Problem $1.(25$ Points$)$ Ann is looking to immunize a sequence of liabilities due by making some

investments that guarantee specific sequences of asset income. The sequence of liabilities are given

by

$L_1=500,L_2=450,L_3=400,L_4=350$

$L_5=300,L_6=250,L_7=200,L_8=150$

where the subscript indicate the time when the liabilities are due $($e$.$g$.,L_1$ is due at time $t=1).$ Ann

can invest $1000$ today, and the investment will provide the asset income $x$ at time $t=2$ and $Y$ at

time $t=3.$ Given an interest rate $i=0\mathrm{.}05,$ find the values of $x$ and $Y$ that Redington immunize

the sequence of liabilities $($HINT: use the two first conditions for Redington immunization$).$ Use the

third condition to check if $x$ and $Y$ Redington immunize the portfolio.