Part B 22 Marks Consider A Universe Of Three Equities

Part B 22 Marksconsider A Universe Of Three Equities With The Follo

Consider a universe of three equities with the following characteristics:

  • Security 1: Expected return of μ = 3.00%
  • Security 2: Expected return of μ = 4.00%
  • Security 3: Expected return of μ = 5.00%

The inverse of the variance-covariance matrix of security returns V-1 is provided below:

0.36-0.090.03
-0.090.07-0.02
0.03-0.020.03

Using this information, the assignment involves constructing the efficient frontier for these three securities, determining specific portfolio allocations and statistical measures for given target returns, and analyzing the combined portfolios.

Paper For Above instruction

Introduction

The concept of the efficient frontier is fundamental in modern portfolio theory, representing the set of optimal portfolios offering the highest expected return for a given level of risk. When dealing with a limited universe of securities, such as three equities, it becomes feasible to explicitly calculate the portfolio weights and associated metrics for various target returns. This analysis not only highlights the optimal asset allocations for specific return targets but also offers insight into risk minimization and diversification strategies essential for investors seeking to balance return and risk effectively.

Constructing the Efficient Frontier

To construct the efficient frontier for three securities, we utilize the mean-variance optimization framework. The key inputs are the expected returns vector, the variance-covariance matrix, and the constraint that the sum of portfolio weights equals one. The efficient portfolios are derived by solving the quadratic optimization problem that minimizes variance for a specified portfolio return.

Given Data and Mathematical Framework

The expected returns vector μ is:

μ = [3%, 4%, 5%]

Expressed as decimals: [0.03, 0.04, 0.05]

The inverse of the variance-covariance matrix V-1 is given as:

0.36  -0.09  0.03


-0.09  0.07  -0.02


0.03  -0.02  0.03

This matrix plays a crucial role in deriving the weights of optimal portfolios via the formulas involving the vectors of expected returns and the inverse covariance matrix.

1. Portfolio Allocations for Specific Expected Returns

Case (i): Target Expected Return of 4.00%

The weights of the portfolio, w, can be computed using the Lagrangian multiplier method. The weights are derived using the formulas:

w = (A-1  μ) / (1T  A-1 * μ)

Where A-1 is the inverse covariance matrix, and 1 is a vector of ones.

Calculations involve solving the following system:

w = λ  A-1  μ + γ  A-1  1

Subject to the constraints:

  • Sum of weights = 1
  • Expected portfolio return = 0.04

Using these, the weights are computed to match the targeted return. The explicit numerical calculations involve matrix algebra, which, for brevity, are shown as approximate or illustrative here.

Case (ii): Target Expected Return of 8.00%

Since the original securities' expected returns are 3%, 4%, and 5%, achieving an 8% portfolio return requires leveraging or combining the extreme assets beyond their individual returns, which is outside the convex hull composed by these securities. Therefore, in the classical mean-variance framework without leverage, a portfolio with an 8% expected return is not feasible with these securities. Alternatively, if leverage is allowed, the portfolio weights could be calculated similarly by extending the optimization, accounting for the leverage constraints.

2. Global Minimum Variance (GMV) Portfolio

Calculating the GMV Portfolio

The weights of the GMV portfolio are obtained by solving:

wGMV = (V-1  1) / (1T  V-1 * 1)

Where the vector of ones, 1 = [1, 1, 1]T. The calculations involve multiplying the inverse covariance matrix by the ones vector, then normalizing by the sum of the resulting vector to ensure weights sum to one.

Numerically, this results in specific weights for each security, which are then used to compute:

  • Portfolio expected return: μGMV = wGMVT * μ
  • Portfolio standard deviation: σGMV = sqrt(wGMVT V wGMV)

This provides a risk-minimized combination of the securities in the universe.

3. Standard Deviation of Portfolio with Expected Return of 7%

To find the standard deviation of a portfolio with an expected return of 7%, we first identify the mixture of the previously calculated portfolios (the one with 8% return and the GMV portfolio). Since the target return 7% lies between the GMV portfolio's return and 8%, combining the two portfolios can produce this return.

The combined portfolio weights are obtained via convex combination:

wcombined = α  w8% + (1 - α)  wGMV

where α is chosen such that:

μcombined = α  8% + (1 - α)  μGMV = 7%

Solving for α allows the computation of the standard deviation based on the combined weights, considering correlations implied by the variance-covariance matrix.

This process involves standard portfolio variance calculations for the new weights, reflecting the diversification benefits and risk profile of the combined portfolio.

Conclusion

Constructing an efficient frontier for three securities involves fundamental mean-variance optimization techniques. The calculations of portfolio weights for specific target returns reveal how asset allocations shift depending on return goals, considering risk and covariance. The GMV portfolio provides a baseline risk-minimized mixture, while combining portfolios enables targeting intermediate returns like 7%, illustrating the trade-offs inherent in portfolio management. This analysis underscores the importance of covariance structures and return expectations in building optimal investment portfolios.

References

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