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Description
The example describes a portfolio optimization problem with parameterized risk/return measures. It is formulated with quadratic constraints and a quadratic objective function.

Source Files

Data Files

portfoliorisk.mos

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

(c) 2013 Fair Isaac Corporation
author: S. Heipcke, Mar. 2013
*********************************************************************!)

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

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)
setparam("xnlp_solver", 0)

! 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

```   