Bootstrapping equation

I have no idea how bootstrapping works. This formula owns me = ( Any explanations? Given the following Treasury data, what is the 1-year spot rate? Maturity YTM Coupon Price 6 months 1.0% 1.0% 100 1 year 1.5% 1.5% 100 18 months 2.5% 2.5% 100 A) 1.50%. B) 1.13%. C) 2.25%. D) 1.51%. The correct answer was A) 1.50%. The bond with 6 months left to maturity has a semiannual discount rate of .01/2 = .005 therefore the 1-year spot rate can be found by solving the following equation: 0.75/1.005 + 100.75/(1 + S1.0/2)2 = 100 Solving for S1.0/2: 100.75/(1 + S1.0/2)2 = 100 - 0.75/1.005 100.75/(1 + S1.0/2)2 = 99.2537 100.75 / 99.25370.5 = (1 + S1.0/2)2 (100.75 / 99.254).5 – 1 = S1.0/2 0.007509 = S1.0/2 2 × 0.007509= S1.0 S1.0 = 0.01502 or 1.5%

If you were instead solving: 0.75/1.005 + 100.75/(1 + S1.0) = 100 would that make it better (I like the second mine better than I like theirs)? The solution aove looks pretty clear. What problems are you having with it?

Why is the pmt 0.75 instead of 0.5 (half of 1%)?

Because the 1-yr bond pays a 0.75 coupon and you are trying to figure out the spot rate for 1-yr. That means you discount the 6-month coupon of the 1-yr bond by the 6 month spot rate and the 1-yr payment by the 1-yr spot rate.

But why are we even calculating anything here… Isnt the second bond… ie. 1 year 1.5% 1.5% 100 giving us the definition of spot rate which is 1.5% ??? Maturity YTM Coupon Price 6 months 1.0% 1.0% 100 1 year 1.5% 1.5% 100 18 months 2.5% 2.5% 100

Joey, thanks a lot for the input! But where does it say that the 1-yr bond pays a 0.75 coupon? I feel like I’m blind or sth…

1 year pays a 1.5 coupon semi annually. That’s .75 each 6-month period.

Damn. 10x. All I could see was the 1% coupon for the 6-months, just ignored the other rate…

BullPow Wrote: ------------------------------------------------------- > But why are we even calculating anything here… > > Isnt the second bond… > > ie. 1 year 1.5% 1.5% 100 > > giving us the definition of spot rate which is > 1.5% ??? > > > Maturity YTM Coupon Price > > 6 months 1.0% 1.0% 100 > > 1 year 1.5% 1.5% 100 > > 18 months 2.5% 2.5% 100 I was thinking like this as well. Can someone explain ?

no… it’s giving the 1-year forward rate . i think it’s a coincidence they are the same in this case

BullPow Wrote: ------------------------------------------------------- > But why are we even calculating anything here… > > Isnt the second bond… > > ie. 1 year 1.5% 1.5% 100 > > giving us the definition of spot rate which is > 1.5% ??? > > > Maturity YTM Coupon Price > > 6 months 1.0% 1.0% 100 > > 1 year 1.5% 1.5% 100 > > 18 months 2.5% 2.5% 100 The second bond gives us the ytm of a coupon bond since it pays a semiannual coupon. The 1 yr spot rate is the interest rate for a zero coupon bond for 1 yr.

Joey - you are wonderful! This was giving me trouble as well, and you made it so simple! Thanks!

thanks Joey

Not sure what I explained there…