Question $4-(35$ points$)$

A company has liabilities of $100$ million due in $10$ years and $200$ million

due in $20$ years. The company's assets consist of a zero coupon bond and an

annuity paid annually in arrears. The zero coupon bond matures in $20$ years,

while the annuity pays cash flows annually with a maturity of $15$ years. The

current interest rate is $5\%$ per annum effective for all terms to redemption.

$(1)$ Determine the present value of the liabilities and calculate the aver$-$

age term to discount of the liabilities.

$(2)$ Determine the cash flows of the two assets: the notional value of the

zero coupon bond and the size of the payments of the annuity that

allow matching the two conditions needed for immunisation against

fluctuations in the interest rate.

$(3)$ Show on a Excel file the level of fluctuation of the interest rate,

denoted as $r,$ that corresponds to a relative difference between

the true value of $P{V}_A,$ denoted as $P{V}_A(hat(r)),$ and that computed by

the first$-$order approximation based on Taylor expansion, denoted as

hat$(PV{)}_A,$ equal to $5\%($i$.$e$.,\{$:$\frac{\text{h}\text{a}\text{t}(PV{)}_A-P{V}_A}{P}{V}_A).$