simple montecarlo

8/19/2019 Simple Montecarlo

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Risk256.comQuantitative Financial Risk Management

Introduction

To construct our Monte Carlo simulation, we need to use Excels !ataTa"le #eatureta"s %rovide a "rie# introduction to !ataTa"les. 'ou are alread' #amiliar wit$ !at$e Monte Carlo ta"s.


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. T$e next twoaTa"les ski% to


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Model !ataTa"le

( (2 ) )

*

2

+

)

5

6

-

*/

**

*2

*+

*)

*5

*6

(2

(2

0um"ers #romt$e 1rst columno# t$e !ataTa"leget su"stitutedinto t$is cell.

T$e result #romt$is cell ends u%in t$e secondcolumn o# t$e!ataTa"le.


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!ata Ta"le

DataTable Introduction

!ataTa"les are not ver' intuitive, "ut t$e' can "e extremel'automaticall' see $ow c$anging t$e in%uts to an e3uation orout%ut. &n t$is exam%le we $ave an extremel' sim%le e3uati

T$e 1rst column o# t$e !ataTa"le contains t$e in%uts. T$esesu"stituted into cell 4. T$ese can "e an't$ing t$at 'ou wanintegers **6.

e set t$e cell at t$e to% o# t$e second column e3ual to t$eadd additional columns wit$ additional out%uts, "ut, #or now,


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use#ul. !ataTa"le allows 'ou toseries o# e3uations will im%act t$en, cell C is sim%l' cell 4 s3uared.

are t$e values t$at will "e. &n t$is exam%le we $ave t$e

ut%ut #rom our e3uation. e couldwe limit ourselves to one out%ut.


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( (2 ) )

*

2

+

)

5

6

-

*/

**

*2

*+

*)

*5

*6

(2

(2

0um"ers #romt$e 1rst columno# t$e !ataTa"leget su"stitutedinto t$is cell.

T$e result #romt$is cell ends u%in t$e secondcolumn o# t$e!ataTa"le.


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!ata Ta"le

DataTable Introduction

0ext we insert our !ataTa"le. First select t$e !ataTa"le rangeasiest wa' to insert t$e !ataTa"le is to go to t$e !ata ta",

Tools "ox, and select !ataTa"le. n e3uation "ox wit$ 9Row is$ould a%%ear. 4ecause t$e in%uts and out%uts o# our !ataTaenter our in%ut cell :4: into t$e 9Column in%ut cell79 "ox ;'cell?, and 'ou are done. Excel s$ould t$en 1lls outwit$ t$e value we would $ave o"tained in cell C i# t$e value

T$is exam%le is a "it contrived. 8iven $ow sim%le our e3uatito enter t$e e3uation into t$e second column o# t$e ta"le dirco%' it down


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, cells F7 82+. &n Excel 2//, t$elick $at nal'sis in t$e !ata%ut cell79 and 9Column in%ut cell79

"le are arranged in columns, weu can t'%e 4, or sim%l' click on t$et$e second column o# t$e !ataTa"lein t$e 1rst column $ad "een in 4.

n is, it would $ave "een easier @ustctl' ;enter it into t$e 1rst cell andwe are dealing wit$ com%lex series


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AB/,* 0;/,*< 0;/,*</.2525* /.6665-) /.6665-

* /.6665-

2 /.6665-

!ataTa"le dumm' + /.6665-

) /.6665-

5 /.6665-

6 /.6665-

/.6665-

- /.6665-

/.6665-

*/ /.6665-

** /.6665-

*2 /.6665-

*+ /.6665-

*) /.6665-

*5 /.6665-

*6 /.6665-

mean /.6665-

standard deviation /./////

!ata


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Ta"le

Very Simple Monte Carlo

&n t$is exam%le, we created a standard normal random varia"le in cell C!ataTa"le to collect *6 draws #rom t$e random varia"le. e $ave calculstandard deviation o# t$is out%ut. Ditting t$e F ke' will recalculate all

Dow did we do t$is T$e 1rst row o# t$e !ataTa"le contains t$e integerst$ese num"ers would "e t$e in%uts #or our !ataTa"le, "ut, in t$is setu%,matter. e could @ust as easil' $ave le#t t$em "lank. T$e to%most cell o8, %oints to our standard normal random varia"le in cell C.

$en we insert t$e !ataTa"le, we select cells F782+, and set t$e 9Colu"lank cell ;in t$is case, cell !*/


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. e t$en use t$eated t$e mean and6 values.

* t$roug$ *6. 0ormall't$eir values dont reall't$e second column, cell

mn in%ut cell79 to an't$e values #rom t$e 1rstvaria"le is com%letel'

&ts as i# we $ad %us$ed

wit$out a !ataTa"le. T$e# t$is a%%roac$.


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&n%uts >ut%ut ;one trial<

!iusion Mean /./5G total log return=td !ev *.//G total return

Hum% IBHum% /.25G

Mean 2.//G

=td !ev +.//G

>ne Trial

t AB/,* AB/,* AB/,* !iusion Hum% Hum%

/

* /.+5) /.*6-6 /.)+* /.2G ).--G /

2 /.+6* /.*5/ /.6/ /.+/G ).-/G /

+ /.6))6 /.++* /./-55 /.)2G +.+/G /

) /.-2++ /.2/ /.)26/ /.-G /.2)G /

5 /.*62 /.))+ /.+ /.+G 2.)/G /

6 /.*-* /.5+6- /./5 /.6+G *.2G /

/./*+5 /.52+ /.6*- 2.*6G +.//G /

- /.*5+2 /.5-) /./-+ /.G *.+)G /

/.2)6 /.2-6* /.-/- /.6+G +.6G /

*/ /.-22 /./225 /.52) /.G -./*G /

** /.5/-) /.2*) /.2/2 /./G +.-+G /

*2 /.+52 /.*/ /.252* /.++G ).62G /

*+ /.**2 /./56 /.5+ *.)/G 6.+G /

*) /./5+ /.*-/ /.+62/ *.26G /.2G /

*5 /.* /.*+ /.-*)/ /.6+G +./G /

*6 /./-) /.)*5 /.5)6/ *.+-G 2./6G /

* /.*)5/ /.*26 /./*2 *./*G 5.)+G /

*- /.65 /.55+5 /.2+/ *.-G *.6/G /

* /.)/+) /./+56 /.+-6 /.*G .)*G /

2/ /.2// /.-+2 /.6*5 /.6G *.)2G /

2* /.-- /.-)*/ /.-// *.22G *.//G /

22 /.2// /.2)2 /.)+5 *.)5G 2.+/G /

2+ /.-5 /.2/2 /.+2-+ /.-2G ).)G /2) /.2662 /.2- /./52 /.5G +.6G /

25 /.-6*/ /.2/65 /.+/ *.*+G ).)6G /

26 /.252 /.-)2 /.5-26 /.)G *.))G /

2 /.+/) /.*+- /.-/* /.)6G 2./G /

2- /./6* /.+6- /.56 /.5G +.//G /

2 /.)2+ /.+-2 /.++) *.62G 2.62G /

+/ /.+26 /.++5) /.6+ /.)/G +.2G /


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+* /.6) /.)2- /./)- /.52G 2.5)G /

+2 /.5/+ /.-5 /.+/ /./G ).2G /

++ /.-52 /.-6-5 /.-5*5 *.*/G *.+6G /

+) /.+655 /.-/5 /.)*) /.2G *.6G /

+5 /.5- /.+222 /.5262 /.25G +.+-G /

+6 /.++- /.*6)5 /.--2* /.+G ).+G /

+ /.55+ /.)2 /.555) /.*G 2.*6G /+- /.2)22 /.)5*) /.+62 /.65G 2.+G /

+ /.5*/ /.+6/5 /.56/+ /./-G +./G /

)/ /.++6 /.+// /.2/5 /.6G 2.-)G /

)* /./5-* /.52- /.-6 *.52G *.-G /

)2 /.)//* /.2)* /.)62 /.2/G /.2*G /

)+ /.))25 /.-65 /.2)++ /./G ).6)G /

)) /.+++) /.+)6 /.*- /.+-G +.*6G /

)5 /.5-6/ /.522 /.6-/) /.2G *.)5G /

)6 /.2)) /.6+/ /.6/5 /.6+G +.+6G /

) /.52/5 /.-6) /.+6 /.*/G *.+G /)- /.2+- /.*-* /.*/)/ /.6)G /.2G /

) /.+5-/ /./5) /.522 /.+*G 6.6-G /

5/ /.-5* /.+--2 /.2-6 *./G 2.-5G /

5* /.+2/ /./5) /.56)* /.)/G 6.-/G /

52 /.66+ /.*+6 /./+5 /.5*G 5.2G /

5+ /.5*+ /./6/) /.5+ /.2-G 6.65G /

5) /.)+) /.522 /.**- /./+G *.-+G /

55 /.// /./5)5 /.2*26 /.5-G 6.-*G /

56 /.2/ /.+/66 /.)22 /.6+G +.52G /

5 /.2++2 /.-+2+ /./-*) /.6-G /.-G /

5- /.** /.+6/ /.+*52 /./G +.//G /

5 /.*)* /./5/- /.+5+2 /.62G 6.*G /

6/ /.)55 /./2+6 /.))25 /./6G .5G /

6* /.*/- /.++6 /.*26+ *.*G 2.-*G /

62 /.*6/2 /.55* /./6+* /.)G *.5G /

6+ /.)2/) /.)/*5 /.)/26 /.*5G 2.5G /

6) /.665) /.565 /.2/ /.)-G /./G /

65 /.6)+* /./+/ /.*-+- /.)2G 5.G /

66 /.)/ /.-* /.5/ /./2G /.)G /

6 /.+62/ /.656 /.-+)5 /.+/G /.G /

6- /.-*2 /.*/)+ /.)+ *.*-G 5.G /

6 /.*/ /.2)55 /.5 /.6+G )./G /

/ /.**55 /.)* /.-/5 *.*5G 2.*G /

* /.//2) /.2+*5 /.5** 2.G ).2/G /

2 /.65+ /.5*5) /.+-2 *.-G *.--G /

+ /.**-5 /.-/25 /.6+- *.*+G /.55G /

) /.*//+ /.))/5 /.56* *.2+G 2.)5G /

5 /.-))* /.566* /.6*+ *./6G *.5/G /


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6 /.++26 /.-)- /.*)*/ /.+-G *./-G /

/.626 /.)/26 /.5+ /.6G 2.)G /

- /.+*6/ /.+-* /.+-66 /.)+G 2.+G /

/.)56 /.2+ /.+6) *.65G +.5-G /

-/ /.6262 /.**** /./)6- /.+G 5.66G /

-* /./56* /.6--* /.6**- *.5)G /.5+G /

-2 /.-/)* /.6/6 /.+*/ /.*G +.2G /-+ /.)/22 /.*/)6 /./25+ /.2/G 5.G /

-) /.))* /.)/ /.22 /.*/G /.)6G /

-5 /.))2) /.+2 /.-56 /./G /./-G /

-6 /.*5++ /.*/+ /.2+*/ /.G ).-6G /

- /.+--- /.*++ /.+)) /.2+G 5.+2G /

-- /./)+ /.-+- /.2 *.62G /.+6G /

- /.-/2- /.-)-) /.622 /./G *./G /

/ /.6-56 /.)56 /./*22 /.5+G 2.++G /

* /.)/++ /.)*6 /./22 /.*G /./5G /

2 /.26 /.)2 /./+* /.65G 2.*G /+ /.)6* /.522 /.-/6- *.66G *.-+G /

) /.6*)2 /.**2/ /.//2- /.+)G 5.65G /

5 /.6)** /./+ /.6/* /.)*G /.+5G /

6 /./2 /.66+ /.)+5+ *.2G /.)6G /

/.-56 /.+65 /.6*+ *.+*G +./2G /

- /.)/ /./+- /.6)26 /./5G .++G /

/.)+/6 /.25** /.25)+ /.*2G )./*G /

*// /.)*6 /.)--) /.55) /./+G 2./G /

*/* /.+5/) /./2++ /.256- /.++G .G /

*/2 /./+ /.2/ /.*/- *.2G +.65G /

*/+ /.226 /.6+* /./**/ /./G /.*5G /

*/) /.+-2+ /.-*)/ /.)+/ /.25G /.6-G /

*/5 /.56 /.)-)/ /.522 *.G 2.*2G /

*/6 /.*+*+ /.6)/+ /.2) *./G /.2G /

*/ /.-) /.//) /.-6-* /.-G 6.)2G /

*/- /.6+-+ /.5*/ /.*++ /.)/G 2.6G /

*/ /.5*6* /.6+6* /.6+) /./G /.6G /

**/ /.252 /.2-* /.*) /.55G +.6G /

*** /.6)+ /.5/ /.2)5+ *.-5G /./G /

**2 /.556 /.*/ /.+ /.2)G /.2+G /

**+ /.2 /.6* /./)2 *.5*G /.2*G /

**) /.+6+ /.//5 /.65 /.+/G ./)G /

**5 /.56 /.*5/ /.-2/ /.5G /.+/G /

**6 /.)6- /.526) /.5-6 /./*G *.-/G /

** /.62-* /./- /.22- /.+-G /.22G /

**- /.5* /.-6) /.-)+5 /.25G *.-G /

** /.6*/ /.656 /.-525 /.++G +.)6G /

*2/ /.*)-) /.**)2 /.5- /.G 5.6*G /


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*2* /.66 /.6556 /.5/* /.)G /.-/G /

*22 /.-22- /.)*2 /.)52 /.-G 2./G /

*2+ /.2+/ /.5/- /.)+-* /.6G *.+G /

*2) /.)+62 /.)22 /.266) /.**G 2.5-G /

*25 /.-/-* /.+2 /.-/+ /.2G 2.-G /

*26 /.*)/ /.62+/ /.555) *.)2G *./6G /

*2 /.66)6 /.-/+ /.-2*+ /.)G /.5G /*2- /.6-/ /.-) /.*/5* /.5)G /.+/G /

*2 /.25+ /./-* /.2+/- *.)G 6.*-G /

*+/ /.--6* /.26+ /.6--) *.26G +.-6G /

*+* /.2)-- /.5*)+ /./-+ /.6+G *.-G /

*+2 /./** /.+5/ /.-* *.)2G 2.-/G /

*++ /.++2 /.)52- /.**5 /.+6G 2.+6G /

*+) /.+5) /.+// /.6)22 *.5G +.5G /

*+5 /.62 /.2* /.266 /.55G +.)G /

*+6 /.26- /.*/ /.+-22 /.5G /.++G /

*+ /.-/-) /.22 /.+*2* /.2G /.))G /*+- /.55 /.*+- /.2+- /.+/G 2./G /

*+ /.*/6 /./6+ /.*/2/ *.+G 6.5G /

*)/ /./* /.5*6 /.2-6 *.+G *.)6G /

*)* /.2)2 /.*665 /.)*) /.65G )./G /

*)2 /.52// /.22+) /.*5/5 /.*/G ).2-G /

*)+ /.6/ /.5-5) /.+*/ *.2G *.+5G /

*)) /.6)5) /./52 /.52* /.)2G 6.-6G /

*)5 /.)/2 /.)6*6 /.6*) /.*-G 2.2G /

*)6 /.+/6) /.+*-+ /.*+ /.)6G +.)2G /

*) /.+-*5 /.25+ /.-/ /.25G +.G /

*)- /.)/ /.-*2+ /./) /./2G /.66G /

*) /.6+2 /.-+5 /.2+ *.-)G *.)G /

*5/ /.6-- /.-* /./55/ /.5*G /.5G /

*5* /./66+ /.))6* /.5*2 *.)5G 2.)*G /

*52 /.)2/ /.5-- /.65 /./+G *.25G /

*5+ /.)6/) /.++ /.+-+ /./5G 2.65G /

*5) /.)56* /.)+5* /.))/ /./6G 2.)G /

*55 /.- /.-55 /.--6 /.-G ).55G /

*56 /.)25 /.**2+ /.62/+ /./2G 5.6)G /

*5 /.5/ /.)5+* /./+ /.+G 2.+5G /

*5- /./5- /./5* /./22 *.52G 6.)G /

*5 /.-/+ /.25+ /.-55 *.*-G +.G /

*6/ /.65 /.)/6* /.--+ /.5*G 2.*G /

*6* /.-6+ /.26/) /.226+ *.*G +.+G /

*62 /.)66 /.)5 /.-26 /./+G 2.*G /

*6+ /.22) /./* /.)/* /.6)G /.)2G /

*6) /.255* /./+// /.6/* /.6*G .6)G /

*65 /.)) /.++- /.)626 /.-G +.25G /


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*66 /.5-26 /.*6 /.*/2 /.26G ).5G /

*6 /.66/ /./+* /.*+ /.-G .+6G /

*6- /.**-5 /.)6 /.)5-2 *.*+G 2.2)G /

*6 /.2)+ /.55/+ /.2*+ /.65G *.62G /

*/ /.*+ /.*) /./5 *./)G 2.2/G /

** /.)+/2 /.5+-* /.// /.*+G *.*G /

*2 /.//)* /.- /.**/) 2.5G /.+*G /*+ /./-)* /./- /.6)-- *.++G /.+5G /

*) /.)/ /./522 /.+*- /./+G 6.-G /

*5 /.26*2 /.*+- /.**6 /.5G ).5G /

*6 /.*)-) /.-5+5 /.++5 /.G *.*6G /

* /.-5-- /.*-- /.*++ *.*+G ).6)G /

*- /.+)6+ /.2)/+ /.26/ /.+5G ).*2G /

* /.)- /.-5 /.-6 /.2G *.)6G /

*-/ /.5)6+ /.5- /.-65 /.*G *.++G /

*-* /./*2 /.*)) /.)/- 2.*G 5.*-G /

*-2 /.-** /.2+) /.)+2 /.+G ).*G /*-+ /.-)/ /.2+-- /./52 *.+/G ).*+G /

*-) /.565 /.65/* /.26 /.2)G /.-)G /

*-5 /.+- /./5-/ /./26 /.2+G 6.*G /

*-6 /.)+ /.)2* /.*+ /.**G 2.2G /

*- /./-55 /.2/*5 /.5/2 *.+2G ).5*G /

*-- /.*26+ /.-+*6 /.-+*6 *./G /.--G /

*- /.6/5+ /./2 /.-++- /.+2G /.+5G /

*/ /.*-+) /.+)* /.)55 /.-5G 2.6G /

** /.)- /.*6-/ /.*22 /.//G ).-G /

*2 /.225* /.+2)2 /.//22 /.*G +.+G *

*+ /.**- /.--/- /./655 /./G *.5)G /

*) /.**)5 /.5))) /.)5- *.*5G *.6G /

*5 /./6+ /.))* /.*)-- *.+-G 2.+-G /

*6 /.*)*6 /.--)- /.6- *./2G *.6/G /

* /./*- /.5)6/ /.5)+6 2./+G *.65G /

*- /.5-)2 /.--+/ /.*// /.26G *.5G /

* /.*2+2 /.-)*) /.+5 *.**G *.//G /

2// /.62*5 /./55 /./22 /.+6G 6.6-G /

2/* /./) /.5**5 /.+*/ *.62G *.*G /

2/2 /.)+*- /.2252 /./6)/ /.*2G ).26G /

2/+ /.*)// /.)25 /.-- *./+G 2.5+G /

2/) /.*+5+ /.+2 /.++)/ *./5G 2.G /

2/5 /.26*- /.*-26 /.+* /.5G ).2G /

2/6 /.--+5 /.*22 /.)*- *.2)G 5.)2G /

2/ /.)+* /.)6 /.+-- /./+G 5.6)G /

2/- /.56/* /.5+ /.22 /.2/G /./6G /

2/ /.)*+ /.)*5/ /.556 /./2G 2.6)G /

2*/ /.-55 /.+*+5 /.-+)6 *.2/G +.)6G /


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2** /.-+2- /.5- /.2)5/ *./2G *.)/G /

2*2 /.6)2 /.++/6 /.)+ /.G +.+*G /

2*+ /.+2/ /.+6*- /.+2 /.22G +./6G /

2*) /.6+52 /.2-+ /.))-2 /.)/G +.6G /

2*5 /./+56 /.5* /.6-+ *.5G *.)6G /

2*6 /.66 /.)2 /.65/ /.56G 2.+G /

2* /.2-)+ /.2) /.-/ /.52G )./2G /2*- /.*+5 /.+-+) /.+/6) *./)G 2.-G /

2* /.2/-* /.6)) /.55)2 /.6G +.)*G /

22/ /.))* /.-- /.5*-- /.*G *.5*G /

22* /.+6*6 /./-6 /.5--) /.+/G 2.//G /

222 /.55+6 /.)5* /.-) /.*-G 2.+6G /

22+ /.+)+ /.+- /.)6 /.22G 2.-)G /

22) /./-5 /.-*-6 /.5-)6 *.+*G /.+G /

225 /.5/-) /.+-*) /.52) /./G 2.*G /

226 /.-5/6 /./-2 /.6-- *./G 6.25G /

22 /.226/ /.2/- /.5-*+ /./G ).)2G /22- /.62* /.)/- /.5)6+ /.6G 2./G /

22 /.-52 /.5/2 /.*52 *.*+G *.-G /

2+/ /.+/// /.-/62 /.)) /.)G /.5G /

2+* /./*) /.22 /./++ /.5-G /.2+G /

2+2 /.2*5 /.)/) /./-5 /.+G 2.6G /

2++ /.6* /.++6+ /.- /.-G +.2G /

2+) /.**2 /.- /.5* /.6*G ).*5G /

2+5 /.62) /.-/2 /.56*5 /.5/G ).*G /

2+6 /.56+ /.*5/ /.*5* /.2*G ).5-G /

2+ /./5/ /.566 /.225 *.5+G *.)2G /

2+- /.)+ /.6*/ /.655 *.6G *.*6G /

2+ /.+)/- /.*2)/ /.2+2 /.+6G 5.)G /

2)/ /. /.*6 /.22- 2.*G 2.*5G /

2)* /.622 /.-6/- /.)55- /.+6G *.25G /

2)2 /.*-6 /.*-55 /.** /.-)G ).6-G /

2)+ /.)5** /.)2) /.+62 /./G 2.2G /

2)) /.+5- /.*55 /.++- /.6-G 5./)G /

2)5 /.26*5 /.2+) /.56+6 /.5G ).*2G /

2)6 /.)/ /.5+56 /.5)6/ /./)G *.+G /

2) /.5-- /.-252 /.*/-+ /.2-G /.-*G /

2)- /.++2 /.*+)5 /.)*)* /.+G 5.+2G /

2) /.555 /.-*+* /.6+/* /.2G /.6G /

25/ /.+/- /.+*+/ /.65) /.)5G +.)6G /

25* /./-* /.)--5 /.-62- *.+G 2./G /

252 /.**5 /.*+2* /.-//- *.*+G 5.+5G /

25+ /.5+2 /.)2 /./26+ /.+G 2.2*G /

25) /.2*) /.*62 /.)*-6 /.56G )./G /

255 /.+2 /.26 /.2226 /.2*G +.6/G /


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256 /.6+25 /./+/ /./6/- /.+G 6.+6G /

25 /.6-5 /.5)5 /.6-5/ /.5G /./G /

25- /./52) /.5+) /.+2+ *.5G *.)G /

25 /.-6 /.+)/+ /.2 *.2*G +.2+G /

26/ /.*5)5 /.)6 /.+2 /.G 2.//G /

26* /.+5-6 /./+-/ /.6/)) /.+*G .+2G /

262 /.25 /.-/2+ /.-)-- /.)-G /.55G /26+ /.*+26 /./++ /./+6 *./6G 6.+5G /

26) /.6*/2 /.+)5* /.265* /.++G +.2/G /

265 /./6-) /.))-) /.+5/) *.))G 2.+G /

266 /.-6+* /.65 /.5/ *.*)G /.6+G /

26 /.+/66 /.- /.2/) /.)6G .*G /

26- /.6*+/ /./2/ /.6*5 /.+)G -.**G /

26 /.*+-+ /.255* /.-+*6 *./)G +.-G /

2/ /./5-+ /.)622 /.)-* *.52G 2.2-G /

2* /.*))2 /.6-*/ /.-*5- *./*G /.5G /

22 /.) /./-2 /.-25 /.2G 5.--G /2+ /.-6- /.-*6 /.-/-5 /.-5G /.2G /

2) /.-2 /./+22 /.)6*- 2.*6G .55G /

25 /.6-/ /.252 /.)/65 /.5G +.6G /

26 /.)+) /.))6 /.6* /.**G 2.)/G /

2 /.-625 /.6-) /.6)- *.*)G /.))G /

2- /./*/ /.2/- /.)25* *.)2G ).)+G /

2 /.65) /.6*+ /.-/) /.)6G *.*+G /

2-/ /.52/ /./-+/ /./2)2 *.*G 6.*6G /

2-* /.)-- /.2*6 /.5** /./2G ).+5G /

2-2 /./)+/ /.-/ /.*)5/ *.6G ).-G /

2-+ /.6*55 /.5) /.6/-/ /.+)G *.6+G /

2-) /.)+)- /.65-* /.56 /.**G /.-G /

2-5 /./2 /.*/5/ /.*-)/ *.+G 5.6G /

2-6 /.6-2) /.2/6 /.562 /.52G 2.2+G /

2- /.-)+* /.*/* /.6)/ *./6G ).6+G /

2-- /.*6+ /.+-5 /.-/ /.+G 2.-G /

2- /.*)) /.-* /.--* *./*G /.+/G /

2/ /.-22 /.25- /.5)/ /.-G +.6*G /

2* /.)/- /.//* /.+5+ /.*-G .+5G /

22 /.5- /.+) /.-2 /.+/G /.*2G /

2+ /.-*2 /.+5)* /.62*- *.*-G +.*2G /

2) /.)5/ /.6+2 /.22/2 /./5G /.)-G /

25 /.2-5/ /.*- /.)-** /.52G ).5+G /

26 /./5 /.6*6) /.25// /.-6G *.**G /

2 /.5*6 /.))+) /.+/2 *.*G 2.)+G /

2- /.++ /./6+ /.*/) /.2*G 6.2G /

2 /.)*2 /.**-2 /.-+) /.*G 5.55G /

+// /.)/2/ /.2- /.6)- /.2/G 2.)/G /


20/60

+/* /.*2/ /.--) /./+/ *./-G ).-*G /

+/2 /.22) /.6--2 /.*- /.5/G /.5+G /

+/+ /.--/ /.++ /.)2 *.2+G 2.-*G /

+/) /.)55 /.25-* /.5/ /.*G +.5G /

+/5 /.*/*) /.-+5+ /.*/ *.22G /.+G /

+/6 /.--5 /.++6 /.5+5 *.26G /.*+G /

+/ /.526 /.6-/* /./6+- /.*2G /.6/G /+/- /.-+/5 /.2)26 /.*-56 *./*G )./G /

+/ /./)*+ /.555/ /.2/*2 *.6G *.5-G /

+*/ /.-) /./+ /.+-/- /.-2G 6.2+G /

+** /.22- /.-*2- /.)55 /.6G /.66G /

+*2 /./* /./))+ /.66- *.+G .**G /

+*+ /.//5- /./625 /.*-/6 2.)G 6.6/G /

+*) /.5)6) /./62 /.)5** /.*G /.+G /

+*5 /.-*5- /.+/ /./6/ /.5G 2.-+G /

+*6 /.6*)+ /.-/ /.*-6) /.+)G ).22G /

+* /./62- /.+-/ /.2-/ *.)-G 2.-6G /+*- /.-66 /.2)2 /.2+)- 2.26G )./+G /

+* /.*+* /.* /.5*- /.-G ).G /

+2/ /.662 /.5655 /.-*6 /.)-G *.5*G /

+2* /.5-5/ /.5) /.*6* /.26G +./-G /

+22 /.)-- /.-) /.-/66 /.//G *.-/G /

+2+ /.)6 /./*26 /./2* *.6G -.*G /

+2) /.-52- /.*/5 /.-55 *.*/G 2./+G /

+25 /.-*+ /.+6*5 /.2+** *.2-G +./6G /

+26 /.6-*5 /.++5* /.+)* /.52G +.2-G /

+2 /.*/2 /./+ /.// *.*G .++G /

+2- /.*/- /.*/ /.5/2) *.*-G 2.*6G /

+2 /.2)/- /.+6 /.*525 /.65G 5.)G /

++/ /.+26+ /.*56 /.6625 /.)/G 2.*+G /

++* /.)*66 /.6)) /.6+52 /.*6G /.-G /

++2 /.26/- /.6 /./-+ /.5G /.)+G /

+++ /.5// /.62+) /.65/) /./5G *./6G /

++) /.)-)6 /.25 /.5-5+ /./*G 5.2G /

++5 /.6-* /.2+* /.6+ /.5)G ).*+G /

++6 /.)2) /.-)/+ /.2*)- /.*+G /.G /

++ /.-6) /.5-/ /.*/ 2.26G *.+)G /

++- /.//6 /./-*6 /.)55 *.+)G 6.*-G /

++ /.//-) /.--6 /.*)* 2.+)G *.6+G /

+)/ /.*5) /.52)+ /.26/ /.--G *.-2G /

+)* /./// /./6+ /./+5 /.5G /.+G /

+)2 /.62/ /.))5 /.-)** /.+6G 2.)*G /

+)+ /.5/62 /.56)/ *.//// /./G *.52G /

+)) /.)*/) /.*55- /.*2 /.*-G 5./)G /

+)5 /.)5 /.+-+ /.*5* /.*G 2.-)G /


21/60

+)6 /.2) /.2)/5 /.+6)+ /.)G ).**G /

+) /.6) /.-+25 /.)2+6 2.)G /.-G /

+)- /.5+/- /.+6 /.**+ /.*+G 2.6G /

+) /.-6// /.6* /.*6* *.*+G /.)+G /

+5/ /.2- /.//// /.)65 /.-/G *6.-6G /

+5* /./2)- /./5 /.))* *.*G 6.2+G /

+52 /.2)/) /.-6*+ /.2) /.65G *.26G /+5+ /./+5) /./25 /.)2 *.6G .-)G /

+5) /.-* /.* /./) *.2G /.2G /

+55 /.-*-5 /.+5 /.+*/ /.6G 2.6/G /

+56 /.//56 /.+56- /./5 2.)G +.*/G /

+5 /.+/6 /.6//- /.**5* /.)5G *.2+G /

+5- /.2)+ /.-5* /.666 /.6+G /.+G /

+5 /.++65 /./)/ /.56** /.+G .22G /

+6/ /.++6/ /.)- /./6+ /.+G /.)G /

+6* /.+)5 /.5 /.)2- /.22G +.+G /

+62 /./-22 /.5655 /.62*5 *.+)G *.5*G /+6+ /.+2/* /.)/ /.2/*- /.)2G /.//G /

+6) /.2++- /.+ /.+22/ /.6-G 2.2G /

+65 /.+62 /.5/2 /.-*5 /.2*G *.5G /

+66 /.565* /.22/- /.*/-- /.2*G ).+*G /

+6 /.**)- /.-6 /.5/-2 *.*5G 6.G /

+6- /.-6* /.- /.226 *.2*G *.52G /

+6 /.2)- /.52 /./2- /.55G +./2G /

+/ /.//)2 /.-2) /.-2/ 2.5G *.)*G /

+* /.2 /.+)) /.-+/- 2./5G 2.5+G /

+2 /./*2 /./)2 /.2-5 *.2-G .*5G /

++ /.566 /.++ /.)56 /.2+G +.-/G /

+) /.5-*6 /.) /.-- /.26G 6.+G /

+5 /.2++ /.5655 /.222* *.)-G *.5/G /

+6 /.5)/ /.6/) /.)+* *.+G /.6-G /

+ /.*6- /.6/2/ /.)65/ /.*G *.22G /

+- /.*-55 /./*-5 /.6+)5 /.-)G -.26G /

+ /.-+-/ /.5-*2 /.-*+ *./)G *.+G /

+-/ /.**- /.6)* /.+/6 *.*)G +.)/G /

+-* /.-6 /.-- /.6-5) /.-2G )./G /

+-2 /./*52 /.665+ /.6*)2 2.**G /.2G /

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+-) /./-) /./5- /.+*+2 /.6/G 6.+/G /

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+- /.*-+ /.2++2 /./*5 /.-/G ).*-G /

+-- /.)*/) /.+**5 /.-/6 /.*-G +.)G /

+- /.2) /./2/2 /.-** *.G -.*5G /

+/ /.-/ /./65 /.256- *.2-G 5.*G /


22/60

+* /.-/5* /.+//+ /.*+6- /.*G +.5G /

+2 /./-2+ /./6/ /.+--+ *.+)G 5.*G /

++ /.*62 /.2/5 /.)/25 /.+G ).)6G /

+) /.-+5 /.*+5 /.-2 /.-+G 5.+/G /

+5 /.*2- /./*) /.2-)6 /.-G -.++G /

+6 /.6- /.//2 /.+*- /.52G */.+)G /

+ /.)*6 /.+5- /.+26 /.*5G +./-G /+- /.6-** /.2/2 /.2-22 /.52G +.66G /

+ /.22* /.+)* /.+*- /.56G +.22G /

)// /.66*6 /.*2** /.2*6 /.)G 5.5*G /

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)/+ /.)* /.6*6 /.--/ *.6G *.*/G /

)/) /./*++ /.*//5 /.-5)5 2.*G 5.-)G /

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)/ /.*6 /.5*)/ /.-22/ /.G *.-G /)/- /.-* /.65- /.)) /.G /.G /

)/ /.*6+/ /./5- /.5/6- /.+G 6.6G /

)*/ /./-/ /.+** /.56/ *.+5G +.)+G /

)** /.)*6 /.6+-- /./-2/ /./2G /.+G /

)*2 /.56 /.)6 /.+*2/ /.5G /./*G /

)*+ /./2/+ /.-/ /./*/) 2.//G ).22G /

)*) /.*+/+ /.+6 /.6666 *./G /.*/G /

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)* /.26) /.6--* /.2-/ /.)-G /.5+G /

)*- /./)+/ /.25- /.++ *.6G +.5G /

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)25 /.*//+ /.+2*2 /./2/5 *.2+G +.+G /

)26 /.*)/6 /.-)6 /.*--/ *./+G *.5G /

)2 /.+/- /.*5- /.--5) /.)5G 2.*+G /

)2- /.*5/ /.) /.-*5/ /.--G /.5/G /

)2 /.)5* /.2-** /.2)- /./G +.)G /

)+/ /.+/ /.**/ /.-652 /.2+G 5.5)G /

)+* /.66-+ /.+)) /.)5+2 /.)G 2.5+G /

)+2 /.+-2 /.)65 /.5 /.2*G 2.*-G /

)++ /.)*)- /.-/) /.255 /.*G /.6+G /

)+) /.-5* /.-** /.-66 2.22G /.65G /

)+5 /.6)) /.*)-6 /./22/ /.56G 5.*+G /


23/60

)+6 /.6*/ /.//-* /.26 /.55G .2*G /

)+ /.+5)- /.+5) /.)26/ /.+2G +./-G /

)+- /.5+56 /./+5 /./) /.*)G 5.6G /

)+ /.6/*- /.*/2 /.22- /.+*G 2./+G /

))/ /.55 /.52* /.5 *.6G +.//G /

))* /.25/) /.-// /.-+/ /.62G /.5+G /

))2 /./+22 /.6)6 /.--- *.-/G /.-6G /))+ /.++2/ /.26) /.-)5 /.+-G +.-G /

))) /.)-+) /.2*-) /.+*6+ /./*G ).++G /

))5 /.266 /.-)22 /.65+ /.5G *./*G /

))6 /.5+/ /.5/5 /.6/* /.*+G *.)G /

)) /.+))5 /./)2 /.)+2+ /.+5G .*6G /

))- /.+6- /.2-+ /.5 /.2*G /.*-G /

)) /.2-66 /.+//) /.2*** /.5*G +.5G /

)5/ /.-/2) /.5*-) /.2*6 /./G *.-6G /

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)5) /.)+)5 /.5-2 /.+-2 /.**G *.+2G /

)55 /.*+52 /.2/*) /.+5-6 *./5G ).5*G /

)56 /.5// /.56*5 /.+55- /./5G *.5)G /

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)5- /.2/)/ /.-/-6 /.*6 /.-G /.62G /

)5 /.*56- /.+2+ /.6)5 /.6G +.++G /

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)* /./ /.+**+ /.*+/6 /.6/G +.)-G /

)2 /.)/ /.5++ /.56- 2.56G *.)5G /

)+ /.5)+- /.2/+ /.-5/- /.*6G ).)+G /

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)6 /.5-* /.-- /.6))+ /.25G ).-6G /

) /.6) /.++2- /.5) /.-*G +.+/G /

)- /./ /.)56 /.5222 /.G /./2G /

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)-/ /.*-- /.-)/5 /.+)5 /.-+G /.G /


24/60

)-* /.266/ /.6555 /.5)2 /.5-G /.-/G /

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)-+ /.26 /./62- /.6*-5 /.)-G 6.6/G /

)-) /.)+ /.5-66 /.+-5+ /.*/G *.+)G /

)-5 /.++26 /.)+2) /.6/ /.+-G 2.5*G /

)-6 /.)) /.)2) /./6 /./*G 2.5+G /

)- /.++5+ /.-26* /.*5/ /.+-G /.-2G /)-- /.65 /.*) /.5/- *.-G /.2G /

)- /.*)*6 /.-*/ /.52 *./2G *./G /

)/ /.5/2 /.//52 /.62+ /./6G .6G /

)* /.**)) /.5*/- /.)5+* *.*5G *.2G /

)2 /.22* /.++5+ /.-2/ /.2G +.2-G /

)+ /.)6+ /.5-/ /./* /.*G *.)*G /

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)5 /.26) /.65/ /./-*2 /.5)G /.*G /

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) /.5+6- /./+2/ /.2/66 /.*)G .56G /)- /.+/2) /.55/ /.+2+ /.)G *.)+G /

) /.5)- /.5* /.2++- /.*-G *.+/G /

5// /.5/- /./55 /.6+ /./G 6.G /

5/* /.)-6/ /./2+2 /.+*** /./*G .-G /

5/2 /./2++ /.*)* /.+--+ *.)G 2.*/G /

5/+ /.2/5 /./6*) /.5)- /.6+G 6.6+G /

5/) /.*/+) /.+2- /.))5 *.2*G 2.-2G /

5/5 /.6 /.-* /.///6 /.--G )./5G *

5/6 /.2)5 /.+) /.552 /.)G /.)6G /

5/ /.2--/ /.6)-/ /./6 /.5*G /.-6G /

5/- /.2*+* /.56 /.2/ /.5G 5.-5G /

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5*/ /.))+/ /.6+)5 /.-) /./G /.G /

5** /.-+) /.-2 /.5- *.2G *.)2G /

5*2 /./-+2 /.-26 /./5*6 *.++G /.-+G /


25/60

>ut%ut Monte Carlo

/.2/G mean /.2/G/.2/G std dev /.//G

min /.2/G

max /.2/G

!ataTa"le

r Trial Total Return

/.2/G

/.2G * /.2/G

/.+/G 2 /.2/G

/.)2G + /.2/G

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*.*+G 25 /.2/G

/.)G 26 /.2/G

/.)6G 2 /.2/G

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*.62G 2 /.2/G

/.)/G +/ /.2/G

More Complicated Mont

&n t$is simulation we $averandom log returns ;colum%eriod.

T$e !ataTa"le t$en runs t$ideviation, mean and max #

s "e#ore we could $ave dentire model and %asted itwould $ave "een ver' diJsee w$at was going on, an

note o# caution7 !ataTa"llarge ;lots o# trials


26/60

/.52G +* /.2/G

/./G +2 /.2/G

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/.2G +) /.2/G

/.25G +5 /.2/G

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2.G * /.2/G

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*./6G 5 /.2/G


27/60

/.+-G 6 /.2/G

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28/60

/.)G

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29/60

/.26G

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/./+G

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30/60

*./2G

/.G

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31/60

/.+G

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