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APPLICATIONS OF ROBUST STATISTICS IN FINANCE
Transcript of APPLICATIONS OF ROBUST STATISTICS IN FINANCE
APPLICATIONS OF ROBUST STATISTICS IN FINANCE
R. Douglas Martin*
Current and Future Challenges in Robust Statistics Banff Center, Canada
Wed. November 18, 2015
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THREE TOPICS: TWO SHORT, ONE LONG
1. Robust detection of fundamental factor model
exposures outliers (skipped)
2. Robust covariances for mean-variance portfolio
optimization: Classic multivariate outliers model
versus independent outliers in assets (IOA) model
(short)
3. Non-parametric versus parametric expected shortfall:
the influence function approach
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1. ROBUST DETECTION OF FUNDAMENTAL
FACTOR MODEL EXPOSURES OUTLIERS*
1. The fundamental factor model context
2. The data for this example
3. The MCD estimator
4. The results
* Joint work with Chris Green, Washington State Investment Board and
UW Statistics PhD Candidate
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1. Fundamental Factor Model
1 t t t t r B f ε
Exposures matrix (N x K)
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Factor returns (K x 1)
Copyright R. Douglas Martin
1 1 1( , , , , )t q q K B b b b b
Industry exposures, each entry a 1 or 0 Continuous risk factor exposures
Work-horse of commercial portolio optimization and risk management
software (MSCI Barra, Axioma, Northfield).
For example: firm size, E/P, B/M, momentum, leverage, etc.
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2. The Data
5 Copyright R. Douglas Martin
Number of companies:
Fundamental data (exposures)
- Large-cap stocks (from CRSP database)
- Size/ME (log. mkt. cap.), B/M, E/P, momentum (Compustat)
- 325 months of data, Dec. 1985-Dec. 2012
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2. The MCD Estimator
6 Copyright R. Douglas Martin
• Cerioli’s IRMCD method
• Test for outliers using Cerioli approach and Bonferroni-corrected significance level of 0.025/325 ≈ 0.00008.
• Use a conservative version of MCD that uses approx. 95% of the data to estimate the robust dispersion matrix
• R package: CerioliOutlierDetection
http://cran.r-project.org/web/packages/CerioliOutlierDetection/index.html
• Working Paper
Green and Martin, “Diagnosing the Presence of Multivariate Outliers in Financial Data using Calibrated Robust Mahalanobis Distances” (2015). Available from http://students.washington.edu/cggreen/uwstat/papers/mvoutliersfinance.pdf
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3. The Results
7 Copyright R. Douglas Martin
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2. ROBUST COVARIANCES FOR MEAN-
VARIANCE OPTIMIZATION
Classic Multivariate Outliers Model vs.
Independent Outliers in Assets (IOA) Model
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(1 ) ( , ) t F N H r μ Σ
Model for common factor outliers/market crashes
The Two Outliers Models
tT NR rtable of returns with rows
, , , 1, 2, , , 1, 2, ,, t i i i M t t i T i Nr r t
Single Factor Market Model for a Portfolio of Assets:
A market return outlier at time t causes an outlier in all assets (to various degrees), i.e., an outlying row.
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Model for independent outliers across assets (IOA) Alqallaf, Van Aelst, Yohai and Zamar (2009)
(AVYZ, 2009) 1 (0) i iB Let if asset is (is not) an outlier.
1 2 , , , ( )N i iB B B P B Assume are independent with
Probability of an outlier-free row : For and the probability of a clean row is .36
tr 1(1 )
i
N
i
, , , 1, 2, , , 1, 2, ,, t i i i M t t i T i Nr r t
Single Factor Market Model for a Portfolio of Assets:
Asset specific (risk) time t outliers tend to be independent
.05 20N
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Empirical Study of IOA Model Validity
Four market-cap groups of 20 stocks, weekly returns
1997 – 2010 in three regimes:
– 1997-01-07 to 2002-12-31
– 2003-01-07 to 2008-01-01
– 2008-01-08 to 2010-12-28
1. Estimate outlier probability for each asset, and hence the probability that a row is free of outliers under the IOA model.
2. Directly estimate the probability that a row is outlier free.
3. Compare results from 1 and 2 across market-caps and regimes.
i
1
1n
i i
1 row
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-0.3
-0.1
0.1
0.3
PLX
S
-0.2
0.0
0.1
0.2
BW
INB
-0.2
0.0
0.1
0.2
HG
IC
2000 2005 2010
Index
-0.1
0.0
0.1
0.2
WT
S
SMALL-CAPS
4 of the 20 Small-Caps for Entire History
Outlier Detection Rule for Counting
optimal 90% efficient bias robust location estimate*
associated robust scale estimate*
Outliers: returns outside of
Probability of normal return being an outlier: 0.5%
* Use lmRob with intercept only in R package robust
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ˆ ˆ ˆ ˆ( 2.83, 2.83)s s
s
15
1 3 5 7 9 11 14 17 20
% OUTLIERS IN EACH ASSET
ASSETS
PE
RC
EN
T
02
46
8
2008 2009 2010 2011
02
46
810
12
Index
CO
UN
T
# OF ASSETS WITH AN OUTLIER
Small-Caps Outliers in Third Regime
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Large-Caps Outliers in Third Regime
2008 2009 2010 20110
510
15
Index
CO
UN
T
# OF ASSETS WITH AN OUTLIER
1 3 5 7 9 11 14 17 20
% OUTLIERS IN EACH ASSET
ASSETS
PE
RC
EN
T
01
23
45
6
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Evaluation of IOA Model for Weekly Returns
1997-01-07 to 2002-12-31 MICRO SMALL MID LARGE
% Clean Rows IOA Model 32 39 46 59
% Clean Rows Direct Count 37 48 58 69
2003-01-07 to 2008-01-01 MICRO SMALL MID LARGE
% Clean Rows IOA Model 33 46 52 57
% Clean Rows Direct Count 37 49 62 66
2008-01-08 to 2010-12-28 MICRO SMALL MID LARGE
% Clean Rows IOA Model 23 31 39 46
% Clean Rows Direct Count 43 48 57 62
Main Result: The differences % Clean Rows Direct Count - % Clean Rows IOA
Model tend to be larger during the first and third time periods, which is apparently
due to the fact that market crashes cause highly correlated outliers in most
assets, for which the IOA model does not hold. However the IOA model seems to
describe much of the behavior, particularly so for the middle time period.
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1.0
1.5
2.0
2.5
gmv.lo
gmv.lo.mcd
gmv.lo.qc
mkt
Cum
ula
tive R
etu
rn
Weekly Returns, Window = 60, Rebalance = Weekly
LONG-ONLY GMV,GMV.MCD, GMV.PW, MKT
-0.1
50.0
0
Weekly
Retu
rn
1998-03-03 2001-07-03 2005-01-04 2008-07-01
Date
-0.5
-0.1
Dra
wdow
n
19
1.0
1.5
2.0 gmv.lo
gmv.lo.mcd
gmv.lo.qc
mkt
Cum
ula
tive R
etu
rn
Weekly Returns, Window = 60, Rebalance = Monthly
LONG-ONLY GMV,GMV.MCD, GMV.PW, MKT
-0.1
50.0
0
Weekly
Retu
rn
1998-03-03 2001-07-03 2005-01-04 2008-07-01
Date
-0.5
-0.1
Dra
wdow
n
20
“Statistics is a science in my opinion, and it is no more a branch of mathematics than are physics, chemistry and economics; for if its methods fail the test of experience – not the test of logic – they will be discarded”
- J. W. Tukey
References
Alqallaf, Konis, Martin and Zamar (2002). “Scalable robust covariance and correlation
estimates for data mining”, Proceedings of the eighth ACM SIGKDD International
Conference on Knowledge Discovery and Data Mining, pp. 14-23. ACM.
Alqallaf, Van Aelst, Yohai and Zamar (2009). “Propagation of Outliers in Multivariate
Data”, Annals of Statistics, 37(1). p.311-331.
Boudt, Peterson & Croux (2008). “Estimation and Decomposition of Downside Risk for
Portfolios with Non-Normal Returns”, Journal of Risk, 11, No. 2, pp. 79-103.
Maronna, Martin, and Yohai (2006). Robust Statistics : Theory and Methods, Wiley.
Martin, R. D., Clark, A and Green, C. G. (2010). “Robust Portfolio Construction”, in
Handbook of Portfolio Construction: Contemporary Applications of Markowitz
Techniques, J. B. Guerard, Jr., ed., Springer.
Martin, R. D. (2012). “Robust Statistics in Portfolio Construction”, Tutorial
Presentation, R-Finance 2012, Chicago.
Scherer and Martin (2005). Modern Portfolio Optimization, Chapter 6.6 – 6.9,
Springer
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3. NON-PARAMETRIC VERSUS PARAMETRIC
EXPECTED SHORTFALL:
The Influence Function Approach*
*Joint work with Shengyu Zhang, PhD, First Vice President, Homestreet Bank Seattle, WA
1. VALUE-AT-RISK
Valued-at-Risk (VaR)
– A capital weighted 1% or 5% tail probability quantile
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VaR is the de facto risk management standard
– Originated in the 1990’s at JP Morgan
– RiskMetrics™ Technical Document (1994)
– Value-at-Risk, Jorion 3 editions: (1996), (2000), (2006)
VaR is simply not informative enough
– Doesn’t tell you anything about the size of losses beyond VaR
1( ; ) ( )q F F ( ; ) ( ; )VaR F W q F
What it is
– The average of the losses beyond VaR
– Much more informative than VaR
– ES a coherent risk measure, VaR is not
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Alternative names for ES
– Conditional Value-at-Visk (CVaR)
– Expected Tail Loss (ETL)
Basel III move to ES instead of VaR
– “Fundamental review of the trading book - second consultative document” (2013). Basel
Committee on Banking Supervision. www.bis.org/publ/bcbs265.htm.
– “Fundamental Review of the Trading Book: Outstanding Issues” (2014). Basel Committee
on Banking Supervision. www.bis.org/bcbs/publ/d305.htm.
2. EXPECTED SHORTFALL (ES)
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Apr 1994 Jan 1996 Jan 1998 Jan 2000 Jan 2002 Jan 2004
-0.1
0-0
.05
0.0
0
ES vs VaR: Event Driven Hedge Funds Index
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ES vs VaR: Event Driven Hedge Funds Index
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1( ; ) ( )q F F
( ; )
1( ; ) ( )
r q F
ES F r dF r
Taking loss as negative for math but positive for some plots
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( ; ) ( | ( ; ))ES F E r r q F
Expected Shortfall Formula
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Non-Parametric Estimators
( )
1( ; )
1 1( ) ( )
n
n
n i
ir q Fn
ES r dF r r
Represent the asymptotic value of any estimator as a functional
on a space of distribution functions:
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( )F
where is the empirical distribution function (edf). nF
ˆ ( )nF
Expected shortfall estimator:
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Parametric Estimators
First get a parametric representation of the risk measure:
( )( , )
zES
Normal distribution ES
t-distribution ES ,
( , , )g
ES s s
, 2 ,
2
2g f q
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Then substitute estimators for the unknown parameters
ES Parametric Estimators
Normal Distribution t-Distribution
( )ˆ ˆ
z
ˆ,ˆ ˆg
s
MLE parameter estimators MLE risk estimator
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OXM Returns
0 100 200 300 400 500
-0.1
5-0
.10
-0.0
50.
00.
050.
10
Normal vs Fat-Tailed Parametric ES/ETL Estimator
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-0.2 -0.1 0.0 0.1 0.2
05
10
15
20
25
30
OXM DAILY RETURNS
1% STABLE ETL vs. NORMAL VAR AND ETL: $1M OVERNIGHT
Normal VaR = $47K
Normal ETL = $51K
Stable ETL = $147K
STABLE DENSITYNORMAL DENSITY
Normal vs Fat-Tailed Parametric ES/ETL Estimator
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Non-Parametric ES IF Formulas
Straightforward but tedious calculations give complex formulas.
But the plots tell the story.
( ; )
1( ; ) ( )
p
p p
r q F
ETL F r dF r
(1 ) , 0 1p rF p F p p
0
( ; ) ( ) pETL
p
dIF r F ETL F
dp
3. ES INFLUENCE FUNCTIONS
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IF of VaR
r
-3 -2 -1 0 1 2 3
01
02
03
04
05
06
07
0
IF of ES
r
-3 -2 -1 0 1 2 3
01
02
03
04
05
06
07
0
p=0.1p=0.05p=0.01
Non-Parametric VaR and ES Influence Functions
Straightforward formula calculations available in paper
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General IF Formula
Parametric Risk Estimators
ˆˆ ( ; ) ( ) ( )IF r F r θ
θ IF
MLE IF Formula 1ˆ( ; , ) ( ) ( ; )IF r F r θ
θ I ψ θ
Risk Measure 1 2( ) ( , , , )K θ
Risk Estimator 1 2ˆ ˆ ˆˆˆ ( ) ( , , , )K θ
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Normal Distribution
t-Distribution
known unknown
2 2( ) ( )( )
2
z rr
,1( ) ( )s
gr r
B C
,
1
, ,
,
( )
1 ( )
( )
s
s
gs
r
r
g r
I
ES IF Formulas for Normal and t-Distributions
2 2
( 1)( )( )
( )
v rr
vs r
2
3 2
1 ( 1)( )( ; , )
( )s
v rr s
s vs s r
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2 2 2
2
2 2 2
, , 2
2 2 2
2
2
1 1 1 1 1 1 1 10
4 2 2 1 2( 3) 3 1)
1 2 0 1 0
3
1 1 1 20
3 1)
v s
l l l
v v v s
l l lI E
v s
l l l
s v s s
v v
v v s v v
s v
s v v s
2 3
v
v
Information Matrix for t-Distribution
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2 2
2 2 2 2
2 2
2
3 2
1 1 1 1 ( ) ( 1)( )log 1
2 2 2 2 2 2 2 ( )
( 1)( )
( )
1 ( 1)( )
( )
s
l
l
l
s
v v r v r
v vs v s v r
v r
vs r
v r
s vs s r
Score Vector for t-Distribution
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IF’s of Normal Distribution VaR and ES MLE’s
Positive returns contribute to risk! A serious short-coming!
These IF are quadratically unbounded. Furthermore:
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IF’s of t-Distribution Risk MLE’s for Known d.o.f. = 5
These influence function are all bounded. But still:
Positive returns contribute to risk! A serious short-coming!
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IF’s of t-Distribution Risk MLE’s for Unknown d.o.f. = 5
IF of VaR
r
-5 -3 -1 0 1 2 3 4 5
-10
01
02
03
04
05
06
07
0
IF of ES
r
-5 -3 -1 0 1 2 3 4 5
-10
01
02
03
04
05
06
07
0
p=0.1
p=0.05
p=0.01
These influence function are unbounded, but only logarithmically.
Positive returns contribute to risk! A serious short-coming!
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IF of ES
r
-10 -8 -6 -4 -2 0 2 4 6 8 10
02
04
06
08
01
00
12
0
Par-Normal
Par-t known d.o.f.
Par-t unknown d.o.f.
IF’s of ES Parametric Estimators (MLE’s)
2.5% tail probability (Basel III recommendation)
t-distribution d.o.f. = 5
log | | for large | |r r
quadratic
bounded
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2
ˆ ˆ( ) ( ; , ) ( )V IF r F dF r
Applicable to both non-parametric and parametric MLE estimators.
Complicated formulas are available in paper.
1/2
ˆˆ( )
ˆ. .( )n
VS E
n
4. ES ESTIMATOR ASYMPTOTIC VARIANCE
Influence function based asymptotic variance:
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0.0 0.1 0.2 0.3 0.4 0.5
01
02
03
04
05
0
Tail Probability
S.E
.
d.o.f.=3
d.o.f.=5
d.o.f.=7
d.o.f.=inf (normal)
Non-Parametric ES S.E.’s for t-Distributions
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0.0 0.1 0.2 0.3 0.4 0.5
05
10
15
20
25
30
Tail Probability
S.E
.
d.o.f.=3
d.o.f.=5
d.o.f.=7
d.o.f.=inf (normal)
Parametric ES S.E.’s for t-Distributions
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0.0 0.1 0.2 0.3 0.4 0.5
01
02
03
04
05
0
Tail Probability
S.E
.
Nonparametric
Parametric unknown d.o.f.
Parametric known d.o.f.
Non-Parametric vs Parametric ES S.E.’s
t-distribution with d.o.f. = 3
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Ratio of S.E. of ES Parametric Estimators to
S.E. of ES Non-Parametric Estimators
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Main Expected Shortfall Messages
Use the influence function to analyze risk and
performance estimators
- Understand data behavior of the estimators
- Get asymptotic standard errors
- Can apply to other risk and performance measures
Parametric estimators are more accurate than non-
parametric estimators, but:
- Defect is that positive returns contribute to downside risk
- Variability price of estimating t-distribution d.o.f. is high
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5. A WAY TO FIX THE PROBLEM
The problem with parametric ES that positive returns
contribute to risk is due to the symmetric nature of the
standard deviation and scale estimators
A potential solution: Use a semi-scale estimator
Initial investigation for normally distributed returns is to
use a semi-standard deviation estimator instead of the
standard deviation (volatility) estimator
1/2
1/22 2
1( ) ( ( ) ) ( ) ( )
i
i
r r
F F r dF r SSD r rn
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IF of Semi-Scale Parametric CVaR
r
IF(r
)
-3 -2 -1 0 1 2 3
0
10
20
CVaR (.01)CVaR (.05)CVaR (.1)
CVaR = ES
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Standard Error of CVaR under Normal Distribution
p (tail probability)
SE
of
Va
R
0.0 0.1 0.2 0.3 0.4 0.5
12
34
Par. CVaR with Std. Dev.Par. CVaR with Semi-SDNonparametric CVaR
CVaR = ES
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6. INEFFICIENCY OF MODIFIED ES
Modified VaR (mVaR) was proposed by Zangari (1996)* to
improve normal VaR by adding skewness and kurtosis corrections
2 3 2 31 1 1[ 1] [2 5 ] [ 3 ]
6 36 24g z z S z z S z z K
( )mVaR g
(Joint work with Rohit Arora, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2692543)
*Zangari, P. (1996). “A VaR methodology for portfolios that include options”,
RiskMetrics Monitor 1, First Quarter, pp. 4–12.
Reduces to usual normal VaR formula under normality
Widely publicized and used in a commercial portfolio product
called AlternativeSoft, but not a very good idea ….
where
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Modified Expected Shortfall (mES)
Modified ES (mES) was proposed and studied by Boudt et al.
(2008)* as a way to improve normal distribution ES when returns
are non-normal.
*Boudt, K., Peterson, B., and Croux, C. (2008). “Estimation and Decomposition of Downside
Risk for Portfolios with Non-normal Returns” , Journal of risk 11, pp. 79–103.
Reduces to usual normal mES formula under normality
2
1 2 31mETL g c SK c SK c K
where are polynomials in . 1 2 3, , c c c g
This turns out to perform even worse than mVaR
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Asymptotic Inefficiency of mES
The parametric ES estimators for normal and t-distributions are
maximum-likelihood estimators (MLE’s), hence they attain the
information lower bound on asymptotic variance
Martin and Rohit (2015)* develop the large sample variance of
mES, and calculate the asymptotic efficiency of mES estimator,
defined as the ratio of the asymptotic standard deviation of the ES
MLE to that of the mES estimator for normal and t-distributions.
The results are on the next slide.
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*Martin, R. D. and Rohit, Arora (2015). “Inefficiency of Modified VaR and ES”, submitted
to Journal of Risk . Will be posted to SSRN.
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Asymptotic Inefficiency of mES
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OPEN QUESTIONS & FUTURE RESEARCH
Will robust semi-scale bound the undesirable influence
of positive returns on parametric risk estimators?
Need results for skewed t-distributions
Need correct asymptotic variances and standard
errors in the presence of serial correlation
- Recall Lo (2002) results for Sharpe ratio
- Incipient work on this with Xin Chen
How do you back-test ES estimators?