The annual return on Portfolio $x$ varies each year being $-2\%$ with probability $0\mathrm{.}25,4\%$ with

probability $0\mathrm{.}35$ and $8\%$ with probability $0\mathrm{.}4.$ Annual returns on Portfolio $x$ are independent

of each other.

The annual return on Portfolio $Y$ also varies each year as follows: $50\%$ of the annual growth

factor each year is guaranteed to be $1\mathrm{.}035$ and the other $50\%$ of the annual growth factor is

log$-$normally distributed with mean $1\mathrm{.}06$ and standard deviation $0\mathrm{.}03.$ Annual returns on

Portfolio $Y$ are independent of each other.

a$)$ Calculate the mean and standard deviation of the accumulated value of $10,000$ invested in

Portfolio $x$ for $10$ years.

b$)$ Calculate the mean and standard deviation of the accumulated value of $10,000$ invested in

Portfolio $Y$ for $10$ years.

c$)$ A man must make a liability payment of $10,000$ in $10$ years time. How much should he

invest in Portfolio $Y$ for the $10$ years to be $($at least$)75\%$ sure of meeting this liability

payment.