Bootstrapping equation

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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?
 
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!
 
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