Question2 When borrowing and lending at the risk-free rate are allowed, which capital allocation line (CAL) should the investor choose to combine with the efficient frontier? I. The one with the highest Sharpe ratio.II. The one which passes through the GMVP.III. The one with the steepest slope. Select one alternative: a. I and II b. I and III c. II and III d. I, II, and III ResetMaximum marks: 1 Flag question undefined单项选择题
A
a. I and II
B
b. I and III
C
c. II and III
D
d. I, II, and III

Question

Question2 When borrowing and lending at the risk-free rate are allowed, which capital allocation line (CAL) should the investor choose to combine with the efficient frontier? I. The one with the highest Sharpe ratio.II. The one which passes through the GMVP.III. The one with the steepest slope. Select one alternative: a. I and II b. I and III c. II and III d. I, II, and III ResetMaximum marks: 1 Flag question undefined单项选择题

A

a. I and II

B

b. I and III

C

c. II and III

D

d. I, II, and III

BlackTom AI · Accepted Answer

The investor should choose the CAL that maximizes the Sharpe ratio because that CAL has the steepest slope and is tangent to the efficient frontier.

1) Relevant concepts
- Efficient frontier: set of portfolios offering maximum expected return for a given risk (standard deviation).
- Risk-free asset: allows borrowing/lending at rate r_f, producing straight-line combinations with any risky portfolio (a CAL).
- Capital Allocation Line (CAL): line from r_f to a risky portfolio with slope S = (E[r_p] − r_f)/σ_p.
- Sharpe ratio: S = (E[r_p] − r_f)/σ_p; a higher Sharpe ratio means more excess return per unit risk.
- Tangency (market) portfolio: the risky portfolio at which the CAL is tangent to the efficient frontier; it maximizes the Sharpe ratio.
- Global minimum variance portfolio (GMVP): the portfolio on the efficient frontier with the lowest variance (smallest σ), not generally the same as the tangency portfolio.

2) Step-by-step reasoning
- For any risky portfolio p, the CAL from r_f has slope S_p = (E[r_p] − r_f)/σ_p.
- When combining r_f with risky portfolios, the investor can choose any CAL; the best one is the one that gives the highest expected return for each level of risk — equivalently, the CAL with the largest slope S.
- Geometrically, the optimal CAL is tangent to the efficient frontier at the portfolio that maximizes S (the tangency or market portfolio).

3) Intermediate/calculation steps
- Define S_p = (E[r_p] − r_f)/σ_p. The investor optimizes over portfolios p to find p* such that S_p* = max_p S_p.
- The resulting CAL: E[r] = r_f + S_p* · σ, which is the steepest possible line from r_f touching the efficient frontier.

4) Why the correct answer is correct
- Statement I (highest Sharpe ratio): True — the tangency portfolio maximizes the Sharpe ratio, so the CAL chosen must be the one with the highest Sharpe.
- Statement III (steepest slope): True — Sharpe ratio equals the CAL slope, so the optimal CAL is the steepest.
Thus I and III are correct.

5) Why the other options are incorrect
- Statement II (passes through the GMVP): False — the GMVP minimizes variance but does not generally maximize the Sharpe ratio. The tangency portfolio is typically not the GMVP except in special cases (e.g., particular r_f or return structure). Therefore any option including II (a, c, d) is incorrect.

6) Conclusion
The correct choice is b. I and III. This matches the provided answer.

类似问题

Question1.3 An investor’s optimal risky allocation y* is required to identify the______: Optimal Complete Portfolio C. Global Minimum Variance Portfolio (GMVP). Optimal risky portfolio P. Capital Allocation Line (CAL). Efficient frontier. ResetMaximum marks: 2.5 Flag question undefined

Constraining MSR portfolio weights to the (0%, 100%) range compared to allowing unconstrained (short-selling) weights tends to:

The Global Minimum Variance (GMV) portfolio is defined as the portfolio that:

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