Two-Asset Portfolio Volatility

Derivatives & Risk

Portfolio Theory · Correlation

Medium

A portfolio invests 50% in Asset A and 50% in Asset B. Both assets have annual volatility 20% and their correlation is 0.5. What is the portfolio's annual volatility?

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Hint 1

Use the two-asset portfolio variance formula $σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}$ .

Hint 2

Plug in $w_{A} = w_{B} = 0.5$ , $σ_{A} = σ_{B} = 0.20$ , and $ρ = 0.5$ , then take the square root.

Show worked solution

1. Write the variance formula
$σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}$ .
$σ_{p}^{2} = w_{A}^{2} σ_{A}^{2} + w_{B}^{2} σ_{B}^{2} + 2 w_{A} w_{B} ρ σ_{A} σ_{B}$
2. Substitute values
$σ_{p}^{2} = (0.5)^{2} (0.2)^{2} + (0.5)^{2} (0.2)^{2} + 2 (0.5) (0.5) (0.5) (0.2) (0.2) = 0.01 + 0.01 + 0.01 = 0.03$ .
$σ_{p}^{2} = 0.01 + 0.01 + 0.01 = 0.03$
3. Take the square root
$σ_{p} = 0.03 \approx 0.1732$ , that is, about 17.32%.
$σ_{p} = 0.03 \approx 0.1732$

Answer: sqrt(0.03)

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Two-Asset Portfolio Volatility

1. Write the variance formula

2. Substitute values

3. Take the square root

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