Problem 293 of 333MediumLinear AlgebraP293

Hedging one asset with two others

Three assets have standardized daily returns $X_1, X_2, X_3$ (mean 0, variance 1) with correlation matrix
$C = \begin{pmatrix} 1 & 0.8 & 0.5 \\ 0.8 & 1 & 0.7 \\ 0.5 & 0.7 & 1 \end{pmatrix}$
You are long asset 3 and want to hedge it with assets 1 and 2, choosing $\beta_1, \beta_2$ to minimize the variance of $X_3 - \beta_1X_1 - \beta_2X_2$ .
- Find the Cholesky factor $L$ with $C = LL^T$ .
- Using $L$ , find the fraction of asset 3's variance that no hedge can remove, and $\det C$ .
- Find $\beta_1$ and $\beta_2$ . Does the hedge buy or sell asset 1?
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