Explanation

When two assets have a correlation coefficient of +1.0, it means they are perfectly positively correlated. In this case, the portfolio's standard deviation (risk) is simply the weighted average of the individual assets' standard deviations.

Mathematical Proof:

The formula for portfolio variance with two assets is:

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂

Where:

• σ_p = portfolio standard deviation
• w₁, w₂ = weights of assets 1 and 2
• σ₁, σ₂ = standard deviations of assets 1 and 2
• ρ₁₂ = correlation coefficient between assets 1 and 2

When ρ₁₂ = +1.0:

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂(1) σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂

This can be factored as: σ_p² = (w₁σ₁ + w₂σ₂)²

Taking the square root: σ_p = w₁σ₁ + w₂σ₂

This shows that the portfolio standard deviation is exactly equal to the weighted average of the individual asset standard deviations.

Key Points:

1. Perfect positive correlation (ρ = +1.0) means the assets move in perfect unison.
2. No diversification benefits - the portfolio risk cannot be reduced below the weighted average.
3. Portfolio risk is linear - it's simply the weighted sum of individual risks.

Comparison with other correlation values:

• ρ < +1.0: Portfolio risk is less than weighted average (diversification benefit)
• ρ = +1.0: Portfolio risk equals weighted average (no diversification)
• ρ = -1.0: Portfolio risk can be reduced to zero with proper weights (perfect diversification)

Therefore, for a correlation coefficient of +1.0, portfolio risk is equal to the weighted average of the risk of the two assets in the portfolio.