FRM Part I · FRM Exam Part I · Properties of Interest Rates
Continuously compounded zero rates are 3% for 1 year and 4% for 2 years. A 3-year bond with face value 100 pays a 5% annual coupon and is priced at par (100). What is the implied 3-year zero rate, continuously compounded?
The implied 3-year zero rate is about 4.94% continuously compounded. After discounting the two earlier coupons at the 1- and 2-year zero rates, 90.53 remains as the present value of the 105 final payment, and solving for the rate gives 4.94%. The 5.00% par yield is not the zero rate.
- A4.94%Correct
- B5.07%
- C5.00%
- D4.50%
Explanation
The PV of the first two coupons is 5e^(-0.03) + 5e^(-0.08) = 4.853 + 4.616 = 9.469. The final cash flow 105 must be worth 100 - 9.469 = 90.531, so e^(-3r) = 0.86221 and r = 0.14826/3 = 4.94%. The 5.07% option is the same value expressed with annual compounding. The 5.00% option is the par yield, which is not a zero rate.
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