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Portfolio optimization
Type: QCQP
Rating: 2 (easy-medium)
Description: The example describes a portfolio optimization problem with parameterized risk/return measures. It is formulated with quadratic constraints and a quadratic objective function.
File(s): portfoliorisk.mos, portfoliorisk_graph.mos
Data file(s): portfoliorisk.dat

portfoliorisk.mos
(!*********************************************************************
   Mosel NL examples
   =================
   file portfoliorisk.mos
   ``````````````````````
   Portfolio optimization with parameterized risk/return measures
   Quadratic constraints / objective.

   (c) 2013 Fair Isaac Corporation
       author: S. Heipcke, Mar. 2013, rev. Jun. 2023
*********************************************************************!)
    
model "portfoliorisk"
  uses "mmxnlp", "mmsystem"

  parameters
    MAXFRAC = 1.0                         ! Max. fraction per asset: in ]0,1]
  end-parameters	

  declarations
    YEARS : range                         ! Set of time periods 
    ASSETS: set of string                 ! Set of assets
    RET: array(YEARS, ASSETS) of real     ! Return of an asset per year
    LAMBDA: list of real                  ! Ordered list of risk aversion values
    NYEARS: integer                       ! Size of set YEARS
    NASSETS: integer                      ! Size of set ASSETS
    Mean: array(ASSETS) of real           ! Average reward of an asset
    Sdev: array(ASSETS) of real           ! Standard deviation of an asset
    Risk: array(YEARS, ASSETS) of real    ! Risk measure
    Var:  array(ASSETS, ASSETS) of real   ! Variance/covariance matrix
    Corr:  array(ASSETS, ASSETS) of real  ! Correlation coefficient
  
    frac: array(ASSETS) of mpvar          ! Investment fraction of an asset
    totalreturn: mpvar                    ! Expected return
    totalrisk: mpvar                      ! Risk measure (total variance)
    TotalCorr: nlctr                      ! Total correlation
    ObjDef: nlctr                         ! Objective function
  end-declarations

! Read data from file
  initializations from 'portfoliorisk.dat'
    RET  LAMBDA
  end-initializations

  finalize(YEARS); finalize(ASSETS)
  NYEARS:= YEARS.size
  NASSETS:= ASSETS.size

! Calculate derived data
  ! Mean return
  forall(a in ASSETS) Mean(a):= sum(y in YEARS) RET(y,a)/NYEARS
  ! Risk measure
  forall(y in YEARS, a in ASSETS) Risk(y, a):= RET(y,a) - Mean(a)
  ! Standard deviation        
  forall(a in ASSETS) 
    Sdev(a):= sqrt(sum(y in YEARS) Risk(y,a)^2)/NYEARS
  ! Variance/Covariance
  forall(a1, a2 in ASSETS) 
    Var(a1,a2):= (sum(y in YEARS) Risk(y,a1)*Risk(y,a2))/NYEARS
  ! Correlation coefficient (normalized covariance)
  forall(a1, a2 in ASSETS) 
    Corr(a1,a2):= Var(a1,a2)/sqrt(Var(a1,a1)*Var(a2,a2))

  writeln("NASSETS=", NASSETS)
  writeln("NYEARS=", NYEARS) 
  forall(a in ASSETS)
    writeln(strfmt(a,-11), ": Mean=", Mean(a), ", StdDev=", Sdev(a))

! State variable bounds and initial values
  forall (a in ASSETS) do
    0 <= frac(a); frac(a) <= MAXFRAC  ! Spend between 0% and MAXFRAC% per asset
    setinitval(frac(a),1.0/NASSETS)
  end-do  

! Auxiliary variables for the definition of the objective function
  totalreturn = sum(a in ASSETS) Mean(a)*frac(a)
  totalreturn is_free
  totalrisk = (sum(y in YEARS) (sum(a in ASSETS) Risk(y,a)*frac(a))^2)/NYEARS
  totalrisk is_free
  TotalCorr := sum(a1,a2 in ASSETS) Corr(a1,a2)*frac(a1)*frac(a2)

! Spend all the capital
  SpendAll:= sum(a in ASSETS) frac(a) = 1

!  setparam("xnlp_verbose", true)
  
! In this example we will use a local solver, since it can be time consuming to solve it to global optimality 
  setparam("xprs_nlpsolver", 1)
  setparam("xnlp_solver", 0)     ! Solver choice: Xpress NonLinear (SLP)

! Objective 1: Maximize a combination of "return - weighted risk"
  writeln("Objective 1:")
  forall(l in LAMBDA) do
    ObjDef:= totalreturn - l*totalrisk
    maximize(ObjDef)

    writeln(" lambda =", strfmt(l,4), ": Mean yield = ", totalreturn.sol, 
            ", StdDev = ", sum(a in ASSETS) Sdev(a)*100.0*getsol(frac(a)),
            ", Obj = ", getobjval, ", Corr = ", TotalCorr.sol)
    write(" "*13)
    forall(a in ASSETS | frac(a).sol>0.001)
      write("  ", a, ":", strfmt(frac(a).sol,5,3), "%")
    writeln
  end-do

! Objective 2: Minimize total (co)variance with parameterized return target
  writeln("Objective 2:")
  ObjDef:=  sum(a1, a2 in ASSETS) Var(a1,a2)*frac(a1)*frac(a2)
  FAC:=50
  RRANGE:= ceil(min(a in ASSETS) FAC*Mean(a))..floor(max(a in ASSETS) FAC*Mean(a))
  forall(r in RRANGE) do
    ReturnTarget:= totalreturn >= r/FAC
    minimize(ObjDef)
    if getprobstat=XPRS_OPT then
      writeln(" r = ", strfmt(r/FAC,4), ": Mean yield = ", totalreturn.sol, 
              ", StdDev = ", sum(a in ASSETS) Sdev(a)*100.0*getsol(frac(a)),
              ", Variance = ", getobjval, ", Corr = ", TotalCorr.sol)
      write(" "*13)
      forall(a in ASSETS | frac(a).sol>0.001)
        write("  ", a, ":", strfmt(frac(a).sol,5,3), "%")
      writeln
    else
      writeln(" r = ", strfmt(r/FAC,4), ": Infeasible")
    end-if  
  end-do

end-model
portfoliorisk_graph.mos
(!*********************************************************************
   Mosel NL examples
   =================
   file portfoliorisk_graph.mos
   ````````````````````````````
   Portfolio optimization with parameterized risk/return measures
   Quadratic constraints / objective.

   - Graphical representation of results -   

   (c) 2013 Fair Isaac Corporation
       author: S. Heipcke, Mar. 2013, rev. Jun. 2023
*********************************************************************!)
    
model "portfoliorisk"
  uses "mmxnlp", "mmsystem", "mmsvg"

  parameters
    MAXFRAC = 1.0                         ! Max. fraction per asset: in ]0,1]
  end-parameters	

  declarations
    YEARS : range                         ! Set of time periods 
    ASSETS: set of string                 ! Set of assets
    RET: array(YEARS, ASSETS) of real     ! Return of an asset per year
    LAMBDA: list of real                  ! Ordered list of risk aversion values
    NYEARS: integer                       ! Size of set YEARS
    NASSETS: integer                      ! Size of set ASSETS
    Mean: array(ASSETS) of real           ! Average reward of an asset
    Sdev: array(ASSETS) of real           ! Standard deviation of an asset
    Risk: array(YEARS, ASSETS) of real    ! Risk measure
    Var:  array(ASSETS, ASSETS) of real   ! Variance/covariance matrix
    Corr:  array(ASSETS, ASSETS) of real  ! Correlation coefficient
  
    frac: array(ASSETS) of mpvar          ! Investment fraction of an asset
    retsol: array(real) of real           ! Solution values per lambda
    sdevsol: array(real) of real          ! Standard deviation per lambda
    corrsol: array(real) of real          ! Correlation per lambda
    retsol2: dynamic array(integer) of real   ! Solution values per expected return
    sdevsol2: dynamic array(integer) of real  ! Standard deviation per expected return
    corrsol2: dynamic array(integer) of real  ! Correlation per expected return
    totalreturn: mpvar                    ! Expected return
    totalrisk: mpvar                      ! Risk measure (total variance)
    TotalCorr: nlctr                      ! Total correlation
    ObjDef: nlctr                         ! Objective function
  end-declarations

! Read data from file
  initializations from 'portfoliorisk.dat'
    RET  LAMBDA
  end-initializations

  finalize(YEARS); finalize(ASSETS)
  NYEARS:= YEARS.size
  NASSETS:= ASSETS.size

! Calculate derived data
  ! Mean return
  forall(a in ASSETS) Mean(a):= sum(y in YEARS) RET(y,a)/NYEARS
  ! Risk measure
  forall(y in YEARS, a in ASSETS) Risk(y, a):= RET(y,a) - Mean(a)
  ! Standard deviation        
  forall(a in ASSETS) 
    Sdev(a):= sqrt(sum(y in YEARS) Risk(y,a)^2)/NYEARS
  ! Variance/Covariance
  forall(a1, a2 in ASSETS) 
    Var(a1,a2):= (sum(y in YEARS) Risk(y,a1)*Risk(y,a2))/NYEARS
  ! Correlation coefficient (normalized covariance)
  forall(a1, a2 in ASSETS) 
    Corr(a1,a2):= Var(a1,a2)/sqrt(Var(a1,a1)*Var(a2,a2))

  writeln("NASSETS=", NASSETS)
  writeln("NYEARS=", NYEARS) 
  forall(a in ASSETS)
    writeln(strfmt(a,-11), ": Mean=", Mean(a), ", StdDev=", Sdev(a))

! State variable bounds and initial values
  forall (a in ASSETS) do
    0 <= frac(a); frac(a) <= MAXFRAC  ! Spend between 0% and MAXFRAC% per asset
    setinitval(frac(a),1.0/NASSETS)
  end-do  

! Auxiliary variables for the definition of the objective function
  totalreturn = sum(a in ASSETS) Mean(a)*frac(a)
  totalreturn is_free
  totalrisk = (sum(y in YEARS) (sum(a in ASSETS) Risk(y,a)*frac(a))^2)/NYEARS
  totalrisk is_free
  TotalCorr := sum(a1,a2 in ASSETS) Corr(a1,a2)*frac(a1)*frac(a2)

! Spend all the capital
  SpendAll:= sum(a in ASSETS) frac(a) = 1

!  setparam("xnlp_verbose", true)
! In this example we will use a local solver, since it can be time consuming to solve it to global optimality 
  setparam("xprs_nlpsolver", 1)
  setparam("xnlp_solver", 0)
  
! In this example we will use a local solver, since it can be time consuming to solve it to global optimality 
  setparam("xprs_nlpsolver", 1)

! Objective 1: Maximize a combination of "return - weighted risk"
  writeln("Objective 1:")
  forall(l in LAMBDA) do
    ObjDef:= totalreturn - l*totalrisk
    maximize(ObjDef)

    retsol(l):= totalreturn.sol
    sdevsol(l) := sum(a in ASSETS) Sdev(a)*100.0*getsol(frac(a))
    corrsol(l):= TotalCorr.sol
    writeln(" lambda =", strfmt(l,4), ": Mean yield = ", retsol(l),
            ", StdDev = ", sdevsol(l),
            ", Obj = ", getobjval, ", Corr = ", TotalCorr.sol)
    write(" "*13)
    forall(a in ASSETS | frac(a).sol>0.001)
      write("  ", a, ":", strfmt(frac(a).sol,5,3), "%")
    writeln
  end-do

! Objective 2: Minimize total (co)variance with parameterized return target
  writeln("Objective 2:")
  ObjDef:=  sum(a1, a2 in ASSETS) Var(a1,a2)*frac(a1)*frac(a2)
  FAC:=50
  RRANGE:= ceil(min(a in ASSETS) FAC*Mean(a))..floor(max(a in ASSETS) FAC*Mean(a))
  forall(r in RRANGE) do
    ReturnTarget:= totalreturn >= r/FAC
    minimize(ObjDef)
    if getprobstat=XPRS_OPT then
      retsol2(r):= totalreturn.sol
      sdevsol2(r) := sum(a in ASSETS) Sdev(a)*100.0*getsol(frac(a))
      corrsol2(r):= TotalCorr.sol
      writeln(" r = ", strfmt(r/FAC,4), ": Mean yield = ", retsol2(r),
              ", StdDev = ", sdevsol2(r),
              ", Variance = ", getobjval, ", Corr = ", TotalCorr.sol)
      write(" "*13)
      forall(a in ASSETS | frac(a).sol>0.001)
        write("  ", a, ":", strfmt(frac(a).sol,5,3), "%")
      writeln
    else
      writeln(" r = ", strfmt(r/FAC,4), ": Infeasible")
    end-if  
  end-do


!**************** Graphical representation of results ****************
  
  YFACT:=0.05
  svgsetgraphviewbox(0.9,0,0.5,11*YFACT)
  svgsetgraphlabels("Expected return", "Standard deviation")
  svgaddgroup("PlotE", "Efficient frontier Obj1", svgcolor(225,100,100))
  svgaddgroup("PlotR", "Efficient frontier Obj2", svgcolor(100,100,225))
  svgaddgroup("PlotEC", "Correlation Obj1", svgcolor(175,50,50))
  svgaddgroup("PlotRC", "Correlation Obj2", svgcolor(50,50,175))
  svgaddgroup("PlotA", "Properties of each asset", svgcolor(100,100,100))
  CFAC:= 10      ! Scaling factor for correlation graphs
  
! Draw the efficient frontier and correlation graphs for objective 1
  forall(l in LAMBDA) svgaddpoint("PlotE", retsol(l), sdevsol(l)*YFACT);
  svgaddline("PlotE", sum(l in LAMBDA) [retsol(l), sdevsol(l)*YFACT])
  svgaddline("PlotEC", sum(l in LAMBDA) [retsol(l), CFAC*corrsol(l)*YFACT])

! Graphs from second objective
  forall(r in RRANGE | exists(retsol2(r))) svgaddpoint("PlotR", retsol2(r), sdevsol2(r)*YFACT);
  svgaddline("PlotR", sum(r in RRANGE | exists(retsol2(r)) ) [retsol2(r), sdevsol2(r)*YFACT])
  svgaddline("PlotRC", sum(r in RRANGE | exists(retsol2(r)) ) [retsol2(r), CFAC*corrsol2(r)*YFACT])

! Draw position of assets
  forall(a in ASSETS) do
    svgaddpoint("PlotA", Mean(a), Sdev(a)*100.0*YFACT)
    svgaddtext("PlotA", Mean(a), (Sdev(a)*100.0-0.5)*YFACT, a)
  end-do

! Scale the size of the displayed graph
  svgsetgraphpointsize(2)
  svgsetgraphscale(500)

  svgsave("portfoliorisk.svg")
  svgrefresh
  svgwaitclose("Close browser window to terminate model execution.", 1)

end-model
Parent Topic Mosel language: examples of nonlinear optimization

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