Financial Mathematics (FIN3073) Assignment 1
A soft copy of hand-written solution, completed independently, should be submitted by 5 PM Oct 08 (Tuesday)
Question 1 (Total 10 marks)
The six-month and one-year zero rates are both 10% per annum. For a bond that has a life of 18 months and pays a coupon of 8% per annum (with semiannual payments and one having just been made), the yield is 10.4% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with semiannual compounding.
Question 2 (Total 10 marks)
An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per annum with a) annual compounding, b) semiannual compounding, c) monthly compounding and d) continuous compounding.
Question 3 (Total 10 marks)
Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6.5%, and 7% respectively. What is the two-year par yield?
Question 4 (Total 10 marks)
Suppose that the 9-month and 12-month LIBOR rates are 2% and 2.3%, respectively. What is the forward LIBOR rate for the period between 9 months and 12 months? What is the value of an FRA where 3% is received and LIBOR is paid on $10 million for the period? All rates are quarterly compounded. Assume that LIBOR is used as the risk-free discount rate.
Question 5 (Total 20 marks)
A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of each year.
a) What is the bond’sprice? (5 marks)
b) What is the bond’s duration? (5 marks)
c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield. (5 marks)
d) Recalculate the bond’s price on the basis of a 10.8% per annum yield and verify that the result is in agreement with your answer to (c). (5 marks)
Question 6 (Total 20 marks)
Consider an annual bond with 3 years before maturity, which pays 8% annual coupon on the bond face value $100. The zero rates for one year, two years, and three years are 9%, 10%, and 11% respectively.
(a) Calculate the bond price. (4 marks)
(b) Explain what bond yield is. Then, find which one of the following is the bond yield: 9.28%, 9.87%, 10.42%, 10.89%, 11.12%, or 11.39%? Show calculations. (4 marks)
(c) Calculate the bond duration for this bond. (4 marks)
(d) If you find from your market that the bond yield is higher than what you calculated above, by half a percentage point, how would you revise your bond valuation? (4 marks)
(e) The “par yield of a bond” is defined to the coupon rate that makes the bond price equal to the bond par value. Calculate the par yield of a 3-year annual bond. (4 marks)
Question 7 (Total 20 marks)
The following table gives the relevant information of four bonds.
|
Bond #
|
Time to Maturity (years)
|
Semiannual Coupon ($)
|
Bond Price (% of par)
|
|
1
|
0.5
|
0.0
|
97
|
|
2
|
1.0
|
0.0
|
95
|
|
3
|
1.5
|
8.0
|
99
|
|
4
|
2.0
|
10.0
|
98
|
a) Calculate the zero rates for maturities of 6 months, 12 months, 18 months, and 24 months. (7 marks)
b) What are the forward rates for the 6-month periods: months 7 to 12, months 13 to 18, months 19 to 24? (7 marks)
c) Estimate the price and the yield of a two-year bond providing a semiannual coupon of 7% per annum. (6 marks)