The price at time t of a portfolio of stocks, St$,$ follows a lognormal distribution with

parameters $=0\mathrm{.}09$t and $\backslash$sigma $2=0\mathrm{.}04$t$,$ where t is the time from now measured in years

and S$0=1.$ An investor needs to repay a loan of $$150,000$ exactly $5$ years from now.

$($i$)$ Determine the amount the investor would need to invest in the portfolio at

time t $=0$ to give an $80\%$ probability of having at least enough to repay the

loan at time t $=5.[4]$

The investor has $$100,000$ available, which they invest in the portfolio today.

$($ii$)$ Calculate the following risk measures applied to the difference at time t $=5$

between the value of the investor$$s portfolio and the amount required to repay

the loan:

$($a$)95\%$ value at risk

$($b$)$ Expected shortfall or surplus.

$[5]$

$($iii$)$ Discuss what conclusions and actions the investor may draw from your

answers to parts $($i$)$ and $($ii$).$