Question

REQUIRED Let the continuously compounded zero interest rates for 6, 12 and 18 months be: r05-4%, ri -5%, and r1.5-5.9%, p.a. respectively. Calculate the prices of a 6-month zero-coupon note a 1-year bond with 7% annual coupon rate (semi-annual payment), and a 15-year coupon bond with 3% annual coupon rate (semi-annual payment). Assume a bond face value of £100 a) (7 marks) b) Calculate the annualised yield to maturity for each security from question (a) and express it both in terms of semi-annual rates as well as continuously compounded rates. In what circumstances can you interpret the yield to maturity as a (certain) rate of return on a bond? (8 marks) c) Consider the term structure of zero rates stated in question (a). If an FRA can deliver a 6% p.a. (continuously compounded) 6-month rate commencing in 1 year, devise an arbitrage strategy. What is the present value of the profit? (10 marks) d) Using the relevant information from question (a), calculate the value of a forward rate agreement that enables the holder to earn a semi-annually compounded interest of 6.2% for a 6-month period starting in 6-months on a principal of £10 million? (5 marks) Is the rate of return on a coupon bond certain if it is going to be held until maturity? Motivate (5 marks) e) your answer f) You hold a portfolio of two coupon bonds with the same maturity and different coupon rates The total value of the portfolio is divided equally between two long positions on these bonds Describe which factor or factor(s) affect their price sensitivity. Finally, if a small parallel rise of the zero rates is imminent, which bond and why it would be ideal to sell if you could only sell one of these two bonds (10 marks) g) Consider a decreasing term structure of zeros rates. Briefly explain why the yield to maturity of a coupon bond that matures at time T is always greater than the zero rate for that maturity (5 marks)

Answer 1

Answer:

Question A)

Given

6 months continuous compound rate=4% pa

6 months effective rate = e^(4%)-1=4.08% pa

6 months effective rate r=2.04% per semi annual

Price of 6 month zero coupon bond

Coupon C=0

face value of Bond A= 100

Price of bond P= A/(1+)^n

r= interest rate

n= periods

P=100/(1+2.04%)^1=98

12 months continuous compound rate=5% pa

12 months effective rate = e^(5%)-1=5.13% pa

12 months effective rate r=2.56% per semi annual

Price of 12 month 7% coupon bond

Coupon C=7%*100/2=3.5 per semi annual

face value of Bond A= 100

Price of bond P=C/(1+r)^1+C/(1+r)^2+ A/(1+r)^2

r= interest rate

P=3.5/(1+2.56%)^1+3.5/(1+2.56%)^2+ 100/(1+2.56%)^2=101.81

18 months continuous compound rate=5.9% pa

18 months effective rate = e^(5.9%)-1=6.08% pa

18 months effective rate r=3.04% per semi annual

Price of 18 month 3% coupon bond

Coupon C=3%*100/2=1.5 per semi annual

face value of Bond A= 100

Price of bond P=C/(1+r)^1+C/(1+r)^2+C/(1+r)^3+ A/(1+r)^3

r= interest rate

P=1.5/(1+3.04%)^1+1.5/(1+3.04%)^2+1.5/(1+3.04%)^3+100/(1+3.04%)^3=95.65