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Question $1$

California Public Employees' Retirement System $($CaIPERS$)$ manages a pension fund for public sector employees in the state of California. CalPERS' assets consist of contributions made by employers and employees. CalPERS invests the contributions in order to generate the funds it needs to pay out retirement benefits in the future. With about $2$ million participants in its pension fund, CalPERS' manages assets worth over $$400$ billion and paid out about $$24$ billion in benefits in $2019.$

In $2016,$ CalPERS determined that it would target an expecied rate of return from its investments of $7\%.$ Suppose CaIPERS believes that the CAPM holds and adopts a simple approach toward constructing its investment portfolio to achieve its expected return target $-$ it will only invest in the S&P $500$ index $($market portfolio$)$ and $3-$month treasury bills $($risk$-$free asset$).$ The $3-$month T$-$bills are expected to yield $2\mathrm{.}70\%($i$.$e$.$ the annual risk$-$free rate is $2\mathrm{.}70\%)$ and the S&P $500$ index is expected to return $6\mathrm{.}60\%$ more than the $3-$month T$-$bill yield $($i$.$e$.$ the expected market risk premium is $6\mathrm{.}60\%)$

a$)$ What would CaIPERS' portfolio allocation have to be $($portfolio weights$)-$ i$.$e$.$ what fraction of its portfolio would it need to invest in the S&P $500$ index and what fraction would it need to invest in $3-$month T$-$bills $-$ in order for it to meet its expected return target of $7\%?$ What would CalPERS' portfolio beta need to be in order to have an expected return of $7\%?$