(!******************************************************
Mosel Example Problems
======================
file folioqpgraph.mos
`````````````````````
TYPE: Portfolio optimization problem (Quadratic Programming problem)
DIFFICULTY: 3
FEATURES: small QP problem, solved repeatedly with modified
constraints, graphical representation of the results,
use of `min' and `max'
DESCRIPTION: An investor is evaluating ten different securities (`shares').
He estimates the return on investment for a period of one
year. To spread the risk he wishes to invest at most
30% of the capital into any share. He further wishes to
invest at least half of his capital in North-American
shares and at most a third in high-risk shares. How should
the capital be divided among the shares to minimize the
risk whilst obtaining a certain target yield? The investor
adopts the Markowitz idea of getting estimates of the
variance/covariance matrix of estimated returns on the
securities.
FURTHER INFO: `Getting Started with Xpress', Chapter 7 `Quadratic
Programming'.
Similar problem:
`Applications of optimization with Xpress-MP',
Section 13.7 `Mean variance portfolio selection'
(c) 2008 Fair Isaac Corporation
author: S. Heipcke, 2003, rev. Sep. 2017
*******************************************************!)
model "Portfolio optimization with QP"
uses "mmxprs", "mmnl", "mmsvg" ! Use Xpress Optimizer with QP solver
parameters
MAXVAL = 0.3 ! Max. investment per share
MINAM = 0.5 ! Min. investment into N.-American values
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
RET: array(SHARES) of real ! Estimated return in investment
VAR: array(SHARES,SHARES) of real ! Variance/covariance matrix of
! estimated returns
SOLRET: array(range) of real ! Solution values (total return)
SOLDEV: array(range) of real ! Solution values (average deviation)
end-declarations
initializations from "folioqpgraph.dat"
RISK RET NA VAR
end-initializations
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
MinAm:= sum(s in NA) frac(s) >= MINAM
! Spend all the capital
SpendAll:= 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
else
writeln("No solution for target return >= ", ct, "%")
break
end-if
target += 1
end-do
! Drawing a graph to represent results (`GrS') and data (`GrL' & `GrH')
declarations
DEV: array(SHARES) of real ! Standard deviation
NAMES: array(SHARES) of string ! Names of shares
end-declarations
initializations from "folioqpgraph.dat"
DEV NAMES
end-initializations
svgaddgroup("GrS", "Solution values", SVG_BLACK)
svgaddgroup("GrL", "Low risk", SVG_GREEN)
svgaddgroup("GrH", "High risk", SVG_RED)
forall(r in 1..ct) svgaddpoint("GrS", SOLRET(r), SOLDEV(r));
svgaddline("GrS", sum(r in 1..ct) [SOLRET(r), SOLDEV(r)])
forall (s in SHARES-RISK) do
svgaddpoint("GrL", RET(s), DEV(s))
svgaddtext("GrL", RET(s)+1, 1.3*(DEV(s)-1), NAMES(s))
end-do
forall (s in RISK) do
svgaddpoint("GrH", RET(s), DEV(s))
svgaddtext("GrH", RET(s)-2.5, DEV(s)-1, NAMES(s))
end-do
! Scale the size of the displayed graph
svgsetgraphscale(10)
svgsetgraphpointsize(2)
svgsetgraphlabels("Expected return", "Standard deviation")
svgsave("folioqpgraph.svg")
svgrefresh
svgwaitclose("Close browser window to terminate model execution.", 1)
end-model
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