Assume now that the investor has to pay a random sum of money $($the liability$)$ to a creditor at the end of the year which can be written as $R15,000(1+x),$ where $x$ is a normal random variable with mean $6\%$ and standard deviation $12\%.x$ has a correlation of $0\mathrm{.}8$ with the return on A and $0\mathrm{.}4$ with the return on B$.$ Surplus is defined as the value of the portfolio less the liability at the end of the year.

ii$)$ Assume the investor has R$15,000$ at the start of the year. Determine the portfolio that will result in the lowest standard deviation of surplus.

$(05$ marks$)$

iii$)$ Assume that the investor has R$16,000$ at the start of the year and that she wishes to construct a portfolio $($the "least risk portfolio"$)$ that minimises the probability that the surplus is negative.

$(05$ marks$)$