3 - Risky and Risk-free asset allocation

Question 1

What is the formula for utility in terms of expected return and risk?

Answer 1

U = E(r) - (1/2) * A * σ², where U is utility, E(r) is the expected return, A is the investor’s risk aversion coefficient, and σ² is the variance of the portfolio return.

Question 2

How do you calculate the expected return of a two-asset portfolio?

Answer 2

E(rC) = ω₁ * E(r₁) + ω₂ * E(r₂), where E(rC) is the expected return of the portfolio, ω₁ and ω₂ are the weights of assets 1 and 2 in the portfolio, and E(r₁) and E(r₂) are their respective expected returns.

Question 3

What is the variance formula for a two-asset portfolio?

Answer 3

σ²_C = ω₁² * σ₁² + ω₂² * σ₂² + 2 * ω₁ * ω₂ * σ₁ * σ₂ * ρ₁₂, where σ²_C is the portfolio variance, σ₁² and σ₂² are the variances of assets 1 and 2, ω₁ and ω₂ are the asset weights, and ρ₁₂ is the correlation between the returns of the two assets.

Question 4

How do you express the expected return of a portfolio with a risky and a risk-free asset?

Answer 4

E(rC) = y * E(rP) + (1 - y) * rf, where E(rC) is the expected return of the complete portfolio, y is the proportion allocated to the risky asset, E(rP) is the expected return of the risky portfolio, and rf is the risk-free rate.

Question 5

How is the expected return of a complete portfolio calculated using the capital allocation line (CAL)?

Answer 5

E(rC) = rf + y * [E(rP) - rf], where rf is the risk-free rate, y is the allocation to the risky portfolio, E(rP) is the expected return of the risky portfolio, and E(rC) is the expected return of the complete portfolio.

Question 6

What is the formula for the portfolio risk (standard deviation) with risky and risk-free assets?

Answer 6

σ_C = y * σ_P, where σ_C is the standard deviation of the complete portfolio, y is the proportion allocated to the risky asset, and σ_P is the standard deviation (risk) of the risky portfolio.

Question 7

What is the slope of the Capital Allocation Line (CAL)?

Answer 7

Slope (CAL) = (E(rP) - rf) / σ_P, where E(rP) is the expected return of the risky portfolio, rf is the risk-free rate, and σ_P is the standard deviation of the risky portfolio.

Question 8

How can you express the expected return of a complete portfolio using the slope of the CAL?

Answer 8

E(rC) = rf + Slope * σ_C, where E(rC) is the expected return of the complete portfolio, rf is the risk-free rate, Slope is the slope of the CAL, and σ_C is the portfolio’s standard deviation.

Question 9

How is the optimal allocation to a risky asset calculated?

Answer 9

y* = (E(rP) - rf) / (A * σ²_P), where y* is the optimal proportion of the portfolio allocated to the risky asset, E(rP) is the expected return of the risky portfolio, rf is the risk-free rate, A is the risk aversion coefficient, and σ²_P is the variance of the risky portfolio.

Question 10

What is the formula for utility in terms of expected return and variance?

Answer 10

U = E(r) - (1/2) * A * σ², where U is utility, E(r) is the expected return, A is the investor’s risk aversion, and σ² is the variance of the portfolio’s return.

Question 11

How do you express the expected return in terms of utility and risk?

Answer 11

E(r) = U + (1/2) * A * σ², where E(r) is the expected return, U is utility, A is the risk aversion coefficient, and σ² is the variance of the portfolio.

Question 12

What is the formula for the Sharpe ratio?

Answer 12

Sharpe Ratio = (E(rP) - rf) / σ_P, where E(rP) is the expected return of the risky portfolio, rf is the risk-free rate, and σ_P is the standard deviation of the risky portfolio (risk).

Question 13

What does the first-order condition (FOC) for optimal portfolio selection represent?

Answer 13

It represents the point where marginal utility equals marginal risk. (y*)

Question 14

What does a risk-averse investor prefer in terms of utility?

Answer 14

Higher utility with lower risk.

Question 15

How do risk-neutral investors value risk?

Answer 15

They are indifferent to risk and focus only on expected return.

Question 16

What does the Mean-Variance Criterion state?

Answer 16

Investors should choose portfolios with the highest expected return for a given level of risk.

Question 17

How is the capital market line (CML) different from the CAL?

Answer 17

CML uses the market portfolio instead of a single risky asset.

Question 18

What is the primary measure of risk used in value at risk (VaR)?

Answer 18

VaR measures the potential loss over a specific time frame at a given confidence level.

Question 19

What does Expected Shortfall (ES) measure in risk management?

Answer 19

ES measures the average loss in scenarios that exceed the VaR threshold.

Question 20

What are Black Swan events?

Answer 20

Extremely rare, unpredictable events with massive impact.

Question 21

What happens to the utility if the investor’s risk aversion (A) increases?

Answer 21

Utility decreases as risk aversion increases, given the same risk level.

Question 22

How does the variance of a two-asset portfolio change if the correlation (ρ₁₂) between the assets is -1?

Answer 22

The portfolio can be fully hedged, reducing the variance to zero.

Question 23

How does increasing the proportion of the risky asset (y) affect the standard deviation (σ_C) of the complete portfolio?

Answer 23

The standard deviation increases as y increases, leading to higher risk.

Question 24

How does the Sharpe ratio influence the slope of the Capital Allocation Line (CAL)?

Answer 24

A higher Sharpe ratio results in a steeper slope, indicating a better risk-return trade-off.

Question 25

Why does a risk-neutral investor set A (risk aversion coefficient) to zero?

Answer 25

A risk-neutral investor is indifferent to risk, so they don’t penalize variance in the utility function.

Question 26

How does increasing the risk-free rate (rf) affect the expected return of the complete portfolio (E(rC))?

Answer 26

The expected return of the complete portfolio increases as rf increases.

Question 27

How does adding more risky assets with low correlation to a portfolio affect overall portfolio risk?

Answer 27

It reduces overall portfolio risk due to diversification.

Question 28

What does it mean when an investor’s optimal allocation to the risky asset (y*) is greater than 1?

Answer 28

The investor is borrowing to leverage and invest more in the risky asset.

Question 29

What happens to the Capital Market Line (CML) when the risk-free rate increases?

Answer 29

The CML shifts upward, offering a higher return for the same risk level.

Question 30

How does risk aversion (A) affect the optimal allocation to the risky asset (y*)?

Answer 30

Higher risk aversion (A) decreases the optimal allocation to the risky asset.

Question 31

How does the optimal allocation to the risky asset (y*) change if the excess return of the risky asset (E(rP) - rf) increases while the investor’s risk aversion (A) remains constant?

Answer 31

The optimal allocation (y*) increases as the excess return rises, assuming constant risk aversion.

Question 32

Why does the inclusion of a risk-free asset in a portfolio improve the Sharpe ratio compared to a portfolio with only risky assets?

Answer 32

The risk-free asset reduces portfolio risk without affecting the expected return, improving the Sharpe ratio.

Question 33

How do you interpret the first-order condition (FOC) for utility maximization in terms of marginal utility and marginal risk?

Answer 33

The FOC ensures that the marginal utility gained from additional return equals the marginal disutility from the increased risk.

Question 34

What is the impact of high risk aversion on the shape of an investor’s indifference curves?

Answer 34

High risk aversion results in steeper indifference curves, indicating a strong preference for low risk.

Question 35

How does increasing the correlation (ρ₁₂) between two risky assets in a portfolio affect its diversification benefit?

Answer 35

As correlation increases, the diversification benefit decreases, leading to higher portfolio risk.

Question 36

Why does the utility function of a risk-averse investor penalize variance more heavily than that of a risk-neutral investor?

Answer 36

A risk-averse investor places more value on reducing risk, so variance is weighted more heavily in their utility function.

Question 37

How does leveraging (y* > 1) affect both expected return and risk in a portfolio?

Answer 37

Leveraging increases both the expected return and the risk (standard deviation) of the portfolio proportionally.

Question 38

How does the variance of a portfolio’s return (σ²_C) change as more assets are added with negative correlations?

Answer 38

Adding more assets with negative correlations reduces the portfolio’s variance, enhancing diversification.

Question 39

Why do passive strategies along the Capital Market Line (CML) provide the same return for different levels of risk tolerance?

Answer 39

The CML represents a linear relationship between risk and return, so different investors can adjust their allocation but still lie on the same efficient frontier.

Question 40

How does the utility level change for a risk-averse investor when faced with non-normal return distributions with fat tails?

Answer 40

The utility decreases due to the higher probability of extreme losses, which the risk-averse investor heavily penalizes.

Question 41

How does an investor’s utility change if they move from a portfolio on the Capital Allocation Line (CAL) to one below it?

Answer 41

Utility decreases because portfolios below the CAL offer lower returns for the same level of risk.

Question 42

How does a change in risk aversion (A) affect the optimal portfolio allocation when multiple risky assets are involved?

Answer 42

Higher risk aversion shifts the optimal allocation towards less risky assets, reducing exposure to higher-risk assets.

Question 43

Why does an investor with low risk aversion prefer a portfolio with higher expected return but higher variance?

Answer 43

Low risk aversion means the investor is more focused on return, accepting higher variance for potentially greater rewards.

Question 44

How does a decrease in correlation (ρ₁₂) between two risky assets improve the efficiency of a portfolio?

Answer 44

A decrease in correlation reduces portfolio variance, improving the risk-return trade-off and creating a more efficient portfolio.

Question 45

Why might a risk-averse investor prefer a lower Sharpe ratio investment over a higher one under certain conditions?

Answer 45

A lower Sharpe ratio investment may offer more predictable, stable returns, which a highly risk-averse investor values over higher potential returns with more volatility.

Question 46

How would an investor’s optimal allocation to risky assets change in response to an increase in expected return volatility (σ²_P)?

Answer 46

The optimal allocation to risky assets would decrease, as the investor would require higher compensation for taking on increased risk.

Question 47

How does an investor’s utility level differ between normal and non-normal return distributions, particularly in scenarios involving black swan events?

Answer 47

Utility decreases with non-normal distributions due to the increased likelihood of extreme losses, especially in black swan events.

Question 48

Why does diversification reduce risk without necessarily reducing the expected return?

Answer 48

Diversification lowers portfolio variance by combining assets with imperfect correlations, but the expected return remains weighted by individual asset returns.

Question 49

How do indifference curves and the Capital Allocation Line (CAL) help an investor determine their optimal portfolio?

Answer 49

The optimal portfolio is at the point where the highest indifference curve is tangent to the CAL, balancing risk and return based on the investor’s preferences.

Question 50

Why might an investor choose to borrow funds and invest more than 100% in the risky asset?

Answer 50

An investor with low risk aversion may borrow to leverage their position, aiming to maximize returns by investing more in the higher-return, risky asset.

Question 51

What is the free-rider benefit in finance?

Answer 51

It refers to the advantage investors get by benefiting from the research or strategies of others without directly contributing to the cost.

Question 52

How do passive investors experience the free-rider benefit?

Answer 52

Passive investors rely on market efficiency maintained by active investors, benefiting from price discovery without incurring research costs.

Question 53

Why might the presence of too many free-riders undermine market efficiency?

Answer 53

If too many investors rely on others, there may not be enough active participants to ensure proper price discovery, weakening market efficiency.

Question 54

How does the free-rider problem affect the sustainability of active and passive investment strategies in the long run?

Answer 54

If too many investors free-ride, the reduced incentive for active research could lead to inefficiencies, potentially making passive strategies less effective in accurately tracking market performance.

Question 55

Formula for the expected return of a portfolio given y, rf, and the risk premium?

Answer 55

E(rC) = rf + y * risk_premium, where E(rC) is the expected return of the portfolio, y is the proportion invested in the risky asset, rf is the risk-free rate.