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At 11:18 AM -0600 6/19/98, Neil Harrington wrote:
>
>Do you or anyone have the formula for the Sharpe ratio? Or a reference to
>calculating it from an article or book?
Below is a more recent explanation I prepared for another purpose. Sorry it
is so long but at the time I needed to include examples to make the points
clear.
I think it is correct but I would appreciate it if anyone who sees an error
would notify me.
Bob Fulks
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Sharpe Ratio
The Sharpe Ratio is a measure of the risk-adjusted return of an investment.
It was derived by Prof. William Sharpe of Stanford Univ. who was one of
three economist who received the Nobel Prize in Economics in 1990 for their
contributions to what is now called "Modern Portfolio Theory".
The calculation is pretty straightforward. You invest money in some
investment. You then calculate the value of your investment (including the
initial investment plus the profit/loss) periodically, say for example,
every month. You then calculate the percentage return in every month.
Then calculate the average monthly return over some number of months, say
for example, 24 months, by averaging the returns for the 24 months. You
also calculate the standard deviation of the monthly returns over the same
period.
Then annualize the numbers by:
> Multiplying the average monthly return by 12
> Multiplying the standard deviation of the monthly returns
by the square root of 12
You also need a number for the "risk-free return" which is the annualized
return currently available on "risk-free" investments. This is usually
assumed to be the return on a 90-day T-Bill which is now about 5% per year.
You now calculate the "Excess return" which is the annualized return
achieved by your investment in excess of the risk-free rate of return
available. This is the extra return you get by assuming some risk. (Risk is
measured by the standard deviation of the returns, which is actually the
"variability" of the returns.)
<Excess_return> = <Annualized_annual_return> - <Risk_free_return>
Then you calculate the Sharpe Ratio as follows:
Sharpe = <Excess_return> / <Annualized_standard_deviation_of_returns>
which gives you the Sharpe Ratio of the past returns over the past 24 months.
This is pretty straightforward when you invest in stocks or mutual funds
not using margin. If you use margin, or invest in futures contracts, it is
a little more complicated. An example below will illustrate this.
If the investment was in buying and holding a mutual fund, you will get a
number between about 0.5 and 3. These numbers are available for most mutual
funds on the Morningstar web site at http://www.morningstar.net/
(subscription required). They say that a Sharpe Ratio of over 1.0 is
"Pretty good". Outstanding funds achieve something over 2.0.
If you are "investing" in a system for trading, you still measure the value
of your account with the profit/loss resulting from the trades. You are, in
effect sampling the value of the equity curve (plus the initial investment
as defined above). An example will clarify this.
As above, a Sharpe Ratio of a system of over 2.0 is considered very good.
Sharpe Ratios above 3.0 are outstanding. (The Sharp Ratio reported by
services such as Future Truth are calculated in some other way and get
other numbers.)
Assume we established an account in 6/96 and bought 5000 SPDRs (S&P
depository receipts). The total value of what we bought would be about
$335,000 at that time [$67 x 5000]. If our account increased on average
$72,000 per year, the average return would be $6,000 per month or about
1.80% per month of the original $335,000. Annualized, this would be about
21.5% [1.80% * 12]. Assume the standard deviation of monthly returns in our
account is 2.4%. Annualizing this we get 8.31%. [2.4 * SQRT(12)].
Excess return (excess over risk-free return) is 21.5% - 5.0% = 16.5%.
Sharpe Ratio = 16.5% / 8.31% = 1.99 which is not bad.
Now consider the use of margin. Assume we bought twice as many SPDRs as
above but borrowed half the money from our broker as a margin loan. Our
investment would be the same but now our monthly returns will be twice as
great so our annualized return would be 43% [2 * 21.5%] before interest on
the margin loan. Assuming margin interest at 5%, our net return would be
38% [43% - 5%]. The excess return is now 33% [38% - 5%]. The standard
deviation of returns would also double to 16.62%. So the Sharpe Ratio
becomes:
Sharpe Ratio = 33% / 16.62% = 1.99 which is the same as above.
Thus, by increasing the leverage, we have increased the returns and the
risk (= variability of returns = standard deviation) but have not changed
the Sharpe Ratio. Thus, the "risk-adjusted return" is the same.
Using a moderate level of leverage increases the return and risk but leaves
the Sharpe Ratio unchanged. But with very high leverage, the situation gets
much worse. The standard deviation of returns continues to increase as the
leverage is increased, but the returns may not continue to increase
proportionately because losses begin to hurt more that the gains help:
> If, without leverage, our investment loses 10% in one month,
it would require 11% return to get back to where we started.
[90% of 111% = 100%]
> But with a leverage of 2 to 1, this investment would lose 20%
in that same month. Then it requires 25% to get even.
[80% of 125% = 100%]
> With leverage of 5 to 1, this investment would lose 50% in
that month. Then it would require 100% return to get even again.
[50% of 200% = 100%]
> With leverage of 10 to 1, this investment would lose 100% in
that month and we would be broke.
So with higher and higher leverage, the standard deviation continues to
increase and the variations in monthly returns bias the returns lower than
we would otherwise expect. This lowers the risk-adjusted return and the
Sharpe ratio. So the Sharpe Ratio is roughly independent of leverage only
so long as the standard deviation doesn't get too high.
Futures have an inherent leverage of as much as 10 to 1 which is why it is
so easy to go broke trading them. We have to severely limit the monthly
(daily) losses, (e g: lower the standard deviation of returns or increase
the Sharpe Ratio) to avoid going broke.
As you can see a trading system with a smoothly increasing equity curve
will have very consistent monthly returns, a low standard deviation of
returns, and a high Sharpe Ratio. This greatly reduces the chances of going
broke. Smooth equity curves are good. Choppy equity curves are risky. So we
should always optimize our trading system for the highest Sharpe Ratio.
With futures, the calculation is similar to the above except that the
"margin loan" is interest free. (It is built into the futures price as part
of the "fair value" calculation.) It is a little more confusing since you
are really using margin inherently. Again an example.
Assume that on 6/96, we purchased five S&P contracts. We are really buying
about $1,675,000 worth of equity. [5 * $670 * 500 big points]. Assume the
margin requirement was $33,500 per contract or $167,500 for 5 contracts.
Assume we want to allow twice this in our account to cover drawdowns so we
establish an account of $335,000 (just as in the above example). We will
use exactly the same position trading system we used for trading the SPDRs.
This is exactly the same as was trading the SPDRs on margin with two
changes:
> The leverage is now 5 to 1 instead of 2 to 1.
> There is no interest on the "margin loan" inherent in the
futures contract.
> The broker pays us interest on the margin we have on deposit.
(Should, but maybe doesn't.)
With no margin, the return on our system trading SPDRs was 21.5%. Trading
futures, the return will be 5% less than this since the price of S&P
futures decreases to the S&P cash index nearing expiration (e g: "premium"
decreases to zero) with the "fair value" calculation. It turns out that the
effective return would be 16.5% before applying the leverage multiplier.
But we make 5% on the margin we have on deposit with the broker. The
calculations now become:
Annualized return = 16.5% * 5 = 82.5%
Interest income on margin = 5%
Total return = 82.5% + 5% = 87.5%
Excess return = 87.5% - 5% = 82.5%
Annualized standard deviation = 8.31% * 5 = 41.55
Sharpe Ratio = 1.99 (which is the same as above)
You can see that the interest we earn on the margin amount equals the
risk-free rate so the two amounts cancel. Thus, the 82.5% we make can be
used as the "excess return" if we are getting interest on our margin
deposit. (There are other ways of getting to this same conclusion but I use
this way to be consistent with the above examples.)
But this will not be the case because, as explained above, with a standard
deviation that high, the amount we lose in down months will not be made up
for by the amount we gain in the up months so the return and Sharpe Ratio
will be somewhat lower in practice (with everything else being equal). We
can reduce this effect by using less leverage, which means having more
capital in the account.
Thus, trading futures is no different than trading SPDRs if we use the same
leverage in both cases. We would get the same result if we traded SPDRs
with a leverage of 5 to 1.
(These conclusions assumes that the price of the futures exactly tracks the
price of the SPDRs except for the "premium"; that commissions, slippage,
etc., are all the same. We know is not the case, that futures prices are
more volatile but it makes the principles easier to understand in this
simple example.)
The benefit, and danger, of futures is the inherent leverage that is built in.
The Sharpe Ratio, which is a reward to risk ratio, is independent of the
leverage we use so long as the standard deviation is small. Increasing the
leverage increases the risk and reward proportionately. But as the standard
deviation gets larger, the benefit of leverage starts decreasing. In fact,
at some point, increasing leverage further decreases the return because the
amount lost in bad months (or days) is not made up for in good months
(days).
Understanding the Sharpe Ratio of your trading system is fundamental to
understanding the risk involved in trading it.
(Additional caveats for the fussy. If the returns follow a "normal
distribution" it should not matter whether you sample the equity curve
daily, weekly, monthly, etc. The process of annualizing the numbers should
give the same result. But such is not the case since the distribution of
returns and not quite "normal". The distribution typically has higher peaks
and fatter tails than does a normal distribution. Work by Edgar Peters
indicates that this is the result of a fractal behavior with roots in chaos
theory. But most of the work in Modern Portfolio Theory still assumes the
normal distribution as I have above.)
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