$($a$)$ As the Financial Risk Manager of a company you have been asked to hedge the interest rate risk for a liability of $1,000,000$ that is due to be paid in exactly $4$ years. Today is $29/05/2015,$ hence the payment is due on $29/05/2019.$ You have several options to hedge your liability by investing in different UK gilts with different maturities:

$1.$ You can perfectly hedge the payment by buying some quantity of a zero$-$coupon UK treasury bond with the same maturity as your liability. $($apply duration and zero coupon$)$

$2.$ You can hedge the payment by buying some quantity of a zero$-$coupon UK treasury bond with a maturity of $2$Y$.$

$3.$ You can hedge the payment by buying some quantity of a zero$-$coupon UK treasury bond with a maturity of $5$Y$.($portfolio bond$)$

$4.$ You can hedge the payment by buying a combination of the zero$-$coupon UK treasury bonds from Part $(2)$ and Part $(3),$ such that the resulting portfolio has the same market value and the same duration as the liability at the time of the purchase.

$[$Hint: Portfolio $($Bond A and Bond B$)$ Duration $=(\%$ in A $*$ Duration of A$)+(\%$ in B $*$ Duration of B$)]$

Start by calculating the current market value $($present value$)$ and determine the duration of the liability on $29/05/2015.$ Next, compute the current prices of the three zero$-$coupon bonds and determine their durations. Compute the number of bonds you need to purchase for each hedging strategy.

$($b$)$ One year later on $30/05/2016,$ you re$-$assess your hedging options. Re$-$calculate the market value of your liability and of each of the four bond positions on $30/05/2016.$ Which of the four hedging options has best tracked the liability$$s market value, which has done the worst job? Discuss and explain your findings. What financial theory can be linked to this?