You work for a consulting firm. The University of Exeter $($UE$)$ has just hired your team to provide recommendations regarding the composition of its endowment fund. Due to the recent pandemic, UE has been relying heavily on its endowment fund for its budgetary needs over the past four years. After discussions with UE$$s Chief Financial Officer $($CFO$),$ your manager has identified several key questions that require further research. Your task is to conduct relevant analyses to address these questions.

UE and your team have decided to invest in the following five stocks: Caterpillar Inc. $($CAT$),$ Johnson and Johnson $($JNJ$),$ Walmart $($WMT$),$ Nvidia $($NVDA$),$ and Proctor and Gamble $($PG$).$ The analysis will be conducted assuming a risk$-$free rate of $0\%$ and no exchange rate fluctuations. Trading fees and transaction costs are negligible given the size of UE$$s portfolio. Assume UE uses historical averages as proxies for expected returns unless otherwise instructed. For portfolio construct purpose, use historical monthly returns from January $2016$ to December $2020($note that you need prices on $1/1/2021$ to calculate returns for December $2020).$ For portfolio performance evaluation purpose, use monthly returns from January $2021$ to December $2022.$

$1.$ UE$$s current risky portfolio invests in CAT, JNJ$,$ and WMT$.$ UE$$s CFO is considering expanding its investment universe to include NVDA and PG$.$ Assuming no trading restrictions, plot the efficient frontier using the three stocks that UE currently invests in$.$ Compare it with the efficient frontier generated using all five proposed stocks. Do you recommend UE to expand its investment universe? Why or why not?

$2.$ Assuming no trading restrictions, how should UE construct its risky portfolio using the aforementioned five stocks? Refer to this portfolio as P$1.$

$3.$ How should UE construct its risky portfolio if it is not allowed to short sell? Refer to this portfolio as P$2$ and report its weights.

$4.$ Explain whether the weights for P$1$ and P$2$ make economic sense.

$5.$ The CFO is also considering managing its risky portfolio passively by following the market wisdom. Utilising the market capitalisation data as of $31/12/2022,$ the weights for the passive portfolio $($denoted as P$3$ hereafter$)$ are as follows: CAT: $7\mathrm{.}35\%$ JNJ: $27\mathrm{.}32\%$ WMT: $22\mathrm{.}70\%$ NVDA: $21\mathrm{.}40\%$ PG: $21\mathrm{.}23\%$

Based on these weights and an assumed monthly market return of $0\mathrm{.}7\%,$ calculate each stock$$s expected return implied by the market.

After reviewing the implied expected returns, your team believes that some adjustments are necessary. Based on your team$$s research, a downward revision of $0\mathrm{.}15\%$ is warranted for WMT$$s expected monthly return, given the bleak outlook for major retail chains. Conversely, NVDA$$s expected monthly return should be increased by $0\mathrm{.}10\%$ considering the promising potential of graphic processing units. Form an active portfolio, denoted as P$4,$ that incorporates these views and report its weights$\mathrm{.}6.$ Compare the weights for P$3$ and P$4.$ Do their differences make economic sense?$7.$ Evaluate the performance of P$1$ ~ P$4$ using monthly returns from January $2021$ to December $2022.$ Provide investment recommendations tailored to UE$$s situation, taking into account the short sale constraint.