(!******************************************************
Mosel Example Problems
======================
file folioqpr.mos
`````````````````
Modeling a small QP problem
to perform portfolio optimization.
-- 1. QP: minimize variance
2. MIQP: limited number of assets --
-- Using R to calculate covariance matrix --
(c) 2015 Fair Isaac Corporation
author: S.Lannez, Jul. 2015, rev. Sep. 2017
*******************************************************!)
model "Portfolio optimization with QP/MIQP"
uses "mmxprs", "mmnl"
uses "r" ! Use R functions
parameters
MAXVAL = 0.3 ! Max. investment per share
MINAM = 0.5 ! Min. investment into N.-American values
MAXNUM = 4 ! Max. number of different assets
TARGET = 9.0 ! Minimum target yield
end-parameters
declarations
SHARES = 1..10 ! Set of shares
RISK: set of integer ! Set of high-risk values among shares
NA: set of integer ! Set of shares issued in N.-America
DATES: set of string ! Historical dates
RET: array(SHARES) of real ! Estimated return in investment
VAR: array(SHARES,SHARES) of real ! Variance/covariance matrix of
! estimated returns
OPEN: array(SHARES,DATES) of real ! Historical share value at market opening
CLOSE: array(SHARES,DATES) of real ! Historical share value at market closing
end-declarations
initializations from "folioqp.dat"
RISK RET NA
end-initializations
! Load historical values to compute the covariance
initializations from "folioqphist.dat"
OPEN CLOSE
end-initializations
! **** Perform some statistics using R ****
! Copy array to R environments
Rset('open',OPEN)
Rset('close',CLOSE)
! Print covariance of share value at market openings
writeln("Covariances at market openings:")
Rprint('cov(t(open))')
! Calculate and retrieve covariance of mean value
Rgetarr('cov(t((open+close)/2))',VAR)
declarations
frac: array(SHARES) of mpvar ! Fraction of capital used per share
end-declarations
! **** First problem: unlimited number of assets ****
! Objective: mean variance
Variance:= sum(s,t in SHARES) VAR(s,t)*frac(s)*frac(t)
! Minimum amount of North-American values
sum(s in NA) frac(s) >= MINAM
! Spend all the capital
sum(s in SHARES) frac(s) = 1
! Target yield
sum(s in SHARES) RET(s)*frac(s) >= TARGET
! Upper bounds on the investment per share
forall(s in SHARES) frac(s) <= MAXVAL
! Solve the problem
minimize(Variance)
! Solution printing
writeln("With a target of ", TARGET, " minimum variance is ", getobjval)
forall(s in SHARES) writeln(s, ": ", getsol(frac(s))*100, "%")
! **** Second problem: limit total number of assets ****
declarations
buy: array(SHARES) of mpvar ! 1 if asset is in portfolio, 0 otherwise
end-declarations
! Limit the total number of assets
sum(s in SHARES) buy(s) <= MAXNUM
forall(s in SHARES) do
buy(s) is_binary
frac(s) <= buy(s)
end-do
! Solve the problem
minimize(Variance)
writeln("With a target of ", TARGET," and at most ", MAXNUM,
" assets,\n minimum variance is ", getobjval)
forall(s in SHARES) writeln(s, ": ", getsol(frac(s))*100, "%")
! Round integer values and resolve
fixglobal(true)
minimize(Variance)
writeln("With all binary variables rounded to the nearest integer:")
forall(s in SHARES) writeln(s, ": ", getsol(frac(s))*100, "%")
end-model
|
(!******************************************************
Mosel R Example Problems
========================
file folioqpgraphr.mos
``````````````````````
Modeling a small QP problem
to perform portfolio optimization.
Minimize variance subject to different target return.
-- Using R to calculate covariance matrix --
-- Graphical output via R --
(c) 2015 Fair Isaac Corporation
author: S.Lannez, Jul. 2015, rev. Sep. 2017
*******************************************************!)
model "Portfolio optimization with QP, graphical output"
uses "mmxprs", "mmnl"
uses "mmsystem"
uses "r" ! Use R functions
parameters
MAXVAL = 0.3 ! Max. investment per share
MINAM = 0.5 ! Min. investment into N.-American values
GRDEVICE = "png" ! Set to 'png' or 'pdf' to change the type of document.
end-parameters
declarations
SHARES = 1..10 ! Set of shares
RISK: set of integer ! Set of high-risk values among shares
NA: set of integer ! Set of shares issued in N.-America
DATES: set of string ! Historical dates
RET: array(SHARES) of real ! Estimated return in investment
VAR: array(SHARES,SHARES) of real ! Variance/covariance matrix of
! estimated returns
OPEN: array(SHARES,DATES) of real ! Historical share value at market opening
CLOSE: array(SHARES,DATES) of real ! Historical share value at market closing
SOLRET: array(range) of real ! Solution values (total return)
SOLDEV: array(range) of real ! Solution values (average deviation)
end-declarations
initializations from "folioqp.dat"
RISK RET NA
end-initializations
! Load historical values to compute the covariance
initializations from "folioqphist.dat"
OPEN CLOSE
end-initializations
! **** Perfom some statistics using R ****
! Copy array to R environments
Rset('open',OPEN)
Rset('close',CLOSE)
! Print covariance of share value at market openings
writeln("Covariances at market openings:")
Rprint('cov(t(open))')
! Calculate and retrieve covariance of mean value
Rgetarr('cov(t((open+close)/2))',VAR)
! **** Mathematical model ****
declarations
frac: array(SHARES) of mpvar ! Fraction of capital used per share
end-declarations
! Objective: mean variance
Variance:= sum(s,t in SHARES) VAR(s,t)*frac(s)*frac(t)
! Minimum amount of North-American values
sum(s in NA) frac(s) >= MINAM
! Spend all the capital
sum(s in SHARES) frac(s) = 1
! Upper bounds on the investment per share
forall(s in SHARES) frac(s) <= MAXVAL
! Solve the problem for a range of returns: this is the efficient frontier
target:= min(s in SHARES) RET(s)
RMAX:= max(s in SHARES) RET(s)
while(target < RMAX) do
Return:= sum(s in SHARES) RET(s)*frac(s) >= target ! Target yield
minimize(Variance) ! Solve the problem
if (getprobstat = XPRS_OPT) then ! Save the optimal solution value
ct+=1
SOLDEV(ct):= getobjval
SOLRET(ct):= target
writeln("Solution for target ", target, "%")
else
writeln("No solution for target return >= ", target, "%")
break
end-if
target += 1
end-do
! **** Drawing graphs to represent results and data ****
declarations
DEV: array(SHARES) of real ! Standard deviation
NAMES: array(SHARES) of string ! Names of shares
t: text
end-declarations
initializations from "folioqpgraph.dat"
NAMES ! DEV
end-initializations
! Compute the standard deviation
forall(s in SHARES) DEV(s) := sqrt(VAR(s,s))
! Setting up the R environment
Reval('require(graphics)') ! plot() and heatmap() functions
Reval('require(grDevices)') ! heatmap dependency
if (GRDEVICE.size>0) then
Reval(GRDEVICE+'("folioqpgraphr%03d.'+GRDEVICE+'")') ! Open graphical file
end-if
! **** Drawing the efficient frontier ****
! Copy arrays to R
Rset('solval',SOLRET)
Rset('soldev',SOLDEV)
! Plot the graphs
Reval('plot(solval,soldev,color="black",type="l",'+
'main="Portfolio Return",'+
'xlab="Solution Value",ylab="Solution Variance")')
Reval('points(solval,soldev,pch=21,cex=1,col="dark red")')
if (GRDEVICE.size=0) then ! Wait for user entry
Reval('dev.flush()') ! Print plots
writeln("Press enter to continue...")
dummy := readtextline(t)
end-if
! Clean up arrays
Reval('rm(solval)')
Reval('rm(soldev)')
! **** Drawing the shares characteristics ****
! Copy arrays to R
Rset('ret',RET)
Rset('dev',DEV)
Rset('names',NAMES)
Rset('risk',RISK)
Rset('shares',SHARES)
! Create the color vectors
Reval('h <- c()')
Reval('h[shares] <- "green"')
Reval('h[risk] <- "red"')
! Plot the graphs
Reval('plot(ret,dev,col=h,bg=h,type="p",pch=21,'+
'main="Shares",'+
'xlab="Share Value",ylab="Share Deviation",'+
'xlim=c(-5,max(ret)+5),ylim=c(min(dev)-5,max(dev)+5))')
Reval('legend("topleft",c("high risk","low risk"),'+
'col=c("red","green"),pch=21)')
Reval('text(ret,dev,names,pos=3)')
if (GRDEVICE.size=0) then ! Wait for user entry
Reval('dev.flush()') ! Print plots
writeln("Press enter to continue...")
dummy := readtextline(t)
end-if
! Clean up arrays
forall(ar in {"ret","dev","risk","shares"})
Reval('rm('+ar+')')
! **** Drawing the correlation heat map ****
Reval('z <- cor(t(open+close)/2)') ! Compute the correlation matrix
Reval('rownames(z) <- names') ! Set the name of the rows
Reval('colnames(z) <- names') ! Set the name of the columns
Rprint('symnum(z)') ! Textual representation of the correlation
Reval('heatmap(z,Rowv=NA,Colv=NA,symm=TRUE,main="Share Value Correlation")')
if (GRDEVICE.size=0) then ! Wait for user entry
Reval('dev.flush()') ! Print plots
writeln("Press enter to continue...")
dummy := readtextline(t)
end-if
Reval('dev.off()') ! Close current document
end-model
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