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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 42 "Optimal Investing with th
e Markowitz Model" }}{PARA 19 "" 0 "" {TEXT -1 55 "by Jason Schattman,
 Waterloo Maple, Inc., October 2000 " }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 427 "Supp
ose your stock broker passes you a hot stock tip.  You're tempted to s
ell some of your current investments to buy this stock.  However, doin
g so will add risk to your portfolio because the stock's price jumps o
r falls wildly during any given month.  The reallocation will also cos
t you some commission fees.  Is the potential increase in return obtai
ned by buying the stock worth the added risk and costs?  More generall
y, " }{TEXT 260 87 "how should you allocate investment capital among t
housands of investment opportunities?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 587 "To address these questions, we start
 by stating the two competing goals of investment: (1) long-term growt
h of net worth, and (2) low risk.  A good portfolio grows consistently
 without wild short-term fluctuations in value.  Investing solely in t
echnology stocks will probably yield greater long-term return than inv
esting solely in utilities but will subject the investor's portfolio t
o stomach-churning roller coaster rides with every quarterly earnings \+
report.  Our question then becomes, how do you choose a portfolio that
 optimally balances long-term return against short-term risk?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 237 "The idea
 is to allocate capital among investments that have good long-term pro
spects individually but balance each other out in the short-term.  For
 instance, during months when dotcoms go up, utilities tend to go down
, and vice versa.  " }{TEXT 257 20 "The Markowitz Model," }{TEXT -1 
343 " proposed in 1952 by Harry Markowitz, is an optimization model fo
r balancing the expected return and risk of a portfolio.  The model us
es the statistical variance of a stock's price as the measure of its r
isk and its expected return as the measure of its long-term prospects.
  The decision variables are the amounts you invest in each asset.  " 
}{TEXT 259 233 "The objective is to minimize the overall variance of t
he portfolio's return, subject to the constraints that (1) the expecte
d return of the portfolio is at least some target level, and (2) you d
on't invest more capital than you have." }{TEXT -1 68 "  You can also \+
add constraints that forbid selling the assets short." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We implement the Marko
witz Model using the " }{TEXT 256 20 "NonlinearProgramming" }{TEXT -1 
9 " package." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "Setting up the \+
Markowitz Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 37 "We first load the necessary packages." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "libname:=\"C:/mylib/nl
p\",libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(Nonlin
earProgramming);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%%)OptimizeG%5Pri
malDualLogBarrierG%4UnconstrainedNewtonG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "with(LinearAlgebra): with(linalg): " }}{PARA 7 "" 1 "
" {TEXT -1 104 "Warning, the previous binding of the name GramSchmidt \+
has been removed and it now has an assigned value\n" }}{PARA 7 "" 1 "
" {TEXT -1 80 "Warning, the protected names norm and trace have been r
edefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 86 "Suppose for simplicity that we will alloc
ate our investment capital among four assets." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "numAssets :
= 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*numAssetsG\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Our decis
ion variables are how much of each asset to buy." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "buy := Vect
or( [seq(x[i], i=1..numAssets)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$buyG-%'RTABLEG6$\"()=v8-%'MATRIXG6#7&7#&%\"xG6#\"\"\"7#&F/6#\"\"#7#&
F/6#\"\"$7#&F/6#\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 46 "Constraint #1:  We cannot invest more cap
ital " }{TEXT 262 1 "c" }{TEXT -1 14 " than we have." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "budgetCon
straint := add(buy[i],i=1..numAssets) <= c;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%1budgetConstraintG1,*&%\"xG6#\"\"\"F*&F(6#\"\"#F*&F(6
#\"\"$F*&F(6#\"\"%F*%\"cG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 99 "We assume we have estimates for the expec
ted rates of return for each asset over the given horizon." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "exp
ectedRates := Vector( [seq(r[i], i=1..numAssets)] );" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%.expectedRatesG-%'RTABLEG6$\"(+&)3\"-%'MATRIXG6#7&
7#&%\"rG6#\"\"\"7#&F/6#\"\"#7#&F/6#\"\"$7#&F/6#\"\"%" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "The total expe
cted return of the portfolio is the dot product of the buy amounts and
 the expected rates of return." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "expectedReturn := Multiply( \+
Transpose(expectedRates), buy  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%/expectedReturnG,**&&%\"xG6#\"\"\"F*&%\"rGF)F*F**&&F(6#\"\"#F*&F,F/F*
F**&&F(6#\"\"$F*&F,F4F*F**&&F(6#\"\"%F*&F,F9F*F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "Constraint #2:  T
he total expected return must meet our target, which can be expressed \+
as a \"goal rate\" times our initial capital " }{TEXT 258 1 "c" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "returnConstraint := expectedReturn >= c * goalRa
te;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1returnConstraintG1*&%\"cG\"
\"\"%)goalRateGF(,**&&%\"xG6#F(F(&%\"rGF.F(F(*&&F-6#\"\"#F(&F0F3F(F(*&
&F-6#\"\"$F(&F0F8F(F(*&&F-6#\"\"%F(&F0F=F(F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 193 "To derive the variance
 of the portfolio's return, we need the matrix of covariances for the \+
assets' returns.  The (i,j) element of the matrix is the covariance of
 the returns of assets i and j." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Q := Matrix( [ seq( [seq( co
v[i,j], j=1..numAssets)], i=1..numAssets)] );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"QG-%'RTABLEG6$\")/1X<-%'MATRIXG6#7&7&&%$covG6$\"\"
\"F1&F/6$F1\"\"#&F/6$F1\"\"$&F/6$F1\"\"%7&&F/6$F4F1&F/6$F4F4&F/6$F4F7&
F/6$F4F:7&&F/6$F7F1&F/6$F7F4&F/6$F7F7&F/6$F7F:7&&F/6$F:F1&F/6$F:F4&F/6
$F:F7&F/6$F:F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 263 107 "The objective is to minimize the variance of the p
ortfolio's total return, subject to constraints #1 and #2" }{TEXT -1 
24 ".  It can be shown that " }{XPPEDIT 18 0 "Var(Sum(x[i]*r[i],i = 1 \+
.. n)) = x^t*Q*x;" "6#/-%$VarG6#-%$SumG6$*&&%\"xG6#%\"iG\"\"\"&%\"rG6#
F.F//F.;F/%\"nG*()F,%\"tGF/%\"QGF/F,F/" }{TEXT -1 57 ", where Q is the
 covariance matrix of the random  vector " }{TEXT 261 4 "r.  " }{TEXT 
-1 26 "We now form this quantity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "buy_T := Transpose( buy );" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&buy_TG-%'RTABLEG6$\"(KVd\"-%'VECT
ORG6#7&&%\"xG6#\"\"\"&F.6#\"\"#&F.6#\"\"$&F.6#\"\"%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "variance := expand( Multiply( buy_T, Mult
iply( Q, buy ) ) );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)varianceG,B*
&&%$covG6$\"\"\"F*F*)&%\"xG6#F*\"\"#F*F**(F,F*&F(6$F*F/F*&F-6#F/F*F**(
F,F*&F(6$F*\"\"$F*&F-6#F8F*F**(F,F*&F(6$F*\"\"%F*&F-6#F>F*F**(F3F*&F(6
$F/F*F*F,F*F**&&F(6$F/F/F*)F3F/F*F**(F3F*&F(6$F/F8F*F9F*F**(F3F*&F(6$F
/F>F*F?F*F**(F9F*&F(6$F8F*F*F,F*F**(F9F*&F(6$F8F/F*F3F*F**&&F(6$F8F8F*
)F9F/F*F**(F9F*&F(6$F8F>F*F?F*F**(F?F*&F(6$F>F*F*F,F*F**(F?F*&F(6$F>F/
F*F3F*F**(F?F*&F(6$F>F8F*F9F*F**&&F(6$F>F>F*)F?F/F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Notice th
at the variance is a quadratic function of the decision variables." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 
"Solving the model with nonlinear programming" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 207 "We're now ready to rea
d in data and solve the optimization problem.  Let's suppose the four \+
assets under consideration have estimated rates of return over the nex
t year of 5%, 10%, 15% and 30%, respectively." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r := <.05, \+
.10, .15, .30>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'RTABLEG6$
\"(7g6\"-%'MATRIXG6#7&7#$\"\"&!\"#7#$\"#5F07#$\"#:F07#$\"#IF0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 "A
lthough asset 1 has a lower expected return, it's price also has a low
er variance (.08) and is negatively correlated with the other assets' \+
prices (-.05).  Assets 3 and 4 have high expected returns but high var
iances (.35 each) and are positively correlated (.06) with each other.
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 130 "cov := << .08,-.05,-.05,-.05> | \n        <-.05, .16
,-.02,-.02> | \n        <-.05,-.02, .35, .06> | \n        <-.05,-.02, \+
.06, .35>>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$covG-%'RTABLEG6$\"(K
/c\"-%'MATRIXG6#7&7&$\"\")!\"#$!\"&F0F1F17&F1$\"#;F0$F0F0F67&F1F6$\"#N
F0$\"\"'F07&F1F6F:F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 82 "We now have a well defined objective function (
variance) that we wish to minimize." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "variance;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,6*$)&%\"xG6#\"\"\"\"\"#F)$\"\")!\"#*($\"#5F-F)F&F)&
F'6#F*F)!\"\"*($F0F-F)F&F)&F'6#\"\"$F)F3*($F0F-F)F&F)&F'6#\"\"%F)F3*&$
\"#;F-F))F1F*F)F)*($F=F-F)F1F)F6F)F3*($F=F-F)F1F)F;F)F3*&$\"#NF-F))F6F
*F)F)*($\"#7F-F)F6F)F;F)F)*&FGF))F;F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Let's assume we will not
 settle for a total expected annual return rate less than 10%, and we \+
have $10,000 to invest." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "goalRate:=.10; c :=10000;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)goalRateG$\"#5!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"cG\"&++\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 41 "Our two constraints are now well define
d." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "returnConstraint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
1$\"'++5!\"#,*&%\"xG6#\"\"\"$\"\"&F&*&$\"#5F&F+&F)6#\"\"#F+F+*&$\"#:F&
F+&F)6#\"\"$F+F+*&$\"#IF&F+&F)6#\"\"%F+F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "budgetConstraint;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#1,*&%\"xG6#\"\"\"F(&F&6#\"\"#F(&F&6#\"\"$F(&F&6#\"\"%F(\"&++\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 230 "F
inally, we solve the optimization problem generated from the above dat
a.  We first set the information level to 2, meaning we want the algor
ithm to report all iterations.  If we only wanted the final answer, we
 would use level 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 63 "infolevel['Optimize']:=2: infolevel['Prim
alDualLogBarrier']:=2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 32 "The parameters to the procedure " }{TEXT 
273 0 "" }{TEXT 274 14 "Optimize(...) " }{TEXT 275 0 "" }{TEXT -1 33 "
are as follows: the objective is " }{TEXT 264 8 "variance" }{TEXT -1 
13 ", we want to " }{TEXT 265 8 "minimize" }{TEXT -1 10 ", we have " }
{TEXT 266 1 "4" }{TEXT -1 41 " decision variables, the constraints are
 " }{TEXT 267 37 "budgetConstraint and returnConstraint" }{TEXT -1 22 
", we specifiy we want " }{TEXT 268 11 "nonnegative" }{TEXT -1 85 " de
cision variables (i.e. short selling prohibited), and we choose as sta
rting point " }{TEXT 269 24 "(2500, 2500, 2500, 2500)" }{TEXT -1 172 "
.  That is, our default plan is to invest evenly in all assets.  The p
rimal-dual log-barrier optimization algorithm will start from there an
d search for better allocations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Optimize( variance, min, 4,
 \{budgetConstraint, returnConstraint\}, nonnegative, <c/4, c/4, c/4, \+
c/4>);" }}{PARA 6 "" 1 "" {TEXT -1 120 "OptimizeInteriorPointMethod:  \+
 Constrained   convex   problem: solving with   convex   primal-dual l
og-barrier algorithm" }}{PARA 6 "" 1 "" {TEXT -1 32 "PrimalDualLogBarr
ier:   Minimize" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*$)&%\"xG6#\"\"\"
\"\"#F)$\"\")!\"#*($\"#5F-F)F&F)&F'6#F*F)!\"\"*($F0F-F)F&F)&F'6#\"\"$F
)F3*($F0F-F)F&F)&F'6#\"\"%F)F3*&$\"#;F-F))F1F*F)F)*($F=F-F)F1F)F6F)F3*
($F=F-F)F1F)F;F)F3*&$\"#NF-F))F6F*F)F)*($\"#7F-F)F6F)F;F)F)*&FGF))F;F*
F)F)" }}{PARA 6 "" 1 "" {TEXT -1 167 "PrimalDualLogBarrier:   subject \+
to   [0 <= x[3], 0 <= x[1], 0 <= x[2], 0 <= x[4], 0 <= -1000.00+.5e-1*
x[1]+.10*x[2]+.15*x[3]+.30*x[4], 0 <= -x[1]-x[2]-x[3]-x[4]+10000]" }}
{PARA 6 "" 1 "" {TEXT -1 38 "PrimalDualLogBarrier:   Starting point" }
}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\"(/M^\"-%'MATRIXG6#7&7#$\"%+D\"\"!F+F+F+
" }}{PARA 6 "" 1 "" {TEXT -1 66 "InitialBarrierParameter:   Initial ba
rrier parameter   167497.9303" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDu
alLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")_/U<-%
'VECTORG6#7&$\"3z6^H8V\"G7$!#:$\"3ar!3h5[`D\"!#9$\"3CELz\"HQC'>F0$\"3[
sQ*pC^\\'>F0" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")Sv9<-%'VECTORG6#7&$\"3Q
Pa_$\\Tt(R!#:$\"3KG\"zAV(G`7!#9$\"3'p*o*>(Gg$*=F0$\"3dsRehqk1>F0" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\"(Whk\"-%'VECTORG6#7&$\"3+^m7u19*p#!#9$
\"3)y[96d4y1#F-$\"3WA]\"oqz^N\"F-$\"3U$)QM!Gzr+#F-" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")ws`<-%'VECTORG6#7&$\"3+^;)[]I.e$!#9$\"3/)[RlaZ4>#F-$\"3
iC-:'H\"o\\$*!#:$\"3_$)))zB1o#e\"F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "Pr
imalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"'
/k#)-%'VECTORG6#7&$\"3'4l%)[ga%eO!#9$\"39)[W#)zbt?#F-$\"3%\\AD+\"[.#4*
!#:$\"3_$)eSc#)=Q:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarr
ier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")StG<-%'VECTORG6
#7&$\"37,>&*Q\"*ziO!#9$\"32[k#GT#z2AF-$\"3OD_C'=/b1*!#:$\"3YLX_*z\"*R`
\"F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(Sh-\"-%'VECTORG6#7&$\"3CcEu-.
*Hm$!#9$\"3#p<T84%y2AF-$\"3Ov*3cv2O1*!#:$\"3cy0!edWP`\"F-" }}{PARA 6 "
" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6$\")%)R*z\"-%'VECTORG6#7&$\"3ysf8o(**Hm$!#9$
\"3+&f'yRKy2AF-$\"3K&4ZT1-N1*!#:$\"32&y&3u9tL:F-" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")3C>=-%'VECTORG6#7&$\"3_LLwU-+jO!#9$\"3!>(p1%=$y2AF-$\"3
1x8YAk\\j!*!#:$\"38RE@$zIP`\"F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "Primal
DualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")Sne<
-%'VECTORG6#7&$\"3wP$emE+Im$!#9$\"3OEd\\!=$y2AF-$\"3!zz')\\7'\\j!*!#:$
\"3&496&e2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")_R8=-%'VECTORG6#7&$
\"3[**)4yE+Im$!#9$\"3#)f&>2=$y2AF-$\"3k;xp1h\\j!*!#:$\"3]_g*fvIP`\"F-
" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")W*)==-%'VECTORG6#7&$\"3[&44yE+Im$!#
9$\"3l)R].=$y2AF-$\"3U@NY2h\\j!*!#:$\"3_l$RivIP`\"F-" }}{PARA 6 "" 1 "
" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'RTABLEG6$\"(;z6\"-%'VECTORG6#7&$\"3soh!yE+Im$!#9$\"3&G4q.=$y2AF-$
\"3'3Jqv5'\\j!*!#:$\"3^ErIc2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "Prim
alDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")_!
py\"-%'VECTORG6#7&$\"3#R`/yE+Im$!#9$\"3jj5Q!=$y2AF-$\"3Q#fYv5'\\j!*!#:
$\"3%e^WjvIP`\"F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier
:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(_rV\"-%'VECTORG6#7
&$\"3OQV!yE+Im$!#9$\"3CRCQ!=$y2AF-$\"3=ePa2h\\j!*!#:$\"3'*z\"\\jvIP`\"
F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"'k!f*-%'VECTORG6#7&$\"3s*G/yE+Im$!
#9$\"3\\$y#Q!=$y2AF-$\"3K`Ia2h\\j!*!#:$\"3qX.Nc2tL:F-" }}{PARA 6 "" 1 
"" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'RTABLEG6$\"(#>[8-%'VECTORG6#7&$\"3WxU!yE+Im$!#9$\"3VpGQ!=$y2AF-$
\"35xGa2h\\j!*!#:$\"3?P1Nc2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "Prima
lDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")[\"
)>=-%'VECTORG6#7&$\"3_wU!yE+Im$!#9$\"3*[(GQ!=$y2AF-$\"3slGa2h\\j!*!#:$
\"3Rb1Nc2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")!)3>=-%'VECTORG6#7&$
\"31wU!yE+Im$!#9$\"3ixGQ!=$y2AF-$\"3/gGa2h\\j!*!#:$\"3[k1Nc2tL:F-" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\")s-q<-%'VECTORG6#7&$\"31wU!yE+Im$!#9$\"
3)*yGQ!=$y2AF-$\"3ydGa2h\\j!*!#:$\"3.p1Nc2tL:F-" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")CYE=-%'VECTORG6#7&$\"31wU!yE+Im$!#9$\"3WzGQ!=$y2AF-$\"3
kcGa2h\\j!*!#:$\"3Ir1Nc2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDu
alLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")K_8=-%
'VECTORG6#7&$\"31wU!yE+Im$!#9$\"3*)zGQ!=$y2AF-$\"3]bGa2h\\j!*!#:$\"3Ws
1Nc2tL:F-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(3pT\"-%'VECTORG6#7&$\"3
1wU!yE+Im$!#9$\"3*)zGQ!=$y2AF-$\"3]bGa2h\\j!*!#:$\"3ns1Nc2tL:F-" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\")o2\"z\"-%'VECTORG6#7&$\"3ka=zn-+jO!#9$
\"3Fr4Z!=$y2AF-$\"37ZFO2h\\j!*!#:$\"3H'3\\mvIP`\"F-" }}{PARA 6 "" 1 "
" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'RTABLEG6$\"(7j=\"-%'VECTORG6#7&$\"3?p)[!o-+jO!#9$\"3P=/)4=$y2AF-$
\"3KZ%>.6'\\j!*!#:$\"3**p>(fvIP`\"F-" }}{PARA 6 "" 1 "" {TEXT -1 47 "P
rimalDualLogBarrier:   Global optimum found at" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<&/&%\"xG6#\"\"%$\"3**p>(fvIP`\"!#9/&F&6#\"\"$$\"3KZ%>.
6'\\j!*!#:/&F&6#\"\"\"$\"3?p)[!o-+jOF+/&F&6#\"\"#$\"3P=/)4=$y2AF+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 183 
"Thus, among all allocations with expected return of 10% or higher, in
vesting $3663 in asset 1, $2208 in asset 2, $906 in asset 3 and $1534 \+
in asset 4 is the one with minimum variance." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "What if we revise our \+
estimates?  Suppose we expect asset 4 to plummet next year.  Let's run
 the model again with the new return data.  We'll use a new starting p
oint (c/3, c/3, c/3, 0) that's closer to our expectations of the optim
al allocation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "r[4] := -.20;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%\"rG6#\"\"%$!#?!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "expectedReturn; returnConstraint;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,*&%\"xG6#\"\"\"$\"\"&!\"#*&$\"#5F*F'&F%6#\"\"#F'F'*&$\"#:F*F'&F%6#
\"\"$F'F'*&$\"#?F*F'&F%6#\"\"%F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#1$\"'++5!\"#,*&%\"xG6#\"\"\"$\"\"&F&*&$\"#5F&F+&F)6#\"\"#F+F+*&$\"#
:F&F+&F)6#\"\"$F+F+*&$\"#?F&F+&F)6#\"\"%F+!\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "Optimi
ze( variance, min, 4, \{budgetConstraint, returnConstraint\}, nonnegat
ive, <c/3,c/3,c/3,0>);" }}{PARA 6 "" 1 "" {TEXT -1 120 "OptimizeInteri
orPointMethod:   Constrained   convex   problem: solving with   convex
   primal-dual log-barrier algorithm" }}{PARA 6 "" 1 "" {TEXT -1 32 "P
rimalDualLogBarrier:   Minimize" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*
$)&%\"xG6#\"\"\"\"\"#F)$\"\")!\"#*($\"#5F-F)F&F)&F'6#F*F)!\"\"*($F0F-F
)F&F)&F'6#\"\"$F)F3*($F0F-F)F&F)&F'6#\"\"%F)F3*&$\"#;F-F))F1F*F)F)*($F
=F-F)F1F)F6F)F3*($F=F-F)F1F)F;F)F3*&$\"#NF-F))F6F*F)F)*($\"#7F-F)F6F)F
;F)F)*&FGF))F;F*F)F)" }}{PARA 6 "" 1 "" {TEXT -1 167 "PrimalDualLogBar
rier:   subject to   [0 <= -1000.00+.5e-1*x[1]+.10*x[2]+.15*x[3]-.20*x
[4], 0 <= x[3], 0 <= x[1], 0 <= x[2], 0 <= x[4], 0 <= -x[1]-x[2]-x[3]-
x[4]+10000]" }}{PARA 6 "" 1 "" {TEXT -1 38 "PrimalDualLogBarrier:   St
arting point" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")W!\\x\"-%'MATRIXG6#7&7#
$\"3/+++LLLLL!#9F+F+7#$\"\"!F1" }}{PARA 6 "" 1 "" {TEXT -1 66 "Initial
BarrierParameter:   Initial barrier parameter   192450.0984" }}{PARA 
6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6$\")C+[=-%'VECTORG6#7&$\"3%R8!=^+7HL!#9$\"3!
*e'H^-S8L$F-$\"3.Rj(z=YBL$F-$\"3s]j<$y8&f>!#<" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")?7O=-%'VECTORG6#7&$\"3[XQ?lk@NL!#9$\"3(34$)*pZ0LLF-$\"3
b9z[B]rGLF-$!3Q(RtI%*GIZ)!#=" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDua
lLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(sMZ\"-%
'VECTORG6#7&$\"3eCSzYYgSL!#9$\"340w\\r.FMLF-$\"3F%QLw*oOBLF-$!3ytioSz!
3w$!#=" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\");+b<-%'VECTORG6#7&$\"3cN8it\"
*\\VL!#9$\"3!pKV>RW!RLF-$\"3)RuA\")ze\">LF-$!3+j,%3F\\=!))!#=" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\")K$)>=-%'VECTORG6#7&$\"3XY_CCP%=L$!#9$
\"3wQlV.lk`LF-$\"3NqFU*46MJ$F-$!3_nfkDpoTY!#=" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\"(_dT\"-%'VECTORG6#7&$\"3=AwlLqi;L!#9$\"3Ce@tM&*G*Q$F-$\"
3;pMK3Zd&H$F-$!3I$oG,./(\\v!#=" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalD
ualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")s_Z<-
%'VECTORG6#7&$\"3eKZ<*oVTD$!#9$\"3_Z33![0+]$F-$\"3'p&=#=s)QWKF-$!3!Q4i
cgNCx\"!#=" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")cv>=-%'VECTORG6#7&$\"3u
Nh$>1,$fJ!#9$\"3u*)e\"y.\"\\$o$F-$\"35_JU,oMdJF-$!3]4vslA-ws!#>" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\"(CK7\"-%'VECTORG6#7&$\"3wN'\\PO4%RJ!#9$
\"3!*R1MZv*)>PF-$\"33-/OEM@SJF-$\"3o!\\sxE\"=9g!#>" }}{PARA 6 "" 1 "" 
{TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%'RTABLEG6$\")/z:=-%'VECTORG6#7&$\"3Gw*\\%RCq@J!#9$\"3_cjTH4YcPF-$\"3
2'G2jIr<7$F-$\"3]^EP%[@48(!#?" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDu
alLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(/#3:-%
'VECTORG6#7&$\"3')Qs@EXH@J!#9$\"3Mcoc\"p,tv$F-$\"3\"f.E%3DN@JF-$\"3p^E
#49[dr&!#?" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")cGr<-%'VECTORG6#7&$\"3
\")QeX=Wn?J!#9$\"3=JP0OPeePF-$\"316S3,Gr?JF-$\"3(=l(*>rn\\S$!#?" }}
{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\")S!fx\"-%'VECTORG6#7&$\"33O()z\"o^.7$!#
9$\"3=c&4'GmDfPF-$\"3_j*oxYv.7$F-$\"3)>l(f\\Hrs>!#?" }}{PARA 6 "" 1 "
" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'RTABLEG6$\")CeF=-%'VECTORG6#7&$\"3(fKq;^$=?J!#9$\"3w)o:7\")4'fPF-
$\"3KQmb.x>?JF-$\"38_w/%\\ZC7\"!#?" }}{PARA 6 "" 1 "" {TEXT -1 21 "Pri
malDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(S
\\Z\"-%'VECTORG6#7&$\"3'fos)4\"3+7$!#9$\"3#*GJnn0)*fPF-$\"3O=HHR/,?JF-
$\"3!=_w\\j&o'Q\"!#@" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarr
ier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")%Gd#=-%'VECTORG
6#7&$\"3&H0V7R++7$!#9$\"3SCk**G!***fPF-$\"3abfvH0+?JF-$\"3O!>_wph`&z!#
B" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(%o\"4\"-%'VECTORG6#7&$\"3uM!>%>++
?J!#9$\"3WP.,_****fPF-$\"3)Qy5e-++7$F-$\"3S;>-]5:#Q%!#C" }}{PARA 6 "" 
1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'RTABLEG6$\")_bF=-%'VECTORG6#7&$\"39A'=B+++7$!#9$\"3sux'y*****
fPF-$\"3'376.+++7$F-$\"3im\">FY)*4O#!#D" }}{PARA 6 "" 1 "" {TEXT -1 
21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLE
G6$\"(Gv8\"-%'VECTORG6#7&$\"3qs)p/+++7$!#9$\"3O;2t******fPF-$\"3)z'*=.
+++7$F-$\"3!*o;p4Hka7!#E" }}{PARA 6 "" 1 "" {TEXT -1 47 "PrimalDualLog
Barrier:   Global optimum found at" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
<&/&%\"xG6#\"\"%$\"3!*o;p4Hka7!#E/&F&6#\"\"\"$\"3qs)p/+++7$!#9/&F&6#\"
\"#$\"3O;2t******fPF2/&F&6#\"\"$$\"3)z'*=.+++7$F2" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "The model predict
s we should invest $0 in asset 4, as expected.  The optimal allocation
 among the other three takes up the slack." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "Given that we expect asset 4 t
o fall so drastically, we might consider selling it short.  What happe
ns to our optimal allocation if we allow short selling?  To find out, \+
we run the model again, but this time change the " }{TEXT 270 11 "nonn
egative" }{TEXT -1 14 " parameter to " }{TEXT 271 5 "free." }{TEXT -1 
72 "  This lifts the restriction that the decision variables be nonneg
ative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "Optimize( variance, min, 4, \{budgetConstraint, retur
nConstraint\}, free, <c/4,c/4,c/4,c/4>);" }}{PARA 6 "" 1 "" {TEXT -1 
120 "OptimizeInteriorPointMethod:   Constrained   convex   problem: so
lving with   convex   primal-dual log-barrier algorithm" }}{PARA 6 "" 
1 "" {TEXT -1 32 "PrimalDualLogBarrier:   Minimize" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,6*$)&%\"xG6#\"\"\"\"\"#F)$\"\")!\"#*($\"#5F-F)F&F)&F
'6#F*F)!\"\"*($F0F-F)F&F)&F'6#\"\"$F)F3*($F0F-F)F&F)&F'6#\"\"%F)F3*&$
\"#;F-F))F1F*F)F)*($F=F-F)F1F)F6F)F3*($F=F-F)F1F)F;F)F3*&$\"#NF-F))F6F
*F)F)*($\"#7F-F)F6F)F;F)F)*&FGF))F;F*F)F)" }}{PARA 6 "" 1 "" {TEXT -1 
123 "PrimalDualLogBarrier:   subject to   [0 <= -x[1]-x[2]-x[3]-x[4]+1
0000, 0 <= -1000.00+.5e-1*x[1]+.10*x[2]+.15*x[3]-.50*x[4]]" }}{PARA 6 
"" 1 "" {TEXT -1 38 "PrimalDualLogBarrier:   Starting point" }}{PARA 
6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6$\")O*Hv\"-%'MATRIXG6#7&7#$\"%+D\"\"!F+F+F+
" }}{PARA 6 "" 1 "" {TEXT -1 62 "InitialBarrierParameter:   Initial ba
rrier parameter   150000." }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLo
gBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(OeE\"-%'VE
CTORG6#7&$\"39,](o:OAc(!#:$\"3E+vox'*)*z?!#9$\"3)\\(oaN#fy3$F0$!38_7yq
YtEVF-" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")_F!z\"-%'VECTORG6#7&$\"3mpgGP
-bc;!#9$\"3s/5JZTdu<F-$\"35s_=SXLw=F-$!3lG^jj3SA7F-" }}{PARA 6 "" 1 "
" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'RTABLEG6$\")7wi<-%'VECTORG6#7&$\"3q(p5;1*\\l%)!#:$\"3YZ+hhv7g#*F-
$\"3s?F:A*G&p**F-$!3sG,52'3tl\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 21 "Pri
malDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")7
1=<-%'VECTORG6#7&$\"3+)p5$[*36B(!#:$\"3-Z+h+'[Q3)F-$\"31@Fvz=NG))F-$!3
mGY$)H1'Hd\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")SOh<-%'VECTORG6#7&$
\"3)zp!HOl/zx!#:$\"3`Y]'*HwcT#)F-$\"3_@FZmM#)*o)F-$!3vG1Y_!fH]\"!#9" }
}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\"'!3%)*-%'VECTORG6#7&$\"3y(prr'\\Q5y!#:$
\"35'p[e#)*QY#)F-$\"3+@(ziWVWn)F-$!3wypJe]7(\\\"!#9" }}{PARA 6 "" 1 "
" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%'RTABLEG6$\")Od;=-%'VECTORG6#7&$\"3rZi!3KbF\"y!#:$\"3C'*[*3+?pC)F-
$\"3)3Jz9mtNn)F-$!3'))oh$[#[n\\\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 21 "P
rimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"
)7Wl<-%'VECTORG6#7&$\"3T#Hl\\e+H\"y!#:$\"3y5dv'QdpC)F-$\"33M-sZ!HNn)F-
$!3%>U::6Fn\\\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrie
r:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"))[^u\"-%'VECTORG6
#7&$\"357mV(y3H\"y!#:$\"3emSoR'fpC)F-$\"3E8]E#yENn)F-$!3K(Hk[(fs'\\\"!
#9" }}{PARA 6 "" 1 "" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")Gq<=-%'VECTORG6#7&$\"3Cp7kE#4H\"y
!#:$\"3[v5xi(fpC)F-$\"3kSc`lm_t')F-$!3yoOd:fs'\\\"!#9" }}{PARA 6 "" 1 
"" {TEXT -1 21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'RTABLEG6$\"'!3%*)-%'VECTORG6#7&$\"3IYV\\l#4H\"y!#:$\"3;%H.)z(fpC
)F-$\"3%fqN/mENn)F-$!3g$Qs<\"fs'\\\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 
21 "PrimalDualLogBarrier:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLE
G6$\")#*GB=-%'VECTORG6#7&$\"3EeUGj#4H\"y!#:$\"3K`#oOxfpC)F-$\"3!HbPFmE
Nn)F-$!3g\\$[=\"fs'\\\"!#9" }}{PARA 6 "" 1 "" {TEXT -1 47 "PrimalDualL
ogBarrier:   Global optimum found at" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#<&/&%\"xG6#\"\"\"$\"3EeUGj#4H\"y!#:/&F&6#\"\"#$\"3K`#oOxfpC)F+/&F&6#
\"\"$$\"3!HbPFmENn)F+/&F&6#\"\"%$!3g\\$[=\"fs'\\\"!#9" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "The mode
l recommends we sell asset 4 short by $1496.  Why does it recommend su
ch small investments in the other assets?  Recall we set our target re
turn to 10%.  The model finds the allocation of " }{TEXT 272 16 "minim
um variance" }{TEXT -1 196 " that will achieve this target.  Apparentl
y, the short sale of asset 4 by itself is almost enough to get a 10% t
otal expected return.  Allocating more funds to the others would raise
 the variance." }}}}}{MARK "3 17 0 0" 246 }{VIEWOPTS 1 1 0 1 1 1803 1 
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