CFAI Reading 42 Practice Problem 2

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Muskie

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This question deals with bootstrapping and spot rates, you are given Exhibit 1 and 2 then asked which of three bond prices is correct.
Exhibit 1
Three-Year, €100 par, 3.00% Coupon, Annual Pay Option-Free Bond
Exhibit 2
Yield to Maturity Par Rates for One-, Two-, and Three-Year Annual Pay Option-Free Bonds
One-year Two-year Three-year
1.25% 1.50% 1.70%

The answer is C 103.7815 Euros

What I don’t understand is why Year 2 spot rate (z2): 100 = 1.5/1.0125 + 101.5/(1+z2)^2 = 0.015019 why is the coupon 1.5? Is that the BEY is that the Two-year YTM, it says in exhibit one a 3.0% coupon so I want to use 3 as the coupon.

I’ve looked elsewhere online and I haven’t come up with a solid explination.
 
I guess no one else started with the new and changed material… I swear it is in my digital copy.
 
Muskie wrote:This question deals with bootstrapping and spot rates, you are given Exhibit 1 and 2 then asked which of three bond prices is correct.
Exhibit 1
Three-Year, €100 par, 3.00% Coupon, Annual Pay Option-Free Bond
Exhibit 2
Yield to Maturity Par Rates for One-, Two-, and Three-Year Annual Pay Option-Free Bonds
One-year Two-year Three-year
1.25% 1.50% 1.70%
The answer is C 103.7815 Euros
What I don’t understand is why Year 2 spot rate (z2): 100 = 1.5/1.0125 + 101.5/(1+z2)^2 = 0.015019 why is the coupon 1.5? Is that the BEY is that the Two-year YTM, it says in exhibit one a 3.0% coupon so I want to use 3 as the coupon.
I’ve looked elsewhere online and I haven’t come up with a solid explination.
Click to expand...
Remember what a par curve is: the coupon rate for bonds priced at par. Thus, the par rate for a given bond is its YTM (a bond will be priced at par if its coupon equals its YTM).
So, a 2-year bond priced at par will have a 1.50% coupon. A 3-year bond priced at par will have a 1.70% coupon.
The spot rates are derived like this:
s1 = 1.25%
100 = 1.50 / 1.0125 + 101.50 / (1 + s2)²
(1 + s2)² = 101.50 / (100 − 1.50 / 1.0125) = 1.030263
1 + s2 = 1.015019
s2 = 1.5019%
100 = 1.70 / 1.0125 + 1.70 / 1.015019² + 101.70 / (1 + s3)³
(1 + s3)³ = 101.70 / (100 − 1.70 / 1.0125 − 1.70 / 1.015019²) = 1.052023
1 + s3 = 1.017049
s3 = 1.7049%
Take it from here.
 
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