3.30.2005

Q: Bootstrap the spot rate for...

...the 4th coupon payment period, given the following market data for $100 par Treasury bonds

Period Years Annual Yield to Maturity (BEY)(%) Price Spot Rate (BEY) (%)
1 .5 3.0   3.0
2 1.0 3.3   3.3
3 1.5 3.5 100 3.5
4 2.0 4.1 100 xxx

A

  • 2-year Treasury coupon = 4.1 / 2 = 2.05
  • 1st period cash flow = 2.05 / (1+3/2/100) = 2.02
  • 2st period cash flow = 2.05 / (1+3.3/2/100)^2 = 1.98
  • 3st period cash flow = 2.05 / (1+3.5/2/100) ^3= 1.95
  • 4st period cash flow = (100+2.05) / (1+R/2/100) ^4 = Price - 1st period cash flow - 2st - 3rd = 100 - 2.02 - 1.98 - 1.95 = 94.05. In Excel's rate function, enter FV = 102.055, PV =-94.05, Nper = 4, and one gets R =2%.
  • annualize the spot rate, one should get 4%

Bonus Points

  • The basic principle underlying the bootstrapping method is that the value of a Treasury coupon security should be equal to the value of the package of zero-coupon Treasury securities that duplicates the coupon bond's cash flows.
  • Since the bond is at par the coupon will be the annual YTM multiplied by par.
  • Since spot rates are normally quoted on an annual basis, the calculation needs to adjusted to a semiannual rate.

Category: C++ Quant > Debt > Valuation

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