CFA Level IPortfolio ManagementHard
An investor combines Asset A (standard deviation = 10%) and Asset B (standard deviation = 20%) in equal 50/50 weights. What is the minimum possible portfolio standard deviation achievable given these weights, assuming the correlation between the assets can range from −1 to +1?
- A10%
- B15%
- C0%
- D5%
Show answer & explanationAnswer & explanation
Correct answer: D. 5%
When correlation equals −1, portfolio standard deviation equals |w_A×σ_A − w_B×σ_B| = |0.5×10% − 0.5×20%| = |5% − 10%| = 5%. This represents the theoretical minimum risk achievable with these weights, since perfect negative correlation allows the assets' movements to offset each other most fully (though not entirely, since the weights are unequal in risk terms).
Why the other options are wrong
- A. This equals Asset A's standalone standard deviation, not the diversified minimum.
- B. This equals the weighted-average (ρ=+1, no diversification benefit) portfolio standard deviation, the maximum not minimum.
- C. Zero risk would only be achieved if the weighted standard deviations were exactly equal, which they are not here.
Correlation and Diversification Extremes
Portfolio standard deviation ranges between |wAσA − wBσB| (at ρ = −1) and wAσA + wBσB (at ρ = +1), illustrating the maximum diversification benefit at perfect negative correlation.
- ρ = +1: no diversification benefit, SD = weighted average
- ρ = −1: maximum diversification benefit, SD = |difference of weighted SDs|
- ρ = 0: partial diversification benefit, SD < weighted average
Memory trick: Opposites cancel: at ρ=−1, risks fight and shrink to their difference.