In a stochastic model for rates of return, $1+$ i$1,...,1+$ in $$ LN$(\backslash$mu $,\backslash$sigma $2)$ with mean exp$(0\mathrm{.}03125)$ and variance exp$(0\mathrm{.}125)$ exp$(0\mathrm{.}0625).$ An agent has an investment of $1$ AUD at time $0$ and will receive its accumulation S$10$ back at time $10.$ They are analysing the variability of the investment under the fixed rate model and the varying rate model.

$($i$)$ What is the distribution of S$10$ under the fixed rate model and the varying rate model respectively?

$($ii$)$ Compute cv$($S$10)$ under the fixed rate model and the varying rate model.

$($iii$)$ Compute VaR$0\mathrm{.}9($S$10)$ VaR$0\mathrm{.}1($S$10)$ under the fixed rate model and the varying

rate model.

$($iv$)$ Based on your results in $($ii$)$ and $($iii$),$ which model gives a more risky position of S$10,$ the fixed rate model or the varying rate model? Briefly explain the intuition behind your observation.