Portfolio Variance

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Understanding Portfolio Variance

Portfolio variance quantifies the risk of a portfolio by measuring how the returns of its constituent assets fluctuate together. It incorporates each asset’s variance and the covariances (or correlations) between assets. Because assets can move differently relative to one another, a portfolio’s variance is not simply the weighted average of individual variances.

What portfolio variance shows
– Captures total volatility arising from individual asset risk and their co-movements.
– Falls when assets have low or negative correlations, illustrating the benefit of diversification.
– Is related to standard deviation: standard deviation = √(variance), and is commonly used to express portfolio volatility.

Key formulae
– Two-asset portfolio variance:
Var(p) = w1^2 σ1^2 + w2^2 σ2^2 + 2 w1 w2 Cov(1,2)
where Cov(1,2) = ρ12 σ1 σ2 and ρ12 is the correlation coefficient.

  • General / matrix form for n assets:
    Var(p) = wᵀ Σ w
    where w is the weights vector and Σ is the covariance matrix (Σij = Cov(i,j)).

Why covariance and correlation matter
– Covariance measures how two asset returns move together. Positive covariance increases portfolio variance; negative covariance can reduce it.
– Correlation (ρ) standardizes covariance to a range between −1 and 1, making it easier to see how strongly assets co-move.
– Lower correlations among assets allow risk reduction without necessarily sacrificing expected return.

Example (two assets)
– Stock A: weight = 33.3% (0.3333), σA = 20% (0.20)
– Stock B: weight = 66.7% (0.6667), σB = 10% (0.10)
– Correlation ρAB = 0.85

Compute:
– Var(p) ≈ (0.3333^2 × 0.20^2) + (0.6667^2 × 0.10^2) + 2×0.3333×0.6667×0.20×0.10×0.85 ≈ 0.01644
– Standard deviation = √0.01644 ≈ 0.1282 → 12.82% volatility

Role of Modern Portfolio Theory (MPT)
– MPT frames portfolio construction as a trade-off between return and risk (volatility).
– The efficient frontier represents portfolios that maximize expected return for a given level of risk.
– MPT emphasizes diversification: combining assets with low or negative correlations can lower portfolio variance and improve the risk-return profile.

Interpreting variance and standard deviation
– Variance is a squared measure, useful in algebraic and matrix calculations.
– Standard deviation (the square root of variance) is preferred for interpretation because it is in the same units as returns.
– A higher standard deviation indicates greater expected dispersion of returns (more volatility/risk).

Practical steps to compute portfolio variance
1. Collect historical return series for each asset.
2. Calculate each asset’s standard deviation and the covariance (or correlation) matrix.
3. Decide portfolio weights (w).
4. Compute Var(p) = wᵀ Σ w. For interpretability, take √Var(p) to get the portfolio standard deviation.
5. Use software (Excel, R, Python, portfolio tools) to handle larger portfolios and compute covariance matrices efficiently.

Key takeaways
– Portfolio variance measures total portfolio risk by combining individual variances and covariances.
– Diversification works because correlations between assets can reduce total variance.
– Use Var(p) = wᵀ Σ w for general calculations; convert to standard deviation for interpretation.
– For multi-asset portfolios, practical computation requires a covariance matrix and software tools.

Frequently asked questions
– Q: What’s the difference between variance and standard deviation?
A: Variance is the average squared deviation of returns; standard deviation is its square root and is easier to interpret as a volatility measure in return units.

Conclusion
Portfolio variance is a foundational metric for quantifying and managing portfolio risk. By understanding and applying covariance, correlation, and the matrix formulation Var(p) = wᵀ Σ w, investors and portfolio managers can construct portfolios that better balance expected return against volatility through diversification.