Understanding the Sharpe Ratio How to Create High-Sharpe-Ratio Investments Market Technician’s Association 28 th Annual Seminar May 14, 2004 Bob Fulks

Understanding the Sharpe Ratio How to Create High-Sharpe-Ratio Investments

Market Technician’s Association

28th Annual Seminar

May 14, 2004

Bob Fulks

May 14, 2004 Bob Fulks © 2004 2

Topics

Introduction Understanding the Sharpe Ratio Measuring the Sharpe Ratio of an Investment Measuring the Sharpe Ratio of a Trading System Creating High-Sharpe-Ratio Investments Summary

May 14, 2004 Bob Fulks © 2004 3

Measuring Investment Performance

Prior to the 1950’s Investors measured performance only by their returns Risks were known to exist but not quantified

No objective way to measure investment advisor’s performance regarding risk

“Rules of Thumb” became common “% of stocks in your portfolio should equal your age”

Harry Markowitz (1952) Proposed that “variability of returns” was equally

important Began the field of Modern Portfolio Theory (MPT)

May 14, 2004 Bob Fulks © 2004 4

The Sharpe Ratio

Prof. William Sharpe (Stanford) Proposed the “Sharpe Ratio” as the best

measure of worth of an investment (1966) Derived from Modern Portfolio Theory Measures a “risk-adjusted” return Now widely used and misused…

Markowitz & Sharpe shared a Nobel Prize in Economics (1990) for their work

May 14, 2004 Bob Fulks © 2004 5

ÊÊÊÊDow Jones Industrial AverageÊÊÊÊ Linear Scale

0

2000

4000

6000

8000

10000

12000

14000

1930 1940 1950 1960 1970 1980 1990 2000

Dow Jones Industrial AverageLinear Scale

ÊÊÊÊDow Jones Industrial AverageÊÊÊÊ Log Scale

10

100

1000

10000

100000

1930 1940 1950 1960 1970 1980 1990 2000

Dow Jones Industrial AverageLog Scale

Logarithmic Price Charts

7% per year

7% per year

5% per year5% per year

May 14, 2004 Bob Fulks © 2004 6

Topics


May 14, 2004 Bob Fulks © 2004 7

$10,000

$100,000

$1,000,000

0 5 10 15 20 25 30 35Years

Investment XInvestment YInvestment ZPerfect 15%

Consider Three Investments

All average 15% per year return over 30 years.Which would you prefer?

All average 15% per year return over 30 years.Which would you prefer?

May 14, 2004 Bob Fulks © 2004 8

Annual Returns

X

Y

Z

15%-40%

-20%

0%20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Returns of investment X are much more consistent

Returns of investment X are much more consistent

May 14, 2004 Bob Fulks © 2004 9

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-45% -30% -15% 0% 15% 30% 45% 60% 75%

Return

Investment X

Investment Y

Investment Z

Variability = Standard Deviation

Variability = 20%

Variability = 10%

Variability = 6.7%

“Bell-shaped curve” (Normal Distribution)“Bell-shaped curve” (Normal Distribution)

(Actual distribution is not strictly “normal” but it is a good approximation.)

May 14, 2004 Bob Fulks © 2004 10

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Investments in Two Dimensions

X Y Z

How do we assess the desirability of each?How do we assess the desirability of each?

PreferredRegion

Variability

Retu

rn

Every investment can be represented by a point in the two dimensions

Every investment can be represented by a point in the two dimensions

May 14, 2004 Bob Fulks © 2004 11

Desirability = “Utility”

Utility = Return – 0.5 * A * Variability^2 “A” is “Risk Aversion Factor”

Typical Values of “A” Risk-neutral person 0.0 Futures trader 1.0 Young engineer 2.5 Conservative investor 10 Elderly widow 60

= Standard Deviation^2 = “Variance”

= Standard Deviation^2 = “Variance”

May 14, 2004 Bob Fulks © 2004 12

Determining “A”

$10,000

$100,000

$1,000,000

0 5 10 15 20 25 30 35Years

Investment XInvestment YInvestment ZPerfect 15%

-40%

-20%

0%20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-40%

-20%

0%

20%

40%

60%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

15%

10%

5%

0%

15% CD14% CD13% CD12% CD11% CD10% CD9% CD8% CD7% CD6% CD5% CD4% CD3% CD2% CD1% CD

All have beenaveraging

15% returns InvestmentInvestmentInvestmentInvestmentInvestmentInvestmentInvestmentInvestmentInvestment

InvestmentInvestmentInvestmentInvestmentInvestmentInvestment

May 14, 2004 Bob Fulks © 2004 13

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Util

ity

“Utility” of the Three Investments

X Y ZFutures trader

A = 1.0

Young engineerA = 2.5

Conservative investorA = 10

Elderly widowA = 60

Risk-neutralA = 0

Utility = Return – 0.5 * A * Variability^2Utility = 15% - 0.5 * 2.5 * 20% * 20% = 10%

Utility = Return – 0.5 * A * Variability^2Utility = 15% - 0.5 * 2.5 * 20% * 20% = 10%

Variability

Uti

lit

y

Utility decreases as Variability increases

Utility decreases as Variability increases

May 14, 2004 Bob Fulks © 2004 14

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Position Sizing (Using Leverage)

X Y Z

100% in T-Bills

50% T-Bills / 50% in Z

100% in Z

Dividing Account Between T-Bills and Investment ZDividing Account Between T-Bills and Investment Z

Variability

Retu

rn

200% in Z(using margin)

All possible combinations lie on a straight line.

Where is the best point to be?

All possible combinations lie on a straight line.

Where is the best point to be?

May 14, 2004 Bob Fulks © 2004 15

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

/ Utility

Position Size for Maximum Utility

X Y Z Return

Utilitywidow

Optimum

Optimum

Utility – Young engineer

Utility = Return – 0.5 * A * Variability^2Utility = Return – 0.5 * A * Variability^2

Variability

Utility has a maximum atsome value of Variability

Utility has a maximum atsome value of Variability

Utility – Conservative investor

May 14, 2004 Bob Fulks © 2004 16

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

/ Utility

Optimum Return Points

X Y Z

Conservative InvestorOptimum Return = 10 * Variability^2 + 5%

Conservative InvestorOptimum Return = 10 * Variability^2 + 5%

Utility – Conservative investor

XY

Z

Variability

Optimum return points forany value of “A” all lie ona curve which does not

depend upon the investment

Optimum return points forany value of “A” all lie ona curve which does not

depend upon the investment

May 14, 2004 Bob Fulks © 2004 17

Optimizing Position Size Graphically

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Retu

rn

Variability

An Investment VehicleAn Investment Vehicle

Optimum position size =intersection of the two

Optimum position size =intersection of the two

Optimum Position Size CurveDepends only upon Investor’s

Risk Aversion Factor “A”Optimum Return = A * Variability^2 + 5%

Optimum Position Size CurveDepends only upon Investor’s

Risk Aversion Factor “A”Optimum Return = A * Variability^2 + 5%

Slope of Line = Sharpe RatioDepends only upon Investment

Slope of Line = Sharpe RatioDepends only upon Investment

T-BillsT-Bills

May 14, 2004 Bob Fulks © 2004 18

Optimizing Leverage Graphically

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Retu

rn

Accepting morevariability riskthan optimum

Variability

Achieving lessreturn thanoptimum

All points not on the optimum curve achieve either less return

or more risk than optimum

All points not on the optimum curve achieve either less return

or more risk than optimum T-BillsT-Bills

May 14, 2004 Bob Fulks © 2004 19

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Sharpe Ratio = Slope of the Line

Risk-free Return

Excess Return

Sharpe Ratio = Excess Return / Variability

Sharpe Ratio = Excess Return / Variability

X Y Z

Variability

Sharpe = 0.5Sharpe = 0.5

Sharpe = 1.0Sharpe = 1.0Sharpe = 1.5Sharpe = 1.5

T-BillsT-Bills

May 14, 2004 Bob Fulks © 2004 20

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Re

turn

Optimum Return for Maximum Utility

Optimum point dependsonly upon the Sharpe Ratio

of the investment andRisk Aversion Factor “A”

Optimum point dependsonly upon the Sharpe Ratio

of the investment andRisk Aversion Factor “A”

Optimum Return = A * Variability^2 + 5%Optimum Return = A * Variability^2 + 5%



T-BillsT-Bills


May 14, 2004 Bob Fulks © 2004 21

Optimum return =

(Sharpe^2) / A + risk-free rate

Examples:

(Extreme returns would require unrealistic leverage so we would limit the leverage used and accept lower than optimal returns in those cases)

Return to Maximize Utility

Investor A 0.5 1 1.5 2 3Futures Trader 1 30% 105% 230% 405% 905%Young Engineer 2.5 15% 45% 95% 165% 365%

Conservative Investor 10 7.5% 15% 27.5% 45% 95%Elderly Widow 60 5.4% 6.7% 8.8% 11.7% 20%

Sharpe Ratio

May 14, 2004 Bob Fulks © 2004 22

What is a Good Sharpe Ratio?

Factor 0.5 1 1.5 2 3% of Years that Return < T-Bills 31% 16% 7% 2.3% 0.13%Investment Quality Poor Decent Good Great Super

Sharpe Ratio

Distribution of Returns

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Multiple of Excess Return

Sharpe = 0.5

Sharpe = 1.0

Sharpe = 1.5

Sharpe = 2.0

Sharpe = 3.0

(Based upon a “normal distribution” which is an approximation.)

May 14, 2004 Bob Fulks © 2004 23

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30% 35% 40%

Variability

Ret

urn

Investments with Equal Sharpe Ratios

Variability

Conservative InvestorOptimum Leverage

Young EngineerOptimum Leverage

InvestmentsEqual Sharpe Ratios

InvestmentsEqual Sharpe Ratios

Conservative InvestorOptimum = 10 * Variability^2 + 5%

Conservative InvestorOptimum = 10 * Variability^2 + 5% Young Engineer

Optimum = 2.5 * Variability^2 + 5%

Young EngineerOptimum = 2.5 * Variability^2 + 5%

T-Bills

Elderly WidowOptimum = 60 * Variability^2 + 5%

Elderly WidowOptimum = 60 * Variability^2 + 5%

Investments with equalSharpe Ratios are

equally useful

Investments with equalSharpe Ratios are

equally useful

May 14, 2004 Bob Fulks © 2004 24

0%

100%

200%

300%

400%

500%

600%

0% 50% 100% 150% 200% 250% 300% 350% 400% 450% 500%

Variability

Ret

urn

“Money Management” (Futures)

Young EngineerYoung Engineer

Futures TraderFutures Trader “Optimal ƒ Point”(Maximum Terminal Wealth)

“Optimal ƒ Point”(Maximum Terminal Wealth)

Retu

rn

Conservative InvestorConservative Investor

As position size increasesa single bad loss can

deplete the total account

Variability

May 14, 2004 Bob Fulks © 2004 25

Derivation of Equations

S = (R - F) / V

U = R – 0.5 A V2

R = S V + F

U = S V + F – 0.5 A V2

dU/dV = S – A V = 0

At maximum U = Uo:

Vo = S / A

Ro = S2 / A + F

Ro = A V2 + F

Where:

S = Sharpe Ratio

R = Return

F = Risk-free Rate

A = Risk aversion factor

V = Variability

U = Utility

Ro = Optimum Return

May 14, 2004 Bob Fulks © 2004 26

Fundamentals - Summary

Both Return and Variability are equally important The fundamental worth of an investment is it’s

Sharpe Ratio (not it’s return) Investments with equal Sharpe Ratios are equally

useful (produce the same optimum return) Optimum return increases as Sharpe Ratio squared Process:

Determine your Risk Aversion Factor “A” Maximize Utility by adjusting leverage for optimum

return:

Expected return = (Sharpe^2) / A + risk-free rate

May 14, 2004 Bob Fulks © 2004 27

Caveats

Charts have used past data Unfortunately we must invest

in the future…

So must estimate future Sharpe Ratio from past data

Never blindly use Sharpe Ratios without checking the equity curve

The curves shown at left have identical Sharp Ratios = 1

Must also consider the trend

Equity Curves

10,000

100,000

1,000,000

0 5 10 15 20 25 30

Years

Y

May 14, 2004 Bob Fulks © 2004 28

Topics


May 14, 2004 Bob Fulks © 2004 29

Sharpe Ratio of an Investment

Sharpe Ratio:

Questions: How to define “Return” How to compute “Excess Return” How to “annualize” measurements

Annualized Excess Return

Annualized Standard Deviation of Returns=

May 14, 2004 Bob Fulks © 2004 30

How to Define “Return”

Account value is sampled at equal time intervals (annually, monthly, weekly, etc.)

If investment performance is perfectly consistent, returns for every time intervals should be equal, so that:

Standard Deviation of Returns = zero Sharpe Ratio = excess_return / zero = infinite

Return for the total period should equal the sum of the returns for each time interval

Why? Two reasons…

May 14, 2004 Bob Fulks © 2004 31

1. Annualizing is simple

Annualized return = 12 * Average monthly return = 52 * Average weekly return = 253 * Average daily return

Annualized standard deviation of returns = 12 * Average monthly standard deviation = 52 * Average weekly standard deviation = 253 * Average daily standard deviation

No “Compounding” Required

May 14, 2004 Bob Fulks © 2004 32

2. Distributions tend toward “normal”

Central Limit Theorem (paraphrased) If annual return is equal to the sum of

periodic (weekly, monthly) returns, then the probability distribution of the annual return will tend to be a “normal distribution” almost regardless of what the distribution of the periodic returns is.

May 14, 2004 Bob Fulks © 2004 33

Distributions

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

0 5 10 15 20 25 30 35 40 45 50

Value

1 Period

2 Periods

3 Periods

4 Periods

5 Periods

Normal

Normal

Normal

Central Limit Theorem - Example

Monthly return is uniform distribution

= 0%, 1%, 2% …9%,10%

Equal distribution

= 1/11 = 9.09% Distribution

becomes “normal” after a few months

May 14, 2004 Bob Fulks © 2004 34

Year Value Profit Return T-Bill xReturn

0 10,0001 11,500 1500 15.0% 5.0% 10.0%2 13,000 1500 15.0% 5.0% 10.0%3 14,500 1500 15.0% 5.0% 10.0%4 16,000 1500 15.0% 5.0% 10.0%5 17,500 1500 15.0% 5.0% 10.0%6 19,000 1500 15.0% 5.0% 10.0%

90% Sum: 90.0% 60.0%Average: 15.0% 10.0%StDev: 0.0% 0.0%Sharpe: #DIV/0!

Case 1 – Fixed Trade/Account Size

10,000

12,000

14,000

16,000

18,000

20,000

0 1 2 3 4 5 6Years

Account grows linearly

Straight line on linear chart

Value increases 1500 (15% of original $10,000) per period

Return for the total time equals the sum of the returns for all time intervals

Standard Deviation = Zero Sharpe Ratio =

(15% - 5%) / Zero = Infinite

Investor withdraws profits each period

May 14, 2004 Bob Fulks © 2004 35

Year Value Profit Return T-Bill xReturn

0 10,0001 11,500 1500 15.0% 5.0% 10.0%2 13,000 1500 13.0% 5.0% 8.0%3 14,500 1500 11.5% 5.0% 6.5%4 16,000 1500 10.3% 5.0% 5.3%5 17,500 1500 9.4% 5.0% 4.4%6 19,000 1500 8.6% 5.0% 3.6%

90% Sum: 67.9% 37.9%Average: 11.3% 6.3%StDev: 2.4% 2.4%Sharpe: 2.63

Case 1 – Fixed Trade/Account Size

10,000

12,000

14,000

16,000

18,000

20,000

0 1 2 3 4 5 6Years

Account grows linearly

Straight line on linear chart

Value increases 1500 (15% of original $10,000) per period

Returns calculated wrong and not all equal

Return for the total time not equals the sum of the returns for all time intervals

Standard Deviation Zero Sharpe Ratio Infinite

Investor withdraws profits each period

Error

May 14, 2004 Bob Fulks © 2004 36

Case 2 – Scaling Trade Size

Example: Performance of your money manager

Trade size increases as account grows Consistent returns creates exponentially

increasing account values Return_in_interval =

Value_end_of_interval

Value_start_of_intervalNatural_logarithm of:

May 14, 2004 Bob Fulks © 2004 37

Equations (i.e.: Monthly Terms)

12321 ... RRRRRyear

11

12

2

3

1

2

0

1

0

12 .....1V

V

V

V

V

V

V

V

V

VR

11

12

2

3

1

2

0

1

0

12 ln.....lnlnlnln)1ln(V

V

V

V

V

V

V

V

V

VR

11

lnlnln

ii

i

ii VV

V

VR

May 14, 2004 Bob Fulks © 2004 38

Case 1 – Scaling Trade Size

Account grows exponentially

Straight line on logarithmic chart

Natural logarithm of Value increases (15%) per period

Return for the total time equals the sum of the returns for all time intervals

Standard Deviation = Zero Sharpe Ratio =

(15% - 5%) / Zero = Infinite

Consistent 15%Rate of Increase

$10,000

$100,000

0 1 2 3 4 5 6 7 8 9 10Years

Investor reinvests profits each period

Year Value LN(Value) Return T-Bill xReturn

0 10,000 9.211 11,618 9.36 15.0% 5.0% 10.0%2 13,499 9.51 15.0% 5.0% 10.0%3 15,683 9.66 15.0% 5.0% 10.0%4 18,221 9.81 15.0% 5.0% 10.0%5 21,170 9.96 15.0% 5.0% 10.0%6 24,596 10.11 15.0% 5.0% 10.0%

LR: 90% Sum: 90.0% 60.0%AR: 146% Average: 15.0% 10.0%

StDev: 0.0% 0.0%Sharpe: #DIV/0!

May 14, 2004 Bob Fulks © 2004 39

Calculating “Excess Return”

If you tie up money you must deduct the “risk-free” interest rate on that money to get the “excess return”

Examples (assuming risk-free rate = 5%): $100,000 stock portfolio or $100,000 managed account:

Subtract 5% of $100,000

$20,000 futures account making $100,000 trades: Subtract 5% of $20,000

Futures account using a bond portfolio as collateral, making $100,000 trades

Subtract nothing. (No money tied up)

May 14, 2004 Bob Fulks © 2004 40

Topics


May 14, 2004 Bob Fulks © 2004 41

Sharpe Ratio of a Trading System

A trading system has two components Market Timing System

When to go Long, Short, or Flat

Position Sizing System (“Money Management”)

What size is each position vs. time

The two are quite different Ideally we should measure

performance of each separately

MT

P

E

P

Position Sizing

Market Timing

Price Data

Equity Data

Price Data

PS

May 14, 2004 Bob Fulks © 2004 42

Algorithm to Calculate Return

Measures Sharpe Ratio of the Market Timing system

Normalizes Return to Trade Size (“Invest”)

Eliminates position size as a factor in the measurement

New Position?

Invest = Shares * Price

PrevAccValInvest

AccValPrevAccVal

* LnReturn =

Invest = PrevAccVal

Yes

No

May 14, 2004 Bob Fulks © 2004 43

SharpeMeasure Indicator

Timing Sharpe = 2.78Timing Sharpe = 2.78

Trade Size as a Percent of Account Size

Cumulative Returns

Distribution of ReturnsOne bar for each sampleOne bar for each sample

May 14, 2004 Bob Fulks © 2004 44

Shares = ($10000 + NetProfit) / Price


Cumulative Returns


Trade Size as a % of Account Size

May 14, 2004 Bob Fulks © 2004 45

Shares = 100


Cumulative Returns



May 14, 2004 Bob Fulks © 2004 46

Shares = $100,000 / Price


Cumulative Returns



May 14, 2004 Bob Fulks © 2004 47

Shares = 1000 / AverageTrueRange


Cumulative Returns



May 14, 2004 Bob Fulks © 2004 48

Measuring Sharpe Ratio - Summary

Investment Fixed trade/account size:

Scaling trades with account size:

Trading System Optimize Market Timing Sharpe Ratio first Then add Position Sizing and optimize overall Trading

System Sharpe Ratio

1

lni

ii V

VR

eAccountSiz

VVR iii

1

May 14, 2004 Bob Fulks © 2004 49

Cautions!

Beware of quoted Sharpe Ratio numbers you see. They are probably not correct.

Prof. Sharpe described the concept, not the details of how to define everything. (Actually he didn’t even call it “Sharpe Ratio”)

There is a lot of confusion and lots of people measure it incorrectly. (TradeStation’s reported value is wrong!)

May 14, 2004 Bob Fulks © 2004 50

Topics


May 14, 2004 Bob Fulks © 2004 51

Creating High-Sharpe Ratio Investments

They do not occur in nature! Buy/Hold and an “Index Fund” are very poor

investments

Must be created using “Financial Engineering” Improve “Index Fund” Based Investments

with Market Timing Use Market Neutral Portfolios Use Dynamic Asset Allocation Use Other Zero-Beta Spreads Putting it all together

May 14, 2004 Bob Fulks © 2004 52

$3,47812.1%10.2%

$1

Red denotes my annotations

May 14, 2004 Bob Fulks © 2004 53

$30.35(Less than

1% ofpreviouschart!)

4.5%(Does not includetransaction costs

or state taxes)

$1

Red denotes my annotations

May 14, 2004 Bob Fulks © 2004 54

S&P 500Growth at Excess Return Rate

1000

10000

100000

1/35 1/40 1/45 1/50 1/55 1/60 1/65 1/70 1/75 1/80 1/85 1/90 1/95 1/00 1/05

Date

Val

ue

Buy/Hold - Growth At Excess-Return Rate

Real-Excess-Return = Real-Total-Return less T-Bill Return

Sharpe = 0.26

Sharpe = 0.62

Sharpe = (0.41)

Sharpe = 1.57

Sharpe = 0.18

Sharpe = (0.70)

May 14, 2004 Bob Fulks © 2004 55

Buy/Hold Conclusions - Inflation Adjusted

Over 67 years, buying the S&P 500 index has been a very poor investment. Sharpe Ratio only 0.26

However, there are some very good and very bad intervals... Can we identify those

intervals? Isn’t that Market Timing?

BH

E

P

Buy/Hold

Equity Data

Price Data

May 14, 2004 Bob Fulks © 2004 56

“But market timing doesn’t work”

Conventional wisdom: “Everybody know market timing doesn’t

work” Elliot Spitzer would have you believe it’s

illegal… “Missing the best 1% of days in the market

wipe cout 90% of the returns”“…so you had better stay in no matter what…”

True, but…

May 14, 2004 Bob Fulks © 2004 57

Market Timing Study

Invest $1000 in the S&P 500 index in 1/5/70 On 4/1/01 (7902 trading days later): Value =

$11,839 Average gain = 8.2%/yr

Missing the best 1% (79) days: Value = $894 Average loss = 0.3%/yr

Missing the worst 1% (79) days: Value = $206,445 Average gain = 17.8%/yr

What would the account be worth if we avoided all the down days over the 30+ years?

May 14, 2004 Bob Fulks © 2004 58

Market Value vs. Days in Market

$1,894,433,971,569,410

($1.89 million billion)

$11,839$11,839

May 14, 2004 Bob Fulks © 2004 59

Market Timing - Conclusions

Market timing has enormous potential… But is hard to do

Requires a disciplined process… A “trading system”

Let’s look at some examples… The “Murphy Model” A pattern-based system

May 14, 2004 Bob Fulks © 2004 60

“Murphy Model” Trading System

Based on an idea from John Murphy 50 day & 150 day moving averages if Price > both then be Long if Price < both then be Short otherwise be out of the market (Somewhat modified to reduce whipsawing)

May 14, 2004 Bob Fulks © 2004 61

Murphy Model – Nasdaq Comp Index


Averages 3 trades per year over almost 33 years

Fixed $100,000 Trade Size

May 14, 2004 Bob Fulks © 2004 62

Murphy ModelPerformance

1,000

10,000

100,000

1,000,000

01/70 01/75 01/80 01/85 01/90 01/95 01/00 01/05

Date

Murphy Model – Nasdaq Comp Index

Adding Position Sizing Trade Size = Account Value

Increases Performance Account grows

exponentially Return: 14.8% / Yr. Sharpe Ratio: 1.5

Much better than Buy/Hold

Conclusion: Even simple systems can

be effective

14.8% / Yr.

Buy/Hold

Trade Size = Account Size

May 14, 2004 Bob Fulks © 2004 63

Pattern-Based Trading Systems

Some systems based upon repeatable patterns

Eugene Fama, “Efficient Market Hypothesis”. “All information on markets is widely available so

the market is efficient and the price chart should be a “random walk” with no tradable patterns.

After 40 years no one has proven the hypothesis

Now even economists are beginning to find tradable “patterns”…

May 14, 2004 Bob Fulks © 2004 64

Nasdaq 100 Index – Daily Values

Do you see any patterns?Do you see any patterns?

May 14, 2004 Bob Fulks © 2004 65

Nasdaq 100 Index – 6 Bars/day

Now do you see any patterns?Now do you see any patterns?

May 14, 2004 Bob Fulks © 2004 66

Market Patterns - Example

Why do markets trade in channels?Why do markets trade in channels?

Bottom of ChannelBottom of Channel

Top of ChannelTop of Channel

May 14, 2004 Bob Fulks © 2004 67

Market Timing System


May 14, 2004 Bob Fulks © 2004 68

Summary - Market Timing

Is very effective for achieving high Sharpe Ratios

Systems can be hard to design Day trading is a full-time job Resources:

Commodity Trading Advisors “Market timer” Money Managers

www.MoniResearch.com newsletter www.SAAFTI.com (Trade Association)

MT

E

P

Market Timing

Price Data

Equity Data

May 14, 2004 Bob Fulks © 2004 69

Market Neutral Portfolios

So an Index fund is a poor investment 2% to 3% real after-tax return over 78 years Daily changes can exceed 2% to 3% Sharpe Ratio = 0.26 (before taxes) “Full of sound and fury, signifying (almost)

nothing…”

And most investments tend to follow the market indices to some extent.

So why not get rid of the market dependence? Result is a “Market Neutral Portfolio”

May 14, 2004 Bob Fulks © 2004 70

A Modern “Asset Allocated” Portfolio

May 14, 2004 Bob Fulks © 2004 71

Past Performance of the Portfolio

Fund portfolio is highly correlated with S&P 500 Index

Clearly doing better than the Index

What if we subtract out the S&P 500 Index component…

Result would be “Market Neutral”

Modern Fund Portfolio

S&P 500 Index

May 14, 2004 Bob Fulks © 2004 72

Creating a “Market Neutral” Portfolio

0%

2%

4%

6%

8%

10%

12%

14%

0% 5% 10% 15% 20% 25%

Variability

Re

turn

100% Funds0% Short

Sharpe = 0.34

0% Funds100% Short

65% Funds35% ShortSharpe =

0.78

A * Variability^2 + 3%(A = 10)

Funds

MarketNeutral9.2%

7.3%

12.4%7.7%

May 14, 2004 Bob Fulks © 2004 73

Traditional = Index + Market Neutral

TraditionalPortfolio

IndexFund

Minus =MarketNeutralPortfolio

AnyTraditionalPortfolio

IndexFund

= PlusMarketNeutralPortfolio

OptimizeSeparately

Optimizing thisis difficult

If:

Then:

Optimize withMarket Timing

May 14, 2004 Bob Fulks © 2004 74

Optimizing Market Neutral Portfolios

The “Single Index” Model of Returns (Sharpe) Strategy to Optimize Market Neutral Portfolio:

Make “Alpha” as high as possible Make “Beta” = zero Minimize the “Noise” term

ReturnPortfolio = Alpha + Beta * ReturnIndex + Noise

Market

Neutral

Term

MarketDependen

tTerm

AllElse

May 14, 2004 Bob Fulks © 2004 75

Single Stock

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return_Index

Ret

urn

_Sto

ck

Returns: Stock vs. Market Index

Slope = “Beta”Slope = “Beta”

Intercept = “Alpha”Intercept = “Alpha”

Daily ChangesDaily Changes

“Best Fit” linearregression line

“Best Fit” linearregression line

Return_Stock = Alpha + Beta * Return_Index + Noise

May 14, 2004 Bob Fulks © 2004 76

Traditional Portfolio

Position Sizing is very complex because Price Data are correlated

Markowitz optimization is hard to use

Too many estimates required

Result very sensitive to assumptions

Diversification decreases “Noise” term

Position Sizing

P P P P P

E

BH Buy/Hold

Equity Data

Price Data

May 14, 2004 Bob Fulks © 2004 77

Unhedged Portfolio

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return_Index

Ret

urn

_Po

rtfo

lio

Returns: Portfolio vs. Market Index

“Noise” decreases as thesquare root of number of

stocks in the portfolio

“Noise” decreases as thesquare root of number of

stocks in the portfolio

Slope = “Beta”Slope = “Beta”

Intercept = “Alpha”Intercept = “Alpha”

Single Stock

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return_Index

Ret

urn

_Sto

ck

May 14, 2004 Bob Fulks © 2004 78

Adding the “Neutralizer” Tool

“Neutralizer” tool Adds short position in

Index Cancels “Beta” of

Portfolio

Eliminates most day-to-day portfolio fluctuations

Increases Sharpe Ratio

Position Sizing

P P P P P

I

E

BH

N

Buy/Hold

Equity Data

Price Data

NeutralizerIndex Data

May 14, 2004 Bob Fulks © 2004 79

Hedged Portfolio

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return_Index

Ret

urn

_Hed

ged

Adding the “Neutralizer” Tool

AlphaAlpha

Hedged to Beta = 0(Market Neutral)

Hedged to Beta = 0(Market Neutral)

Unhedged Portfolio

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

-10% -8% -6% -4% -2% 0% 2% 4% 6% 8% 10%

Return_Index

Ret

urn

_Po

rtfo

lio

May 14, 2004 Bob Fulks © 2004 80

$0

$500,000

$1,000,000

$1,500,000

$2,000,000

$2,500,000

$3,000,000

07/02 08/02 09/02 10/02 11/02 12/02 01/03 02/03 03/03 04/03 05/03 06/03 07/03

Portfolio

Long

Short

EndPoints

Return: 240%Std. Dev.: 9.2%Sharpe: 22.2

Example: Stock Portfolio

Past performance of our 6/19/03 portfolio as designedPast performance of our 6/19/03 portfolio as designed

May 14, 2004 Bob Fulks © 2004 81

Portfolio Performance - 2003

I used this method all last year

Worked well most of the time

Problem: Group of our stocks

began dropping faster than the hedge

Solution: Move “Neutralizer”

the stock level

In-sample Out-of-sample

Design Date

Portfolio

Hedge

Stocks

May 14, 2004 Bob Fulks © 2004 82

First Improvement

“Neutralize” each stock separately

Resulting price data mostly uncorrelated

Position Sizing become much simpler

Position Sizing

E

P

I

BH

N

P

N

P

N

P

N

P

N

Buy/Hold

Equity Data

May 14, 2004 Bob Fulks © 2004 83

Combined

Stock

Market

Daily DataDaily Data

May 14, 2004 Bob Fulks © 2004 84

Weekly DataWeekly Data

May 14, 2004 Bob Fulks © 2004 85

Observations

Neutralized price curve: Is much smoother Tends to trend better

Problem: Does not trend consistently in either direction

Solution: Add a “Market Timing” system for each stock

May 14, 2004 Bob Fulks © 2004 86

Second Improvement

Add Market Timing system for each stock

Simple since neutralized stocks trend nicely

Position Sizing now combines Equity Data

Tends to be uncorrelated

Sharpe Ratio increases as the square root of the number of stocks

Position Sizing

E

P

I N

P

N

P

N

P

N

P

N

MT MT MT MT MT

Equity Data

MarketTiming

May 14, 2004 Bob Fulks © 2004 87

Summary – Optimizing Market Neutral

For each stock, mutual fund, or ETF: Neutralize to Beta = 0 by shorting an index-

based vehicle (Bear-Fund, Futures, ETFs, Options, etc.)

Apply a trading system to create rising equity curve

Combine resulting equity curves with a “Position Sizing” process Tends to be simple since all components are

uncorrelated The diversification further increases Sharpe Ratio

May 14, 2004 Bob Fulks © 2004 88

Dynamic Asset Allocation (“DAA”)

“Asset Allocation” well known technique to reduce portfolio risk

How can we improve it? Solution:

“Dynamic Asset Allocation” between asset classes

Vary the mix adaptively to maximize Sharpe Ratio

Use short positions to become market neutral

May 14, 2004 Bob Fulks © 2004 89

May 14, 2004 Bob Fulks © 2004 90

Dynamic Asset Allocation (“DAA”)

Chart shows which asset classes did best in each year

Intended message: “Hold a little bit of each asset

oclass all the time”

But why do that when you can measure which is doing better? Zero Beta Spreads

PL

N

PS

P

May 14, 2004 Bob Fulks © 2004 91

Zero Beta Spreads

Calculate zero beta spread of two indices

Trend is obvious – we now want to be:

Long S&P 400 Short S&P 500

Spread much smoother than either component

Sharpe Ratio = 1.6

May 14, 2004 Bob Fulks © 2004 92

Other Zero-Beta Spreads

There are many possible Zero-Beta spreads

Exchange-traded funds are very useful Can sell short as well as long Inherent diversification vs. stocks reduces

portfolio “Variability”

May 14, 2004 Bob Fulks © 2004 93

Exchange-Traded Fund Spread Map

Long Position

Short

Posi

tion

Sharpe > 2.5

Sharpe > 2.0

Sharpe > 1.5

May 14, 2004 Bob Fulks © 2004 94

Putting It All Together

Position Sizing

E

P

I N

P

N

P

N

P

N P

MT MT MT MT MT

Equity Data

PL

N

PS PL

N

MT MT

Stocks or ETFs“DAA”

SpreadsETF

Spreads

PS

IndexMarketTiming

May 14, 2004 Bob Fulks © 2004 95

But “The Devil is in the Details…”

Beta = zero But which Beta? Beta varies over time

Cancel Beta with a market index But which market index?

Improve parameter measurements Classical techniques have many flaws Signal Processing techniques are much

better

May 14, 2004 Bob Fulks © 2004 96

Parameter Measurements

Alpha

Beta

Classical Techniques Equal weighted data Lag = half of window width Noise from old data leaving

window Nulls in frequency response No smoothing

Signal Processing Approach Emphasize recent data Less lag No noise from old data No nulls in frequency

response Smoothed

May 14, 2004 Bob Fulks © 2004 97

Summary

Neutralizer removes overall market fluctuations and most correlation

Trading system improves Sharpe Ratio and removes most remaining correlation

Position sizing is much simpler since all components are uncorrelated

Diversification further increases Sharpe Ratio as the square root of the number of components used

Final portfolio is Market Neutral = “Absolute Returns”

May 14, 2004 Bob Fulks © 2004 98

Conclusions

The Sharpe Ratio is the best quality measure of an investment

High-Sharpe-Ratio investments are hard to find but can be created using “Financial Engineering”

These techniques aren’t “rocket science” but are beyond the capabilities of the average investor

Opportunity for Advisors & Mutual/Hedge Funds?

Further reading: Prof. Sharpe’s web site: http://www.wsharpe.com “Investments” Bodie, Kane, & Marcus

My email: [email protected]

May 14, 2004 Bob Fulks © 2004 99

Documents

Understanding the Sharpe Ratio How to Create High-Sharpe-Ratio Investments Market Technician’s Association 28 th Annual Seminar May 14, 2004 Bob Fulks