Why Distributions

Matter

By Peter Urbani

16 Jan 2012

My ventures are not in one bottom trusted,Nor to one place; nor is my whole estate Upon the fortune of this present year;Therefore, my merchandise makes me not sad.

Spoken by Antonio in Act I, Scene I, Merchant of Venice, William Shakespeare, circa 1650

Why Distributions Matter

Rembrandt’s Christ in the storm on the lake of Galilee – the cover illustration of Peter Bernstein’s excellent Against the God’s – The Remarkable Story of Risk

Diversification

Diversification is the key principle upon which Modern Portfolio Theory (MPT) is built - although the concept of not putting all of one’s eggs into one basket dates all the way back to biblical times.

Central to this is the concept of Correlation as measure of dependence between assets

Key Assumptions

Correlation ( the standardised covariance between assets ) as the measure of dependence between assets

Normally distributed returns. The assumption that asset returns are normally distributed about their means.

They make an ASS out of U and ME

The Problem with Assumptions

Normality Testing

Normality Testing

ETF's

Normal

Not-Normal

Hedge Funds

Normal

Not-Normal

Assumed NormalAssumed Normal

Normal VaR

Normal CVaR

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Assumed Normal FundPDF (Normal)

If not Normal then What ? - Modified

VaR ( Modified ) lower for small positive skew

Cornish Fisher - Modification

VaR ( Modified ) higher for small negative skew

Cornish Fisher - Modification

VaR ( Modified ) same as VaR ( Normal ) for no skew

Assumed NormalAssumed Normal

Normal VaR

Normal CVaR

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Assumed Normal FundPDF (Normal)

Assumed ModifiedAssumed Normal and Modified Distributions

Normal VaR

Normal CVaR

Modified CVaR -

Modified VaR -

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Assumed Normal FundPDF (Normal)

Assumed ModifiedNormal Fund PDF(Modified)

Cornish Fisher - Modification1.3.2 Properties

The qualitative properties of the Cornish-Fisher expansion are:

If is a sequence of distributions converging to the standard normal distribution , the Edgeworth- and Cornish-Fisher approximations present better approximations (asymptotically for ) than the normal approximation itself.

The approximated functions and are not necessarily monotone.

has the ``wrong tail behavior'', i.e., the Cornish-Fisher approximation for -quantiles becomes less and less reliable

for (or ).

The Edgeworth- and Cornish-Fisher approximations do not necessarily improve (converge) for a fixed and increasing order of approximation, .

For more on the qualitative properties of the Cornish-Fisher approximation see (Jaschke; 2001). It contains also an empirical analysis of the error of the Cornish-Fisher approximation to the 99%-VaR in real-world examples as well as its worst-case error on a certain class of one- and two-dimensional delta-gamma-normal models:

http://fedc.wiwi.hu-berlin.de/xplore/tutorials/xfghtmlnode8.html

Problems with Modified VaR – Not MonotoneModified VaR as a function of Skewness

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

-6 -4 -2 0 2 4 6

Skewness

VaR

( M

od

ifie

d )

Modified VaR @ CL 99.00%Modified VaR @ CL 95.00%CurrentCurrent

http://discussions.ft.com/alchemy/forums/edhec-risk-forum/hedge-fund-risk-management-models-for-the-return-distribution/

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Modified CDF

Normal CDF

0.00%

500.00%

1000.00%

1500.00%

2000.00%

2500.00%

-30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0%

Modified PDF

Normal PDF

-6

0

6

12

18

-6 -4 -2 0 2 4 6

↔Skew

↕ Kurt

S

GoodZ Z

S S

Z

Problems with Modified VaR – Bad Tail Behaviour

Problems with Modified VaR – Bad Tail Behaviour

0.00%

10.00%

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30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

90.00%

100.00%

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Modified CDF

Normal CDF

0.00%

500.00%

1000.00%

1500.00%

2000.00%

2500.00%

-30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0%

Modified PDF

Normal PDF

-6

0

6

12

18

-6 -4 -2 0 2 4 6

↔Skew

↕ Kurt

S

GoodZ Z

S S

Z

How Prevalent is this problem ?VERY

ETF's

OK

WARNING: DegenerateCornish Fisher. CDFw ill turn in tails

WARNING: DegenerateCornish Fisher. CDFw ill turn in body

Hedge Funds

OK

WARNING: DegenerateCornish Fisher. CDFw ill turn in tails

WARNING: DegenerateCornish Fisher. CDFw ill turn in body

Public Function CFRegionWarning(ByVal Skew As Double, Kurt As Double) As String

Dim a As Double Dim b As Double Dim c As Double Dim Q As Double Dim R As Double Dim Denom As Double

Denom = 3 * Kurt - 4 * (Skew ^ 2)

If Denom > 0 Then a = 12 * Skew / Denom b = (10 * Skew ^ 2 - 9 * Kurt + 72) / Denom c = -12 * Skew / Denom Q = (a ^ 2 - 3 * b) / 9 R = (2 * (a ^ 3) - 9 * a * b + 27 * c) / 54 If R ^ 2 > Q ^ 3 Then CFRegionWarning = "" 'Its in Well Behaved Region Else CFRegionWarning = "WARNING: Degenerate Cornish Fisher. CDF will turn in body (S)" End If Else CFRegionWarning = "WARNING: Degenerate Cornish Fisher. CDF will turn in tails (Z)" End If

End Function

VBA code to check Cornish Fisher - Modification

If not Normal or Modified then What ?

ETF's

Gumbel (Max) 7.8%

Gumbel (Min) 8.2%

Johnson (Lognormal) 13.5%

Johnson (Unbounded) 0.6%

Mixture of Normals 20.1%

Modified Normal 35.8%

Normal 12.9%

Uniform 1.0%

Hedge Funds

Gumbel (Max) 23.3%

Gumbel (Min) 13.2%

Johnson (Lognormal) 13.2%

Johnson (Unbounded) 2.6%

Mixture of Normals 15.9%

Modified Normal 21.2%

Normal 6.9%

Uniform 3.7%

Best Fitting Distributions

Assumed NormalAssumed Normal

Normal VaR

Normal CVaR

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Assumed Normal FundPDF (Normal)

Assumed ModifiedAssumed Normal and Modified Distributions

Normal VaR

Normal CVaR

Modified CVaR -

Modified VaR -

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Assumed Normal FundPDF (Normal)

Assumed ModifiedNormal Fund PDF(Modified)

Best FittingBest Fit and Assumed Normal and Modified Distributions

Best Fit VaR -

Best Fit CVaR -

Normal VaR

Normal CVaR

Modified CVaR -

Modified VaR -

-40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%

Best Fit Fund PDF(Gumbel (Min))

Assumed Normal FundPDF (Normal)

Assumed ModifiedNormal Fund PDF(Modified)

Goodness of FitCummulative Probability Distributions

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

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Best Fit Fund CDF(Gumbel (Min))

Assumed Normal FundCDF (Normal)

Empirical CDF Fund

P-P Plot (Showing Goodness of Fit for Fund)

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

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

Series1

Fund Best Fit (Gumbel (Min))

Fund Assumed (Normal)

Relationship Between Assets

-4.00%

-2.00%

0.00%

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6.00%

8.00%

10.00%

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FUND B

FUN

D A

Linear Regression

Relationship between Fund A and B

Assumed Bi-Variate Normal Copula

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-6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00%

FUND B

FUN

D A

Assumed Bi-variateNormal Copula Lines

Linear Regression

Best Fit Bi-Variate Copula

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-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

-6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00%

FUND B

FUN

D A

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

-20.00%

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0.00%

10.00%

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50.00%

-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00%

FUND D

FUN

D C

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Example – Johnson Lognormal - Normal

Example – Normal - Normal

-15.00%

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FUND F

FUN

D E

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Example – Modified Normal - Normal

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2.00%

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6.00%

8.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

FUND H

FUN

D G

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Example – Mix Normals – Modified Normal

-40.00%

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0.00%

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FUND J

FUN

D I

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Example – Mod Normal – Mod Normal

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30.00%

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FUND L

FUN

D K

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Relationship Between Assets2011 Asset Class Correlations

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30.00%

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-1.00 -0.50 0.00 0.50 1.00

Correlation to S&P500

To

tal

Re

turn

s (

% )

Relationship Between Assets2011 Asset Class Correlations

Volatility

US Dollar Oil

Gold

US Real EstateUS High Yield Bonds

Emerging Markets Bonds

International Government Bonds

US Government Bonds

US Total Bond Market

US Bonds

Global Bond Index

Emerging Market Equities

World Equities

Cash US Equities

CTA'sHedge Funds

-20.00%

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15.00%

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-1.00 -0.50 0.00 0.50 1.00

Correlation to S&P500

To

tal R

etu

rns

( %

)

Correlations at multi-decade highs

Best Fit and Pearson Correl Pairs with CAGR > 7%

Base Fund Lowest Correlation via Best Fit Lowest Correlation via Pearson Correl

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United States Oil Fund

Corn

Fund

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

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DB Crude Oil Long ETN

Corn

Fund

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Best Fit Pair CORN - OLO Pearson Pair CORN - USO

-6.00%

-4.00%

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SPECTRUM Lg Cap U.S. Sector ETN

Barc

lays

ETN

+ S

&P

VEQ

TOR E

TN

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00%

Ultra Consumer Services

Barc

lays

ETN

+ S

&P

VEQ

TOR E

TN

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Best Fit Pair VQT - UCC Pearson Pair VQT - EEH

-10.00%

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15.00%

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-10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

SPDR Barclays Capital High Yield Bond ETF

US

Treasu

ry L

ong B

ond B

ull

ETN

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00%

iBoxx $ HY Corp Bond Fund

US

Treasu

ry L

ong B

ond B

ull

ETN

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Best Fit Pair DLBL - HYG Pearson Pair DLBL - JNK

-15.00%

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-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

-20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%

SPDR DJ Wilshire Global Real Estate ETF

Wils

hire

US

REI

T ET

F

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

-30.00%

-20.00%

-10.00%

0.00%

10.00%

20.00%

30.00%

40.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

FTSE NAREIT Real Estate 50 Index Fund

Wils

hire

US

REI

T ET

F

Best Fit Bi-variateCopula Lines

Best Fit Regression

Linear Regression

Best Fit Pair WREI - FTY Pearson Pair WREI - RWO

Other Measures of Dependence

• Spearman’s Rank Order Correlation

• Lin’s Concordance measure

• Copula methods

• Distance measures

• Mutual Information and other entropy based measures

Conclusions• Distributions do differ from Normal at least 15 – 20% of the time and up to

30 – 40% of the time depending on the data set being used – Test them

• The Cornish Fisher modification is not strictly monotone and should probably not be used at confidence levels above 95%

• The Cornish Fisher modification has poor tail behaviour almost half of the time – CHECK

• Correlation is a limited and linear measure of dependence only

• Non-Linear Copula based methods offer significant promise in helping to find better diversification and pairs trading opportunities

Pietro (‘Peter’) Urbani (45)• Chief Investment Officer (CIO) – Infiniti Capital $3bn Fund of Hedge Funds Group

Head of Quantitative Research – Infiniti Capital

• CEO – KnowRisk Consulting – Asset Consulting

• Head of Investment Strategy – Fairheads Asset Managers HNW Trust and Investment BoutiqueHead of Research – Fairheads Asset Managers

• Head of Portfolio Management – Nexus Securities

• Senior Portfolio Manager – Commercial Union – Superfund

• Equities Dealer – Junior Portfolio Manager – Mathison & Hollidge Stockbrokers

http://nz.linkedin.com/in/peterurbani