(!*********************************************************************
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
*********************************************************************!)
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)
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
|
(!*********************************************************************
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. Sep. 2017
*********************************************************************!)
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)
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)
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
|
