Question4 Which of the following statement(s) is(are) true regarding the selection of a portfolio from those that lie on the capital allocation line? I. Less risk-averse investors will invest more in the risk-free security and less in the optimal risky portfolio than more risk-averse investors.II. More risk-averse investors will invest less in the optimal risky portfolio and more in the risk-free security than less risk-averse investors.III. Investors choose the portfolio that maximizes their expected utility. Select one alternative: a. I and III b. III only c. II and III d. II only e. I only ResetMaximum marks: 1 Flag question undefined单项选择题
A
a. I and III
B
b. III only
C
c. II and III
D
d. II only
E
e. I only

Question

Question4 Which of the following statement(s) is(are) true regarding the selection of a portfolio from those that lie on the capital allocation line? I. Less risk-averse investors will invest more in the risk-free security and less in the optimal risky portfolio than more risk-averse investors.II. More risk-averse investors will invest less in the optimal risky portfolio and more in the risk-free security than less risk-averse investors.III. Investors choose the portfolio that maximizes their expected utility. Select one alternative: a. I and III b. III only c. II and III d. II only e. I only ResetMaximum marks: 1 Flag question undefined单项选择题

A

a. I and III

B

b. III only

C

c. II and III

D

d. II only

E

e. I only

BlackTom AI · Accepted Answer

More risk-averse investors allocate more to the risk-free asset and less to the optimal risky portfolio; all investors choose the mix that maximizes their expected utility.

1. Relevant knowledge points and concepts
- Capital Allocation Line (CAL): combinations of a risk-free asset and a single optimal risky portfolio.  
- Mean–variance utility: U = E[r] − 0.5 A Var(r), where A is risk aversion.  
- For investors facing the same risky opportunity set, the optimal risky portfolio (tangent portfolio) is the same for all; only the weight w in that risky portfolio varies with A.

2. Complete reasoning process (step-by-step)
- Let r_f be the risk-free return, r_p the return of the optimal risky portfolio with expected excess return μ_p = E[r_p] − r_f and variance σ_p^2.
- The investor chooses weight w on the risky portfolio to maximize utility: U(w) = E[r_c] − 0.5 A Var(r_c) = r_f + w μ_p − 0.5 A w^2 σ_p^2.
- Take derivative w.r.t. w and set to zero: dU/dw = μ_p − A w σ_p^2 = 0.

3. Intermediate calculation steps
- Solve for w*: μ_p − A w* σ_p^2 = 0 → w* = μ_p / (A σ_p^2).
- This shows w* is inversely proportional to A: larger A (more risk aversion) → smaller w* (less invested in risky portfolio) and more in risk-free.

4. Why the correct answer is correct
- Statement II: “More risk-averse investors will invest less in the optimal risky portfolio and more in the risk-free security than less risk-averse investors.” This directly follows from w* = μ_p/(A σ_p^2). True.
- Statement III: “Investors choose the portfolio that maximizes their expected utility.” The derivation used utility maximization; thus true.

5. Why other options are incorrect
- Statement I is the opposite of II (claims less risk-averse invest more in risk-free), so I is false.  
- Any option including I is therefore wrong. Option b (III only) omits II which is true. Option d (II only) omits III which is true. Option e (I only) is false.