The multivariate normal

Idea: One leg has printed

If index move $X$ and sector move $Y$ are jointly normal, a few numbers answer everything.

• $E[X]=2$, $\mathrm{Var}(X)=25$
• $E[Y]=1$, $\mathrm{Var}(Y)=36$
• $\mathrm{Cov}(X,Y)=18$, so $\rho =\frac{18}{5\times 6}=\frac{3}{5}$

Formula: Any mix stays normal

$a^{T}R\sim N(a^T\mu ,a^T\Sigma a)$

Every fixed combination of jointly normal variables is normal, with mean and variance from μ and Σ.

Formula: Conditional mean

$E[Y\mid x]={\mu }_Y+\left(\frac{\mathrm{Cov}}{\mathrm{Var}(X)}\right)\times (x-{\mu }_X)$

The forecast is a straight line in x; its slope is the regression slope.

Formula: Conditional variance

$\mathrm{Var}(Y\mid x)=\mathrm{Var}(Y)-\frac{{\mathrm{Cov}}^2}{\mathrm{Var}(X)}$

It does not depend on x: a surprise moves the forecast, not the uncertainty.

Example: Forecast the sector

$E[X]=2$, $E[Y]=1$, $\mathrm{Var}(X)=25$, $\mathrm{Cov}(X,Y)=18$. The index prints $12$. What is the sector forecast?

1. Surprise: $12-2=10$
2. Slope: $\frac{18}{25}=0.72$
3. Forecast: $1+0.72\times 10=8.2$

Quick check: Uncertainty left

$\mathrm{Var}(Y)=36$, $\mathrm{Var}(X)=25$, $\mathrm{Cov}(X,Y)=18$. What is $\mathrm{Var}(Y\mid X)$?

Idea: In standard units

The forecast moves $\rho$ times the move you saw, counted in standard deviations.

• Index: $\frac{10}{5}=2$ standard deviations up
• Sector forecast: $0.6\times 2=1.2$ up
• Range: $8.2-2\times 4.8=-1.4$ to $8.2+2\times 4.8=17.8$

Common mistake: Normal columns, not a normal pair

With a random sign $S$, $Y=S\times X$ is normal and uncorrelated with $X$, yet $Y^2=X^2$.

Instead Ask whether the pair is jointly normal before using any of these rules.

Idea: Where the model stops

Real returns have heavier tails and fall together in a crash more than the model allows.

• Uncorrelated means independent only for a jointly normal pair
• Treat the formulas as a first answer, and say so