Problem 1)
a) Using an interest rate of R=4%, show that as m becomes very large, that (1 + R/m)m approaches eR
b) Using an Rm of 5% and m = 10 check that the two formulas on the Conversion Formulas slide are correct. (First calculate Rc and then recalculate Rm using the Rc you obtained).
Problem 2)
a) Find the price of a 7 year semi-annual bond with a coupon of 6% (annualized) and a continuously compounded yield of 5%.
b) Now use Goal Seek to calculate the yield given the price you obtained in part a). You should obtain 5%. Hint: Use Goal Seek to find the rate that sets the difference between your newly calculated price using a “guess yield” and the price from part a) to zero.
Problem 3)
Using the four “zero rates” from the Example slide, use Goal Seek to find the Par Yield of a two year semi-annual bond. Hint: Discount each cash flow using the appropriate zero rate, calculate the price, and then find the coupon that sets this price equal to $100 using Goal Seek.
Problem 4)
Consider another two year semi-annual bond with a coupon of 5% (annualized) and a price of $105. You are given that the .5 year zero rate is 2.0%, 1 year zero rate is 2.1% and 1.5 year zero rate is 2.2%. Use Goal Seek to “bootstrap” to find the 2-year zero rate.
Problem 5)
Suppose you are given the following zero rates:
Find the one year forward rates for years 2 thru 5. (T2 – T1 = 1 for all four calculations).
Problem 6)
a) Consider a 5-year semiannual bond with a coupon of 5% (annualized) and a yield of 6.00% Calculate its price and then its duration.
b) Now calculate the price again by changing the yield to 6.50%. Show that
holds approximately. What is the error?
c) Now calculate the Convexity. Repeat part b) using
Does this reduce the error?
Sample Solution