Question $1$

California Public Emplovees' Retirement System $($CalPERS$)$ 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

billion in benefits in $2019.$

In $2016,$ CalPERS determined that it would target an expected rate of return from its investments of $7\%.$ Suppose

CalPERS 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 CalPERS' 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 $7\%?$

b$)$ If the S&P $500$ index return has a volatility of $15\mathrm{.}30\%($i$.$e$.$ roughly equal to the S&P $500$ index's historical

annual volatility$),$ compute the volatility of CalPERS' portfolio.

The $3-$month treasury yield increases to $5\mathrm{.}25\%.$ As a result of a potential decline in economic activity, S&P $500$ index is

now expected to return $5\mathrm{.}00\%$ more than the $3-$month treasury yield $($i$.$e$.$ the expected market risk premium is now

$5\mathrm{.}00\%).$ Lastly, as a result of expectations for increased inflation going forward and recent declines in the stock

market, CalPERS will now require a return of $11\mathrm{.}75\%$ from its portfolio in order to meet its liability obligations.

CaIPERS decides to revise its investment approach as follows. It will continue to hold the S&P $500$ index $($market

portfolio$)$ but will no longer hold $3-$month T$-$bills $($risk$-$free asset$)$ in its portfolio. In addition to the S&P $500$ index,

CalPERS will now invest in a portfolio of either low$-$grade$/$high$-$yield bonds $("$junk bonds"$)$ or private equity. The S&P

$500$ index is still expected to have a volatility of $15\mathrm{.}30\%$ and CalPERS has the following information on junk bonds and

private equity:

c$)$ What would CalPERS' new portfolio allocation $($portfolio weights$)$ have to be in order to achieve its required

expected return of $11\mathrm{.}75\%$ if in addition to the S&P $500$ index CalPERS invested in:

i$.$ Junk bonds

ii$.$ Private equity

d$)$ Compute CalPERS' portfolio volatility in each case $($i$.$e$.$ cli, and c$)$ii$.).$ Should CalPERS invest in the portfolio

that includes junk bonds $($c$)$i$,)$ or private equity $($c$)$ii$.)?$