Question $4(19$ marks$)$

The following presents the performance of stock and bond funds under various scenarios. The T$-$bill rate is $3\%.$

$\backslash$table$[[$Scenario$,$Probability,$\backslash$table$[[$Rate of return of$],[$stock fund $(\%)]],\backslash$table$[[$Rate of return of$],[$bond fund $(\%)]]],[$Boom$,0\mathrm{.}3,35,9],[$Normal$,0\mathrm{.}5,19,5],[$Recession$,0\mathrm{.}2,-8,12]]$

a$.$ Calculate the expected returns of the stock and bond funds. $(2$ marks$)$

b$.$ Calculate the standard deviations of the stock and bond funds. $(2$ marks$)$

c$.$ Calculate the correlation coefficient of the stock and bond funds. $(3$ marks$)$

d$.$ Calculate the weight in stock fund, expected return and standard deviation of the optimal risky portfolio. $(6$ marks$)$

e$.$ Calculate the Sharpe ratio for the optimal risky portfolio. $(2$ marks$)$

f$.$ Suppose Mandy decides to invest in the optimal risky portfolio a proportion $($y$)$ of her total investment budget so that her overall portfolio will have an expected rate of return of $8\%.$

i$.$ Determine the proportion y$.(2$ marks$)$

ii$.$ Calculate the standard deviation of Mandy's portfolio. $(2$ marks$)$