Please give a full explanation and show workings please. Thanks

8. (a) Explain the concept of Macaulay duration and explain the relationship between Macaulay duration and: (i) bond maturity (ii) interest rates (iii) bond coupon rate (7 marks) (b) Calculate the price and Macaulay duration of a five-year 5% coupon bond where the market interest rate is 5%. Assume the par value of the bond is $1000 and coupons are paid annually. (6 marks) (c) An investor decides to construct a bond portfolio made up of $20,000 in a bond with a Macaulay duration of 5 years and $30,000 in a three-year zero- coupon bond (par value = $1,000). What is the Macaulay duration of this bond portfolio? (4 marks) (d) Estimate, using modified duration, the change in the price of the five-year 5% coupon bond (described in part (b)) if the market interest rate decreases from 5% to 4%. (4 marks) (e) Explain why the modified duration measure only gives good estimates of price changes when the change in the market interest rate being considered is small. (4 marks)