comparing sharpe ratios: so where are the p-values?. opdyke -- jsm2006 - activity 468.pdf ·...

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Comparing Sharpe Ratios: So Where are the p-values?

J.D. Opdyke, DataMineIt

JSM Activity #468: Confidence Intervals and Hypothesis Testing Contributed - Papers


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1. Introduction: Why Consider the Sharpe Ratio?2. The Commonly Used Sharpe Ratio: 2 Definitions3. Asymptotic Distribution of Under iid Normality4. Asymptotic Distribution of Generally5. Small Sample Bias Adjustment for

6. Asymptotic Distribution of7. Simulation Study Under Skewness & Leptokurtosis8. Application to Large Growth Mutual Funds9. Conclusions

Contents

!SR!SR

!SR! ! !( )= −diff b aSR SR SR


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1. Why Consider ?• Ubiquitous in Financial Analysis• Funds and Fund Managers worldwide are

continuously ranked according to their Sharpe Ratios. This ordering means little without statistical inference: how certain are we that one samples really is larger than that of another? Comparisons via ranking are implicit pairwise hypothesis tests:

• Important theoretical foundations: CAPM & MPT0 : v. : b a A b aH SR SR H SR SR≤ >

!SR

!SR


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i. Using StdDev of EXCESS returns

2. Two Common Definitions

! ˆˆµσ

= ee

e

SR( )

( )

1

2

1

ˆ ,

,

ˆˆ

1

µ

µσ

=

=

=

= −

−=

−

∑

∑

T

ett

e

et t ft

T

et et

e

R

TR R R

R

T

where

See Jobson & Korkie (1981), Memmel (2003), Sharpe (1994)

# time periods, period's risk free rate,

period's return

==

=ft

t

TRR


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ii. Using StdDev of returns


! ˆ ˆˆ

µ µσ−

= s fs

s

SR

# time periods, period's risk free rate,

period's return

==

=ft

t

TRR

( )

1

1

2

1

ˆ ,

ˆ ,

ˆˆ

1

µ

µ

µσ

=

=

=

=

=

−=

−

∑

∑

∑

T

tt

s

T

ftt

f

T

t st

s

R

T

R

T

R

TSee Christie (2005), Lo (2002)


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• As an empirical matter, if the risk-free rate is not actually constant, it will be nearly so, making , so for all practical purposes, definition ii) is appropriate.


ˆ ˆs ftσ σ"


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• Jobson & Korkie (1981) under iid normality:

• Lo (2002) presented same result, but misread by many as iid generally, not NORMAL iid(normality implied in a footnote). See Getmanskyet al. (2004), Hennard & Aparicio (2003), Lee (2003), McLeod & van Vurren (2004), and Pinto & Curto (2005).

3. Asymptotic Distribution of Under iid Normality

!SR

!( ) 210, 12

− +

∼a

T SR SR N SR


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• Lo (2002) uses variance of estimated variance for a normal distributionwhen using delta method to get variance of .

• More generally,

3. Asymptotic Distribution of Under iid Normality

!SR

!( )2 42σσ =Var

!( )2 44σ µ σ= −Var

!SR


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• Using the more general result leads to

• Mertens (2002) presents this, but he does not generalize beyond iid returns, as is done by Christie (2005).

4. Asymptotic Distribution of Generally

!SR

!( )2

344 30, 1 1

4

a SRT SR SR N SR µµσ σ

− + − − ∼


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• Christie (2005) uses a GMM approach to obtain:

• This is quite unwieldy, but valid under very general conditions, requiring only stationary and ergodic returns. It thus allows for time-varying conditional volatilities, serial correlation, and even non-iid returns.

• However, this is identical to Mertens (2002)!


!SR

!( ) ( ) ( ) ( ) ( ) ( )2 2 22 2

44 3 2

2 3 44

µ σ µ µµσσ σ σ

− − − − − − = − + − +

t ft t t ft t tSR R R R R R R RSR SRVar T SR E

© J.D. Opdyke

DDaattaaMMIInneeIItt

Equivalence of Asymptotic Distributions of Christie (2005) and Mertens (2002)

Under only the requirements of stationarity and ergodicity, Christie (2005) derives (C21),

¶( ) ( )( ) ( ) ( ) ( )2 222 2

44 3 2

2 3 44

µ σµσσ σ σ

− − − − − − = − + − +

t ft t t ft t ft t ftSR R R R R R R R R R SRSR SRVar TSR E (C21)

which can be simplified as below13:

( )( ) ( )2 2 2244 3 2

23

4t f t t f t t f f

R R R R R R R R RSR SR E SR E Eµµ

σσ σ σ

− − − − + = + − ⋅ − ⋅ +

since 2 2 2tE R σ µ = + ,

( )( )2 2 2 2 2224

4 3 2

2 23

4t f t t f f

R R R R R RSR SR E SRµ µ σ µ µµ

σ σ σ

− − + + − + = + − ⋅ − +

3 2 2 2 222 24

4 3

2 23 1

4t t t t f t f fR R R R R R R RSR SR E SR SR

µ µ µ µµσ σ

− + − + − = + − ⋅ − + +

since 3 2 33 3tE R µ σ µ µ = + + ,

( ) ( )2 3 2 2 3 2 2 2 2234

4 3

3 2 21 3

4f f fR R RSR SR

µ µσ µ µ σ µ µ σ µ µ µµσ σ

+ + − + + − + + − = + + − ⋅

2 3 2 3 3 2 2 2234

4 3

3 2 21 3

4f f fR R RSR SR

µ µσ µ µσ µ µ σ µ µµσ σ

+ + − − + − − + = + + − ⋅

2 2 2234

4 3

3 21 3

4fRSR SR

µ µσ µσ σµσ σ

+ − − = + + − ⋅

234

4 31 3

4fRSR SR

µµµσσ σ

− = + + − ⋅ +

2234

4 31 34

SR SR SRµµσ σ

= + + − −

234

4 31 14

SR SR µµσ σ

= + − −

So ¶( )2

344 31 1

4SRVar TSR SR µµ

σ σ = + − −

which is Merten’s (2002) result, and that derived in Appendix A.

13 As previously mentioned, even if the variance of the risk-free rate is not literally zero, as is often the case, as a practical empirical matter it can be treated as zero, and its arithmetic mean used as the presumed constant rate (so above, let ˆft f fR Rµ= = ). Covariances of the risk-free rate with fund returns, too, can be treated as zero as an empirical matter. Mathematically, these assumptions are necessary for the above simplification.

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•

• Note that only SR, skewness, and kurtosis of the returns determine the distribution of .

• So asymptotically, no moments beyond the fourth affect the distribution of . This is shown by distribution in Graph 1. Consistent with the empirical evidence, it shows that assuming normality could be a poor basis for inference.


!SR

!SR

Consider!( )

234

4 30, 1 14

a SRT SR SR N SR µµσ σ

− + − − ∼

!SR


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0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Prob

abili

ty

Laplace

Normal

APD (λ=1.35, α=0.7, η3=-1.882, η4=5.191)

GRAPH 1Distribution of by Distribution of Returns !SR

1.0, 30SR T= =

! −SR SR


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• However, with this more mathematically tractible derivation we can see now that, in addition to stationarity and ergodicity, converging third and fourth moments are required for the convergence of the asymptotic distribution of .

• An apparently non-trivial number of financial instruments have diverging fourth moments, so this is important to note.


!SR

!SR


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• Because is convex, its estimator will be biased due to Jensens inequality:

• Christie (2005) obtains a 2nd order Taylor series expansion of about , and then a 1st order expansion of about to obtain the distribution of . However, like Lo (2002), he uses Var( ) = , valid only under normality!

5. Small Sample Bias of !SR

! ( ) [ ] [ ]( ) ( )ˆ ˆ ˆ ˆ, , ,µ σ µ σ µ σ ≥ = E SR SR E E SR

2σ 2σσ

σ2σ 42σ

!SR

SR


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• Christie (2005) obtains:

which not surprisingly, resembles the asymptotic distribution under normal returns.

• Using the more appropriate Var( ) = ,

which not surprisingly, resembles the asymptotic distribution generally.


! ( ) ( ) 1 1ˆ ˆ, , 12

µ σ µ σ = + E SR SR

T

2σ 44µ σ−

! ( ) ( )4

41ˆ ˆ, , 14

µ σµ σ µ σ

= +

E SR SRT


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• So for small sample estimates of , divide by

to obtain an approximately unbiased estimate.

• Simulations show this works very well


44ˆ ˆ11

4

µ σ + T

!SR!SR


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• are asymptotically unbiased normallydistributed variables, so by CLT, their linear combination is asymptotically unbiased & normal.

• Statistic = , Ho: ,

6. Asymptotic Distribution of

! !( ) ( ) ! !( ) − − − = − = b a b ab aVar SR SR SR SR Var SR SR

!( ) !( ) !( ) ! !( )2 ,= + −diff b a b aVar SR Var SR Var SR Cov SR SR

! ! & a bSR SR

! !( ) ( )− − −b a b aSR SR SR SR

!diffSR

=b aSR SR


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• Jobson & Korkie (1981) solved for normal iid returns; improved upon by Memmel (2003).

• But we now know returns are not normal.

• Vinod & Morey (2000) used the bootstrap and the double bootstrap, but this is computationally intensive, and bootstrap-based variance estimates are notoriously poor under asymmetric heavy tails, and even symmetric heavy tails (see Rocke & Downs, 1981, Gosh et al., 1984, and Salibian-Barrera, 1998), so caution is warranted with this approach.

6. Asymptotic Distribution of !diffSR


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• Previous work on distribution of a single used the delta method (Jobson & Korkie, 1981; Lo, 2002; and Memmel, 2003), so why not use it to derive the two-sample statistic?

• If a function is continuous and continuously differentiable (loosely speaking), then the delta method obtains its variance if the random variables it uses are asymptotically normal.

!SR



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( ) ( )( ) ( )

2, 3 1 ,22

, 1 ,22 2

2 2

34

3 1 ,2 4

41 ,2 3 4

,

,

a a b a a b

a b b b a b

b b a a a

a b b b

a b

a b b

Cov

Cov

σ σ µ µσ σ µ µ

µ µ µ σ

µ µ µ σ

σ σ

σ σ

Ω = =− −

So by delta, , where!( )'∂ ∂ = Ω ∂ ∂

difff fVar SRu u

( ),µ µ σσ

= =SR f ( )2 2, , ,µ µ σ σ= a b a bu

Variance/covariance matrix of u


, , and


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delta method yields:2

4 34 31 1

4a a a

diff aa a

SRVar SRµ µσ σ

= + − − +

( )( ) ( )( )2 22 ,2µ = − − a b E a E a b E b = 2nd central

moment of joint distribution


where

24 34 31 1

4b b b

bb b

SR SRµ µσ σ

+ − −

2 ,2 1 ,2 1 ,2, 2 2 2 2

1 12 14 2 2

a b b a a ba ba b a b

a b b a a b

SR SR SR SRµ µ µ

ρσ σ σ σ σ σ

− + − − −

and ( )( ) ( )( )21 ,2a b E a E a b E bµ = − −


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• When returns are normal iid,

and

reduces to Jobson & Korkie (1981), as shown below.

• Note also that when and , so the entire covariance term disappears, as it should.

4 34 3 1,23, 0, 0µ σ µ σ µ= = =

( )2 2 22 ,2 ,1 2 , so µ ρ σ σ= +a b a b a b diffVar

2 2, 2 ,20, a b a b a bρ µ σ σ= =


1,2 0µ =

© J.D. Opdyke


Variance of the Difference Between Two Sharpe Ratios

If aSR and bSR are the respective Sharpe ratios for the returns (Rat and Rbt) of funds “a” and “b,” then use the “delta

method”14 (see Greene, 1993, and Stuart & Ord, 1994) to obtain the asymptotic variance of ¶ ¶( ) ( )− − −a b a bSR SR SR SR :

Assuming 2 0σ =f , which is always essentially, if not literally true, ( )2,fRSR f

µµ σ

σ

−= = , so let ( )2 2, , ,µ µ σ σ= a b a bu

and ( )2 2ˆ ˆ ˆ ˆ ˆ, , ,µ µ σ σ= a b a bu , then ( ) ( )ˆ 0,− Ω∼T u u N where Ω is the variance-covariance matrix of u :

( ) ( )( ) ( )

2, 3 1 ,22

, 1 , 2 34 2 2

3 1 ,2 4

2 2 41 ,2 3 4

,

,

a a b a a b

a b b b a b

a b a a a a b

a b b a b b b

Cov

Cov

σ σ µ µ

σ σ µ µ

µ µ µ σ σ σ

µ µ σ σ µ σ

Ω = − −

where ( ), ,σ =a b Cov a b , ( ) ( )3 23 ,a a a aE a Covµ µ µ σ = − = ,

( ) ( )3 23 ,b b b bE b Covµ µ µ σ = − = (see Mertens, 2002), ( )( ) ( )2 2

1 , 2 ,a b a b a bE a b Covµ µ µ µ σ = − − = ,

and ( )( ) ( )2 21 , 2 ,b a b a b aE b a Covµ µ µ µ σ = − − = (see Espejo & Singh, 1999). Now,

¶ ¶( ) ( )( ) ( )0,− − − ∼a b a b diffT SR SR SR SR N Var , '∂ ∂ = Ω ∂ ∂

difff f

Varu u

, ( ) ( )

3 31 1

, , ,2 2a f b f

a b a b

R Rfu

µ µ

σ σ σ σ

− −∂ = − − ∂

,

( ) ( ) ( ) ( ) ( ) ( )3 1 , 2 1 , 2 3 3 1 , 2, ,4 3 3 4 4 31 1

2 2 2 2 2 2a a f a b b f b a a f b b f a a f b a a fa b a b

diffa b a ba a b a b b a a b

R R R R R RVar

µ µ µ µ µ µ µ µ µ µ µ µσ σ

σ σ σ σσ σ σ σ σ σ σ σ σ

− − − − − −= − − + − + + − − +

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2 24 2 2 2 2 44 41 ,2 3

6 3 3 3 4 3 3 6

, ,

4 4 2 2 4 4a a a f a f b f a b a f b f a b b b b fa b b f b b f

a a b a b b a b b

R R R Cov R R Cov RR Rµ σ µ µ µ σ σ µ µ σ σ µ σ µµ µ µ µ

σ σ σ σ σ σ σ σ σ

− − − − − − − −− −+ − + − − +

( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )2 24 4 2 2

4 43 3 1 , 2 1 ,2, 4 4 3 3 6 6 3 3

,2 2

4 4 2a a a f b b b f a f b f a ba a f b b f b a a f a b b f

a ba b a b a b a b a b

R R R R CovR R R R µ σ µ µ σ µ µ µ σ σµ µ µ µ µ µ µ µρ

σ σ σ σ σ σ σ σ σ σ

− − − − − −− − − −= − − − + + + + −

( ) ( ) ( )4 4 2 22 24 41 , 2 1 ,23 3, 3 3 2 2 4 4 2 2

,2 2

4 4 2a a b b a bb a a ba b a b a b

a b a b a ba b b a a b a b a b

CovSR SR SR SRSR SR SR SR

µ σ µ σ σ σµ µµ µρ


− − = − − − + + + + −

Since ( ) ( ) [ ]( ) [ ]( ) [ ]( )4 2 22 2 2 4 4 2 24,σ σ σ µ σ σ σ σ = = = − = − − = − − − a a a a a a a aVar Cov E a E a E a E a a E a ,

14 The delta method is a widely used technique that provides an asymptotic approximation of the variance of a particular function (see Greene, 1993, pp.297-298, and Stuart & Ord, 1994, p.350). It is valid as long as the random variables used in the function are asymptotically normal, and the function is (loosely speaking) continuous and continuously differentiable. The former assumption is true in this case, since the sample mean and the sample variance are asymptotically normal. The latter assumption clearly is violated if the variance of returns is zero. This will never actually occur in practice using real data samples, but if the variance approaches zero, making the Sharpe ratio highly nonlinear, delta method estimates will become unstable, as correctly noted by Vinod & Morey (2000). However, this scenario, too, arguably will affect few, if any cases in practice, as the variances of the returns of most, if not all funds or stocks that would be of enough interest to be subjected to Sharpe ratio comparisons are quite far from zero; if they were not, there would be nothing to compare! Still, it is important to note the limitations of analytical methods relied upon in any study, in case their domain of application changes. Jobson & Korkie (1981), Lo (2002), Memmel (2003), and Mertens (2002) all use the delta method in their studies of Sharpe ratios, thus supporting its practical use here.

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© J.D. Opdyke

DDaattaaMMIInneeIItt then ( ) [ ]( ) [ ]( )2 22 2 2 2 2 2

2 ,2,σ σ σ σ µ σ σ = − − − = − a b a b a b a bCov E a E a b E b , where 2 ,2µ a b is the joint second central

moment of the joint distribution of a and b. The same result can be obtained using Stuart & Ord’s (1994) (pp.457-458) result of ( ) ( )2 2 2

2 ,2 1 ,1ˆ ˆ, 2 1σ σ κ κ= + −a b a b a bCov n n , where 2 ,2κ a b is the second joint cumulant of the joint distribution of

a and b, and 1 ,1κ a b is the first joint cumulant, equal to the first joint central moment, 1 ,1µ a b , which is the covariance.

Dropping the n coefficients due to the use of the estimators 2 2ˆ ˆ,σ σa b for 2 2,σ σa b yields

( )2 2 2 2 22 ,2 1 ,1 2 ,2 1 ,1 2 ,2 ,, 2 2 2σ σ κ κ κ µ κ σ= + = + = +a b a b a b a b a b a b a bCov . Recognizing that the joint cumulant also can be

expressed in terms of central moments, 2 2 2 22 ,2 2 ,2 2 ,0 0,2 1 ,1 2 ,2 ,2 2κ µ µ µ µ µ σ σ σ= − × − = − −a b a b a b a b a b a b a b (see Stuart &

Ord, 1994, p.107, and Smith, 1995), we have:

( )2 2 2 2 2 2 2 2 22 ,2 1 ,1 2 ,2 , , 2 ,2, 2 2 2σ σ κ κ µ σ σ σ σ µ σ σ= + = − − + = −a b a b a b a b a b a b a b a b a bCov . Thus,

2 22 21 , 2 1 ,2 2 ,23 3 4 4

,3 3 2 2 4 4 2 21

2 1 1 24 4 2

b a a b a b a ba b a a b bdiff a b a b a b a b

a b b a a b a b a b

SR SRVar SR SR SR SR SR SR

µ µ µ σ σµ µ µ µρ


−= − − + + + − + − − −

2 2

1 , 2 1 , 2 2 ,23 3 4 4,3 3 2 2 4 4 2 22 1 1 2 1

4 4 4b a a b a ba b a a b b a b

a b a b a ba b b a a b a b a b

SR SR SR SRSR SR SR SR

µ µ µµ µ µ µρ


= − − + + + − + − − + −

So analogous to the variance of the distribution of a single ¶SR , (6), the variance of the difference between two ¶SRs is

24 34 31 1

4a a a

diff aa a

SRVar SR

µ µ

σ σ

= + − − +

2

4 34 31 1

4b b b

bb b

SRSR

µ µ

σ σ

+ − −

2 ,2 1 , 2 1 , 2, 2 2 2 2

1 12 1

4 2 2a b b a a ba b

a b a ba b b a a b

SR SRSR SR

µ µ µρ

σ σ σ σ σ σ

− + − − −

Note that when 2 2, 2 ,20, ρ µ σ σ= =a b a b a b , 1 , 2 0a bµ = , and 1 , 2 0b aµ = , so the entire covariance term of diffVar disappears,

as it should.

Minimum variance unbiased estimators of 1 , 2a bµ , 1 , 2b aµ , & 2 ,2µ a b are the respective h-statistics 1 ,2a bh , 1 ,2b ah , &

2 ,2a bh , where ( ) ( )2 21,2 0,1 1,0 0,2 1,0 0,1 1,1 1,22 2 1 2h s s ns s s s n s n n n = − − + − − , and 2,2h =

( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 20,1 1 , 0 0,21,0 0,11 ,01 ,1 1,1 1,01,2 0,1 2,0 0,2 2,0 0,12,1 2,23 4 2 2 3 2 2 3 2 3 2 2 3 2 3 = − + + − − − − + + − − − − + + − +

s s ns s ns s s n s n n s s s s n s s n n s s n n n s

( ) ( )( ) 3 2 1 − − − n n n n , where ,x ys are the simple power sums of ,1=

= ∑n

yxx y i i

i

s a b (see Rose & Smith, 2002, pp.259-260).

This derivation is valid under iid returns, but because the one-sample estimator (6), derived using the same (delta) method (a la Mertens, 2002), was shown in Appendix B to be valid under the more general conditions afforded by its (identical) GMM derivation (a la Christie, 2005), we suspect those more general conditions of stationarity and ergodicity are the only requirements for the two-sample estimator of (13) as well. Proving this is the topic of continuing research.

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© J.D. Opdyke


Equivalence of Vardiff with Memmel (2003) and Jobson & Korkie (1981)

Under iid normality, Memmel’s (2003) correction of Jobson and Korkie’s (1981) variance of the two-sample s tatistic for the difference between two Sharpe ratios is:

( )2 2 2, ,

12 2 2

2ρ ρ= = − + + −a b a b a b a bVar TV SR SR SR SR

Under normality, TV is identical to diffVar , as shown below:

2 21 , 2 1 , 2 2 ,24 3 4 3

,4 3 2 4 3 2 2 21 1 1 1 2 14 4 4

b a a b a ba a a b b b a bdiff a a b b a b

a a b a b b a b a b

SR SR SR SRVar SR SR SR SR

µ µ µµ µ µ µρ


= + − − + + + − − + − + −

Under iid normality, 33 0µ σ = , 1,2 0µ = , 4

4 3µ σ = , & ( )2 2 22 ,2 ,1 2µ ρ σ σ= +a b a b a b (see Stuart & Ord, 1994, p.105), so

[ ] [ ]( )2 2 2 2 22 2 ,

, 2 2

1 21 3 1 0 0 1 3 1 0 0 2

4 4 4a b a b a ba b a b

diff a ba b

SR SR SR SRVarρ σ σ σ σ

ρσ σ

+ − = + − − + + + − − + − + =

2 2

2, ,2 2 2

2 2 4ρ ρ

= + + − + a b a b

a b a bSR SR SR SR

2 2 2, ,

12 2 2

2a b a b a b a bSR SR SR SR TVρ ρ = − + + − = , which is Memmel’s (2003) result.

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©J.D. Opdyke

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• Limitations of delta: Estimates become unstable when function is highly nonlinear. This would occur for if approaches zero. Also, obviously is not continuous or continuously differentiable at

• However, this is not a problem for practical usage, since there would be nothing to compare if was close to zero! The problem in practice is not too little variance, but rather, too much (see Christie, 2005).

2σ

2σ

2 0.σ =!SR

!SR



©J.D. Opdyke

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• Asymptotics are fine, but how doesperform under realistic data conditions

returns that are:i. Leptokurtotic (i.e. heavy tailed)

ii. Asymmetric

iii. Strongly (positively) correlated with each other

iv. Based on finite sample sizes

7. Simulation Study

diffVar


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• Use Komunjers (2006) Asymmetric Power Distribution (APD) for simulation study testing empirical level and power.

• APD has skewness (α) and kurtosis (λ) parameters and nests the normal, Laplace, asymmetric (2-piece) normal, and asymmetric Laplace. These all have been used extensively in the empirical finance literature.

7. Simulation Study

© J.D. Opdyke


0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5

APD, a = 0.9 a = 0.7 a = 0.5

0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5

APD, a = 0.9 a = 0.7 a = 0.5 ? = 2.00 (Normal at a = 0.5) ? = 1.75

0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5

APD, a = 0.9 a = 0.7 a = 0.5

0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5

APD, a = 0.9 a = 0.7 a = 0.5 ? = 1.50 ? = 1.25

0

0.2

0.4

0.6

0.8

-5 -4 -3 -2 -1 0 1 2 3 4 5

APD, a = 0.9 a = 0.7 a = 0.5 ? = 1.00 (Laplace at a = 0.5)

Figure 2: Asymmetric Power Distribution by a by ? (all densities standardized so that Variance = 1.0)

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© J.D. Opdyke


Simulation Distributions: APD of Komunjer (2006)

Komunjer (2006) gives the density of the asymmetric power distribution (APD) below:

( )( )

( ) ( )

1, ,

1, ,

exp if 0,1 1

exp if 0,

1 1 1

λλα λ α λ

λ

λλα λ α λ

λ

δ δλ α

δ δλ α

− ≤ Γ +

= − >

Γ + −

u u

f uu u

where 0 < a < 1, ? > 0, and ( )( ),

2 1

1

λλ

α λ λλ

α αδ

α α

−≡

+ −

The a parameter controls skewness, with symmetry at a = 0.5, and ? controls kurtosis, such that when a = 0.5, ? = 8 ? the uniform distribution, ? = 1.0 ? the Laplace distribution (with variance = 2.0), and ? = 2.0 ? the normal distribution (with variance = 0.5). When a ? 0.5, ? = 1.0 ? the Asymmetric Laplace distribution of Kozubowski & Podgorski (1999), and ? = 2.0 ? the two-piece normal distribution (see Johnson, Kotz & Balakrishnan, 1994, vol. 1 p.173 and vol. 2 p.190). Thus does APD allow simultaneous control over skewness and kurtosis, nesting the normal and Laplace densities, and asymmetric versions of each, as well as any “in between” combination of asymmetry and kurtosis .

Location and scale are accommodated via the simple transformation: θ φ≡ +X U

APD moments are given by:

( ) ( )( )( )

( ) ( )1 1

,

1 1 11 λ

α λ

λ α αλ δ

+ +Γ + − + −=

Γ

r r rr

r

rE U (see Table III below). So for example,

( ) ( )( ) ( ) 1

,

21 2

1λ

α λ

λα δ

λ−Γ

= −Γ

E U and ( )( ) ( ) ( ) [ ]

( )

2 222,2

3 1 1 3 3 2 1 2

1λ

α λ

λ λ α α λ αδ

λ−

Γ Γ − + − Γ − =Γ

Var U

To standardize the APD for the simulations presented in this study, U is modified by u’ = u / sqrt[Var(u)] (because, for example, when a = 0.5 and ? = 1.0, Var(U) = 2.0, and when a = 0.5 and ? = 2.0, Var(U) = 0.5).

Table III: Skewness ? 3 and Kurtosis ? 4 of APD by Values of a and ?

Special-case Nested Distribution a ? Skewness ? 3 Kurtosis ? 4 Asymmetric Laplace 0.1 / 0.9 1.00 ± 2.2311 6.6485 0.1 / 0.9 1.25 ± 1.9870 5.0165 0.1 / 0.9 1.50 ± 1.8415 4.1686 0.1 / 0.9 1.75 ± 1.7457 3.6595 Two-piece normal 0.1 / 0.9 2.00 ± 1.6784 3.3243 Asymmetric Laplace 0.3 / 0.7 1.00 ± 2.1867 7.4726 0.3 / 0.7 1.25 ± 1.9474 5.6383 0.3 / 0.7 1.50 ± 1.8048 4.6853 0.3 / 0.7 1.75 ± 1.7109 4.1131 Two-piece normal 0.3 / 0.7 2.00 ± 1.6450 3.7363 Laplace (variance = 2.0) 0.5 1.00 0.0000 6.0000 GPD 0.5 1.25 0.0000 4.5272 GPD 0.5 1.50 0.0000 3.7620 GPD 0.5 1.75 0.0000 3.3026 Normal (variance = 0.5) 0.5 2.00 0.0000 3.0000

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N = 10,000 simulations run for over 5,200 combinations of:

• Sample size (T = 15, 30, 50, 100, 300)• configurations; sizes of • Skewness (range used = η3 = ±2.23)• Kurtosis (range used = η4 = ±7.47)• Correlation between the two series of

returns ( 0.00, 0.25, 0.50, 0.75)

7. Simulation Study

2/µ σ !SR

,ρ =a b


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Many results, but key results include:• Under concurrent skewness and leptokurtosis at

least as extreme as typical returns (used α = 0.7 & λ = 1.35 for skewness & kurtosis of η3 = -1.88 & η4 = 5.19, respectively; see Haas et al., 2005; Cajigas & Urga, 2005; Cappiello et al., 2003; and Vinod, 2005)

• Under positive correlation at least as extreme as typical returns (IMPORTANT! NOT carefully examined in literature on a 2-sample tests)

7. Simulation Study - Results

!SR


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

Prob

abili

tyGRAPH 3

APD Real World Simulated Returns(α = 0.7, λ = 1.35, η3 = -1.88, η4 = 5.19)

Standardized Returns, r

E [ r ] = -0.496


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• As shown in Graph 4, excellent convergence to nominal level α under real world skewness and leptokurtosis of APD simulated returns, even for large values of .


a bSR SR=


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0.00

0.05

0.10

0.15

0 50 100 150 200 250 300

T Periods

Rej

ectio

n R

ate

0.00.21.03.0

GRAPH 4Level of 2-sample Estimator

Under Real World APD Returns by ρ by SR by TSRa = SRb =

0.00

0.05

0.10

0.15

0 50 100 150 200 250 300

T Periods

Rej

ectio

n R

ate

0.00.21.03.0

0.00

0.05

0.10

0.15

0 50 100 150 200 250 300

T Periods

Rej

ectio

n R

ate

0.00.21.03.0

0.00

0.05

0.10

0.15

0 50 100 150 200 250 300

T Periods

Rej

ectio

n R

ate

0.00.21.03.0

ρ = 0.75

ρ = 0.25

ρ = 0.50

ρ = 0.00

SRa = SRb =

SRa = SRb = SRa = SRb =


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• Under realistic skewed and leptokurtotic APD-simulated returns, note the dramatic increases in power under strong, positive correlation, which is typical for Sharpe ratio comparisons in practice.


© J.D. Opdyke

DDaattaaMMIInneeIItt Figure 5a: Power – SRa=0.0, SRb=0.1 Figure 5b: Power – SRa=0.0, SRb=0.2

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ectio

n R

ate

? = 0.00 0.25 0.50 0.75

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ectio

n R

ate

? = 0.00 0.25 0.50 0.75 Figure 5c: Power – SRa=0.0, SRb=0.5 Figure 5d: Power – SRa=0.2, SRb=0.4

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ecti

on

Rat

e

? = 0.00 0.25 0.50 0.75

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ectio

n R

ate

? = 0.00 0.25 0.50 0.75 Figure 5e: Power – SRa=1.0, SRb=1.5 Figure 5f: Power – SRa=3.0, SRb=3.5

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ectio

n R

ate

? = 0.00 0.25 0.50 0.75

0.00

0.25

0.50

0.75

1.00

0 50 100 150 200 250 300

# T Periods

Rej

ecti

on

Rat

e

? = 0.00 0.25 0.50 0.75

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• Christie (2005) finds that, when based on actual returns data, statistically significant differences between Sharpe ratios remain elusive due to large variances.

• However, finds statistically significant differences between the Sharpe ratios of several randomly selected large growth mutual funds right off the bat!

8. Using Actual Returns - Mutual Funds

diffVar


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• For the period 09/01 through 08/06, take the monthly returns of:– Fidelitys Contrafund (FCNTX)– Janus Growth & Income (JAGIX)– Vanguard Growth Index (VIGRX)– 90-day Treasury Bill rate (divided by 12).

• Only Fidelity approaches statistically significant positive excess returns, as indicated by (one-sided p-values of 0.074, 0.320, and 0.477, respectively).


0>SR


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• However, the strong, positive correlations between them give greater precision, and thus, greater power to detect differences between their Sharpe Ratios.

• Ho: -Fidelity ≤ -Janus, p = 0.047Ho: -Fidelity ≤ - Vanguard, p = 0.030Ho: -Janus ≤ -Vanguard, p = 0.195


diffVar

SR SRSR SR

SRSR


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0.050.190.140.010.060.20

0.0300.77-Fidelity ≤ -Vanguard

0.90

0.86

Pearsons Sample

Correlation Coefficient

r

0.477-Vanguard ≤ 0

0.195-Janus ≤ -Vanguard

0.047-Fidelity ≤ -Janus

0.320-Janus ≤ 0

0.074-Fidelity ≤ 0

Opdyke (2006)one-sided p-valueNull Hypothesis, Ho:


SR

!SR!

diffSR,

SRSRSRSRSR

SRSRSR


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Major contributions of this study:

• It generalized the only useable version of the asymptotic distribution of to very realistic conditions, requiring only stationary and ergodicreturns with converging 3rd & 4th moments.

• It derived an easily used 2-sample statistic for( ) that nests the normal iid derivation of Jobson & Korkie (1981) and has excellent level control under real-world data conditions (i.e. asymmetric, leptokurtotic, and highly correlated returns based on finite samples).

9. Conclusions

!SR

! !b aSR SR−


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• Power of the 2-sample statistic is generally modest, but it increases dramatically under strong, positively correlated returns: since most Sharpe Ratio comparisons are apples-to-apples, this is the rule rather than the exception!

• Therefore, as it would be used in practice, the statistic appears to have GOOD power, as demonstrated by a comparison of actual mutual fund returns.

9. Conclusions


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Selected References:• Cajigas & Urga, 2005• Cappiello et al., 2003• Christie, 2005• Getmansky et al., 2003• Gosh et al., 1984• Haas et al., 2005• Hennard & Aparicio, 2003• Jobson & Korkie, 1981• Johnson, Kotz, & Balakrishnan,

1994• Komunjer, 2006• Kozubowski & Podgorski, 1999

• Lee, 2003• Lo, 2002• McLeod & van Vurren, 2004• Memmel, 2003• Mertens, 2002• Pinto & Curto, 2005• Rocke & Downs, 1981• Salibian-Barrerra, 1998• Sharpe, 1966, 1975, 1994• Vinod, 2005• Vinod & Morey, 2000


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Acknowledgments:• I express thanks to Keith Ord, Andrew

Clark, and Hrishikesh Vinod for useful comments, and sincere appreciation to Steve Christie for spotting an error in an earlier draft, and to Geri S. Costanza for numerous and valuable insightful discussions.


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Contact:• J.D. Opdyke, President

[email protected] (ph) ∑∑∑∑ 781-639-6463 (ph/fax)

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