If you have par rates for a bond and bootstrap the spot rates, would discounting the cash flows under EITHER par or spot rates produce the same market value.
An example from CFAI (Reading 44, Problem 2) , a three year $100 par at 3.0% coupon is trading at $103.7815.
The YTM/par rates are 1.25%, 1.50%, and 1.70% for 1, 2, and 3 years. If I take the bond above and discount cash flows by the 3-year par rate of 1.7%, I get a value of $103.771.
However, if I extract the spot rates from the par rates (not shown here) and discount the cash flows, I get the $103.7815 shown above.
Is something missing in my logic?
Only if you do the bootstrapping correctly.
Would that mean the CFAI example provided did not bootstrap it correctly?
I got:
- 1-year spot rate = 1.25%
- 2-year spot rate = 1.5019%
- 3-year spot rate = 1.7049%
The value when I discount using the spot rates = $103.7814.
And when I discounted by the YTM I, too, got $103.7711.
The par rate gives the YTM for par bonds, but not the YTM for all bonds; that shouldn’t come as a surprise. (For example, what’s the YTM of a 3-year, zero-coupon bond?)
Thanks, so only bonds that trade are par will you get the same value when you discount by par rates or bootstrapped spot rates?
So the takeway is to always use spot rates to obtain the arbitrage free price of a bond.
No.
The takeaway is that you can use spot rates from the spot curve, or the YTM _ for that particular bond _, but, unless the bond sells at par, you cannot use the YTM from the par curve.