Question

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)

please answer the questions according to the marks appointed per question and sub question. this is a banking and finance question so please answer it accordingly.

Answer 1

Please note : The words Duration mean Macaulay Duration in this answer

a) The Macaulay Duration gives the weighted average time in which the bond price is repaid.

The Calculation of Macaulay Duration is

Duration = {(t * CF:/(1+y)+)/P t=1

where CF_t is the Cashflow from the bond at time t (including principal repayment) and P is the price of the bond, y is the Yield to maturity

All else remaining the same

1. Generally, the Duration of a bond increases with the maturity of the bond i.e. a bond with higher maturity will have higher Duration

2. Duration has an inverse relationship with Interest rates (Yield to maturity is assumed to be linked to Interest rates). Hence, an increase in interest rate will decrease the Duration

3. Duration has an inverse relationship with coupon rates Hence, an increase in coupon rate will decrease the Duration.

b) As the coupon rate is the same as interest rate, the bond is trading at par

ie Price of the bond P=$1000

From the Duration formula

D = (1*50/1.05+2*50/1.05^2+3*50/1.05^3+4*50/1.05^4+5*1050/1.05^5)/1000

= 4.545951

So, Duration of the bond is 4.55 years

c) Duration of a portfolio is the weighted average duration of constituents

and Duration of zero coupon bond is the same as its maturity

So,

Duration of portfolio = 20000/(20000+30000)*5 years + 30000/(20000+30000)*3 years

=3.8 years

d)

Modified Duration = Duration/(1+y) = 4.55/1.05 =4.33 years

So, if the market interest rates decreases from 5% to 4%, the price of the bond increases by

= (difference in yields)*modified duration  

=(5%-4%)* 4.33

= 4.33%

So, from the modified duration formula, the price should increase by 4.33% i.e. it should be $1043.33

e) Modified Duration is an approximate measure and the price does not linearly change with respect to change in Interest rates. Hence the measure works best over small changes in interest rates.