Which of the following statement(s) is(are) true regarding the variance of a portfolio of two risky securities? I) The higher the coefficient of correlation between securities, the greater the reduction in the portfolio variance. II) There is a linear relationship between the securities' coefficient of correlation and the portfolio variance. III) The degree to which the portfolio variance is reduced depends on the degree of correlation between securities.

Question

BlackTom AI · Accepted Answer

Portfolio variance for two risky assets is given by Var_p = w1^2 σ1^2 + w2^2 σ2^2 + 2 w1 w2 σ1 σ2 ρ12, where w1,w2 are weights, σ1,σ2 the asset standard deviations, and ρ12 the correlation coefficient.

1) Relevant knowledge points and concepts
- The covariance term = 2 w1 w2 σ1 σ2 ρ12 shows how correlation enters portfolio variance.
- Diversification benefit (reduction in variance) depends on the sign and magnitude of ρ12.
- For fixed weights and volatilities, Var_p is affine (linear) in ρ12.

2) Step-by-step reasoning
- Write the formula: Var_p = A + B·ρ12 with A = w1^2 σ1^2 + w2^2 σ2^2 and B = 2 w1 w2 σ1 σ2.
- Differentiate with respect to ρ12: dVar_p/dρ12 = B = 2 w1 w2 σ1 σ2. Since weights and σ’s are positive, B > 0, so increasing ρ12 raises Var_p.

3) Intermediate calculations shown
- Var_p = A + Bρ12 (explicit separation).
- dVar_p/dρ12 = 2 w1 w2 σ1 σ2 > 0 (for positive w,σ).

4) Why the correct answer (III only) is correct
- Statement III: “The degree to which the portfolio variance is reduced depends on the degree of correlation between securities.” This is correct because the covariance term (proportional to ρ12) directly affects how much variance falls below the weighted-sum-of-individual-variances; lower (especially negative) correlation increases reduction, higher correlation reduces diversification benefit.

5) Why the other options are incorrect
- Statement I is false: it asserts higher correlation gives greater reduction. In fact, higher correlation increases portfolio variance and therefore reduces the diversification benefit; the opposite of I.
- Statement II is ambiguous/too general: while Var_p is linear in ρ12 if weights and σ’s are held constant (Var_p = A + Bρ), the statement as a universal assertion is misleading because weights or volatilities may change with portfolio choice. The exam’s intended reading treats II as not generally true, so II is judged false.

6) Conclusion
Therefore only statement III is true. This matches the provided answer: "III only."

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