4. Markowitz Flashcards
What is the formula for portfolio return?
rp = wD * rD + wE * rE, where wE = 1 - wD
What is the formula for portfolio expected return?
E(rp) = wD * E(rD) + wE * E(rE)
What is the range of the correlation coefficient?
-1 ≤ ρ ≤ 1
How is Sharpe ratio calculated?
S = (E(r) - rf) / σ
What is the formula for portfolio variance with two assets?
σ²p = wD² * σ²D + wE² * σ²E + 2 * wD * wE * Cov(rD, rE)
How do you calculate the variance of a portfolio with perfect positive correlation?
σP = wE * σE + wD * σD
What is the formula for the minimum-variance portfolio variance for two assets?
σ²A = wD² * σ²D + (1 - wD)² * σ²E + 2 * wD * (1 - wD) * σD * σE * ρDE
What does the Sharpe ratio measure in portfolio management?
It measures the risk-adjusted return of a portfolio.
How is the optimal allocation to the risky portfolio determined?
y = (E(rP) - rf) / (A * σ²P)
What happens to portfolio variance as the number of stocks (n) increases in naive diversification?
Portfolio variance decreases due to the averaging effect and reduced firm-specific risk.
How does the correlation coefficient between two assets affect portfolio risk?
The lower the correlation coefficient, the greater the potential diversification benefit, reducing portfolio risk.
Why is the minimum-variance frontier important in portfolio optimization?
It represents the set of portfolios with the lowest risk for a given level of return, helping investors optimize their choices.
How do equally weighted portfolios differ in risk compared to optimally weighted portfolios?
Equally weighted portfolios may not minimize risk due to lack of consideration for individual asset variances and covariances, unlike optimally weighted portfolios.
What role does the risk-free rate (rf) play in determining the optimal complete portfolio?
The risk-free rate is used to calculate the optimal proportion allocated to risky assets, balancing risk and return.
How does a separation property simplify portfolio construction?
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.
Explain how the Sharpe ratio guides the selection of the optimal risky portfolio.
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.
Analyze the impact of diversification on reducing firm-specific risk.
Diversification spreads investments across uncorrelated or negatively correlated assets, significantly reducing firm-specific risk, while market risk remains unchanged.
Evaluate the effect of increasing the correlation coefficient on the minimum-variance portfolio.
As the correlation coefficient increases, the benefits of diversification diminish, leading to higher portfolio variance at the minimum-variance portfolio.
Discuss why the opportunity set of debt and equity funds is not linear in the presence of imperfect correlation.
Imperfect correlation allows the combination of debt and equity to create curved opportunity sets, showing non-linear risk-return tradeoffs due to diversification benefits.
How can the efficient frontier help in identifying the best portfolio for a risk-averse investor?
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.
What is the goal of constructing an optimal risky portfolio?
To maximize the Sharpe ratio by finding the best combination of risky assets based on their risk-return characteristics and correlations.
Why is diversification important in portfolio management?
Diversification reduces overall portfolio risk by combining assets that do not move perfectly in sync, lowering the impact of individual asset volatility.
How does Markowitz’s problem approach portfolio optimization?
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.
What is the significance of the minimum-variance portfolio?
It represents the portfolio with the lowest possible risk for a given set of assets, forming the starting point of the efficient frontier.
How does correlation between assets affect portfolio variance?
Lower correlation reduces portfolio variance due to better diversification, while higher correlation results in less risk reduction.
Why is the separation property useful for investors?
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.
What is the theoretical basis of the capital allocation line (CAL)?
CAL represents the risk-return combinations achievable by mixing a risk-free asset with a risky portfolio, guiding investors to their optimal allocation.
How do risk preferences influence the choice of an optimal complete portfolio?
Investors with low risk aversion allocate less to risky assets, while those with higher risk tolerance allocate more, balancing their preferences with potential returns.
Why does naive diversification not guarantee risk minimization?
It ignores asset correlations and weights equally, potentially resulting in higher risk than a portfolio optimized based on variance and covariance.
What does the efficient frontier represent in portfolio theory?
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.
How does the inclusion of a risk-free asset change the efficient frontier?
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.
Why is the tangency portfolio considered optimal on the CML?
The tangency portfolio maximizes the Sharpe ratio, representing the highest risk-adjusted return achievable when combining risky and risk-free assets.
How does portfolio size impact diversification benefits?
Larger portfolios typically reduce firm-specific risk more effectively, as risk is spread across a greater number of uncorrelated or negatively correlated assets.
What theoretical assumption underlies Markowitz portfolio optimization?
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.
How is the naive diversification strategy theoretically limited?
It oversimplifies risk reduction by ignoring asset correlations and variances, which are crucial for constructing an optimal portfolio.
