Folio - performing simple statistics with R

(!******************************************************
   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 --
  
   !!! This example requires an installation of R, see the
   !!! chapter 'R' of the 'Mosel Language Reference' for
   !!! compatible versions and setup instructions
   
  (c) 2015 Fair Isaac Corporation
      author: S.Lannez, Jul. 2015, rev. Sep. 2022
*******************************************************!)

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
 fixmipentities(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 --
  
   !!! This example requires an installation of R, see the
   !!! chapter 'R' of the 'Mosel Language Reference' for
   !!! compatible versions and setup instructions
   
  (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