Homework 4
You have collected data on monthly return rates of 10 securities, the data is provided in excel file.
1 Portfolio Analysis and Efficient Frontier
This section deals with estimating expected return rates, covariance matrices, and constructing efficient frontiers under various constraints and scenarios.
1.1 Expected Return Rates and Covariance Matrix
Estimate the expected return rates and the covariance matrix of the return rates.
1.2 Efficient Frontier with Shorting Allowed
Determine the mean-variance efficient frontier, allowing for short selling. Em-ploy the Two-Fund Theorem, using the minimum variance portfolio as one of the funds.
1.3 Efficient Frontier without Shorting
Analyze the efficient frontier by calculating 10 points on it for the case when shorting is not allowed. Compare with the scenario where shorting is allowed.
1.4 Adding a Riskless Asset
Integrate a riskless asset returning 0.005 each month into the problem. Describe the efficient frontier by identifying the market portfolio according to the One-Fund Theorem.
1.5 Efficient Frontier without Shorting with a Riskless Asset
Examine the efficient frontier (by calculating 10 points) for the scenario when shorting is not allowed, including the riskless asset. Compare with the case where shorting is allowed alongside the riskless asset.
2 Portfolio Performance Evaluation
2.1 Value at Risk for Equally Likely Returns
Given 12 realizations as equally likely scenarios, evaluate three distinct portfolios considering an initial investment of $100,000.
• Portfolio 1: Investment solely in asset 9.
• Portfolio 2: Uniformly distributed investment across all assets.
• Portfolio 3: Equal investments in assets 2, 3, 6, 8, and 9, with no invest-ment in other assets.
Calculate the expected return and the Value at Risk (VaR) for three port-folios at risk levels α = 0.1, 0.2, and 0.3
2.2 Value at Risk for Normal Distribution
You assume that the data come from a joint normal distribution (you estimated the means and the covariances in part 1). For the same portfolios, compute the Value at Risk (VaR) at risk levels α = 0.1, 0.2, and 0.3.
2.3 Average Value at Risk
For the same portfolios, assuming equally likely scenarios, compute the Average Value at Risk (AVaR) at risk levels α = 0.1, 0.2, and 0.3.