# Fixed Income - The Arbitrage-Free Value of an Option-Free Bond Question

I was wondering how the author got the spot rates in the book? I tried figuring out but I haven’t gotten why they obtained the spot rates of 3.015% and 1.40533. Thank you The Arbitrage-Free Value of an Option-Free Bond The yield to maturity for a benchmark one-year annual-pay bond is 2%, for a benchmark two-year annual-pay bond is 3%, and for a benchmark three-year annual-pay bond is 4%. A three year, 5% coupon, annual-pay bond with the same risk and liquidity as the benchmarks is selling for \$102.7751 today (time zero) to yield 4%. Is this value correct for the bond given the current term structure? Solution: The first step in the solution is to find the correct spot rate (zero-coupon rates) for each year’s cash flow.3 The spot rates are 2%, 3.015%, and 4.055%. The correct arbitrage-free price for the bond, then, is P0 = 5/1.02 + 5/1.030152 + 105/1.040553 = \$102.8102 To be arbitrage-free, each cash flow of a bond must be discounted by the spot rate for zero-coupon bonds maturing on the same date as the cash flow. Discounting early coupons by the bond’s yield to maturity gives too much discounting with an upward sloping yield curve and too little discounting for a downward sloping yield curve. The bond is mispriced by \$0.0351 per \$100 of par value.

I’ll explain on the 3.015% example: Start with pricing a 2-year bond using the 2-year yield (3%):

(5/1.03)+(105/1.03^2)=103.8269.

The logic behind the arbitrage free valuation is you need a rate at which it would make no difference whether you invest in 1st and 2nd spot or 2yr upfront. You know the first rate’s spot is 2% (this is given), so just find the missing rate for the second year:

103.8269=(5/1.02)+(105/(1+r)^2) -> (1+r)^2 equals 1.06141 so r=3.025%

Do the same for 3rd year rate and you should get 4.055%

Is the only way to convert from the par yield to spot yield to go through the work you have outlined, or can it be done with the calculator?

For 1 or 2 time periods, its ok, but beyond that, can get very time consuming

There’s no easier way to do it.

You’re correct: it can get time-consuming.

I don’t seem to get it. To compute the spot rate, I did (1+ .03/2)^2. And I’m getting 3.0225% Help please.

Take a look at the article I wrote on par rates, spot rates, and forward rates: http://financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/.

About ⅓ of the way down starts the section on bootstrapping; that’s what you want.

Yay. Thank you.

You’re welcome.

I thought about making a post the other day just to thank you for that article S2000. It really helped me understand the ‘why’ for the par stuff. Thats a rough couple of chapters in the curriculum that could really use some clarification. I skipped FI at the L1 (time) so I thought it may just be me until all of the recent posts.

You’re quite welcome.

Glad it was a help.