Regimes make fat tails

Daily returns are $r_t = \sigma_t z_t$ where $\sigma_t = 0.8\%$ with probability 0.9 and $2.5\%$ with probability 0.1, independent of $z_t$. (a) With $z_t \sim N(0,1)$, compute the unconditional daily standard deviation and excess kurtosis. (b) Repeat with $z_t$ a unit-variance Student-$t$ with 5 degrees of freedom (kurtosis $3 + 6/(\nu-4)$). (c) Relate the answers to Proposition 2.13(c) and to Cont's stylized fact (7).