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At 05:12 PM 6/19/98 -0400, you wrote:
>>Do you or anyone have the formula for the Sharpe ratio? Or a reference to
>>calculating it from an article or book?
>>Neil
>I don't have a formula, but I believe it is the average monthly return
>divided by the standard deviation of of monthly returns. I'ts a reward to
>risk ratio.
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There follows two good posts published here previously on the Sharpe Ratio,
one by Bob Brickey and one by Bob Fulks. The Sharpe Ratio is not as simple
as first appears and it is a widely misused ratio in the financial
community. I think it is included in the new add-on product by Rina Systems
now marketed by Omega Research but I'm not sure how they handle the
anomalies related to margin. Anyway, here are two good posts on the subject.
Bob Brickey's post is first as follows;
"There have been several posts about the "Sharpe Ratio" recently, so I
decided to make some remarks about it.
The Sharpe Ratio is a widely used and often misused measure of investment
performance. It was introduced by William F. Sharpe in 1966 as a measure
for the performance of mutual funds. He currently is a Professor of Finance
at Stanford University's Graduate School of Business. In 1990 he received
the Nobel Prize in Economic Sciences.
Sharpe proposed the term "Reward-to-Variability-Ratio" to describe his ratio
in 1966 and again when he wrote about it in 1975. While his measure has
gained considerable popularity, the name he gave it has not. Other authors
writing about it have called it the Sharpe Ratio, the Sharpe Measure, the
Sharpe Index, and other things. More importantly, many who have written
popular books about it, such as Treynor and Black [1973], Rudd and Clasing
[1982], and Grenold [1989], corrupted the original formula so all values are
positive -- even those for which the mean differential return is negative!
That obviously obscures important information concerning performance.
Misuse is especially wide-spread in the securities investment community,
where few participants are well-educated in statistics. In the investment
arena, a number of authors associated with BARRA (a major supplier of
analytic tools and databases) have used the term "information ratio" to
describe a ratio that is equivalent to the "Sharpe Ratio." Others have used
the same term to encompass the ratio of the mean to the standard deviation
of the distribution of the return on a single investment, such as a fund or
a benchmark. While such a "return information ratio" may be useful as a
descriptive statistic, it lacks a number of the key properties of what might
be termed a "differential return information ratio" and may in some
instances lead to wrong decisions.
That is the way the "Sharpe Ratio" often is misused by traders to score
individual futures trading systems. Sharpe has been annoyed by the misuse
and abuse of his method and has published papers about it. One appeared in
"The Journal of Portfolio Management" in the fall of 1994.
His method is sound in the application for which it was originally proposed.
Broadly defined, it is the ratio of the expected value of a zero-investment
strategy to the standard deviation of that strategy. It is closely related
to the t-statistic used in statistics to measure the statistical
significance of mean differential return. In fact, the t-statistic is
exactly equal to the "Sharpe Ratio" times the square root of T (the number
of return periods used for the calculation). If historic "Sharpe Ratios"
for a set of funds are computed using the same number of observations, the
Sharpe Ratios will be proportional to the t-statistics of the means.
There are both ex ante and ex post measures. The ex ante Sharpe Ratio takes
into account both the expected differential return and the associated risk,
while the ex post version takes into account both the average differential
return and the associated variability. Neither incorporates information
about previous realizations of its own return, which, of course, is a very
important consideration in practical trading.
Whether measured ex ante or ex post, it is essential that the Sharpe Ratio
be computed using the mean and standard deviation of a "differential return,
" or more broadly, the return on what is termed a zero investment strategy.
Used any other way the result is not a true "Sharpe Ratio" and the
indications can be misleading.
The problem with application of the "Sharpe Ratio" to futures trading, is
futures trading does not require an "investment." A margin deposit usually
is required, but a margin deposit is not the same as an investment, because
losses are not limited to the amount of margin deposited and because the
amount of margin deposited has no relationship to the amount that will be
"earned" or lost. Margin is nothing but a guarantee.
Even if for the purpose of comparing different trading systems margin
amounts are assumed to be "investments," a "Sharpe Ratio" could give
misleading indications, because the resulting "ratio" would be unrelated to
the true situation. Two traders with different amounts of margin on deposit
would have different "Sharpe Ratios" trading exactly the same trading system
, even though their profits and losses were identical.
Calculated correctly, the "Sharpe Ratio" for any futures trading system that
makes money is infinite, regardless of the amount of profit returned. The
"Sharpe Ratio" for any futures trading system that loses money is minus
infinity, regardless of the amount of money lost. That is because the
required investment is zero.
-Bob Brickey
Scientific Approaches
sci@xxxxxxxxxx
_________________________________________________________________________
Bob Fulks post follows:
At 1:15 PM -0500 11/1/97, Karl wrote:
>Bob,
>Could you give me the formula for calculating the Sharpe Ratio? I
>found your post interesting and would like to calculate the ratio for some
>of the systems I have developed.
>Many thanks.
>Karl
-------------
At 10:13 AM -0800 11/1/97, Dennis Holverstott wrote:
>> Timer Trend III R-Breaker
>> Trading System Results
>> Annualized Return 25% 31%
>
>Anyone know how they come up with these numbers? 25% of what? 31% of
>what? Margin, 2x margin, margin + maxDD, what?
>
>-Dennis
--------------
The formula is:
(Actual.Return - Risk.free.return) / Standard.deviation.of.returns
(All figures annualized.)
I calculate the log return which is more accurate than simple returns. Log
returns take into account the effects of compounding whereas simple returns
do not. They are about the same for small values of returns.
I use 5.00% for the risk-free return which is the current yield on the 90
day T-Bill.
The formulas below are Excel spreadsheet formulas. The data are the S&P cash
values as an example of the calculations.
Simple return for the first row below would be: (278 - 247) / 247 = 12.55%
Log return for the first row below would be: LN(278 / 247) = 11.82%
Yr S&P LogReturn
87 247 = Beginning value
88 278 12% = LN(278 / 247)
89 353 24% = LN(353 / 278)
90 330 -7% etc.
91 417 23%
92 439 5%
93 466 6%
94 459 -2%
95 616 29%
96 741 18%
Average 12.21% = Average(column above)
Std Deviation 11.71% = STDEVP(column above)
Sharpe Ratio 0.62 = (12.21% - 5.00%) / 11.71%
Average 12.98% = EXP(12.21%) - 1
Std Deviation 12.42% = EXP(11.71%) - 1
Sharpe Ratio 0.64 = (12.98% - 5.00%) / 12.42%
The first set of values are based upon log returns.
The second set is the result of converting the logarithms back to ratios.
I don't know which is correct but the differences are usually slight.
When converted back, the 12.98% average you get is equal to the value you
would get by calculating the average return based upon the first and last
values of the S&P.
12.98% = (741 / 247) ^ (1 / 9) - 1
where:
741 is the last value
247 is the starting value
9 is the number of periods.
The above table uses annual figures. If you use quarterly figures you do the
same calculations but multiply the Average by 4 and the Std Deviation by
SQRT(4) to annualize it.
If you use monthly figures you do the same calculations but multiply the
Average by 12 and the Std Deviation by SQRT(12) to annualize it.
I guess that to be rigorous the above constants actually should be:
(1 + Average%) ^ 4 - 1 and (1 + StdDev%) ^ SQRT( 4) - 1
(1 + Average%) ^ 12 - 1 and (1 + StdDev%) ^ SQRT(12) - 1, respectively
All of this is correct for a buy/hold strategy. The figures in the column
labeled "S&P" include both the initial investment plus the equity build-up
over time.
For calculations for a trading system you need a little pre-processing. The
Equity values given by TradeStation show only the total profit and do not
include the initial investment, so you have to add the initial investment
back into the figures before calculating the returns. The figures in the
column labeled "Equity" below were taken from the graph in the article. The
"Beginning Value" I used was the cost of one contract since the article
gave the return for trading one contract. (The Sharpe Ratio calculation is
pretty insensitive to the "Beginning Value" you use.)
The figures in the column labeled "Acct" are the total of the "Beginning
Value" added to the equity build-up. They are used to calculate returns in
the next column as above.
The Sharpe Ratio is independent of margin so margin does not need to be
included in the calculation. (Using margin, you increase your returns but
also increase your standard deviation of returns by a corresponding amount.)
Data for Time Trend III
Yr Equity Acct LogReturn
87 24.7 Beginning Value = Cost of one contract
88 8 32.7 28% = LN(32.7 / 24.7)
89 37 61.7 63% etc.
90 37 61.7 0%
91 78 102.7 51%
92 78 102.7 0%
93 80 104.7 2%
94 67 91.7 -13%
95 122 146.7 47%
96 160 184.7 23%
Average 22.35% = Average(column above)
Std Deviation 25.46% = STDEVP(column above)
Sharpe Ratio 0.68 = (22.35% - 5.00%) / 25.46%
Average 25.05% = EXP(22.35%) - 1
Std Deviation 29.00% = EXP(25.46%) - 1
Sharpe Ratio 0.69 = (25.05% - 5.00%) / 29.00%
There is a question of what you do with the cash when you are out of the
money with a trading system. You probably should include money market
interest during this period to be rigorous. I don't know what FuturesTruth
does.
Hope this is useful. Perhaps some mathematician in the audience can point
out any errors in the above.
Bob Fulks
--
Bob Fulks
bfulks@xxxxxxxxxxx
__________________________________________________________________
Isn't trading great? ;-)
Michael Paauwe
mpaauwe@xxxxxxxxxx
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