Reference no: EM132833986

Problem 1: Practice with CVX

(a) Suppose we have the following optimization problem:

maximize Σ^m_i=1 -w_ilog w_i
subject to w ≥ 0, Σ^m_i=1 w_i = 1, X^Tw = b

where w ∈ R^m, X ∈ R^mxn and b ∈ Rⁿ. Given that m = 20 and n = 10, we can generate the data for X and b as follows:

m <-20
n <- 10
set . seed (22)
X matrix (rnorm(m*n) ,m, n)
b <- t (X) %*% as. vector ( rep (1 /m, m)) + rnorm(n, 0, 0.005)

Solve this problem, show the code and optimal value of w*.

(b) Solve the following optimization problem

minimize_{x∈R^m,y∈R^m} Σ^m_i=1 y_i

subject to x₁ = 0, x_m, = 1, y₁ = 1, y_m = 1,
(x_i+i - x_i)² + 2(y_i+1 - y_i)² ≤ 0.02², i = 1,.....,m - 1

Set m = 101, plot the curve of (x, y), which consists of a series of points (x₁, y₁), ,(x101, y101).

Problem 2: Portfolio optimization

In this problem, we aim at designing portfolios among following assets:

StocksArray = c('MSFT',' AAPL',' AMZN',' GOOGL',' JNJ',' JPM', 'UNH',' MA',' HD',' INTC','VZ',' K0',' BAC',' X0M',' MRK',' DIS', 'PEP',' CMCSA',' CV X',' ADBE',' CSCO',' NVDA','WMT',' NFLX', 'WFC',' MCD',' ABT',' BMY',' COST' BA',' C',' PM',' NEE',' MDT', 'AMGN','TM0',' LLY',' HON',' ACN',' IBM')

The period we consider is from 01-01-2010 to 31-12-2020. We will use the R package 'portfolioBacktest'. Please check the vignette before answering the questions.

(a) Prepare your data set for backtest. In R, you should create a specified list, resample it into 50 different sub-lists, each of which contains 20 stocks and 252 x 2 days' prices. Hint: use the function 'stockDataDownload' to get the data, then use 'financialDataResample' to resample the data set. (N_sample = 20,T-sample = 252 . 2, num_datasets = 25.)

data <- stockDataDownload ( stock _symbols = StocksArray )

(b) Implement Markowitz portfolio with max position constraint:

minimize_w -w^Tμ + λw^TΣw
subject to ||w||_∞ ≤ μ, 1^Tw = 1, w ≥ 0,

where μ and Σ can be estimated by sample mean and sample covariance matrix. Set λ = 2 and u = 0.04. Do the backtest, show the performance table and compare it with standard Markowitz portfolio (without max position constraint).

(c) Consider a portfolio formualted as

minimize_w -μ^Tw - ∈||W||_∞
subject to w ≥ 0, 1^Tw = 1,

where μ can be estimated by sample means. Try to determine the value of ∈, do the backtest, then show the performance table.

(d) Instead of using sample meanµ and sample covariance matrix Σ to construct the portfolio, here we would use

Σ = ρ₁Σ + (1 - ρ₁) trace(Σ)I_N/N

μ = ρ₂μ + (1 - ρ₂)1.^T_Nμ/N.1_N

where N is the number of asset and I_N is the identity matrix of order N. Use all possible pairs for ρ₁ and ρ₂ in {0, 0.25, 0.5, 0.75, 1), do the backtest for the mean-variance portfolio

minimize -w^Tμ, + λw^TΣw

subject to w ≥ 0, 1^Tw = 1,

under all possible combination of ρ1 and ρ2, ( λ= 0.01), show the performance table and tell which pair of (ρ1, ρ2) gives the best Sharpe Ratio performance.

(e) Implement the index tracking portfolio by simply regressing the dense index

minimize 1/T||r_b - Xb||²₂

subject to 1^Tb = 1, b ≥ 0.

Do the backtest and show the performance table. (Check the slide for the notations in this sub-problem.)

Problem 3: Gradient descent

Consider an unconstrained optimization problem:

minimize f(x) = e^{x₁+3x₂-0.1} + e^{x₁ -3x₂ -0.1} + e^{-x₁ -0.1}

Design a gradient descent method for solving it. You should clearly define the starting point, step size, and stopping criteria. Check whether your results converge to the optimal using CVX.