CFA Level IFixed IncomeHard
The one-year spot rate is 5.00%. A newly issued, annual-pay, 2-year bond with a 6% coupon rate is currently priced at par ($100). Using bootstrapping, calculate the 2-year spot rate.
- A5.00%
- B6.50%
- C6.03%
- D6.00%
Show answer & explanationAnswer & explanation
Correct answer: C. 6.03%
Set the bond's price equal to the present value of cash flows using spot rates: 100 = 6/(1.05) + 106/(1+z2)^2. The first term equals 5.7143, so 106/(1+z2)^2 = 94.2857, giving (1+z2)^2 = 1.12409 and z2 = 6.03%. This bootstrapped 2-year spot rate exceeds the par coupon rate because the 1-year spot rate is below the coupon, requiring a higher rate on the final cash flow to keep the bond at par.
Why the other options are wrong
- A. This is the given 1-year spot rate, not the bootstrapped 2-year rate.
- B. This overstates the spot rate; it does not satisfy the par-pricing equation with the given 1-year spot rate.
- D. This is simply the bond's coupon rate, not the result of discounting cash flows via bootstrapping.
Bootstrapping Spot Rates
A method of deriving zero-coupon (spot) rates sequentially from the par yield curve by using previously solved spot rates to price successive coupon bonds.
- Uses no-arbitrage pricing: bond price = PV of cash flows at spot rates.
- Each new spot rate is solved using the prior year's already-known spot rate(s).
- Spot rates form the basis for valuing off-market bonds and computing forward rates.
Memory trick: Bootstrapping = pulling yourself up rate by rate, using last year's spot rate to solve this year's.