Reference no: EM132817900
Problem 1: Portfolio Optimization
Suppose we want to design a portfolio among following assets:
StocksArray = c(' MSFTY AAPL',' AMZN',' FB',' GOOGY GOOGLY JNJ',' JPM',' VY PG',
'T',' UNH',' MA',' HD',' INTC',' V ZY KO',' BAC',' XOM',' MRK',' DIS',' PFE',
'PEP',' CMCSA',' CV X',' ADBE',' CSCO',' NVDA',' WMT',' NFLX',' CRM',
'WFC',' MCD',' ABT',' BMY',' COST',' BA',' C',' PM',' NEE',' MDT',' ABBV',
'AMGN',' TMO',' LLY',' HON',' ACN',' IBM')
The period we consider is from 01-01-2014 to 31-12-2020. In this problem, you are allowed to use any package.
(a) Please split the data set into training set and test set with 75% data in training set and 25% data in test set.
(b) Please use training data to construct the following portfolios:
• Uniform portfolio.
• Inverted volatility portfolio
• Quintile portfolio.
Apply them on test data, draw the cumulative return and drawdown. (your budget starts from 1 dollar and assuming risk-free rate is zero.)
(c) Estimation of mean vector and covariance matrix is a critical problem in portfolio optimization. Let ∑. be the sample covariance matrix. Instead of using ∑. to construct the portfolios, here we would use
∑. = ρ∑. + (1 - ρ) Trace(∑.)IN/N
where N is the number of asset, IN is the identity matrix of order N, and ρ is a real number between 0 and 1. Try different values of ρ in {0, 0.2, 0.4, 0.6, 0 8, 1}, construct the global minimum variance portfolio (GMVP)
minimize wT∑w
subject to 1Tw = 1, w ≥ 0, and compute the Sharpe ratio on test set.
(d) In this subproblem, instead of estimating E via sample covariance matrix, we assume E is one of M (known) covariance matrices ∑(k) ∈ Sn++, k =1,... , M, in which M = 10 and ∑(k) is computed as
∑(k) = k-1/M-1∑ + M-k Trace(∑) IN/N
where ∑ is the sample covariance matrix, N is the number of asset, and IN is the identity matrix of order N.
We will choose the portfolio weights in order to maximize the expected return, adjusted by the worst-case risk
minimize -μTw + λ maxk=1,...,M (wT∑(k)w)
subject to 1Tw = 1.
Use CVX to solve it, apply the portfolio on test set, then show the Sharpe ratio, annualized return and average drawdown for different λ ∈ {0, 1,4,10,20}.
Problem 2: Gradient Descent
Consider a linear factor model:
xt = α + βft + ∈t, t = 1, 2,... T,
where xt ∈ RN. We estimate α and β by solving the following least-squares problem via gradient descent
minimize ∑Tt=1 ||Xt - α - βft||2
α,β
For the vector of log-returns xt, we'll use
Stocks = c(MSFT' AAPL',' AMZN',' KB' ,GOOG')
starting from 2016-01-01 to 2019-12-31. For the factor ft we'll use the log-return of S&P500 index in the same period. Compare the estimates you obtained to that of CVX.
Hint The objective function is a function of α∈RN and β∈RN
So, there are 2N variables (α1, , αN, , β1,.....βN) to optimize. Given a point (αt, βt), a gradient descent step updates α and β as
![1176_figure.jpg](https://secure.expertsmind.com/CMSImages/1176_figure.jpg)
You should choose step size n and stopping criteria properly so that your algorithm converges to the optimal.
Need to use R language with CVXR package.
Attachment:- Portfolio Optimization.rar