Covariance matrices and portfolios

Idea: Volatilities do not add

A fund holds $60\%$ index at $30\%$ volatility and $40\%$ oil at $40\%$, correlation $0.5$.

• Naive sum: $0.6\times 0.30+0.4\times 0.40=0.34$
• The true risk is lower: the cross term decides

Formula: Portfolio variance

$\mathrm{Var}(w^{T}R)=w^T\Sigma w$

Sum over every pair of assets: weight times weight times their covariance.

Formula: A pair of assets

${\sigma }^2=w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_{2}c$

The cross term carries the covariance c; it vanishes at zero correlation.

Example: Index and oil

Weights $0.6$ and $0.4$, variances $0.09$ and $0.16$, covariance $0.06$. What is the portfolio variance?

1. Squares: $0.36\times 0.09=0.0324$ and $0.16\times 0.16=0.0256$
2. Cross: $2\times 0.6\times 0.4\times 0.06=0.0288$
3. Total: $0.0324+0.0256+0.0288=0.0868$

Common mistake: Adding volatilities

Volatilities add only at correlation $1$. Variances add only at $0$.

Instead Every other case needs the cross term, and a short position flips its sign.

Idea: Not every table is a covariance

A covariance matrix is symmetric and gives $w^T\Sigma w\ge 0$ for every $w$.

• Correlations $0.9$, $0.9$, $0$, unit variances, $w=(-2,1,1)$
• Variance: $6-7.2=-1.2$, so no such assets exist

Formula: The lowest risk mix

$w=\frac{{\sigma }_2^2-c}{{\sigma }_1^2+{\sigma }_2^2-2c}$

The weight on the first asset with the least variance. The same slope sets a hedge.

Quick check: Weight on the index

Index variance $0.09$, oil $0.16$, covariance $0.06$. What weight on the index gives the lowest variance?

Idea: Many assets

Equal weights on $n$ names of variance ${\sigma }^2$ and correlation $\rho$: $\frac{{\sigma }^2}{n}+\left(\frac{n-1}{n}\right)\times \rho {\sigma }^2$.

• $\frac{0.09}{10}+\frac{9}{10}\times 0.018=0.0252$
• Floor as $n$ grows: $0.2\times 0.09=0.018$
• Specific risk diversifies away, shared risk stays