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LOS d, (Part 1): Calculate the variance of an equally-weighted portfolio of n stocks.

Consider an equally-weighted portfolio comprised of seven assets in which the average asset variance equals 0.31 and the average covariance equals 0.27. What is the variance of the portfolio?

Portfolio variance = σ²_p = (1 / n) σ ²₁ + [(n ? 1) / n]cov = [(1 / 7) × 0.31] + [(6 / 7) × 0.27] = 0.044 + 0.231 = 0.275 = 27.5%

Consider an equally-weighted portfolio comprised of five assets in which the average asset standard deviation equals 0.57 and the average correlation between all asset pairs is ?0.21. The variance of the portfolio is closest to:

Portfolio variance = σ²_p = (1 / n) σ ²₁ + [(n - 1) / n]cov

ρ_1,2 = (cov_1,2) / (σ₁ σ₂) therefore cov_1,2 = (ρ_1,2)(σ₁ σ₂) = (?0.21)(0.57)(0.57) = ?0.068

σ² = (0.57)² = 0.32

σ²_p = (1 / 5)(0.32) + (4 / 5)(?0.068) = 0.064 + (?0.0544) = 0.0096 or 1.00%

Consider an equally-weighted portfolio comprised of 17 assets in which the average asset standard deviation equals 0.69 and the average covariance equals 0.36. What is the variance of the portfolio?

Portfolio variance = σ²_p = (1 / n) σ ²₁ + [(n ? 1) / n]cov = [(1 / 17) × 0.48] + [(16 / 17) × 0.36] = 0.028 + 0.339 = 0.367 = 36.7%

Which of the following statements regarding the capital market line (CML) is least accurate? The CML:

A)	implies that all portfolios on the CML are perfectly positively correlated.>

B)	slope is equal to the expected return of the market portfolio minus the risk-free rate.>

C)	dominates everything below the line on the original efficient frontier.>

The slope of the CML = (the expected return of the market ? the risk-free rate) / (the standard deviation of returns on the market portfolio)

Because the CML is a straight line, it implies that all the portfolios on the CML are perfectly positively correlated.

A)	global minimum-variance portfolio as the tangency portfolio.

B)	market portfolio as the global minimum-variance portfolio.

C)	market portfolio as the tangency portfolio.

Which of the following statements most accurately describes the capital allocation line (CAL) and the capital market line (CML)? The market portfolio:

A)	always lies on both the CAL and the CML.

B)	may lie on the CML, but it always lies on the CAL.

C)	may lie on the CAL, but it always lies on the CML.

When a minimum variance frontier is constructed in risk return space (i.e., y-axis = expected return, x-axis = standard deviation), the capital allocation line is the line emanating from the riskless return through the highest point of tangency with the minimum variance frontier. When the point of tangency is the market portfolio, the capital allocation line is the capital market line.

The capital allocation line (CAL) with the market portfolio as the tangency portfolio is the:

The capital market line is the capital allocation line with the market portfolio as the tangency portfolio.

If a risk-free asset is part of the investment opportunity set, then the efficient frontier is a:

A)	straight line called the capital allocation line (CAL).

B)	curve called the minimum-variance frontier.

C)	curve called the efficient portfolio set.

If a risk-free investment is part of the investment opportunity set, then the efficient frontier is a straight line called the capital allocation line (CAL), whether or not risky asset correlations are equal to one. The y-intercept of the CAL is the risk-free rate. The CAL is tangent to the minimum-variance frontier of risky assets.

A)	The CAL is tangent to the minimum-variance frontier.

B)	It runs through the global minimum-variance portfolio.

C)	It is the efficient frontier when a risk-free asset is available.

If a risk-free investment is part of the investment opportunity set, then the efficient frontier is a straight line called the capital allocation line (CAL). The CAL is tangent to the minimum-variance frontier of risky assets; therefore, it cannot run through the global minimum-variance portfolio.

The equation of the capital market line (CML) says that the expected return on any portfolio equals the:

A)	risk-free rate plus the product of the market price of risk and the portfolio's standard deviation.

B)	risk-free rate plus the product of the market risk premium and the market's portfolio standard deviation.

C)	risk-free rate plus the product of the market price of risk and the market's portfolio standard deviation.

The CML is the capital allocation line with the market portfolio as the tangency portfolio. The equation of the CML is:

E(R_P) = R_F + [(E(R_M) – R_F)/s_M] s_p

where:
E(R_M) = the expected return on the market portfolio, M
s_M = the standard deviation of the market portfolio, M
R_F = the risk-free return

The intercept is the risk-free rate, R_F. The slope is equal to [(E(R_T) – R_F) / s_T], where [E(R_T) – R_F] is the expected risk premium on the tangency portfolio.

A)	the expected return on the tangency portfolio divided by the standard deviation of the tangency portfolio.

B)	the expected risk premium on the tangency portfolio divided by the standard deviation of the tangency portfolio.

C)	the inverse of the slope of the security market line.

Because the capital allocation line is a straight line, we can express it as the equation of a straight line (y = mx + b) where the dependent variable, y, is the expected return E(R_p) and the independent variable, x, is the standard deviation s_p:

E(R_P) = R_F + [(E(R_T) – R_F)/s_T] s_p

where:
E(R_T) = the expected return on the tangency portfolio, T
s_T = the standard deviation of the tangency portfolio, T
R_F= the risk-free return

The slope is equal to [(E(R_T) – R_F)/s_T], where [E(R_T) – R_F] is the expected risk premium on the tangency portfolio.

A)	expected market return.

B)	expected return on the tangency portfolio.

C)	risk-free rate.

The capital market line (CML) is the capital allocation line with the market portfolio as the tangency portfolio. The equation of the CML is:

E(R_P) = R_F + [(E(R_M) – R_F)/s_M] s_p

where:
E(R_M) = the expected return on the market portfolio, M
s_M = the standard deviation of the market portfolio, M
R_F = the risk-free return

The intercept is the risk-free rate, R_F. The slope is equal to [(E(R_T) – R_F)/s_T], where [E(R_T) – R_F] is the expected risk premium on the tangency portfolio.

The best possible risk-return trade-off attainable, given the investor’s expectations of expected returns, variances, and covariances, is represented by the:

A)	the slope of the minimum-variance frontier at the global minimum-variance portfolio.

B)	slope of the capital allocation line (CAL).

C)	standard deviation of the market portfolio.

We can interpret the slope coefficient [(E(R_T) ? R_F) / s_T] of the CAL the same way we do the slope of any straight line (it’s the change in E(R_T) for a one unit change in s_T). Thus, it represents the risk-return trade from moving along the CAL and how much additional expected return do we get for a one-unit increase in risk. Because the tangency portfolio T is the best portfolio, the slope of the CAL line represents the best possible risk-return trade-off attainable, given the investor’s expectations of expected returns, variances, and covariances.

Portfolio Management Associates (PMA) provides asset allocation advice for pensions. PMA recommends that all their pension clients select an appropriate weighting of the risk-free asset and the market portfolio. PMA should explain to its clients that the market portfolio is selected because the market portfolio:

A)	maximizes return and minimizes risk.

B)	maximizes return.

C)	maximizes the Sharpe ratio.

The risk and return coordinate for the market portfolio is the tangency point for the capital market line (CML). The CML has the steepest slope of any possible portfolio combination. The slope of the CML is the Sharpe ratio. Therefore, the Sharpe ratio is highest for the market portfolio.

Investment Management Inc. (IMI) uses the capital market line to make asset allocation recommendations. IMI derives the following forecasts:

Samuel Johnson seeks IMI’s advice for a portfolio asset allocation. Johnson informs IMI that he wants the standard deviation of the portfolio to equal one half of the standard deviation for the market portfolio. Using the capital market line, the expected return that IMI can provide subject to Johnson’s risk constraint is closest to:

The equation for the capital market line is:

Johnson requests the portfolio standard deviation to equal one half of the market portfolio standard deviation. The market portfolio standard deviation equals 20%. Therefore, Johnson’s portfolio should have a standard deviation equal to 10%. The intercept of the capital market line equals the risk free rate (5%), and the slope of the capital market line equals the Sharpe ratio for the market portfolio (35%). Therefore, using the capital market line, the expected return on Johnson’s portfolio will equal:

If an investors’ portfolio lies on the capital market line (CML) at the point where the CML touches the efficient frontier then this implies the investor has:

A)	100% of their funds invested in the market portfolio.

B)	less than 100% of their money invested in the market portfolio.

C)	a larger percentage of their money invested in the market portfolio and have loaned the remaining amount at the risk-free rate.

Portfolios that are on the CML where the CML touches the efficient frontier implies that 100% of investors funds should be invested in the market portfolio to achieve greatest utility.

Adrian Jones is the portfolio manager for Asset Allocators, Inc., (AAI). Jones has decided to alter her framework of analysis. Previously, Jones made recommendations among efficient portfolios of risky assets only. Now, Jones has decided to make recommendations that include the risk-free asset. The efficient frontier for Jones has changed shape from a:

Initially, Jones selected only efficient portfolios comprising risky assets. Formally, Jones selected portfolios along the Markowitz efficient frontier (a curve). When Jones decided to add the risk-free asset, her efficient frontier changed from a curve (the Markowitz efficient frontier) to a line (the capital market line). The capital market line starts at the risk-free rate and extends along (tangent to) the Markowitz curve.