The optimal portfolio for any investor lies at the point where their indifference curve (which reflects their risk-return preferences) touches the capital allocation line (CAL). A more risk-averse investor will prefer a portfolio with less risk and therefore accept a lower expected return. Their indifference curves are steeper, not flatter, and the slope of the CAL does not change based on investor preferences—it is determined by market opportunities.

Liquidity refers to how easily a security can be bought or sold without significantly affecting its price. Less liquid securities tend to have wider bid–ask spreads and may experience more volatile pricing. However, brokerage commissions are typically fixed or negotiated independently of a security’s liquidity and are therefore least affected by it.

The capital allocation line represents all possible combinations of a risk-free asset and a portfolio of risky assets. The point where the CAL is tangent to the efficient frontier identifies the optimal risky portfolio because it offers the highest Sharpe ratio (i.e., the best risk-return tradeoff). This is distinct from the optimal investor portfolio, which considers the investor’s individual risk preferences. The global minimum-variance portfolio, meanwhile, has the lowest possible risk but not necessarily the best return for that level of risk.

Risk-averse investors require higher expected returns to compensate for taking on more risk. This behavior creates a positive relationship between risk and return—assets with greater risk tend to offer higher returns over time. A negative or neutral relationship would suggest that investors accept equal or lower returns for more risk, which contradicts the principle of risk aversion.

The global minimum-variance portfolio lies at the left-most point of the minimum-variance frontier, meaning it has the lowest standard deviation and is not dominated by any other portfolio with the same risk. Portfolios Y and Z both have the same standard deviation (17%), but Portfolio Y has a lower expected return, which suggests Portfolio Z dominates it. A dominated portfolio cannot lie on the minimum-variance frontier, making Portfolio Z the least likely to be the global minimum-variance portfolio.

The utility of an investment is given by the formula:

U = E(r) - ( \frac{1}{2} ) × A × σ ^{2}

where E(r) is the expected return, A is the investor’s risk aversion coefficient, and σ² is the variance of returns. For a risk-free asset, the variance (σ²) is zero. This means the second term drops out, leaving U = E(r). Since this formula no longer includes the risk aversion coefficient A, the utility from a risk-free asset is the same for all investors, regardless of whether they are risk-averse, risk-neutral, or risk-seeking.

To calculate the covariance between the returns of two assets, we use the formula:

COV(A,B) = P_{AB} σ_{A} σ_{B}

Why this formula?
Covariance measures how two variables move together. When we have the correlation coefficient (ρ) and standard deviations (σ) of the two assets, this formula helps us find the linear relationship between their returns over time. Correlation standardizes the direction and strength of the relationship, while standard deviations scale it back into actual units.

Step-by-step:

1. Identify the inputs:

• Correlation (ρXY) = 0.50

• Variance of Stock X = 0.36 → Standard deviation (σX) = √0.36 = 0.6

• Variance of Stock Y = 0.16 → Standard deviation (σY) = √0.16 = 0.4

1. Plug values into the formula:

Cov(X, Y) = 0.50 × 0.6 × 0.4
Cov(X, Y) = 0.50 × 0.24
Cov(X, Y) = 0.12

So, the correct answer is 0.12.

A more risk-averse investor requires a significantly higher increase in expected return to accept a marginal increase in risk, which results in a steeper indifference curve. This reflects the investor’s strong preference for safety and their reluctance to trade off return for risk.

In the utility formula, 𝐴 is the risk aversion coefficient. A higher positive A means the investor dislikes risk more. A value of 6 means the investor strongly penalizes risk (σ²), while 0 means the investor is risk neutral, and a negative value like −2 means the investor actually seeks out risk.

To calculate the portfolio’s standard deviation, use the formula:

σ _{p} = √ w ^{2} _{x} σ ^{2} _{x} + w ^{2} _{y} σ ^{2} _{y} + 2w _{2} w _{y} p _{xy} σ _{x} y _{y}

Where:

σ _{p} = √ (0.40) ^{2} (0.125) ^{2} + (0.60) ^{2} (0.078) ^{2} + 2 (0.40)(0.60)(0.35)(0.125)(0.078)

=

9.07%