Asset liability management $($Answer all parts of this question.$)$

A pension fund has to pay $100,000$ for the next three years $($at the end of each year$).$

Suppose that the nominal interest rate is $5\%,$ consisting of a $2\%$ expected inflation

rate and $3\%$ real interest rate. The yield curve is supposed to be flat, and the

expected real interest rate and inflation rate is the same for all maturities. Suppose

the current value of the pension fund's investment portfolio is $300,000.$

$($a$)(3$P$)$ Calculate the liability of the pension fund.

$($b$)(3$P$)$ Calculate the funding ratio and the surplus of the fund.

$($c$)(5$P$)$ Suppose the pension fund would like to implement a cash$-$flow matching

strategy. In which securities would the fund invest, and how much? What

is the cost $($i$.$e$.,$ current price$)$ of this investment portfolio? Suppose that

inflation$-$indexed zero$-$coupons bonds with all possible maturities are available.

$($d$)(5$P$)$ Now assume that the fund would like to implement duration matching

instead. What is the duration of the pension fund's liability? Now assume

that only a $3-$year inflation$-$indexed zero$-$coupon bond is available. In which

securities would the fund invest, and how much? What is the cost $($i$.$e$.,$ current

price$)$ of this investment portfolio?

$($e$)(4$P$)$ Discuss the problems of cash$-$flow immunization and duration matching.