Portfolio Risk Calculation
by Portfolio risk refers to the uncertainty or variability associated with the returns generated by a portfolio. It measures the possibility that actual portfolio returns may differ from expected returns. Portfolio risk depends on the individual risks of securities as well as the relationship between their returns. In portfolio management, variance and standard deviation are commonly used to measure risk. A well-diversified portfolio may have lower risk because poor performance in one investment can be offset by better performance in another.
Portfolio Variance
Portfolio variance measures the overall variability of portfolio returns. For a two-asset portfolio, the formula is:
σp² = W₁²σ₁² + W₂²σ₂² + 2W₁W₂Cov₁₂
Where W₁ and W₂ are the portfolio weights, σ₁² and σ₂² are the individual variances, and Cov₁₂ is the covariance between the two assets. This formula demonstrates that portfolio risk depends on individual asset risks as well as the interaction between their returns.
Portfolio Standard Deviation
Portfolio standard deviation is obtained by taking the square root of portfolio variance:
σp = √σp²
It is generally easier to interpret than variance because it is expressed in the same percentage units as investment returns. A higher standard deviation indicates greater volatility and uncertainty, while a lower standard deviation suggests relatively stable portfolio returns. Investors use standard deviation to compare the riskiness of different portfolios and determine whether the level of risk is suitable for their investment objectives.
Numerical Calculation of Portfolio Risk
Consider a portfolio consisting of two assets, A and B. Suppose an investor allocates 60% of the funds to Asset A and 40% to Asset B. The standard deviation of Asset A is 10%, the standard deviation of Asset B is 15%, and the correlation between the two assets is 0.20.
Step 1: Identify the Portfolio Weights
W₁ = 60% = 0.60
W₂ = 40% = 0.40
The total investment weights equal:
0.60 + 0.40 = 1.00
Step 2: Identify Individual Risks
σ₁ = 10% = 0.10
σ₂ = 15% = 0.15
Therefore, the individual variances are:
σ₁² = 0.10² = 0.01
σ₂² = 0.15² = 0.0225
Step 3: Calculate Covariance
Covariance is calculated using:
Cov₁₂ = ρ₁₂ × σ₁ × σ₂
Where ρ₁₂ = 0.20.
Therefore:
Cov₁₂ = 0.20 × 0.10 × 0.15
= 0.003
Step 4: Calculate Portfolio Variance
For a two-asset portfolio:
σp² = W₁²σ₁² + W₂²σ₂² + 2W₁W₂Cov₁₂
Substituting the values:
σp² = (0.60² × 0.01) + (0.40² × 0.0225) + 2(0.60)(0.40)(0.003)
= (0.36 × 0.01) + (0.16 × 0.0225) + 0.00144
= 0.0036 + 0.0036 + 0.00144
= 0.00864
Therefore, Portfolio Variance = 0.00864.
Step 5: Calculate Portfolio Standard Deviation
Portfolio standard deviation is:
σp = √0.00864
σp ≈ 0.09295
Converting into percentage:
Portfolio Risk ≈ 9.30%