Why
Distributions
Matter
By Peter Urbani
16 Jan 2012
Why Distributions Matter
My ventures are not in one
bottom trusted,
Nor to one place; nor is my
whole estate
Upon the fort...
Diversification
Diversification is the key principle upon which
Modern Portfolio Theory (MPT) is built - although
the conc...
Key Assumptions
Correlation ( the standardised covariance
between assets ) as the measure of dependence
between assets
Nor...
The Problem with Assumptions

They make an ASS out of U and ME
Normality Testing
Normality Testing

Hedge Funds

ETF's

Normal

Normal

Not-Normal

Not-Normal
Assumed Normal
Assumed Normal

Assumed Normal Fund
PDF (Normal)

Normal VaR

Normal CVaR

-40.00%

-30.00%

-20.00%

-10.0...
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 Normal
Assumed Normal

Assumed Normal Fund
PDF (Normal)

Normal VaR

Normal CVaR

-40.00%

-30.00%

-20.00%

-10.0...
Assumed Modified
Assumed Normal and Modified Distributions

Assumed Normal Fund
PDF (Normal)

Assumed Modified
Normal Fund...
Cornish Fisher - Modification
1.3.2 Properties
The qualitative properties of the Cornish-Fisher expansion are:

If
is a se...
Problems with Modified VaR – Not Monotone
Modified VaR as a function of Skewness
-10.00%
Modified VaR @ CL 99.00%
Modified...
Problems with Modified VaR – Bad Tail Behaviour
100.00%

18

Modif ied CDF

90.00%

↕

Normal CDF

80.00%

Kurt

70.00%
60...
Problems with Modified VaR – Bad Tail Behaviour
100.00%

18

Modified CDF

90.00%

↕

Normal CDF

80.00%

Kurt

70.00%
60....
How Prevalent is this problem ?
VERY
ETF's

Hedge Funds

OK

OK

WARNING: Degenerate
Cornish Fisher. CDF
w ill turn in tai...
VBA code to check Cornish Fisher - Modification
Public Function CFRegionWarning(ByVal Skew As Double, Kurt As Double) As S...
If not Normal or Modified then What ?
Best Fitting Distributions
ETF's

Hedge Funds

Gumbel (Max) 7.8%

Gumbel (Max) 23.3%...
Assumed Normal
Assumed Normal

Assumed Normal Fund
PDF (Normal)

Normal VaR

Normal CVaR

-40.00%

-30.00%

-20.00%

-10.0...
Assumed Modified
Assumed Normal and Modified Distributions

Assumed Normal Fund
PDF (Normal)

Assumed Modified
Normal Fund...
Best Fitting
Best Fit and Assumed Normal and Modified Distributions
Best Fit Fund PDF
(Gumbel (Min))
Assumed Normal Fund
P...
Goodness of Fit
Cummulative Probability Distributions

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

100%
90%
80%
70%

20%
...
Relationship Between Assets
Relationship between Fund A and B
10.00%

8.00%

6.00%

FUND A

4.00%
Linear Regression
2.00%

0.00%

-2.00%

-4.00%
-6.00...
Assumed Bi-Variate Normal Copula
10.00%

8.00%

6.00%

Assumed Bi-variate
Normal Copula Lines

FUND A

4.00%

2.00%

Linea...
Best Fit Bi-Variate Copula
10.00%
Best Fit Bi-variate
Copula Lines
8.00%
Best Fit Regression

6.00%

Linear Regression

FU...
Example – Johnson Lognormal - Normal
50.00%
Best Fit Bi-variate
Copula Lines
40.00%
Best Fit Regression

30.00%

Linear Re...
Example – Normal - Normal
15.00%
Best Fit Bi-variate
Copula Lines

10.00%

Best Fit Regression

Linear Regression

FUND E
...
Example – Modified Normal - Normal
8.00%
Best Fit Bi-variate
Copula Lines

6.00%

Best Fit Regression

Linear Regression

...
Example – Mix Normals – Modified Normal
30.00%
Best Fit Bi-variate
Copula Lines
20.00%
Best Fit Regression

10.00%

Linear...
Example – Mod Normal – Mod Normal
40.00%
Best Fit Bi-variate
Copula Lines
30.00%
Best Fit Regression
20.00%
Linear Regress...
Relationship Between Assets
2011 Asset Class Correlations
40.00%
30.00%

Total Returns ( % )

20.00%
10.00%
0.00%
-10.00%
...
Relationship Between Assets
2011 Asset Class Correlations
20.00%
15.00%
Total Returns ( % )

10.00%

US Government Bonds

...
Best Fit and Pearson Correl Pairs with CAGR > 7%
Base Fund

Lowest Correlation via Best Fit

Lowest Correlation via Pearso...
30.00%

30.00%

Best Fit Pair CORN - OLO

Pearson Pair CORN - USO

Best Fit Bi-variate
Copula Lines

20.00%

Best Fit Bi-v...
12.00%

12.00%

Best Fit Pair VQT - UCC

10.00%

Best Fit Bi-variate
Copula Lines
10.00%

Pearson Pair VQT - EEH

Best Fit...
25.00%

25.00%

Best Fit Pair DLBL - HYG

Pearson Pair DLBL - JNK

Best Fit Bi-variate
Copula Lines

20.00%

Best Fit Bi-v...
40.00%

20.00%

Best Fit Pair WREI - FTY

Pearson Pair WREI - RWO

Best Fit Bi-variate
Copula Lines

30.00%

Best Fit Bi-v...
Other Measures of Dependence
• Spearman’s Rank Order Correlation
• Lin’s Concordance measure
• Copula methods
• Distance m...
Conclusions
• Distributions do differ from Normal at least 15 – 20% of the time and up to
30 – 40% of the time depending o...
Pietro (‘Peter’) Urbani (45)
•

Chief Investment Officer (CIO) – Infiniti Capital
$3bn Fund of Hedge Funds Group
Head of Q...
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Why distributions matter ( 20 dec 2013 )

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Presentation on deficiencies of the Cornish Fisher modification to the Normal distribution and also on some Best Fit distributions including Gumbel, Johnson Family, Mixture of Normals, Skew-T, 3-Parameter Lognormal etc. Also includes bi-variate Best Fit Copula correlation with applications to Pairs Trading.

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Why distributions matter ( 20 dec 2013 )

  1. 1. Why Distributions Matter By Peter Urbani 16 Jan 2012
  2. 2. Why Distributions Matter 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 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
  3. 3. 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
  4. 4. 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.
  5. 5. The Problem with Assumptions They make an ASS out of U and ME
  6. 6. Normality Testing
  7. 7. Normality Testing Hedge Funds ETF's Normal Normal Not-Normal Not-Normal
  8. 8. Assumed Normal Assumed Normal Assumed Normal Fund PDF (Normal) Normal VaR Normal CVaR -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  9. 9. If not Normal then What ? - Modified VaR ( Modified ) lower for small positive skew
  10. 10. Cornish Fisher - Modification VaR ( Modified ) higher for small negative skew
  11. 11. Cornish Fisher - Modification VaR ( Modified ) same as VaR ( Normal ) for no skew
  12. 12. Assumed Normal Assumed Normal Assumed Normal Fund PDF (Normal) Normal VaR Normal CVaR -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  13. 13. Assumed Modified Assumed Normal and Modified Distributions Assumed Normal Fund PDF (Normal) Assumed Modified Normal Fund PDF (Modified) Normal VaR Normal CVaR Modified VaR Modified CVaR - -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  14. 14. Cornish Fisher - Modification 1.3.2 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 for (or ). -quantiles becomes less and less reliable The Edgeworth- and Cornish-Fisher approximations do not necessarily improve (converge) for a fixed order of approximation, . and increasing 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
  15. 15. Problems with Modified VaR – Not Monotone Modified VaR as a function of Skewness -10.00% Modified VaR @ CL 99.00% Modified VaR @ CL 95.00% Current Current -5.00% VaR ( Modified ) 0.00% 5.00% 10.00% 15.00% 20.00% -6 -4 -2 0 2 4 Skewness http://discussions.ft.com/alchemy/forums/edhec-risk-forum/hedge-fund-risk-management-models-for-the-return-distribution/ 6
  16. 16. Problems with Modified VaR – Bad Tail Behaviour 100.00% 18 Modif ied CDF 90.00% ↕ Normal CDF 80.00% Kurt 70.00% 60.00% S 50.00% 12 40.00% S 30.00% S 20.00% 10.00% 0.00% -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 6 Good Z Z 2500.00% Modified PDF Normal PDF 2000.00% ↔ 0 1500.00% Skew 1000.00% Z 500.00% -6 0.00% -30.0% -6 -20.0% -10.0% 0.0% 10.0% 20.0% 30.0% -4 -2 0 2 4 6
  17. 17. Problems with Modified VaR – Bad Tail Behaviour 100.00% 18 Modified CDF 90.00% ↕ Normal CDF 80.00% Kurt 70.00% 60.00% S 50.00% 12 40.00% S 30.00% S 20.00% 10.00% 0.00% -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 6 Good Z Z 2500.00% Modified PDF Normal PDF 2000.00% ↔ 0 1500.00% Skew 1000.00% Z 500.00% -6 0.00% -30.0% -6 -20.0% -10.0% 0.0% 10.0% 20.0% 30.0% -4 -2 0 2 4 6
  18. 18. How Prevalent is this problem ? VERY ETF's Hedge Funds OK OK WARNING: Degenerate Cornish Fisher. CDF w ill turn in tails WARNING: Degenerate Cornish Fisher. CDF w ill turn in tails WARNING: Degenerate Cornish Fisher. CDF w ill turn in body WARNING: Degenerate Cornish Fisher. CDF w ill turn in body
  19. 19. VBA code to check Cornish Fisher - Modification 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
  20. 20. If not Normal or Modified then What ? Best Fitting Distributions ETF's Hedge Funds Gumbel (Max) 7.8% Gumbel (Max) 23.3% Gumbel (Min) 8.2% Gumbel (Min) 13.2% Johnson (Lognormal) 13.5% Johnson (Lognormal) 13.2% Johnson (Unbounded) 0.6% Johnson (Unbounded) 2.6% Mixture of Normals 20.1% Mixture of Normals 15.9% Modified Normal 35.8% Modified Normal 21.2% Normal 12.9% Normal 6.9% Uniform 1.0% Uniform 3.7%
  21. 21. Assumed Normal Assumed Normal Assumed Normal Fund PDF (Normal) Normal VaR Normal CVaR -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  22. 22. Assumed Modified Assumed Normal and Modified Distributions Assumed Normal Fund PDF (Normal) Assumed Modified Normal Fund PDF (Modified) Normal VaR Normal CVaR Modified VaR Modified CVaR - -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  23. 23. Best Fitting Best Fit and Assumed Normal and Modified Distributions Best Fit Fund PDF (Gumbel (Min)) Assumed Normal Fund PDF (Normal) Assumed Modified Normal Fund PDF (Modified) Best Fit VaR Normal VaR Best Fit CVaR Normal CVaR Modified VaR Modified CVaR - -40.00% -30.00% -20.00% -10.00% 0.00% 10.00% 20.00% 30.00% 40.00%
  24. 24. Goodness of Fit Cummulative Probability Distributions P-P Plot (Showing Goodness of Fit for Fund) 100% 90% 80% 70% 20% Best Fit Fund CDF (Gumbel (Min)) 15% Assumed Normal Fund CDF (Normal) 10% Empirical CDF Fund 0% -25% 50% 40% -5% 30% -10% 20% -15% 10% -20% -30.00% -20.00% -10.00% Fund Best Fit (Gumbel (Min)) Fund Assumed (Normal) 5% 60% 0% -40.00% Series1 0.00% 10.00% 20.00% 30.00% 40.00% -25% -20% -15% -10% -5% 0% 5% 10% 15%
  25. 25. Relationship Between Assets
  26. 26. Relationship between Fund A and B 10.00% 8.00% 6.00% FUND A 4.00% Linear Regression 2.00% 0.00% -2.00% -4.00% -6.00% -4.00% -2.00% 0.00% 2.00% FUND B 4.00% 6.00% 8.00% 10.00%
  27. 27. Assumed Bi-Variate Normal Copula 10.00% 8.00% 6.00% Assumed Bi-variate Normal Copula Lines FUND A 4.00% 2.00% Linear Regression 0.00% -2.00% -4.00% -6.00% -4.00% -2.00% 0.00% 2.00% FUND B 4.00% 6.00% 8.00% 10.00%
  28. 28. Best Fit Bi-Variate Copula 10.00% Best Fit Bi-variate Copula Lines 8.00% Best Fit Regression 6.00% Linear Regression FUND A 4.00% 2.00% 0.00% -2.00% -4.00% -6.00% -4.00% -2.00% 0.00% 2.00% FUND B 4.00% 6.00% 8.00% 10.00%
  29. 29. Example – Johnson Lognormal - Normal 50.00% Best Fit Bi-variate Copula Lines 40.00% Best Fit Regression 30.00% Linear Regression FUND C 20.00% 10.00% 0.00% -10.00% -20.00% -8.00% -6.00% -4.00% -2.00% 0.00% 2.00% FUND D 4.00% 6.00% 8.00% 10.00%
  30. 30. Example – Normal - Normal 15.00% Best Fit Bi-variate Copula Lines 10.00% Best Fit Regression Linear Regression FUND E 5.00% 0.00% -5.00% -10.00% -15.00% -6.00% -4.00% -2.00% 0.00% 2.00% FUND F 4.00% 6.00% 8.00%
  31. 31. Example – Modified Normal - Normal 8.00% Best Fit Bi-variate Copula Lines 6.00% Best Fit Regression Linear Regression FUND G 4.00% 2.00% 0.00% -2.00% -4.00% -15.00% -10.00% -5.00% 0.00% FUND H 5.00% 10.00% 15.00%
  32. 32. Example – Mix Normals – Modified Normal 30.00% Best Fit Bi-variate Copula Lines 20.00% Best Fit Regression 10.00% Linear Regression FUND I 0.00% -10.00% -20.00% -30.00% -40.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% FUND J 10.00% 15.00% 20.00% 25.00% 30.00%
  33. 33. Example – Mod Normal – Mod Normal 40.00% Best Fit Bi-variate Copula Lines 30.00% Best Fit Regression 20.00% Linear Regression FUND K 10.00% 0.00% -10.00% -20.00% -30.00% -40.00% -50.00% -40.00% -30.00% -20.00% -10.00% 0.00% FUND L 10.00% 20.00% 30.00% 40.00% 50.00%
  34. 34. Relationship Between Assets 2011 Asset Class Correlations 40.00% 30.00% Total Returns ( % ) 20.00% 10.00% 0.00% -10.00% -20.00% -30.00% -40.00% -1.00 -0.50 0.00 Correlation to S&P500 0.50 1.00
  35. 35. Relationship Between Assets 2011 Asset Class Correlations 20.00% 15.00% Total Returns ( % ) 10.00% US Government Bonds Gold US Total Bond Market 5.00% Global Bond Index Cash 0.00% US Dollar -5.00% US High Yield Bonds US Bonds CTA's International Government Bonds Volatility US Real Estate Emerging Markets Bonds US Equities Oil Hedge Funds World Equities -10.00% -15.00% Emerging Market Equities -20.00% -1.00 -0.50 0.00 0.50 1.00 Correlation to S&P500 Correlations at multi-decade highs
  36. 36. Best Fit and Pearson Correl Pairs with CAGR > 7% Base Fund Lowest Correlation via Best Fit Lowest Correlation via Pearson Correl
  37. 37. 30.00% 30.00% Best Fit Pair CORN - OLO Pearson Pair CORN - USO Best Fit Bi-variate Copula Lines 20.00% Best Fit Bi-variate Copula Lines 20.00% Best Fit Regression 10.00% Best Fit Regression 10.00% Linear Regression Corn Fund 0.00% Corn Fund 0.00% Linear Regression -10.00% -10.00% -20.00% -20.00% -30.00% -30.00% -40.00% -25.00% -40.00% -20.00% -15.00% -10.00% -5.00% 0.00% DB Crude Oil Long ETN 5.00% 10.00% 15.00% 20.00% -25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% United States Oil Fund 10.00% 15.00% 20.00% 25.00%
  38. 38. 12.00% 12.00% Best Fit Pair VQT - UCC 10.00% Best Fit Bi-variate Copula Lines 10.00% Pearson Pair VQT - EEH Best Fit Bi-variate Copula Lines Best Fit Regression Best Fit Regression 8.00% 8.00% Linear Regression Linear Regression 6.00% Barclays ETN+ S&P VEQTOR ETN Barclays ETN+ S&P VEQTOR ETN 6.00% 4.00% 2.00% 0.00% 4.00% 2.00% 0.00% -2.00% -2.00% -4.00% -4.00% -6.00% -15.00% -6.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% Ultra Consumer Services 20.00% 25.00% 30.00% 35.00% -20.00% -15.00% -10.00% -5.00% 0.00% SPECTRUM Lg Cap U.S. Sector ETN 5.00% 10.00% 15.00%
  39. 39. 25.00% 25.00% Best Fit Pair DLBL - HYG Pearson Pair DLBL - JNK Best Fit Bi-variate Copula Lines 20.00% Best Fit Bi-variate Copula Lines 20.00% Best Fit Regression 15.00% Linear Regression US Treasury Long Bond Bull ETN US Treasury Long Bond Bull ETN 15.00% Best Fit Regression 10.00% 5.00% Linear Regression 10.00% 5.00% 0.00% 0.00% -5.00% -5.00% -10.00% -8.00% -10.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% iBoxx $ HY Corp Bond Fund 6.00% 8.00% 10.00% 12.00% -10.00% -5.00% 0.00% 5.00% SPDR Barclays Capital High Yield Bond ETF 10.00% 15.00%
  40. 40. 40.00% 20.00% Best Fit Pair WREI - FTY Pearson Pair WREI - RWO Best Fit Bi-variate Copula Lines 30.00% Best Fit Bi-variate Copula Lines 15.00% Best Fit Regression 10.00% Linear Regression 10.00% Wilshire US REIT ETF Wilshire US REIT ETF 20.00% Best Fit Regression 0.00% Linear Regression 5.00% 0.00% -10.00% -5.00% -20.00% -10.00% -30.00% -15.00% -15.00% -10.00% -5.00% 0.00% 5.00% FTSE NAREIT Real Estate 50 Index Fund 10.00% 15.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% SPDR DJ Wilshire Global Real Estate ETF 10.00% 15.00% 20.00%
  41. 41. Other Measures of Dependence • Spearman’s Rank Order Correlation • Lin’s Concordance measure • Copula methods • Distance measures • Mutual Information and other entropy based measures
  42. 42. 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
  43. 43. 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 Boutique Head 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

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