### CFA Practice Question

A bond with face value \$1,000 and 4 years to maturity has 5% yearly coupons. The one year, two year, and three year Treasury spot rates are: 4%, 4.25%, and 4.75%. The price of the bond (present value) today is \$1,030. What is the 4 year spot rate?
A. 4.15%
B. 4.19%
C. 4.17%
Explanation: The bond pays yearly coupons of \$50. The present values of the first 3 coupons are 50/(1+0.04), 50/(1+0.0425)^2 and 50/(1+0.0475)^3 that is, 48.08, 46.01, and 43.50 respectively. The final payment is 1,050. The initial investment for the final payment is \$1,030 - 48.08 - 46.01 - 43.50. The 4 year spot rate is the rate that will grow this amount to 1,050 over 4 years.

### User Contributed Comments4

User Comment
rubin1 can somebody explain in detail
chandsingh this is a bootstrapping question. So another way to compute this is what is the final spot rate that can discount the final payment of 1050 so that when it is added to the 48.08, 46.01 and 43.50, it would equal the bond price of \$1030.
You will note from the question, that each cashflow from the bond is discounted by its corresponding spot rate.
safash 48.08+46.01+43.50+1050/(1+z)^4=1030
137.59+1050/(1+z)^4=1030
1050/(1+z)^4=892.41
1050/892.41=(1+z)^4
1.041493=1+z
.041493=z
=4.14%
jjhigdon 1) Draw a timeline with all the coupon payments and the principal payment. 2) Discount payments 1-3 by their applicable spot rates 3) Subtract discounted CFs from Price 4) You then are left with the discounted value of the last coupon and principal payment, solver for r.
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