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Interactive visualization showing how correlation between two assets shapes the portfolio opportunity set and determines diversification benefit. Compare three reference curves (ρ=+1, ρ=0, ρ=−1) with the active correlation. Identify the minimum-variance portfolio, zero-risk portfolio at ρ=−1, and the risk reduction from diversification. Essential CFA Portfolio Management topic.
Asset A
Asset B
Portfolio Setting
Moderate correlation — some diversification benefit.
Expected return is a simple weighted average of individual asset returns. It does NOT depend on correlation or standard deviations. Changing ρ shifts the curve left or right (changing risk) but leaves the y-axis spread unchanged.
The cross-term (2 wA wB ρ σA σB) is the diversification term. When ρ < 1, this term is smaller than the ρ=+1 case, reducing total portfolio variance. This is the mathematical foundation of diversification.
At ρ = +1:
At ρ = 0:
At ρ = −1:
For any ρ ≠ +1:
This is the leftmost point on each opportunity-set curve. Below the MVP, portfolios are dominated — lower return with the same risk.
A positive value indicates that the portfolio achieves lower risk than the simple weighted-average risk — the gain from diversification.
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Avoid these frequent errors
Believing diversification increases expected return. Correlation affects only risk — expected return remains a weighted average regardless of ρ. The opportunity set shifts horizontally, not vertically
Confusing "diversification" with "number of assets." A portfolio of 10 assets with ρ = +1 between all pairs has the same risk as the weighted average. Correlation, not count, determines diversification benefit
Thinking the zero-risk portfolio requires a special condition beyond ρ = −1. At ρ = −1, a zero-risk portfolio always exists for long-only portfolios as long as both assets have positive σ. The weight is simply w*A = σB / (σA + σB)
Confusing the MVP with the zero-risk portfolio. The MVP exists for any ρ < +1 and has positive risk (except at ρ = −1). The zero-risk portfolio is a special point only at ρ = −1
Forgetting that at ρ = +1, the MVP is at the lower-risk asset endpoint, not an interior portfolio. The opportunity set is a line with no interior minimum
Misapplying the diversification benefit formula. Risk reduction should be calculated relative to the ρ=+1 benchmark (weighted average risk), not relative to either individual asset
Strategic insights for success
ρ = +1 questions: portfolio risk = weighted average. Memorize the exact formula: σp = wAσA + wBσB
ρ = −1 zero-risk questions: memorize w*A = σB / (σA + σB). Verify by substituting back into the variance formula
MVP calculation: w*A = (σB² − ρ σA σB) / (σA² + σB² − 2ρ σA σB). The numerator uses σB² (the other asset's variance)
Diversification effect: as ρ decreases from +1 to −1, the frontier shifts left. No change in return axis; only risk axis affected
Common trick: given two portfolios with the same return, lower σ is always preferred. Given same σ, higher return is always preferred. The efficient frontier provides the highest return for each risk level
Risk reduction percentage = (σ at ρ=+1 − σ at actual ρ) / σ at ρ=+1. This is the direct measure of diversification benefit
If the exam asks about the shape of the opportunity set as ρ changes: ρ=+1 → straight line; ρ between +1 and −1 → curve bowing left; ρ=−1 → V-shape touching the y-axis