Please show all the steps:

Question 3. A pension fund has liabilities of $80 and $90 in 12 and 22 years, respectively. The yield for maturities 10,12,20,22 years is found to be 3.0%,3.1%,3.5% and 3.6%, respectively. All bonds are zero-coupon bonds with face value $100. (a) If the pension fund exactly covers its liabilities by holding two bonds, what would be the value of this bond portfolio L ? [1 mark] (b) The bond portfolio in (a) represents the pension fund's liabilities. In particular, the value of the bond portfolio is the present value of the liabilities, and the duration of the bond portfolio is the duration of the liabilities. Find the duration of the liabilities. [2 marks] (c) Suppose the pension wants to hedge its liabilities against a parallel shift in the yield curve s. It buys c units of a bond maturing in 10 years and c units of a bond maturing in 20 years, and this portfolio has value P. Find c such that L, the change in L due to a parallel shift in the yield curve s, is approximately equal to P, the change in P due to a parallel shift in the yield curve s. [2 marks] (d) Compute (PL)=PL separately in the following situations: - A small parallel shift in the yield curve s=0.1%. - A large parallel shift in the yield curve s=1.5%. - A partial hedge due to the pension fund not being well-funded enough so that c is a quarter of the value calculated in (c), and a small parallel shift in the yield curve s=0.1%. Then comment on how well the hedge performs. [4 marks] Question 3. A pension fund has liabilities of $80 and $90 in 12 and 22 years, respectively. The yield for maturities 10,12,20,22 years is found to be 3.0%,3.1%,3.5% and 3.6%, respectively. All bonds are zero-coupon bonds with face value $100. (a) If the pension fund exactly covers its liabilities by holding two bonds, what would be the value of this bond portfolio L ? [1 mark] (b) The bond portfolio in (a) represents the pension fund's liabilities. In particular, the value of the bond portfolio is the present value of the liabilities, and the duration of the bond portfolio is the duration of the liabilities. Find the duration of the liabilities. [2 marks] (c) Suppose the pension wants to hedge its liabilities against a parallel shift in the yield curve s. It buys c units of a bond maturing in 10 years and c units of a bond maturing in 20 years, and this portfolio has value P. Find c such that L, the change in L due to a parallel shift in the yield curve s, is approximately equal to P, the change in P due to a parallel shift in the yield curve s. [2 marks] (d) Compute (PL)=PL separately in the following situations: - A small parallel shift in the yield curve s=0.1%. - A large parallel shift in the yield curve s=1.5%. - A partial hedge due to the pension fund not being well-funded enough so that c is a quarter of the value calculated in (c), and a small parallel shift in the yield curve s=0.1%. Then comment on how well the hedge performs. [4 marks]