"""
Portfolio horizon returns model
Horizon.gms: Portfolio horizon returns model.
Consiglio, Nielsen and Zenios.
PRACTICAL FINANCIAL OPTIMIZATION: A Library of GAMS Models, Section 4.3.1
Last modified: Apr 2008.
"""
from __future__ import annotations
import os
import numpy as np
import pandas as pd
from gamspy import Alias
from gamspy import Card
from gamspy import Container
from gamspy import Equation
from gamspy import Model
from gamspy import Number
from gamspy import Ord
from gamspy import Parameter
from gamspy import Set
from gamspy import Sum
from gamspy import Variable
def BondDataTable():
# Bond data. Prices, coupons and maturities from the Danish market
bond_data_recs = pd.DataFrame(
np.array([
[112.35, 2006, 8],
[105.33, 2003, 8],
[111.25, 2007, 7],
[107.30, 2004, 7],
[107.62, 2011, 6],
[106.68, 2009, 6],
[101.93, 2002, 6],
[101.30, 2005, 5],
[101.61, 2003, 5],
[100.06, 2002, 4],
]),
columns=["Price", "Maturity", "Coupon"],
index=[
"DS-8-06",
"DS-8-03",
"DS-7-07",
"DS-7-04",
"DS-6-11",
"DS-6-09",
"DS-6-02",
"DS-5-05",
"DS-5-03",
"DS-4-02",
],
)
bond_data_recs = bond_data_recs.reset_index().melt(
id_vars="index", var_name="Category", value_name="Value"
)
return bond_data_recs
def main():
m = Container(
system_directory=os.getenv("SYSTEM_DIRECTORY", None),
delayed_execution=int(os.getenv("DELAYED_EXECUTION", False)),
)
Time = Set(
m,
name="Time",
records=[str(time) for time in range(2001, 2012)],
description="Time periods",
)
t = Alias(m, name="t", alias_with=Time)
# SCALARS #
Now = Parameter(m, name="Now", description="Current year")
Horizon = Parameter(m, name="Horizon", description="End of the Horizon")
Now[...] = 2001
Horizon[...] = Card(t) - 1
# PARAMETER #
tau = Parameter(m, name="tau", domain=t, description="Time in years")
# Note: time starts from 0
tau[t] = Ord(t) - 1
# SET
Bonds = Set(
m,
name="Bonds",
records=[
"DS-8-06",
"DS-8-03",
"DS-7-07",
"DS-7-04",
"DS-6-11",
"DS-6-09",
"DS-6-02",
"DS-5-05",
"DS-5-03",
"DS-4-02",
],
description="Bonds universe",
)
i = Alias(m, name="i", alias_with=Bonds)
# SCALARS #
spread = Parameter(
m,
name="spread",
description="Borrowing spread over the reinvestment rate",
)
Budget = Parameter(m, name="Budget", description="Initial budget")
# PARAMETERS #
Price = Parameter(m, name="Price", domain=i, description="Bond prices")
Coupon = Parameter(m, name="Coupon", domain=i, description="Coupons")
Maturity = Parameter(
m, name="Maturity", domain=i, description="Maturities"
)
Liability = Parameter(
m, name="Liability", domain=t, description="Stream of liabilities"
)
rf = Parameter(m, name="rf", domain=t, description="Reinvestment rates")
F = Parameter(m, name="F", domain=[t, i], description="Cashflows")
# Bond data. Prices, coupons and maturities from the Danish market
BondData = Parameter(
m,
name="BondData",
domain=[i, "*"],
records=BondDataTable(),
description="Bonds data",
)
# Copy/transform data. Note division by 100 to get unit data, and
# subtraction of "Now" from Maturity date (so consistent with tau):
Price[i] = BondData[i, "Price"] / 100
Coupon[i] = BondData[i, "Coupon"] / 100
Maturity[i] = BondData[i, "Maturity"] - Now
# Calculate the ex-coupon cashflow of Bond i in year t:
F[t, i] = (
Number(1).where[tau[t] == Maturity[i]]
+ Coupon[i].where[(tau[t] <= Maturity[i]) & (tau[t] > 0)]
)
# For simplicity, we set the short term rate to be 0.03 in each period
rf[t] = 0.04
spread[...] = 0.02
# Initial available budget to buy the matching portfolio
Budget[...] = 803021.814
# 803021.814
# 850000
# PARAMETER #
Liability.setRecords(
np.array([
0,
80000,
100000,
110000,
120000,
140000,
120000,
90000,
50000,
75000,
150000,
])
)
# VARIABLES #
x = Variable(
m,
name="x",
type="positive",
domain=i,
description="Face value purchased",
)
surplus = Variable(
m,
name="surplus",
type="positive",
domain=t,
description="Amount of money reinvested",
)
borrow = Variable(
m,
name="borrow",
type="positive",
domain=t,
description="Amount of money borrowed",
)
HorizonRet = Variable(m, name="HorizonRet", description="Horizon Return")
# EQUATION #
CashFlowCon = Equation(
m,
name="CashFlowCon",
type="regular",
domain=t,
description="Equations defining the cashflow balance",
)
CashFlowCon[t] = (
Sum(i, F[t, i] * x[i])
+ (Budget - Sum(i, Price[i] * x[i])).where[tau[t] == 0]
+ borrow[t].where[tau[t] < Horizon]
+ ((1 + rf[t.lag(1)]) * surplus[t.lag(1)]).where[tau[t] > 0]
== Liability[t].where[tau[t] > 0]
+ surplus[t].where[tau[t] < Horizon]
+ HorizonRet.where[tau[t] == Horizon]
+ ((1 + rf[t.lag(1)] + spread) * borrow[t.lag(1)]).where[tau[t] > 0]
)
HorizonMod = Model(
m,
name="HorizonMod",
equations=[CashFlowCon],
problem="LP",
sense="MAX",
objective=HorizonRet,
)
HorizonMod.solve()
print("HorizonRet: ", round(HorizonRet.toValue(), 3))
print("borrow: ", borrow.toDict())
print("surplus: ", surplus.toDict())
print("x: ", x.toDict())
# Simulation for different values of the initial budget
HorizonHandle = open("HorizonPortfolios_new.csv", "w", encoding="UTF-8")
budget = 778985.948
while budget <= 818985.948:
Budget[...] = budget
HorizonMod.solve()
for ii in i.toList():
horizon_ret = round(HorizonRet.records.level[0], 2)
bond_mat = round(BondData.pivot().loc[ii, "Maturity"], 2)
coupon = Coupon.records[Coupon.records["i"] == ii].value.array[0]
purchased_price = round(
x.records[x.records["i"] == ii].level.array[0]
* Price.records[Price.records["i"] == ii].value.array[0],
3,
)
HorizonHandle.write(
f'{round(budget,2)},{horizon_ret},"{ii}",{bond_mat},{coupon},{purchased_price}'
)
HorizonHandle.write("\n")
for tt in t.toList():
borrow_rec = borrow.records[borrow.records["t"] == tt]
borrow_rec = (
round(borrow_rec.level.array[0], 3)
if not borrow_rec.empty
else 0
)
HorizonHandle.write(f'"{tt}",{borrow_rec}')
HorizonHandle.write("\n")
budget += 10000
HorizonHandle.close()
if __name__ == "__main__":
main()
