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

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BlackTom AI · Accepted Answer

In mean–variance portfolio theory, y* is the fraction of wealth invested in the chosen risky portfolio; therefore it identifies the Optimal risky portfolio P*.

1) Relevant knowledge points and concepts
- Mean–variance (Markowitz) framework: investors choose portfolios to maximize expected return for a given variance (or minimize variance for a given return).  
- Two-fund (mutual fund) separation: all investors hold a combination of the risk-free asset and a single mutual fund—the same optimal risky portfolio P*—but differ in the weight y* placed on that risky fund depending on risk aversion.  
- Expected utility with quadratic utility or mean-variance preferences: U = E[r_C] − 0.5 A Var(r_C), where A is risk aversion.

2) Step-by-step reasoning
Let r_f be the risk-free return, r_P the return on a given risky portfolio P, and y the weight in P (1 − y in the risk-free asset). Then r_C = y r_P + (1 − y) r_f.

3) Intermediate calculation steps
E[r_C] = y E[r_P] + (1 − y) r_f = r_f + y (E[r_P] − r_f).  
Var(r_C) = y^2 Var(r_P).  
Maximize U(y) = r_f + y (E[r_P] − r_f) − 0.5 A y^2 Var(r_P).  
Differentiate w.r.t. y and set to zero:
dU/dy = (E[r_P] − r_f) − A y Var(r_P) = 0.  
Solve for y*: y* = (E[r_P] − r_f) / (A Var(r_P)).  
Thus y* depends on investor risk aversion A and on the chosen risky portfolio P (specifically its expected excess return and variance). The risky portfolio P that all investors pick is the one that maximizes the Sharpe ratio; once P* is chosen, each investor chooses y* by the formula above.

4) Why the correct answer is correct
y* identifies the allocation to the optimal risky portfolio P*, not the complete portfolio or the frontier. By separation, the risky portfolio P* is common to all investors; y* merely scales exposure to P*.

5) Why other options are incorrect
- Optimal Complete Portfolio C*: C* is the combined (y, P*) position including risk-free asset; y* alone does not identify the complete portfolio (you also need P*).  
- Global Minimum Variance Portfolio (GMVP): GMVP minimizes variance regardless of return but is generally different from the tangency/optimal risky portfolio that maximizes Sharpe ratio.  
- Capital Allocation Line (CAL): CAL is a line (set) of possible complete portfolios for a given P; y* is a single weight, not the CAL.  
- Efficient frontier: A set of risky portfolios; y* is a scalar weight, not the frontier.

6) Conclusion
Therefore, y* is used to identify the Optimal risky portfolio P*. This matches the provided correct answer: "Optimal risky portfolio P*."

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