Case-Shiller Example

This is an example that shows that Cvxportfolio can work as well with different asset classes, in this case a real estate portfolio. We use the Case-Shiller index as proxy for the price of housing units in various metropolitan areas in the USA. We impose realistic transaction costs, which are comparable to the annual return on the asset, and we show that multi-period optimization is useful to correctly balance transaction cost and expected risk-adjusted return. This example script is available in the repository.

import os

import cvxportfolio as cvx
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# We test the Multi-Period Optimization model on a real estate
# portfolio for which the transaction cost are large with respect
# to the typical returns.
# We use the Case-Shiller indices as proxies for (residential) real
# estate prices.

# These are monthly time serieses of the Case-Shiller index for 
# a selection of US metropolitan areas. You can find them on the
# website of FRED https://fred.stlouisfed.org/
UNIVERSE = [
    'SFXRSA', # San Francisco
    'LXXRSA', # Los Angeles
    'SEXRSA', # Seattle
    'DAXRSA', # Dallas
    'SDXRSA', # San Diego
    'MIXRSA', # Miami
    'PHXRSA', # Phoenix
    'NYXRSA', # New York
    'CHXRSA', # Chicago
    'ATXRSA', # Atlanta
    'LVXRSA', # Las Vegas
    'POXRSA', # Portland
    'WDXRSA', # Washington D.C.
    'TPXRSA', # Tampa
    'CRXRSA', # Charlotte
    'MNXRSA', # Minneapolis
    'DEXRSA', # Detroit
    'CEXRSA', # Cleveland
    'DNXRSA', # Denver
    'BOXRSA', # Boston
    ]
    
# we assume that the cost of transacting 
# residential real estate is about 5%
LINEAR_TCOST = 0.05

simulator = cvx.MarketSimulator(
    universe = UNIVERSE,
    # we enabled the default data interface to download index
    # prices from FRED
    datasource='Fred', 
    costs = [cvx.TransactionCost(
        a=LINEAR_TCOST,
        b=None, # since we don't have market volumes, we can't use the
                # market impact term of the transaction cost model
        )]
    )

# let's see what a uniform allocation does
result_uniform = simulator.backtest(cvx.Uniform())
print('BACK-TEST RESULT OF UNIFORM (1/N) ALLOCATION')
print(result_uniform)

result_uniform.plot()
if 'CVXPORTFOLIO_SAVE_PLOTS' in os.environ:
    plt.savefig('case_shiller_uniform.png')

# These are risk model coefficients. They don't seem to have
# a strong effect on this example.
NUM_FACTORS = 5
kappa = 0.1

# This is the multi-period planning horizon. We plan for 6 months 
# in the future.
HORIZON = 6

policies = []

# sweep over risk aversion
for gamma_risk in np.logspace(0, 3, 10):
    policies.append(cvx.MultiPeriodOptimization(
        cvx.ReturnsForecast() - gamma_risk * (
        cvx.FactorModelCovariance(num_factors=NUM_FACTORS) 
        + kappa * cvx.RiskForecastError())
             - cvx.TransactionCost(a=LINEAR_TCOST, b=None), 
                 [cvx.LongOnly(applies_to_cash=True)], 
                     planning_horizon=HORIZON,
             )
        )

results = simulator.backtest_many(policies)

print('BACK-TEST RESULT OF MPO WITH HIGHEST (OUT-OF-SAMPLE) PROFIT')
print(results[np.argmax([el.profit for el in results])])

results[np.argmax([el.profit for el in results])].plot()
if 'CVXPORTFOLIO_SAVE_PLOTS' in os.environ:
    plt.savefig('case_shiller_highest_profit.png')

# Plot
fig = plt.figure()
plt.plot(
    [result.excess_returns.std() * np.sqrt(12) for result in results],
    [result.excess_returns.mean() * 12 for result in results],
    'r*-',
    label='Multi-period optimization frontier'
    )
plt.scatter([result_uniform.excess_returns.std() * np.sqrt(12)], 
    [result_uniform.excess_returns.mean() * 12],
    label='Uniform (1/n) allocation'
    )
plt.legend()
plt.title('Back-Test Result (Out-Of-Sample) for Real Estate Portfolio')
plt.xlabel('Excess risk (annualized)')
plt.ylabel('Excess return (annualized)')

if 'CVXPORTFOLIO_SAVE_PLOTS' in os.environ:
    fig.savefig('case_shiller_frontier.png')
else:
    plt.show()

This is the output printed to screen when executing this script. You can see many statistics of the back-tests.

Updating data....................
BACK-TEST RESULT OF UNIFORM (1/N) ALLOCATION

#################################################################
Universe size                                                  21
Initial timestamp                       1988-01-01 00:00:00+00:00
Final timestamp                         2023-08-01 00:00:00+00:00
Number of periods                                             428
Initial value (USDOLLAR)                                1.000e+06
Final value (USDOLLAR)                                  3.880e+06
Profit (USDOLLAR)                                       2.880e+06
                                                                 
Avg. return (annualized)                                     3.8%
Volatility (annualized)                                      2.5%
Avg. excess return (annualized)                              0.8%
Excess volatility (annualized)                               2.7%
                                                                 
Avg. growth rate (annualized)                                3.8%
Avg. excess growth rate (annualized)                         0.8%
                                                                 
Avg. TransactionCost                                          4bp
Max. TransactionCost                                        500bp
                                                                 
Sharpe ratio                                                 0.30
                                                                 
Avg. drawdown                                               -6.8%
Min. drawdown                                              -36.1%
Avg. leverage                                               99.8%
Max. leverage                                              105.3%
Avg. turnover                                                0.4%
Max. turnover                                               50.0%
                                                                 
Avg. policy time                                           0.000s
Avg. simulator time                                        0.002s
Total time                                                 0.686s
#################################################################

BACK-TEST RESULT OF MPO WITH HIGHEST (OUT-OF-SAMPLE) PROFIT

#################################################################
Universe size                                                  21
Initial timestamp                       1988-01-01 00:00:00+00:00
Final timestamp                         2023-08-01 00:00:00+00:00
Number of periods                                             428
Initial value (USDOLLAR)                                1.000e+06
Final value (USDOLLAR)                                  4.865e+06
Profit (USDOLLAR)                                       3.865e+06
                                                                 
Avg. return (annualized)                                     4.5%
Volatility (annualized)                                      3.6%
Avg. excess return (annualized)                              1.5%
Avg. active return (annualized)                              1.5%
Excess volatility (annualized)                               3.8%
Active volatility (annualized)                               3.8%
                                                                 
Avg. growth rate (annualized)                                4.4%
Avg. excess growth rate (annualized)                         1.4%
Avg. active growth rate (annualized)                         1.4%
                                                                 
Avg. TransactionCost                                          1bp
Max. TransactionCost                                        300bp
                                                                 
Sharpe ratio                                                 0.40
Information ratio                                            0.40
                                                                 
Avg. drawdown                                              -12.2%
Min. drawdown                                              -42.2%
Avg. leverage                                               97.8%
Max. leverage                                              100.1%
Avg. turnover                                                0.1%
Max. turnover                                               30.0%
                                                                 
Avg. policy time                                           0.010s
Avg. simulator time                                        0.003s
Total time                                                 5.375s
#################################################################

And these are the figure that are plotted. The result of the cvxportfolio.Uniform policy, which allocates equal weight to all non-cash assets:

This figure is made by the plot() method of cvxportfolio.BacktestResult

And result of the cvxportfolio.MultiPeriodOptimization policy, selected among the efficient frontier below as the one with highest back-tested profit:

This figure is made by the plot() method of cvxportfolio.BacktestResult

And finally, the efficient frontier, which shows that the cvxportfolio.MultiPeriodOptimization policies out-perform the cvxportfolio.Uniform allocation in both risk and reward, including the transaction costs.

Efficient frontier of back-test results, which include transaction costs.