return will be within one standard
deviation of the average. So two thirds of the years, your rate of
return will be between 5% (20% - 15%) and 35% (20% + 15%). That means
one sixth of the years the rate of return will be below 5% and one
sixth of the years it will be above 35%. So, with a Sharpe Ratio of
1.0, one out of every six years we will make a lower return than we
could have made with T-Bills.
This is true regardless of how many contracts you trade. If you trade
more contracts, your average return will be higher and your standard
deviation of returns will be higher but the Sharpe Ratio you measure
will stay the same. So you would still expect to have one year out of
six where you made less than T-Bills. The Sharpe Ratio depends upon
the combination of the trading system and the market but NOT upon the
number of contracts you trade.
Investing in an S&P500 index fund over the past five years
("buy/hold") had a Sharpe Ratio of about one.
Things improve dramatically with higher Sharpe Ratios. With a Sharpe
Ratio of 2, you beat T-Bills about 97.5% of the years. That means
2.5% of the years you make less than T-Bills which is one year out of
every 40. With a Sharpe Ratio of 3, it means you beat T-Bills all
except one year out of every 200 years! (These calculations assume a
"normal" distribution which we know is not quite right but you get
the idea.)
A "decent" trading system will have a Sharpe Ratio of at least 2. I
usually shoot for at least 3. My best ones are over 5 and I have seen
values as high as 10 for really great systems.
The Sharpe Ratio reported by TradeStation and by Future Truth and
several others does not seem to be calculated correctly so be careful.
I have found that many people who quote their Sharpe Ratios do not
really understand it or how to calculate it.
The Sharpe Ratio was named for Prof. William Sharpe who was one of
three people who shared the Nobel Prize in Economics in 1990 for his
contributions to what is now called "Modern Portfolio Theory". He is
now a Professor at Stanford and has a lot of interesting papers on
his web site at <http://www.stanford.edu/~wfsharpe/>.
The Sharpe Ratio has one theoretical flaw - its value does not depend
upon the sequence of returns. As an example, if all the losing trades
occurred first followed by all the winning trades, the measured
Sharpe Ratio would be the same as had all trades occurred randomly.
This flaw is mostly theoretical since it hardly ever is a factor in
real testing but there have been various attempts at fixing it.
The first was in Tushar Chande's "Beyond Technical Analysis" (ISBN
0-471-16188-8). Chapter 6 of the book is dedicated to "Equity Curve
Analysis" and describes an approach based upon using the "standard
error" calculation:
> He defines a "risk return ratio" as equal to slope/standard error. This
measure is similar but is not scaled properly to be independent of the
time or price scales so it is difficult to compare different systems.
The second was in an article by Lars Kester "Measuring System
Performance" (Technical Analysis of Stocks and Commodities, March
1996, pages 46-50). His approach, called the K-Ratio, is similar to
Chande's but with some changes:
> He scales the result by the number of observations, which he
assumes are days, but the TradeStation program in the article
uses trades as the proxy for observations which gives a
different result. It still has no absolute measure of goodness
as does the Sharpe Ratio.
For some time, I have used an improved version of the Sharpe Ratio
(which I have modestly called the F-Ratio or Fulks-Ratio) that I
think fixes all of these issues. I will publish it when I get some
time.
But in practice the Sharpe Ratio gives satisfactory results in almost
all cases.
Bob Fulks
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