"""
## GAMSSOURCE: https://www.gams.com/latest/finlib_ml/libhtml/finlib_MAD.html
## LICENSETYPE: Demo
## MODELTYPE: LP, NLP
## DATAFILES: WorldIndices.gdx
Mean absolute deviation model
MAD.gms: Mean absolute deviation model.
Consiglio, Nielsen and Zenios.
PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 5.3
Last modified: Apr 2008.
"""
from __future__ import annotations
import os
from pathlib import Path
import gamspy.math as gams_math
from gamspy import (
Alias,
Card,
Container,
Equation,
Model,
Ord,
Parameter,
Smax,
Smin,
Sum,
Variable,
)
from gamspy.math import sqr
def main():
gdx_file = str(Path(__file__).parent.absolute()) + "/WorldIndices.gdx"
m = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
load_from=gdx_file,
)
# SETS #
i, l = m.getSymbols(["i", "l"])
# SCALARS #
Budget = Parameter(
m, name="Budget", description="Nominal investment budget"
)
MU_TARGET = Parameter(
m, name="MU_TARGET", description="Target portfolio return"
)
MU_STEP = Parameter(m, name="MU_STEP", description="Target return step")
MIN_MU = Parameter(
m, name="MIN_MU", description="Minimum return in universe"
)
MAX_MU = Parameter(
m, name="MAX_MU", description="Maximum return in universe"
)
Budget[...] = 100.0
# PARAMETERS #
pr = Parameter(m, name="pr", domain=l, description="Scenario probability")
P = Parameter(m, name="P", domain=[i, l], description="Final values")
EP = Parameter(m, name="EP", domain=i, description="Expected final values")
AssetReturns = m.getSymbols(["AssetReturns"])[0]
pr[l] = 1.0 / Card(l)
P[i, l] = 1 + AssetReturns[i, l]
EP[i] = Sum(l, pr[l] * P[i, l])
MIN_MU[...] = Smin(i, EP[i])
MAX_MU[...] = Smax(i, EP[i])
# Assume we want 20 portfolios in the frontier
MU_STEP[...] = (MAX_MU - MIN_MU) / 20
print("MAX_MU: ", MAX_MU.records.value.round(3)[0])
# VARIABLES #
x = Variable(
m,
name="x",
type="positive",
domain=i,
description="Holdings of assets in monetary units (not proportions)",
)
y = Variable(
m,
name="y",
type="positive",
domain=l,
description="Measures of the absolute deviation",
)
z = Variable(m, name="z", description="Objective function value")
# EQUATIONS #
BudgetCon = Equation(
m,
name="BudgetCon",
type="regular",
description="Equation defining the budget contraint",
)
ReturnCon = Equation(
m,
name="ReturnCon",
type="regular",
description="Equation defining the portfolio return constraint",
)
ObjDef = Equation(
m,
name="ObjDef",
type="regular",
description="Objective function definition for MAD",
)
yPosDef = Equation(
m,
name="yPosDef",
type="regular",
domain=l,
description="Equations defining the positive deviations",
)
yNegDef = Equation(
m,
name="yNegDef",
type="regular",
domain=l,
description="Equations defining the negative deviations",
)
BudgetCon[...] = Sum(i, x[i]) == Budget
ReturnCon[...] = Sum(i, EP[i] * x[i]) >= MU_TARGET * Budget
yPosDef[l] = y[l] >= Sum(i, P[i, l] * x[i]) - Sum(i, EP[i] * x[i])
yNegDef[l] = y[l] >= Sum(i, EP[i] * x[i]) - Sum(i, P[i, l] * x[i])
ObjDef[...] = z == Sum(l, pr[l] * y[l])
MeanAbsoluteDeviation = Model(
m,
name="MeanAbsoluteDeviation",
equations=[BudgetCon, ReturnCon, yPosDef, yNegDef, ObjDef],
problem="LP",
sense="MIN",
objective=z,
)
output_csv = '"MAD","Mean"\n'
mu_target = MIN_MU.records.value[0]
while mu_target <= MAX_MU.records.value[0]:
# MU_TARGET = MIN_MU TO MAX_MU BY MU_STEP,
MU_TARGET[...] = mu_target
MeanAbsoluteDeviation.solve()
output_csv += f"{z.records.level.round(3)[0]},{round(MU_TARGET.records.value[0] * Budget.records.value[0],3)},"
x_recs = [str(x_rec) for x_rec in x.records.level.round(3).tolist()]
output_csv += ",".join(x_recs) + "\n"
mu_target += MU_STEP.records.value[0]
# Compute variances and covariances
# for comparison between Mean Variance and Mean Absolute Deviation
# ALIAS
i1 = Alias(m, name="i1", alias_with=i)
i2 = Alias(m, name="i2", alias_with=i)
# PARAMETERS
VP = Parameter(m, name="VP", domain=[i, i])
VP[i, i] = Sum(l, sqr(P[i, l] - EP[i])) / (Card(l) - 1)
VP[i1, i2].where[Ord(i1) > Ord(i2)] = Sum(
l, (P[i1, l] - EP[i1]) * (P[i2, l] - EP[i2])
) / (Card(l) - 1)
print("VP: ", VP.records.value.round(3).tolist())
# EQUATION
ObjDefMV = Equation(
m,
name="ObjDefMV",
type="regular",
description="Objective function definition for Mean-Variance",
)
ObjDefMV[...] = z == Sum([i1, i2], x[i1] * VP[i1, i2] * x[i2])
MeanVariance = Model(
m,
name="MeanVariance",
equations=[BudgetCon, ReturnCon, ObjDefMV],
problem="NLP",
sense="MIN",
objective=z,
)
MeanVariance.solve()
output_csv += '"SD","Mean"\n'
mu_target = MIN_MU.records.value[0]
while mu_target <= MAX_MU.records.value[0]:
MU_TARGET[...] = mu_target
MeanVariance.solve()
z.l[...] = gams_math.sqrt(z.l)
output_csv += f"{z.records.level.round(3)[0]},{round(MU_TARGET.records.value[0] * Budget.records.value[0],3)},"
x_recs = [str(x_rec) for x_rec in x.records.level.round(3).tolist()]
output_csv += ",".join(x_recs) + "\n"
mu_target += MU_STEP.records.value[0]
# SCALARS
lambdaPos = Parameter(
m,
name="lambdaPos",
description="Weight attached to positive deviations",
)
lambdaNeg = Parameter(
m,
name="lambdaNeg",
description="Weight attached to negative deviations",
)
lambdaPos[...] = 0.5
lambdaNeg[...] = 0.5
# EQUATIONS
yPosWeightDef = Equation(
m,
name="yPosWeightDef",
type="regular",
domain=l,
description=(
"Equations defining the positive deviations with weight attached"
),
)
yNegWeightDef = Equation(
m,
name="yNegWeightDef",
type="regular",
domain=l,
description=(
"Equations defining the positive deviations with weight attached"
),
)
yPosWeightDef[l] = y[l] >= lambdaPos * (
Sum(i, P[i, l] * x[i]) - Sum(i, EP[i] * x[i])
)
yNegWeightDef[l] = y[l] >= lambdaNeg * (
Sum(i, EP[i] * x[i]) - Sum(i, P[i, l] * x[i])
)
MeanAbsoluteDeviationWeighted = Model(
m,
name="MeanAbsoluteDeviationWeighted",
equations=[BudgetCon, ReturnCon, yPosWeightDef, yNegWeightDef, ObjDef],
problem="LP",
sense="MIN",
objective=z,
)
output_csv += '"MADWeighted","Mean"\n'
mu_target = MIN_MU.records.value[0]
while mu_target <= MAX_MU.records.value[0]:
MU_TARGET[...] = mu_target
MeanAbsoluteDeviationWeighted.solve()
output_csv += f"{z.records.level.round(3)[0]},{round(MU_TARGET.records.value[0] * Budget.records.value[0],3)},"
x_recs = [str(x_rec) for x_rec in x.records.level.round(3).tolist()]
output_csv += ",".join(x_recs) + "\n"
mu_target += MU_STEP.records.value[0]
lambdaPos[...] = 0.2
lambdaNeg[...] = 0.8
output_csv += '"MADWeighted","Mean"\n'
mu_target = MIN_MU.records.value[0]
while mu_target <= MAX_MU.records.value[0]:
MU_TARGET[...] = mu_target
MeanAbsoluteDeviationWeighted.solve()
output_csv += f"{z.records.level.round(3)[0]},{round(MU_TARGET.records.value[0] * Budget.records.value[0],3)},"
x_recs = [str(x_rec) for x_rec in x.records.level.round(3).tolist()]
output_csv += ",".join(x_recs) + "\n"
mu_target += MU_STEP.records.value[0]
with open("MADvsMV.csv", "w", encoding="UTF-8") as FrontierHandle:
FrontierHandle.write(output_csv)
# Note that, the last two models will yield the same portfolios! See PFO Section 5.2.2.
if __name__ == "__main__":
main()
