4. Markowitz

Question 1

What is the formula for portfolio return?

Answer 1

rp = wD * rD + wE * rE, where wE = 1 - wD

Question 2

What is the formula for portfolio expected return?

Answer 2

E(rp) = wD * E(rD) + wE * E(rE)

Question 3

What is the range of the correlation coefficient?

Answer 3

-1 ≤ ρ ≤ 1

Question 4

How is Sharpe ratio calculated?

Answer 4

S = (E(r) - rf) / σ

Question 5

What is the formula for portfolio variance with two assets?

Answer 5

σ²p = wD² * σ²D + wE² * σ²E + 2 * wD * wE * Cov(rD, rE)

Question 6

How do you calculate the variance of a portfolio with perfect positive correlation?

Answer 6

σP = wE * σE + wD * σD

Question 7

What is the formula for the minimum-variance portfolio variance for two assets?

Answer 7

σ²A = wD² * σ²D + (1 - wD)² * σ²E + 2 * wD * (1 - wD) * σD * σE * ρDE

Question 8

What does the Sharpe ratio measure in portfolio management?

Answer 8

It measures the risk-adjusted return of a portfolio.

Question 9

How is the optimal allocation to the risky portfolio determined?

Answer 9

y = (E(rP) - rf) / (A * σ²P)

Question 10

What happens to portfolio variance as the number of stocks (n) increases in naive diversification?

Answer 10

Portfolio variance decreases due to the averaging effect and reduced firm-specific risk.

Question 11

How does the correlation coefficient between two assets affect portfolio risk?

Answer 11

The lower the correlation coefficient, the greater the potential diversification benefit, reducing portfolio risk.

Question 12

Why is the minimum-variance frontier important in portfolio optimization?

Answer 12

It represents the set of portfolios with the lowest risk for a given level of return, helping investors optimize their choices.

Question 13

How do equally weighted portfolios differ in risk compared to optimally weighted portfolios?

Answer 13

Equally weighted portfolios may not minimize risk due to lack of consideration for individual asset variances and covariances, unlike optimally weighted portfolios.

Question 14

What role does the risk-free rate (rf) play in determining the optimal complete portfolio?

Answer 14

The risk-free rate is used to calculate the optimal proportion allocated to risky assets, balancing risk and return.

Question 15

How does a separation property simplify portfolio construction?

Answer 15

It allows investors to split portfolio decisions into identifying the optimal risky portfolio and choosing a mix of it with the risk-free asset based on individual risk tolerance.

Question 16

Explain how the Sharpe ratio guides the selection of the optimal risky portfolio.

Answer 16

The Sharpe ratio identifies the portfolio with the highest excess return per unit of risk, enabling investors to choose the portfolio that maximizes risk-adjusted performance.

Question 17

Analyze the impact of diversification on reducing firm-specific risk.

Answer 17

Diversification spreads investments across uncorrelated or negatively correlated assets, significantly reducing firm-specific risk, while market risk remains unchanged.

Question 18

Evaluate the effect of increasing the correlation coefficient on the minimum-variance portfolio.

Answer 18

As the correlation coefficient increases, the benefits of diversification diminish, leading to higher portfolio variance at the minimum-variance portfolio.

Question 19

Discuss why the opportunity set of debt and equity funds is not linear in the presence of imperfect correlation.

Answer 19

Imperfect correlation allows the combination of debt and equity to create curved opportunity sets, showing non-linear risk-return tradeoffs due to diversification benefits.

Question 20

How can the efficient frontier help in identifying the best portfolio for a risk-averse investor?

Answer 20

The efficient frontier identifies portfolios with the highest expected return for a given level of risk; risk-averse investors can choose a point that aligns with their tolerance.

Question 21

What is the goal of constructing an optimal risky portfolio?

Answer 21

To maximize the Sharpe ratio by finding the best combination of risky assets based on their risk-return characteristics and correlations.

Question 22

Why is diversification important in portfolio management?

Answer 22

Diversification reduces overall portfolio risk by combining assets that do not move perfectly in sync, lowering the impact of individual asset volatility.

Question 23

How does Markowitz’s problem approach portfolio optimization?

Answer 23

It uses expected returns, variances, and covariances to construct a portfolio that minimizes risk for a given return or maximizes return for a given risk.

Question 24

What is the significance of the minimum-variance portfolio?

Answer 24

It represents the portfolio with the lowest possible risk for a given set of assets, forming the starting point of the efficient frontier.

Question 25

How does correlation between assets affect portfolio variance?

Answer 25

Lower correlation reduces portfolio variance due to better diversification, while higher correlation results in less risk reduction.

Question 26

Why is the separation property useful for investors?

Answer 26

It simplifies portfolio construction by separating the decision into two parts: selecting the optimal risky portfolio and deciding the risk-free allocation based on individual preferences.

Question 27

What is the theoretical basis of the capital allocation line (CAL)?

Answer 27

CAL represents the risk-return combinations achievable by mixing a risk-free asset with a risky portfolio, guiding investors to their optimal allocation.

Question 28

How do risk preferences influence the choice of an optimal complete portfolio?

Answer 28

Investors with low risk aversion allocate less to risky assets, while those with higher risk tolerance allocate more, balancing their preferences with potential returns.

Question 29

Why does naive diversification not guarantee risk minimization?

Answer 29

It ignores asset correlations and weights equally, potentially resulting in higher risk than a portfolio optimized based on variance and covariance.

Question 30

What does the efficient frontier represent in portfolio theory?

Answer 30

It represents the set of portfolios that offer the maximum expected return for a given level of risk or the minimum risk for a given return, guiding optimal portfolio selection.

Question 31

How does the inclusion of a risk-free asset change the efficient frontier?

Answer 31

The efficient frontier becomes a straight line (the Capital Market Line, CML) when combined with a risk-free asset, representing the best risk-return combinations.

Question 32

Why is the tangency portfolio considered optimal on the CML?

Answer 32

The tangency portfolio maximizes the Sharpe ratio, representing the highest risk-adjusted return achievable when combining risky and risk-free assets.

Question 33

How does portfolio size impact diversification benefits?

Answer 33

Larger portfolios typically reduce firm-specific risk more effectively, as risk is spread across a greater number of uncorrelated or negatively correlated assets.

Question 34

What theoretical assumption underlies Markowitz portfolio optimization?

Answer 34

It assumes that investors are rational and risk-averse, aiming to maximize returns for a given level of risk or minimize risk for a desired return.

Question 35

How is the naive diversification strategy theoretically limited?

Answer 35

It oversimplifies risk reduction by ignoring asset correlations and variances, which are crucial for constructing an optimal portfolio.