# bootstrapping

Given the following information, compute the 1-1/2 year spot rate: Maturity Price YTM Spot rate .5 yr. 100 4.25% 4.25% 1.0 yr. 100 4.5% 4.51% 1.5 yr. 100 4.80% 2.0 yr. 100 5.00% a. 4.38% b. 4.51% c. 4.81% d. 4.90%

I would go for (b)4.51%

C?

The answer is C. .5 yr .5 x .0425 x 100 +100 = 102.12 102.12 / (1+z1) = 100 (1 + z1) = 1.0212 z1 = .0212 BEY = .0212 x 2 = .0424 1yr .5x .045 x 100 = 2.25 .5 x .045 x 100 +100 = 102.25 (2.25 / 1 + .0212) + 102.25 / (1 + z2)^2 = 100 2.20 + 102.25 / (1 + z2)^2 = 100 102.25 / (1 + z2)^2 = 97.80 z2 = .02249 1.5 year 2.40 / 1.0212 + 2.40 / 1.02249 + 102.40 / (1 + Z3)^3 = 100 z3 = .024177 .024177 x 2 = .0483 C

We’ve got the YTMs, and because the bonds are at par, the coupons will be the YTM*100*0.5, which you’ve done. Fine. But in the next part, all you need to do is to discount the coupons (2.125, 2.25, 2.4+100) at the spot rates given for each period (S_1 = 4.25%/2, S_2 = 4.51%/2, S_3 = ??), equate to 100, and then quickly solve for the spot rate in period 3 (S_3). You only need to go through the process that you have done when they have only given you forward rates; working with spot rates is simpler.

yep, C is the correct answer…

CFALondon0109 i’m not sure what you’ve done but my method was: 100 = (4.8/1.0425) + (4.8/1.0451^2) + (104.8/(1 + x)^3) and solved for x. the numerators are the ytm*par and each denominator is the equiv spot rate for that period. my answer was: 0.04815 approx: 4.82%

perfect, that’s precisely the way it should be done … cielito

cielito, you’re correct; thanks for the correction However, shouldn’t you use half coupons and half spot rates: 100 = (2.4/1.02125)+(2.4/1.02255^2)+(2.4/[1+x/2]^3)

i thought of that but i assumed as it was based on half years then essentially each coupon is a semi annual coupon?

This is how I would solve it … 100 = (2.4/1.02125)+(2.4/1.02255^2)+(102.4/[1+x/2]^3) Ans = X * 2 - Dinesh S

But YTM is an annual concept, so halving is the way to calc the equiv semi-annual coupons. (Of course the answer that we get from both methods is very similar, but we want to be methodologically correct)

i’ll bear that in mind! thanks!!