Asset Allocation: Matrix Algebra Representation

Given a portfolio (x) composed of assets (A), (B), and (C), and assuming a Certainty Equivalent Return (CER) model, which of the following statements about the portfolio return (R_{p,x}) is most accurate, considering the implications of matrix algebra?

In matrix algebra representation of portfolio optimization, if (x) is a vector of portfolio weights, and (\mathbf{1}) is a conformable vector of ones, the constraint (x'\mathbf{1} = 1) ensures that short selling is strictly prohibited, and all portfolio weights must be non-negative and sum to one.

Formulate the Lagrangian for determining the global minimum variance portfolio (m), given the covariance matrix (\Sigma), and the constraint that the portfolio weights sum to one. Explicitly denote all terms, including the Lagrangian multiplier.

In the context of mean-variance portfolio optimization, the efficient frontier represents a set of portfolios that offer the ________ expected return for a given level of ________, or conversely, the ________ level of risk for a given expected return.

Match the following matrix algebra expressions with their corresponding portfolio optimization interpretations:

Given the alternative derivation of the global minimum variance portfolio, where (m = -\frac{1}{2}\lambda\Sigma^{-1}\mathbf{1}) and (\lambda) is a Lagrange multiplier enforcing (\mathbf{1}'m = 1), deduce the most accurate implication regarding the relationship between the portfolio weights and the inverse of the covariance matrix.

In the context of Markowitz's efficient frontier, the problem of finding a portfolio (x) that minimizes portfolio variance (\sigma_{p,x}^2) subject to achieving a target expected return (\mu_{p}^0) and the budget constraint can always be solved analytically, irrespective of the number of assets or the structure of the covariance matrix.

Describe the process for composing efficient portfolios as convex combinations of two frontier portfolios, (x) and (y), detailing how the resulting portfolio (z) is formulated through the allocation parameter (\alpha).

In computing the tangency portfolio, the objective is to maximize the ________ ratio, which represents the excess expected return per unit of ________.

Match the following concepts with their corresponding mathematical representation in portfolio optimization:

Given a scenario where the first-order conditions for the tangency portfolio are being solved via Lagrangian methods, and assuming the Lagrangian multiplier (\lambda) has been determined, what inferential step is most critical for completing the determination of the tangency portfolio composition, assuming a risk-free rate (r_f)?

If assets (A), (B), and (C) are perfectly positively correlated, then any portfolio constructed from these assets will always lie on the efficient frontier, regardless of the weights assigned to each asset.

Suppose you are given two frontier portfolios, (x) and (y), with known expected returns and variances. Explain how one would compute the expected return and variance of a portfolio (z) that is a convex combination of (x) and (y), highlighting the role of the covariance between (x) and (y) in determining the variance of (z).

The analytic solution for the global minimum variance portfolio involves solving a system of linear equations derived from the first-order conditions of the Lagrangian. If (A_m) represents the matrix containing the covariance matrix and constraint coefficients, and (z_m) represents the vector of portfolio weights and the Lagrange multiplier, then the solution can be expressed as (z_m =) ________.

Match the following matrix algebra concepts with their application in portfolio optimization when constructing efficient portfolios using the Markowitz algorithm.

In computing the tangency portfolio, what specific role does the risk-free rate play in determining the final portfolio composition, assuming the investor aims to maximize the Sharpe ratio?

Assuming that portfolio returns are normally distributed, diversifying a portfolio across a large number of uncorrelated assets will completely eliminate portfolio variance, leading to a risk-free investment.

For a portfolio consisting of assets (A), (B), and (C), represented by weights (x_A), (x_B), and (x_C) respectively, explain how the constraint (x_A + x_B + x_C = 1) is expressed and implemented within the matrix algebra framework of portfolio optimization.

In the context of portfolio optimization, the global minimum variance portfolio is characterized by the ________ possible variance among all feasible portfolios, irrespective of the ________ returns.

Associate each component of the Lagrangian function for the Tangency Portfolio with its corresponding role in the portfolio optimization problem:

Assuming that the risk-free rate is altered exogenously, deduce the implications regarding adjustments to the allocation weights within both the tangency portfolio (t) and a complete portfolio (c) that combines the tangency portfolio and the risk-free asset.

When constructing the efficient frontier using the Markowitz model, if one asset has a demonstrably higher Sharpe ratio than all other assets, then, absent constraints on short selling, the efficient frontier will consist solely of portfolios fully invested in this single asset.

Given two efficient portfolios, (x) and (y), with differing expected returns and variances, describe the rationale behind formulating a portfolio (z) as a convex combination of (x) and (y), explicitly referencing the properties of the resulting portfolio's expected return, variance, and location on the efficient frontier.

The process of using Excel's Solver to determine optimal portfolio weights can be viewed as a ________ method, contrasting with the ________ solutions derived directly from matrix algebra.

Match each method or reference to its appropriate usage concerning portfolio construction and analysis:

Given a scenario where diversification across multiple assets demonstrably fails to enhance the Sharpe ratio beyond that attainable from a single asset, deduce the most plausible explanation, assuming non-restrictive short-selling constraints.

In portfolio optimization, increasing the number of assets in a portfolio invariably leads to a superior Sharpe ratio, provided that transaction costs are negligible and short selling is permitted without constraints.

Assuming two assets, (X) and (Y), exhibit a perfect negative correlation, delineate how an investor can construct a risk-free portfolio by determining the appropriate allocation weights, explicitly addressing the implications for hedging and risk management.

The efficient portfolios from mean-variance optimisation all plot a curve called the ________ ________

Match the concept with their expression:

In the three asset example where you create a portfolio (x) from Microsoft (A), Nordstrom (B) and Starbucks (C), with a return of Rp,x = xARA + xBRB + xCRC, what would happen if Nordstrom becomes highly correlated to Microsoft?