Mr$.$ Simpson is considering to invest his savings in the financial market. As his financial advisor, you collect the following discrete distribution of three financial assets: $(1)$ an equity investment fund tracking the S&P $500$ index $($fund A$),(2)$ an equity investment fund tracking the S&P $500$ ESG index $($fund B$)$ and $(3)$ a bond investment fund tracking the S&P $500$ Investment Grade Corporate Bond Index $($fund C$).$

Based on the risk aversion of Mr$.$ Simpson, you advise that he should buy $2000$ units of fund $A$ at $$10$ per unit, $1000$ unit of fund $B$ at $$20$ per unit and $1000$ units of fund $C$ at $$10$ per unit.

Part $($a$).$ What are the recommended portfolio weights on the three financial assets? $(1$ point$)$

Part $($b$).$ What are the expected return and standard deviation of the recommend portfolio? $(3$ points$)$

Part $($c$).$ Mr$.$ Simpson wonders why he should hold a portfolio of financial assets instead of buying a single asset. Explain to Mr$.$ Simpson the concept and benefit of the diversification effect, and whether the recommended portfolio achieves the diversification effect $(4$ points$)$

Part $($d$).$ What is the Sharpe ratio of the recommended portfolio? If the current yield on the $1-$month Treasury bill is $1\%,$ and the market portfolio has an expected return of $10\%$ with a standard deviation of $12\%,$ is the market portfolio efficient? $(2$ points$).$