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Problems and Solutions in
Portfolio Risk
Brian Peterson
Peter Carl
Kris Boudt
Authors of PerformanceAnalytics
1 July, 2009
eielisal!, S"it#erland
R$Retri%s &orksho!
1 Jul 2009 Meielisalp RMetrics Workshop 2
'utline
• About Performan%e Analyti%s
• Risk easures
• Risk
• A!!endi() Buildin* Blo%ks

+ata

+istributions

,ra!hi%s

-ables
1 Jul 2009 Meielisalp RMetrics Workshop 3
About Performan%eAnalyti%s
• .ibrary of e%onometri% fun%tions for !erforman%e and risk
analysis of finan%ial !ortfolios/
• Aims to be useful to both !ra%titioners and resear%hers
alike/
• Analysis of return streams, "hether distributed normally or
not/
• 0n de1elo!ment sin%e early 2002, first released to CRA3 in
2004/
• 3o" %ontains more than 150 fun%tions and more than
11,000 lines of %ode and 4,600 lines of do%umentation/
• Collaboration, !at%hes and su**estions from users in
industry and a%ademia "orld"ide
1 Jul 2009 Meielisalp RMetrics Workshop 4
PerformanceAnalytics
CRAN Version 0.9.7.1
• Shar!e7s Style Analysis
• Snailtrail %hart
• 8aR Sensiti1ity %hart
• odified 9(!e%ted Shortfall
• ulti1ariate moments and
risk metri%s
• :i*her %o;moments
• Robust data %leanin*
• any fe"er de!enden%ies
• 'ther ne" fun%tions, bu*
fi(es
Release 1.0
• Com!onent 8aR and
• Com!onent 9(!e%ted
Shortfall
• &ra!!ers of 8aR and 9S
fun%tions for %onsisten%y
• Ado!tion of (ts for time
series
• 0m!ro1ed (;a(is handlin* in
%harts
• <ormatted tables in de1i%es
• Ca!ture ratios and other
metri%s
• Bu* fi(es, ne" fun%tions
eielisal! Retri%s &orksho! 6 1 Jul 2009
Performan%e Summary

:o" has the fund$ strate*y$
!ortfolio !erformed in the !ast=

Cumulati1e !erforman%e does not
sho" relati1e !erforman%e "ell,
but *i1es a sense for the o1erall
sha!e/ See chart.CumReturns

Rankin* and relati1e !erforman%e
alon* the "ay is im!ossible to
assess/

+ra"do"ns sho" e(tent of
losses relati1e to !eak >e?uity>/
See chart.Drawdown

8aR throu*h time sho"s ho"
estimation %han*ed/ See
chart.BarVaR

Relati1e !erforman%e sho"s
!eriods of out;!erforman%e,
under;!erforman%e/ See
chart.RelativePerformance
1 Jul 2009 Meielisalp RMetrics Workshop 6
Risk easures
1 Jul 2009 Meielisalp RMetrics Workshop 7
+o"nside Risk
• &hat ha1e @normalA losses been in a do"n !eriod= At "hat le1el should "e set
alarms for re;e1aluation of an asset in a !ortfolio=
• +o losses beha1e differently than *ains= +ra"do"n analysis may be 1aluable/
• arko"it# offers semi1arian%e
• Sortino su**ests that risk may in%lude shortfall from a *oal
• Se1eral do"nside risk measures ha1e been %olle%ted in tale.DownsideRis!
1 Jul 2009 Meielisalp RMetrics Workshop 8
Risk$Re"ard Ratios
• Shar!e first su**ested a ratio of return to risk Bmeasured by
annuali#ed Ce(%essD return o1er annuali#ed standard
de1iationE "hi%h is im!lemented as "har#eRatio
• &illiam Shar!e no" re%ommends $nformationRatio
!referentially to the ori*inal Shar!e Ratio
• any other authors ha1e modified the risk measure in use
to %reate similar ratios
• "ortinoRatio return o1er do"nside de1iation
• %#sidePotentialRatio im!ro1ement of the Sortino ratio
that measures u!side return in *ood runs o1er losses in
dra"do"n !eriods
• Se1eral other modified Shar!e ratios ha1e been !ro!osed,
and are im!lemented in Performan%eAnalyti%s, in%ludin*
usin* any of ,aussian and Cornish <isher modified 8aR
and 9S measures as the risk measure in !arameters to
"har#eRatio.modified
1 Jul 2009 Meielisalp RMetrics Workshop 9
• 8alue at Risk B8aRE has be%ome
a re?uired standard risk measure
re%o*ni#ed by Basel 00 and
i<0+
• Performan%eAnalyti%s 1/0 has a
VaR fun%tion "ith standardi#ed
!arameters for histori%al,
,aussian, kernel, and Cornish
<isher 8aR/
8alue at Risk B8aRE

&ra!!er "ill !robably be e(tended to in%lude onte Carlo, Ske"
Student;t, and *enerali#ed Pareto 8aR estimators soon

Key !arameters are the time series, !F!robabilityG, and the method
desired/

Also allo"s moments to be !assed in to allo" for different estimators
or o!timi#ation

Also allo"s for ar*inal and Com!onent 8aR %al%ulations
eielisal! Retri%s &orksho! 10 1 Jul 2009
8alue at Risk B8aRE

&hat le1el of loss is >normal>=

A s!e%ial %ase of do"nside risk, the
limitations of mean 8alue;at;Risk are
"ell %o1ered

'ther sin*le;instrument 8aR methods,
in%ludin* Cornish <isher, Ske";t,
onte Carlo

9(tensions su%h as 9(!e%ted Shortfall

Com!arison hel!s *ain some insi*ht
into the !erforman%e of the different
8aR measures/ See
chart.VaR"ensitivit&

<or e(am!le, modified BCornish <isherE
8aR %an be lo"erBsmallerE than
traditional 8aR if the asset e(hibits
!ositi1e ske"ness and lo" kurtosis

8aR measures *ain e(tra im!ortan%e
in a !ortfolio B%om!onentE settin*
1 Jul 2009 Meielisalp RMetrics Workshop 11
9(!e%ted Shortfall B9SE
• Conditional Value at Risk (CVaR) also kno"n as Expected
Shortfall (ES) is the mean e(!e%ted loss "hen the loss
e(%eeds the 8aR
• &e ha1e "ra!!ed '" in a manner similar to VaR and
!ro1ide both sin*le;instrument and %om!onent 1ersions for
,aussian, :istori%al, Kernel, and Cornish <isher 9(!e%ted
Shortfall
• Beyond 8aR BVaR.Be&ondE is a related measure of mean
e(!e%ted tail loss that adds 8aR and 9S
1 Jul 2009 Meielisalp RMetrics Workshop 12
&hat Sensiti1ity$Confiden%e akes Sense for Risk=
• Some in1estors, risk mana*ers, or re*ulators "ill ask for
%onfiden%e of 99/99H or e1en hi*her/
• -his means)

'n%e in 56 years for monthly data

'n%e in 20 years for "eekly data

'n%e in 2 years for daily data
• -hese kinds of risk %onfiden%e !robabilities only make
sense if you ha1e a .'- of data, or for hi*h;fre?uen%y data
o1er short hori#ons Bminutes to maybe an hourE
• :i*h %onfiden%e le1els also assume a stable series,
!robably a bad assum!tion o1er the time frames abo1e/
• +e1elo!in* these %onfiden%e le1els by e(tendin*$e(!andin*
your time series throu*h simulation may "ork
• Be"are of hidden distributional assum!tions
1 Jul 2009 Meielisalp RMetrics Workshop 13
+ata Cleanin* and Smoothin*
• 'ut;of;sam!le !redi%tions of return or risk may be ad1ersely
affe%ted by auto%orrelation or outliers in the data
• ,eltner and ,etmansky ha1e !ro!osed different methods of
dealin* "ith auto%orrelation im!lemented as Return.(eltner
and "moothin)$nde*
• &e !ro!ose and test a method in our 2005 JoR !a!er for a
robust method of %leanin* returns to im!ro1e out of sam!le risk
!redi%tions/

identify the returns that are outside the %onfiden%e threshold desired
for the risk measure Be/*/ 96H or 99HE

redu%e the ma*nitude of those outlyin* returns if they are outside the
ran*e of the other returns as identified by the ahalanobis distan%e

redu%tion in ma*nitude kee!s ranked ma*nitude inta%t Bthe lar*est
loss$*ain "ill remain the lar*est loss$*ain, e1en after smoothin*E

see Boudt,Peterson,Crou( B2005E for details or Return/%lean
1 Jul 2009 Meielisalp RMetrics Workshop 14
+ata Cleanin*
• 3ote the smoothin* of outlyin* returns around the Russian finan%ial
%risis/
• Robust smoothin* im!ro1es out of sam!le risk !redi%tions
• >edhe%> data series %leaned by Return.clean 1ia the fun%tion
chart.arVaR
1 Jul 2009 Meielisalp RMetrics Workshop 15
Analy#in* Performan%e in <inan%e
9(tensions to Portfolios of Assets
1 Jul 2009 Meielisalp RMetrics Workshop 16
9(tensions to Portfolios of Assets
• Risk Bud*ets de%om!ose total !ortfolio risk in the risk
%ontribution of ea%h !ortfolio !osition

Risk bud*et te%hnolo*y is relati1ely re%ent B90sE, but already
"ides!read in !ortfolio risk mana*ement
• 3ot as easy as it sounds, sin%e it re?uires)

Additi1ity

A%%ount for di1ersiI%ation B%orrelation, non;linear de!enden%eE

0nter!retability of risk %ontributions
• So in a !ortfolio %onte(t, 8aR is not an ideal risk measure,
be%ause it is not al"ays subadditi1e

8aR of a !ortfolio J Sum of 8aR of the %om!onents

8iolates the idea of !ortfolio di1ersiI%ation BArt#ner et al, 1999E
1 Jul 2009 Meielisalp RMetrics Workshop 17
Prior Attem!ts to e(tend to !ortfolios of assets
• Kse of uni1ariate models to inde!endently e1aluate risk on
indi1idual !ositions

Kni1ariate 8aR

Kni1ariate 9S
• ar*inal 8aR

Contribution at the mar*in may be useful for %om!arin* similar
assets to %hoose the better one to !la%e in a !ortfolio
• +e1elo!ment of Coherent Risk easures BArt#nerE

8aluable addition sets standards to be follo"ed
1 Jul 2009 Meielisalp RMetrics Workshop 18
ar*inal, 0n%remental, and Com!onent 8aR
• Marinal VaR is the %han*e in the !ortfolio 8aR attributable to a
%han*e in !osition at the mar*in/
; 0m!lemented in the VaR "ra!!er or Bde!re%atedE as VaR.+ar)inal
; 3ot !arti%ularly useful, e(%e!t in s!e%ial %ases for rebalan%in* similar
instruments
• Component VaR is the risk %ontribution of ea%h instrument to the
risk of the "hole !ortfolio

0m!lemented by settin* @!ortfolioLmethodA in VaR$'" "ra!!er
• <un%tion returns both additi1e %ontribution and !er%ent
%ontribution
1 Jul 2009 Meielisalp RMetrics Workshop 19
&ra!!ers for 8aR and 9s
• <un%tions ha1e been e(tended to !ro1ide %om!onent risk,
too)
J 8aRBR, !M0/96, methodMNmodifiedN, !ortfolioLmethodMN%om!onentNE
O8aR
C,1D
C1,D ;0/026P4122
O%ontribution
C1D ;0/002Q2Q0P9 ;0/004P5P56P ;0/00Q112469 ;0/00P5Q4612 ;0/002Q996Q2
CPD ;0/00QQ11P94
O!%tL%ontribL8aR
C1D 0/09029220 0/2992Q2Q9 0/12126Q5Q 0/2PPQ2419 0/09Q24092 0/12900Q26
&arnin* messa*e)
0n 8aRBR, ! M 0/96, method M NmodifiedN, !ortfolioLmethod M N%om!onentNE )
no "ei*hts !assed in, assumin* e?ual "ei*hted !ortfolio
1 Jul 2009 Meielisalp RMetrics Workshop 20
Risk Contribution in an 9& Portfolio
• An e?ual "ei*hted !ortfolio of
si( assets
• 0n this %ase, si( hed*e fund
style inde(es from dataBedhe%E
• Si*nifi%ant differen%es in
estimates for %ontribution by
C-A>s
• Portfolio mana*er and risk
mana*ement both ha1e an
interest in %onstrainin*
!er%enta*e risk %ontributions

+oes it lead to different "ei*ht
and risk allo%ations=

&hat is the effe%t on !ortfolio
!erforman%e=
1 Jul 2009 Meielisalp RMetrics Workshop 21
Problems "ith Com!onent Risk for Portfolio '!timi#ation
• 3on;normal e(tensions are not o!timi#able as a %losed;
form solution, so other solutions are re?uired)

Brute for%e o!timi#er

:as ad1anta*e of findin* the true ma(ima$minima, but is 1ery
%om!utationally e(!ensi1e

Random solution s!a%e sear%h

Simulated annealin*

3on;linear o!timi#ations
• ethods of usin* Risk Contributions are not "ell studied as
de%ision makin* %riteria

-y!i%ally ba%k"ards lookin*, des%ri!ti1e

'ur %urrent resear%h is lookin* at "ays of usin* these method to
%onstru%t better !ortfolios
1 Jul 2009 Meielisalp RMetrics Workshop 22
-hin*s .eft -o +o
,urther Research
• Risk Bud*etin*

A!!li%ations of Com!onent Risk to lar*e !ortfolios
• :istori%al !ortfolio frame"ork in R B!a%ka*e) blotterE

-ransa%tions and !ositions "ith !rofit and loss

0nstrument !ro!erties and model

0nterfa%es to data !ro1ider !a%ka*es
• Performan%e attribution and Portfolio Analyti%s B"ith 8iRay or
+iethelm, !erha!s=E
• Pra%ti%al Bayesian a!!li%ations
• Anythin* S'K "ant to "ork "ith us on
!hank "ou for "our Attention

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