Problem $1$

Suppose it is the start of year $2008.$ You have $$100,000$ to invest and decide to invest in a combina$-$

tion of a riskless asset $(f),$ a stock index fund $(S)$ and a long$-$term bond fund $(B),$ with the following

properties:

The correlation of the returns on the long$-$term bond fund and the stock index fund is equal to $0\mathrm{.}3.$

a$)$ If you only consider the risky assets, what is the global minimum variance portfolio? What

is the global minimum variance portfolio if you also consider the risk$-$free asset?

b$)$ The tangency portfolio is $66\%$ in fund $S$ and $34\%$ in fund $B.$ What dollar amounts should be

invested in each of the three asset classes $f,S,$ and $B$ if you would like to achieve $($for the

year $2008)$ an expected return of $8\%$ with the lowest possible standard deviation? What is

the standard deviation of the $($percentage$)$ return on this portfolio?

c$)$ For the year $2008,$ the realized return on $f$ was $0\mathrm{.}03(3\%),$ the realized return on $B$ was $0\mathrm{.}10$

$(10\%)$ and the realized return on $S$ was $-0\mathrm{.}40(-40\%).$ At the end of $2008,$ what are your dollar

holdings of $f,$ of $B$ and of $S,$ and how much money do you have in total? What is your real$-$

ized $($percentage$)$ portfolio return for year $2008?$