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Bootstrapping and Interest Rate Example Problem

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One method used to obtain an estimate of the term structure of interest rates is called bootstrapping. Suppose you have a one-year zero coupon bond with a rate of r1 and a two-year bond with an annual coupon payment of C. To bootstrap the two-year rate, you can set up the following equation for the price (P) of the coupon bond: P=C_1/(1+r_1 )+(C_2+Par value)/(1+r_2 )^2

Because you can observe all of the variables except r2, the spot rate for two years, you can solve for this interest rate. Suppose there is a zero coupon bond with one year to maturity that sells for $949 and a two-year bond with a 7.5 percent coupon paid annually that sells for $1,020. What is the interest rate for two years? Suppose a bond with three years until maturity and an 8.5 percent annual coupon sells for $1,029. What is the interest rate for three years?

Solution Preview

What is the interest rate for two years?
$1,020 = 1,000*7.5%/[1+($1,000-$949)/$1,000] + ($1,000*7.5% + $1,000)/(1 + r2)^2
$1,020 = 75/[1+($51)/$1,000] + ($75 + $1,000)/(1 + r2)^2
$1,020 = 75/[1+5.1%] + ($1,075)/(1 + r2)^2
$1,020 = 75/1.051 + ($1,075)/(1 + ...

Solution Summary

This solution provides assistance with determining the interest rate for each of the problems.