THE SHARPE RATIOby Bob Fulks

---------------

Sharpe Ratio

The Sharpe Ratio is a measure of the risk-adjusted return of an investment.

It was derived by Prof. William Sharpe, now at of Stanford University 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". Prof. Sharpe's web site at http://www-sharpe.stanford.edu/ has

several papers on this topic.

The calculation is pretty straightforward. You invest money in some

investment. You then calculate the value of your investment account

(including the initial investment plus the profit/loss) periodically, say

for example, every month. You then calculate the percentage return in each

month. It doesn't matter what kind of investment. It could be simply buying

and holding a single stock, or trading several different commodities with

several different trading systems. All that matters is the account value at

the end of each 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 receive 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.

Mutual Funds

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.

Trading Systems

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 Sharpe Ratio reported by

services such as Future Truth are calculated in some other way and get

other numbers.)

Buy/Hold Example

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 to 479,000

over the two years (not the real numbers), 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

(The real number over this period was about 0.60. This is just an

example of the calculation method so do not assume these numbers

are correct.)

Buy/Hold Using Margin

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.

The actual figures for SPDRs over this period show this effect clearly. The

table below shows the numbers. The column "Leverage" is the ratio of the

investment to our account equity. A "Leverage" of 2 means we are investing

twice the value of our equity, borrowing the balance as a margin loan.

Annualized

Leverage StdDev Return Sharpe

1 46% 32% 0.58

2 73% 42% 0.50

3 126% 63% 0.46

4 182% 82% 0.42

5 242% 99% 0.39

6 286% 107% 0.36

7 358% 123% 0.33

8 437% 138% 0.30

9 523% 152% 0.28

10 592% 158% 0.26

As you can see, as the leverage is increasing, both the Standard Deviation

and the Return increase, but not at the same rate so the Sharpe Ratio is

decreases.

This is because losses tend to hurt worse than gains help. The following

data will illustrate this:

If, without leverage, an investment loses 10% in one month, it would

require 11% return to get back to where we started. [90% * 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% * 125% = 100%]

Achieving a 20% gain would only get up back to 96% of the original value.

[80% * 120% = 96%]

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% * 200% = 100%]

Achieving a 50% gain would only get up back to 75% of the original value.

[50% * 150% = 75%]

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

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.

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

(and assume that we get the same trading signals in this simplified

example). 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, and that we use the same trading system and that it

gives the same trading signals. It also neglects dividends. We know this is

not the case, that futures prices are more volatile and most trading

systems will give different signals 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.

Conclusions

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, ("Chaos and Order in the Capital

Markets") 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.

There is much controversy on the practice of using standard deviation as

the measure of risk. Clearly, most people would prefer upside risk to

downside risk. But you can argue that upside risk is also bad. Uncertainty

is really the problem since if an investment had consistently more upside

risk, you are losing the opportunity to invest more money in it vs. some

other investment. Many people still use standard deviation but some,

including MorningStar, use a modified version for their risk ratings.

Bob Fulks

June 22, 1998

Lorenzo V. [email protected]

TORNA ALLA HOMEPAGE

1998

---------------

Sharpe Ratio

The Sharpe Ratio is a measure of the risk-adjusted return of an investment.

It was derived by Prof. William Sharpe, now at of Stanford University 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". Prof. Sharpe's web site at http://www-sharpe.stanford.edu/ has

several papers on this topic.

The calculation is pretty straightforward. You invest money in some

investment. You then calculate the value of your investment account

(including the initial investment plus the profit/loss) periodically, say

for example, every month. You then calculate the percentage return in each

month. It doesn't matter what kind of investment. It could be simply buying

and holding a single stock, or trading several different commodities with

several different trading systems. All that matters is the account value at

the end of each 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 receive 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.

Mutual Funds

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.

Trading Systems

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 Sharpe Ratio reported by

services such as Future Truth are calculated in some other way and get

other numbers.)

Buy/Hold Example

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 to 479,000

over the two years (not the real numbers), 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

(The real number over this period was about 0.60. This is just an

example of the calculation method so do not assume these numbers

are correct.)

Buy/Hold Using Margin

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.

The actual figures for SPDRs over this period show this effect clearly. The

table below shows the numbers. The column "Leverage" is the ratio of the

investment to our account equity. A "Leverage" of 2 means we are investing

twice the value of our equity, borrowing the balance as a margin loan.

Annualized

Leverage StdDev Return Sharpe

1 46% 32% 0.58

2 73% 42% 0.50

3 126% 63% 0.46

4 182% 82% 0.42

5 242% 99% 0.39

6 286% 107% 0.36

7 358% 123% 0.33

8 437% 138% 0.30

9 523% 152% 0.28

10 592% 158% 0.26

As you can see, as the leverage is increasing, both the Standard Deviation

and the Return increase, but not at the same rate so the Sharpe Ratio is

decreases.

This is because losses tend to hurt worse than gains help. The following

data will illustrate this:

If, without leverage, an investment loses 10% in one month, it would

require 11% return to get back to where we started. [90% * 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% * 125% = 100%]

Achieving a 20% gain would only get up back to 96% of the original value.

[80% * 120% = 96%]

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% * 200% = 100%]

Achieving a 50% gain would only get up back to 75% of the original value.

[50% * 150% = 75%]

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

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.

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

(and assume that we get the same trading signals in this simplified

example). 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, and that we use the same trading system and that it

gives the same trading signals. It also neglects dividends. We know this is

not the case, that futures prices are more volatile and most trading

systems will give different signals 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.

Conclusions

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, ("Chaos and Order in the Capital

Markets") 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.

There is much controversy on the practice of using standard deviation as

the measure of risk. Clearly, most people would prefer upside risk to

downside risk. But you can argue that upside risk is also bad. Uncertainty

is really the problem since if an investment had consistently more upside

risk, you are losing the opportunity to invest more money in it vs. some

other investment. Many people still use standard deviation but some,

including MorningStar, use a modified version for their risk ratings.

Bob Fulks

June 22, 1998

Lorenzo V. [email protected]

TORNA ALLA HOMEPAGE

1998