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Diversification & Correlation Effect: Interactive Portfolio Risk Chart | EasyCFA | Wall Street Uni

Portfolio Management

Diversification & Correlation Effect

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.

Subtopic: Portfolio Risk and Return
Level: Applied
Status: Featured
Views: 347

Parameters

Formula

E(R_p) = w_A E(R_A) + w_B E(R_B) \quad ; \quad \sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho \sigma_A \sigma_B}

Interactive Chart

PORTFOLIO EXPECTED RETURN

E(Rp) = wA E(RA) + wB E(RB)

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.

PORTFOLIO RISK (VARIANCE)

σ²p = wA²σA² + wB²σB² + 2 wA wB ρ σA σB

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.

SPECIAL CASES

At ρ = +1:

σp = wA σA + wB σB (exact weighted average — no benefit)

At ρ = 0:

σp = sqrt(wA²σA² + wB²σB²) — cross-term disappears

At ρ = −1:

σp = |wA σA − wB σB| — can reach zero when:

w*A = σB / (σA + σB)

MINIMUM-VARIANCE PORTFOLIO (MVP)

For any ρ ≠ +1:

w*A = (σB² − ρ σA σB) / (σA² + σB² − 2ρ σA σB)

This is the leftmost point on each opportunity-set curve. Below the MVP, portfolios are dominated — lower return with the same risk.

DIVERSIFICATION BENEFIT

Risk reduction (%) = (σp at ρ=+1 − actual σp) / σp at ρ=+1 × 100

A positive value indicates that the portfolio achieves lower risk than the simple weighted-average risk — the gain from diversification.

Interactive charts are desktop only

The sliders, formulas and analytics view need more room than a phone screen can give them. Open this chart on a desktop or larger tablet to use the full interactive experience.

Study aids

Working with this chart

The errors candidates make on this topic, and what to carry into the exam.

Common mistakes

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

Exam tips

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

Parameters

Asset A

expectedReturnAExpected Return A%

1120

stdDevAStd Deviation A (σ)%

1830

Asset B

expectedReturnBExpected Return B%

210

stdDevBStd Deviation B (σ)%

515

Portfolio Setting

correlationCorrelation (ρ)

-11

portfolioWeightAAsset A Weight (%)%

0100

Formula

E(R_p) = w_A E(R_A) + w_B E(R_B) \quad ; \quad \sigma_p = \sqrt{w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \rho \sigma_A \sigma_B}

Interactive Chart

Diversification & Correlationρ = 0.30(Positive)w_A = 50.0%Risk Reduction: 16.4%

ρ = +1 (No Div.)ρ = 0 (Partial Div.)ρ = −1 (Max Div.)Active ρ = 0.30MVPSelected Portfolio

Selected Portfolio

Asset A Weight50.0%

Asset B Weight50.0%

Expected Return10.00%

Actual Risk (σ)13.38%

Risk if ρ = +116.00%

Risk Reduction16.4%

MVP (Active ρ = 0.30)

Asset A Weight7.5%

Asset B Weight92.5%

Expected Return6.60%

Min Risk (σ)9.87%

Diversification Across ρ

ρ = +1 MVP risk:10.00%

ρ = 0 MVP risk:9.10%

ρ = −1 MVP risk:0.00%

Moderate correlation — some diversification benefit.

CFA Key Relationships

At ρ = +1, σp = weighted average of individual risks — no diversification benefit. The opportunity set is a straight line.

At ρ = −1, a zero-risk portfolio exists when w*A = σB / (σA + σB). This is the theoretical maximum diversification.

At ρ = 0, σ²p = w²AσA² + w²BσB² — no cross-term. Significant but not maximum diversification benefit.

Lower ρ shifts the frontier leftward only — not upward. Correlation affects risk, not expected return.

The MVP is the leftmost point on each curve. Portfolios below the MVP are inefficient — same risk, lower return.

Diversification benefit = (weighted avg risk − actual risk) / weighted avg risk. Depends on correlation, not asset count alone.

Student Tasks

Set ρ = +1. Verify that the active (violet) curve becomes a straight line and risk reduction = 0%.
Slowly move ρ from +1 toward 0. Observe the frontier bowing left and the MVP risk decreasing.
Set ρ = −1. Locate the zero-risk portfolio (cyan dot) and confirm: w*A = σB / (σA + σB).
Set σA = σB (equal volatilities) with ρ = −1. Verify the zero-risk portfolio is exactly 50% / 50%.
For a target return of 11%, slide ρ and identify which correlation produces the lowest risk portfolio at that return level.