How do we calculate the spot rates when the par rates are given for a bond? For eg, if the par rates of a three year maturity bond are 2.5%, 3% and 3.5% respectively, how do we calculate the spot rates for the corresponding years?
I wrote an article on par rates, spot rates, and forward rates that covers this: http://www.financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/
Full disclosure: as of 4/25 I’ve installed the subscription software on my website, so there’s a charge for viewing the articles.
In a nutshell, you have the par coupon rates from the par curve. The 1-period (1-year in this case) spot rate is the same as the 1-period par rate. To get the 2-period (2-year) spot rate, discount the 2-year par bond’s cash flows using the (known) 1-period spot rate and the (unknown) 2-year spot rate, and solve for the 2-year spot rate, given that the bond sells at par. Now that you have the 2-year spot rate, you can do the same thing with the 3-year par bond to get the 3-year spot rate.
By definition the par rate is the spot rate for the first rate. In your example the par rate of 2.5% if the spot rate for year one.
From there you need to bootstrap - which is a tedious process. By definion a par rate is the coupon rate for a bond trading at par for that maturity. In your example the coupon rate for a 2 year bond would be 3%. Therefore you need to discount year 1 coupon of $3 at 2.5%. Then you need to solve for the rate needed to discount the CF (next coupon + principal) in order to make this bond price at par ($100). Hint - this rate will be higher than the 3%.
Hey Magician - is there a closed form solution for this or do you literally have to bootstrap every rate? For the 30-year spot rate do you have to boostrap out all those rates? I swear I remember learning a closed form solution along the way but cannot remember it.
Each spot rate is calculated using the all previous spot rates (previously calculated), so you need to go step by step. How could you find a formula for that?
Yes: it’s closed-form.
Look at my article.
Took me a little while to get bootstrapping but really once you understand it the whole relationship between par/spot/future rates makes a heck of a lot more sense