# Sharpe model#

```"""
## GAMSSOURCE: https://www.gams.com/latest/finlib_ml/libhtml/finlib_Sharpe.html
## MODELTYPE: NLP
## DATAFILES: Sharpe.gdx

Sharpe model

Sharpe.gms: Sharpe model.
Consiglio, Nielsen and Zenios.
PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 3.3
"""

from __future__ import annotations

import os
from pathlib import Path

import gamspy.math as gams_math
import numpy as np
import pandas as pd
from gamspy import (
Alias,
Container,
Equation,
Model,
Problem,
Sense,
Sum,
Variable,
)

def main():
m = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
)

# SETS #
Assets = m.getSymbols(["subset"])[0]
ii = Alias(m, name="ii", alias_with=Assets)
j = Alias(m, name="j", alias_with=Assets)

# PARAMETERS #
RiskFreeRate, ExExpectedReturns, ExVarCov = m.getSymbols(
["MeanRiskFreeReturn", "MeanExcessRet", "ExcessCov"]
)

# VARIABLES #
x = Variable(
m,
name="x",
type="positive",
domain=ii,
description="Holdings of assets",
)
PortVariance = Variable(
m, name="PortVariance", description="Portfolio variance"
)
d_bar = Variable(
m, name="d_bar", description="Portfolio expected excess return"
)

# EQUATIONS #
ReturnDef = Equation(
m,
name="ReturnDef",
description="Equation defining the portfolio excess return",
)
VarDef = Equation(
m,
name="VarDef",
description="Equation defining the portfolio excess variance",
)
NormalCon = Equation(
m,
name="NormalCon",
description="Equation defining the normalization contraint",
)

ReturnDef[...] = d_bar == Sum(ii, ExExpectedReturns[ii] * x[ii])

VarDef[...] = PortVariance == Sum([ii, j], x[ii] * ExVarCov[ii, j] * x[j])

NormalCon[...] = Sum(ii, x[ii]) == 1

# Objective Function
ObjDef = d_bar / gams_math.sqrt(PortVariance)

# Put strictly positive bound on Variance to keep the model out of trouble:
PortVariance.lo[...] = 0.001

Sharpe = Model(
m,
name="Sharpe",
equations=[ReturnDef, VarDef, NormalCon],
problem=Problem.NLP,
sense=Sense.MAX,
objective=ObjDef,
)
Sharpe.solve()

print("Objective Function Variable: ", round(Sharpe.objective_value, 3))

current_port_variance = 0
results = []
while current_port_variance <= 1:
theta = np.sqrt(current_port_variance / PortVariance.records.level[0])
current_port_return = (
RiskFreeRate.records.value[0] + theta * d_bar.records.level[0]
)
results.append(
[np.sqrt(current_port_variance), current_port_return, theta]
)
current_port_variance += 0.1

# Also plot the tangent portfolio
theta = 1
results.append(
[
np.sqrt(PortVariance.records.level[0]),
RiskFreeRate.records.value[0] + theta * d_bar.records.level[0],
theta,
]
)
SharpeFrontier = pd.DataFrame(
results, columns=["Standard Deviations", "Expected Return", "Theta"]
)
SharpeFrontier.to_csv("SharpeFrontier.csv")

if __name__ == "__main__":
main()
```