Risk Managment & Financial Returns

8/9/2019 Risk Managment & Financial Returns

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Risk Management andFinancial Returns

Elements of Financial Risk Management

Chapter 1



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Why should firms manage risk?

Classic portfolio theory: Inestors caneliminate firm!specific risk "y diersifyingholdings to include many different assets

Inestors should hold a com"ination of therisk!free asset and the market portfolio#

Firms should not $aste resources on riskmanagement% as inestors do not carea"out firm!specific risk#

Modigliani!Miller: &he alue of a firm isindependent of its risk structure#

Firms should simply ma'imi(e e'pected

profits regardless of the risk entailed# Elements of Financial Risk Management


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Eidence on RM practices

*arge firms tend to manage risk moreactiely than small firms% $hich is perhapssurprising as small firms are generallyie$ed to "e more risky#

+o$eer smaller firms may hae limitedaccess to deriaties markets andfurthermore lack staff $ith risk management

skills#



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-oes RM improe Firm .erformance?

&he oerall ans$er to this /uestion appearsto "e 0E#

nalysis of the risk management practicesin the gold mining industry found that share

prices $ere less sensitie to gold pricemoements after risk management#imilarly% in the natural gas industry% "etter

risk management has "een found to result in

less aria"le stock prices# study also found that RM in a $ide group

of firms led to a reduced e'posure to interestrate and e'change rate moements#



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-oes RM improe Firm .erformance?

Researchers hae found that less olatile

cash flo$ result in lo$er costs of capitaland more inestment#

portfolio of firms using RM outperformed

a portfolio of firms that did not% $hen otheraspects of the portfolio $ere controlled for#imilarly% a study found that firms using

foreign e'change deriaties had higher

market alue than those $ho did not#&he eidence so far paints a fairly rosy pictureof the "enefits of current RM practices in thecorporate sector#



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Can RM &echni/ues "e improed?

Eidence on the RM systems in some of thelargest 5## commercial "anks is less cheerful#6n aerage risk forecasts tend to "e oerly

conseratie!perhaps a irtue!"ut at certain

times the reali(ed losses far e'ceeded the riskforecasts# Importantly% the e'cessie lossestended to occur on consecutie days#

6ne is a"le to forecast an e'cessie loss

tomorro$ "ased on the o"seration of ane'cessie loss today#&his serial dependence uneils a potential fla$

in current financial sector RM practices#



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"rief &a'onomy of Risks

Market Risk: the risk to a financial portfoliofrom moements in market prices such ase/uity prices% foreign e'change rates%interest rates and commodity prices#

In financial sector firms market risk should"e managed# E#g# option trading desk#

In nonfinancial firms market risk should

perhaps "e eliminated#



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-efining sset Returns

&he daily geometric or 9log return on anasset is defined as

&he arithmetic return is instead defined as


( ) ( )t t t

S S R lnln 11 −= ++

( ) 1111 −=−=

+++ t t t t t t S S S S S r


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;-efining Returns

&he t$o returns are typically fairly similar% as can

"e seen from

the appro'imation holds "ecause$henis close to 1#

6ne adantage of the log return is that $e caneasily calculate the compounded return for K-days


( ) ( ) ( ) ( ) 11111

1lnlnlnln+++++

≈+==−=t t t t t t t

r r S S S S R

1)ln( −≈ x x x

∑ ∑= =

+−+++++ =−=−=

K

k

K

k

k t k t k t t K t K t t RS S S S R

1 1

1:1 )ln()ln()ln()ln(


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E'ercise sset Returns

6pen file -ata Chapter2-ata#'lsCompute the returns for Close =>. 3. 3


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tyli(ed Facts of sset Returns

We can consider the follo$ing list of so!called styli(ed facts $hich apply to moststochastic returns#

Each of these facts $ill "e discussed indetail in the first part of the "ook#

We $ill use daily returns on the >.3


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tyli(ed Fact 1

-aily returns hae ery little autocorrelation#We can $rite

Returns are almost impossi"le to predictfrom their o$n past#

Fig 1#1 sho$s the correlation of daily

>.3


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1, utocorrelations of -aily >. Returns for *ags 1 through1


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tyli(ed Fact 2

&he unconditional distri"ution of daily returnshae fatter tail than the normal distri"ution#Fig#1#2 sho$s a histogram of the daily

>.3


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E'ercise: tyli(ed Fact 2

5sing same data and function fre/uency%construct 11 e/ually spaced "ins from min=returnto ma'=return% compute pro"a"ilities and graphthe histogram

Remem"er to use C&R**&E&ER to 9dragformula ns$er



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17Figure 1.2

Histogram of Daily S&P 500 eturns an! t"e #ormal Distri$ution

%an 1, 2001 De' 31, 2010

0.0000

0.0200

0.0(00

0.0)00

0.0*00

0.1000

0.1200

0.1(00

0.1)00

#ormal Fre+uen'y

Daily eturn

Pro$a$ility Distri$ution


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E'ercise: tyli(ed Fact ,

Compute mean and standard deiation ofthe returns# 5se function aerage= andstde=

ns$er:

◦ -aily mean of


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tyli(ed Fact 3

@ariance measured for e'ample "y s/uaredreturns% displays positie correlation $ith itso$n past#

&his is most eident at short hori(ons suchas daily or $eekly#

Fig 1#) sho$s the autocorrelation ins/uared returns for the >.3−++ t t R RCorr


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Compute returns s/uared for >.3


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2)Figure 1.3

uto'orrelation of S+uare! Daily S&P 500 eturns

%an 1, 2010 De' 31, 2010

0 10 20 30 0 50 0 0 *0 0 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0

0.5

4ag r!er

uto'orrelation of S+uare! eturns


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tyli(ed Fact 4

E/uity and e/uity indices display negatiecorrelation "et$een ariance and returns#

&his often termed the leeraged effect%

arising from the fact that a drop in stockprice $ill increase the leerage of the firmas long as de"t stays constant# =*gJ-BE

&his increase in leerage might e'plain the

increase ariance associated $ith the pricedrop# We $ill model the leerage effect inChapters , and 3#



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5sing >. 3


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tyli(ed Fact 7

Correlation "et$een assetsappears to "e time arying#

Importantly% the correlation "et$een

assets appear to increase in highlyolatile do$n!markets ande'tremely so during market crashes#

We $ill model this importantphenomenon in Chapter 7#



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tyli(ed Fact 8

Een after standardi(ing returns "ya time!arying olatility measure%they still hae fatter than normal

tails#We $ill refer to this as eidence of

conditional non!normality#It $ill "e modeled in Chapters 4 and

;#



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E'ercise: tyli(ed Fact ;Compute returns $ith lag 3% 1< > 13 for

the data#

+int: 5se the copy!paste function 9smartly

ns$er:



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generi' mo!el of asset returns %P organ


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From asset returns to ortfolio returnsK 9"e /alue of a ortfolio -it" n assets at time t is t"e

-eig"te! a/erage of t"e asset ri'es using t"e

'urrent "ol!ings of ea'" asset as -eig"ts:

• 9"e return on t"e ortfolio $et-een !ay t ;1 an! !ay t

is t"en !e=ne! as -"en

using arit"meti' returns

K ?"en using log returns return on t"e ortfolio is:

))


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@ntro!u'ting t"e Aa ris8 measureAalueatis8 ?"at loss is su'" t"at it -ill only $e

eB'ee!e! pC100 of t"e time in t"e neBt K tra!ing

!aysE

VaR is often !e=ne! in !ollars, !enote! $y VaR

VaR loss is imli'itly !e=ne! from t"e ro$a$ility ofgetting an e/en larger loss as in

#ote $y !e=nition t"at (1G p)100 of t"e time, t"e Loss -ill $e smaller t"an t"e VaR.

lso note t"at for t"is 'ourse -e -ill use VaR $ase! on

log returns !e=ne! as


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)3


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@ntro!u'ing t"e Aa ris8 measureSuose our ortfolio 'onsists of Iust one se'urity

For eBamle an S&P 500 in!eB fun!

#o- -e 'an use t"e is8etri's mo!el to ro/i!e t"e VaR

for t"e ortfolio.

4et VaR P

t+1 !enote t"e p .100 VaR for t"e 1!ay a"ea!return, an! assume t"at returns are normally !istri$ute!

-it" 7ero mean an! stan!ar! !e/iation σ.F%t1# 9"en:

)4


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@ntro!u'ing t"e Aa ris8 measure9a8ing Φ1(∗) on $ot" si!es of t"e re'e!ing e+uation

yiel!s t"e VaR as

@f -e let p > 0.01 t"en -e get Φ1P> Φ1

0.01> ≈ !2#))

@f -e assume t"e stan!ar! !e/iation fore'ast, σ.F%t1 for

tomorro-


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)7Figure 1.

Aalue at is8 (Aa) from t"e #ormal Distri$ution

eturn Pro$a$ility Distri$ution

<

2

,

4

8

1<

12

1,

14

18

Portfolio Return

Return Distribution

1 ! d a y 1 H @ a R J


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@ntro!u'ing t"e Aa ris8 measureonsi!er a ortfolio -"ose /alue 'onsists of 0

s"ares in i'rosoft (S) an! 50 s"ares in JK.

9o 'al'ulate VaR for t"e ortfolio, 'olle't "istori'al

s"are ri'e !ata for S an! JK an! 'onstru't t"e

"istori'al ortfolio seu!o returns

onstru'ting a time series of ast ortfolio seu!oreturns ena$les us to generate a ortfolio /olatility

series using for eBamle t"e is8etri's aroa'"

-"ere

);


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@ntro!u'ing t"e Aa ris8 measure?e 'an no- !ire'tly mo!el t"e /olatility of t"e ortfolio

return, R.F%t1% 'all it σ.F%t1% an! t"en 'al'ulate t"e VaR

for t"e ortfolio as

K ?e assume t"at t"e ortfolio returns are normally!istri$ute!

,


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Dra-$a'8s of Aa

KBtreme losses are ignore! 9"e VaR num$er only tellsus t"at 1 of t"e time -e -ill get a return $elo- t"e

reorte! VaR num$er, $ut it says not"ing a$out -"at

-ill "aen in t"ose 1 -orst 'ases.

VaR assumes t"at t"e ortfolio is 'onstant a'ross t"eneBt K !ays, -"i'" is unrealisti' in many 'ases -"en

K is larger t"an a !ay or a -ee8.

Finally, it may not $e 'lear "o- K an! p s"oul! $e

'"osen.

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