Portfolio Diversification

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📊 Portfolio Diversification: Correlation & Risk

Introduction

This demonstrates how combining assets with different correlations reduces portfolio risk.

Asset	Return	Risk (σ)
Asset A	10%	20%
Asset B	8%	15%

Portfolio: w_A = 0.5, w_B = 0.5

σ_p = √[w_A²σ_A² + w_B²σ_B² + 2w_Aw_Bσ_Aσ_Bρ]

Interactive Calculator

Adjust correlation (ρ):

ρ = -1
ρ = 0
ρ = +1

ρ = +1.00

Risk: 17.5%

Current Calculation:

σ_p = √[0.01 + 0.005625 + 0.015] = 17.5%

With perfect positive correlation (ρ = +1), no diversification benefit.

Diversification Benefit:

Fixed Correlation Cases

Case 1: ρ = +1

Perfect positive correlation

√[0.01 + 0.005625 + 0.015] = 17.5%

17.5%

No diversification

Case 2: ρ = 0

Zero correlation

√[0.01 + 0.005625 + 0] = 12.5%

12.5%

28.6% less risk

Case 3: ρ = -1

Perfect negative correlation

√[0.01 + 0.005625 – 0.015] = 2.5%

2.5%

85.7% less risk

Summary Table

Correlation (ρ)	Portfolio Risk	Diversification	Risk Reduction
+1 (Perfect Positive)	17.5%	None	0%
0 (No Correlation)	12.5%	Moderate	28.6%
-1 (Perfect Negative)	2.5%	Maximum	85.7%

Risk vs Correlation

Key Insights

Lower correlation = Lower risk: As ρ decreases, portfolio risk falls dramatically.

Negative correlation maximizes benefits: When assets move opposite directions, risk reduction is greatest.

Diversification works: Combining uncorrelated assets reduces overall portfolio volatility.

Practical application: Seek assets with low/negative correlation to build efficient portfolios.

// Set current year
document.getElementById(‘currentYear’).textContent = new Date().getFullYear();

// Portfolio calculation function
function calculatePortfolioRisk(correlation) {
const wA = 0.5, wB = 0.5;
const sigmaA = 0.20, sigmaB = 0.15;

const part1 = Math.pow(wA, 2) * Math.pow(sigmaA, 2);
const part2 = Math.pow(wB, 2) * Math.pow(sigmaB, 2);
const part3 = 2 * wA * wB * sigmaA * sigmaB * correlation;

const variance = part1 + part2 + part3;
const portfolioRisk = Math.sqrt(variance);

// Calculate diversification benefit
const maxRisk = 0.175; // ρ = +1
const minRisk = 0.025; // ρ = -1
const diversificationBenefit = ((maxRisk – portfolioRisk) / (maxRisk – minRisk)) * 100;

return {
risk: portfolioRisk,
riskPercent: (portfolioRisk * 100).toFixed(2),
diversificationBenefit: Math.max(0, Math.min(100, diversificationBenefit)),
calculation: `σ_p = √[${part1.toFixed(5)} + ${part2.toFixed(5)} + ${(part3 >= 0 ? ‘+’ : ”) + part3.toFixed(5)}] = ${(portfolioRisk*100).toFixed(2)}%`
};
}

// Update display
function updateDisplay(correlation) {
const result = calculatePortfolioRisk(correlation);

// Update correlation value
const correlationValue = document.getElementById(‘correlationValue’);
correlationValue.innerHTML = `ρ = ${correlation >= 0 ? ‘+’ : ”}${correlation.toFixed(2)}`;

// Update correlation color
if (correlation > 0.3) {
correlationValue.className = ‘correlation-value correlation-positive’;
} else if (correlation 0.15) {
riskBadge.className = ‘risk-badge risk-high’;
} else if (result.risk > 0.05) {
riskBadge.className = ‘risk-badge risk-medium’;
} else {
riskBadge.className = ‘risk-badge risk-low’;
}

// Update calculation
document.getElementById(‘calculation’).innerHTML = result.calculation;

// Update interpretation
const interpretation = document.getElementById(‘interpretation’);
let interpretationText = ”;
if (correlation > 0.7) {
interpretationText = ‘With perfect positive correlation (ρ = +’ + correlation.toFixed(2) + ‘), no diversification benefit.’;
} else if (correlation > 0.3) {
interpretationText = ‘With positive correlation (ρ = +’ + correlation.toFixed(2) + ‘), limited diversification benefit.’;
} else if (correlation > -0.3) {
interpretationText = ‘With low correlation (ρ = ‘ + (correlation >= 0 ? ‘+’ : ”) + correlation.toFixed(2) + ‘), good diversification.’;
} else {
interpretationText = ‘With negative correlation (ρ = ‘ + correlation.toFixed(2) + ‘), maximum diversification benefit.’;
}
interpretation.textContent = interpretationText;

// Update diversification bar
const diversificationBar = document.getElementById(‘diversificationBar’);
const benefitPercent = Math.round(result.diversificationBenefit);
diversificationBar.style.width = benefitPercent + ‘%’;
diversificationBar.textContent = benefitPercent + ‘%’;

// Update bar color
if (benefitPercent < 30) {
diversificationBar.style.background = '#e74c3c';
} else if (benefitPercent < 70) {
diversificationBar.style.background = '#f39c12';
} else {
diversificationBar.style.background = '#27ae60';
}

// Update chart
updateChart(correlation);
}

// Initialize chart
const ctx = document.getElementById('riskChart').getContext('2d');
let riskChart;

function initializeChart() {
// Generate data points
const correlations = [];
const risks = [];
for (let rho = -1; rho {
return Math.abs(parseFloat(label) – correlation) {
button.addEventListener(‘click’, function() {
const correlation = parseFloat(this.getAttribute(‘data-correlation’));
document.getElementById(‘correlationSlider’).value = correlation * 100;
updateDisplay(correlation);
});
});

// Initialize on load
window.addEventListener(‘load’, function() {
initializeChart();
updateDisplay(1);
});