Exercises - Constrained Optimization

Exercises - Constrained Optimization#

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Data#

All the analysis below applies to the data set,

  • data/spx_weekly_returns.xlsx

  • The file has weekly returns.

  • For annualization, use 52 periods per year.

Consider only the following 10 stocks…

TICKS =  ['AAPL','NVDA','MSFT','GOOGL','AMZN','META','TSLA','AVGO','BRK/B','LLY']

As well as the ETF,

TICK_ETF = 'SPY'

Data Processing#

import pandas as pd

INFILE = '../data/spx_returns_weekly.xlsx'
SHEET_INFO = 's&p500 names'
SHEET_RETURNS = 's&p500 rets'
SHEET_BENCH = 'benchmark rets'

FREQ = 52

info = pd.read_excel(INFILE,sheet_name=SHEET_INFO)
info.set_index('ticker',inplace=True)
info.loc[TICKS]
name mkt cap
ticker
AAPL Apple Inc 3.008822e+12
NVDA NVIDIA Corp 3.480172e+12
MSFT Microsoft Corp 3.513735e+12
GOOGL Alphabet Inc 2.145918e+12
AMZN Amazon.com Inc 2.303536e+12
META Meta Platforms Inc 1.745094e+12
TSLA Tesla Inc 9.939227e+11
AVGO Broadcom Inc 1.148592e+12
BRK/B Berkshire Hathaway Inc 1.064240e+12
LLY Eli Lilly & Co 7.332726e+11 
rets = pd.read_excel(INFILE,sheet_name=SHEET_RETURNS)
rets.set_index('date',inplace=True)
rets = rets[TICKS]

bench = pd.read_excel(INFILE,sheet_name=SHEET_BENCH)
bench.set_index('date',inplace=True)
rets[TICK_ETF] = bench[TICK_ETF]

1 Constrained Optimization for Mean-Variance#

Continue working with the data above. Suppose we want to constrain the weights such that

  • there are no short positions beyond negative 20%, \(w_i\ge -.20\) for all \(i\)

  • none of the positions may have weight over 35%, \(w_i \le .35\) for all \(i\).

  • all the asset weights must sum to 1

Furthermore,

  • The targeted mean return is 20% per year.

  • Be careful; the target is an annualized mean.

Consider using the code below as a starting point.

1.1.#

Report the weights of the constrained portfolio.

Report the mean, volatility, and Sharpe ratio of the resulting portfolio.

1.2.#

Compare these weights to the assets’ Sharpe ratios and means.

Do the most extreme positions also have the most extreme Sharpe ratios and means?

Why?

1.3.#

Compare the bounded portfolio weights to the unbounded portfolio weights (obtained from optimizing without the inequality constraints, keeping the equality constraints.)

Report the mean, volatility, and Sharpe ratio of both.

Code Help#

The minimize function will be how we optimize.

from scipy.optimize import minimize

Build the objective functions.

Before doing this, you will need to define

  • TARGET_MEAN

  • FREQ

  • cov

  • mean

# def objective(w):        
#     return (w.T @ cov @ w)

# def fun_constraint_capital(w):
#     return np.sum(w) - 1

# def fun_constraint_mean(w):
#     return (mean @ w) - TARGET_MEAN

Build the constraints

  • sum of weights add to one

  • weighted average of means is the target mean

# constraint_capital = {'type': 'eq', 'fun': fun_constraint_capital}
# constraint_mean = {'type': 'eq', 'fun': fun_constraint_mean}

# constraints = ([constraint_capital, constraint_mean])

Build the upper and lower bounds on each asset.

You will need to use the minimize function along with these contraints, bounds, and an initial guess.

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