Lecture 12 summary · April 9, 2027
Predictable risk with an unpredictable return
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The economic question
A zero predictable mean can coexist with predictable risk. A variance recursion does not establish mean-return predictability, and a tail quantile needs a distribution.
Central relation
Individual problem
Omega is 0.15, alpha 0.1, beta 0.75, current variance two, and latest surprise negative three. Find next and long-run variance, then convert percentage-point returns to decimals.
Worked reasoning
Next variance is 2.55 and long-run variance one. Dividing returns by one hundred divides variances and omega by ten thousand. Alpha and beta are dimensionless and unchanged.
A consequential failure
Variance predictability is called mean predictability, or return units change without the variance intercept.
Repair
Separate mean and variance, enforce the domain, check exact unit invariance, and declare the innovation distribution for a risk quantile.