Portfolio Optimization

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Questions and Answers

What is the constraint in the minimization problem to find the optimal portfolio weights?

wB = 1 - wA (correct)

wA + wB = 1

wA / wB = 1

wA - wB = 0

What is the formula for the minimum variance portfolio (MV) for two assets?

σB2 - ρAB σA σB

σA2 + σB2 + 2ρAB σA σB

σA2 + σB2 - 2ρAB σA σB (correct)

σA2 - σB2 + ρAB σA σB

What is the slope of the capital allocation line (CAL) that connects the risk-free asset and any other risky asset X?

(E(rX) + rF) / σX

(E(rX) + rF) / σP

(E(rX) - rF) / σP

(E(rX) - rF) / σX (correct)

What is the Sharpe ratio formula?

(E(rP) - rF) / σP Signup and view all the answers

What is the point that separates the efficient and inefficient portfolios?

Minimum variance portfolio Signup and view all the answers

What is the capital allocation line?

A straight line that connects the efficient frontier and the risk-free asset Signup and view all the answers

What is the expected return and standard deviation of the minimum variance portfolio?

10% and 18% Signup and view all the answers

What is the correlation between assets S and R?

0 Signup and view all the answers

What is the optimal portfolio of risky assets dependent on according to the two-fund theorem?

Investor's risk preference Signup and view all the answers

What is the same for two investors sharing the same perceptions as to expected returns, variances, and return correlations of assets, but differing in their willingness to take risks?

Efficient frontier Signup and view all the answers

What is the purpose of the Taylor approximation in generating indifference curves?

To approximate the utility function Signup and view all the answers

What does the allocation between the risk-free and the tangency portfolio depend on?

Risk preferences of the investor Signup and view all the answers

What is the efficient frontier a part of?

Mean-Variance diagram Signup and view all the answers

What is the term for the portfolio that is the optimal combination of risky assets?

Optimal portfolio Signup and view all the answers

What is used to generate the indifference curves?

Taylor approximation Signup and view all the answers

What is the purpose of the Capital Allocation Line?

To allocate assets between risk-free and tangency portfolio Signup and view all the answers

What is the purpose of finding the tangency portfolio?

To determine the optimal weights of a portfolio Signup and view all the answers

What is the formula to find the optimal weight of asset A in the tangency portfolio?

[E (rA ) − rf ] σB2 − [E (rB ) − rf ] ρAB σA σB Signup and view all the answers

What is the efficient frontier?

A curve that shows the optimal portfolios with the highest Sharpe ratios Signup and view all the answers

What is the Sharpe ratio?

A measure of the reward-to-risk ratio of a portfolio Signup and view all the answers

What is the purpose of the capital allocation line?

To find the capital allocation between the risk-free asset and the tangency portfolio Signup and view all the answers

What is the point of tangency on the efficient frontier?

The point where the capital allocation line is tangent to the efficient frontier Signup and view all the answers

What is the formula to find the weight of asset B in the tangency portfolio?

1 - wA,T Signup and view all the answers

What is the condition for finding the optimal weights of the tangency portfolio?

wA,T + wB,T = 1 Signup and view all the answers

Study Notes

Minimum Variance Portfolio

The minimum variance portfolio can be found by solving the minimization problem: min wA2 σA2 + wB2 σB2 + 2wA wB · ρAB · σA σB subject to wB = 1 − wA
The solution to this problem is wA,MV = (σB2 − ρAB σA σB) / (σA2 + σB2 − 2ρAB σA σB)

Capital Allocation Line (CAL)

A straight line connecting the risk-free asset to any point along the efficient frontier represents a risk-reward tradeoff involving a particular mix of assets
The slope of the CAL is E (rX ) − rf / σP, also known as the Sharpe ratio

Tangency Portfolio

The tangency portfolio is the point of tangency on the efficient frontier
The optimal weights of the tangency portfolio can be found by solving the maximization problem: max (E (rT ) − rf ) / σT subject to wB = 1 − wA
The solution to this problem is wA,T = [E (rA ) − rf ] σB2 − [E (rB ) − rf ] ρAB σA σB / ([E (rA ) − rf ] σB2 + [E (rB ) − rf ] σA2 − [E (rA ) − rf + E (rB ) − rf ] ρAB σA σB)

Example Problem

Find the weights of the tangency portfolio given the following assets: Risk-free (6%, 0), Risky S (14%, 20%), Risky R (8%, 15%), with a correlation of 0 between S and R

Separation Theorem

The efficient frontier is the same for all investors, but the proportions invested in the available assets will be different
The two-fund theorem or separation theorem states that the optimal portfolio of risky assets can be identified separately from an investor's risk preference

Investor Utility Functions

Investors with different risk preferences can have different indifference curves
The allocation between the risk-free and the tangency portfolio depends on the risk preferences of the investor

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Description

Optimize portfolio by minimizing variance. Learn how to calculate efficient and inefficient portfolios with given formulas and variables.