# Introduction: As discussed in the text and the PowerPoints, the Yield Curve should be constructed using theoretical rates, which represent the spot rates corresponding to each maturity. These rates an

Question

Introduction: As discussed in the text and the PowerPoints, the Yield Curve should be constructed using theoretical rates, which represent the spot rates corresponding to each maturity. These rates and corresponding calculated prices are referred to as “Spot Rates” (or pure discount rate) and “Implied Zeroes” (or zero coupon prices). The purpose of this mini-case is to illustrate how to calculate these implied prices and spot rates. Ideally, the calculated spot rates are the rates that should be used to discount the respective cash flows, instead of using for example a single yield to maturity when determining the value of a bond.

The Problem: Consider the problem of finding the pure discount bond prices from the coupon prices that are available. Table 1 gives data for three bonds for a period of three years.

Table 1: Coupon Bond Prices and Coupon Payments. (Notice that the Cash Flows in years 1 through Year 3 already reflect the coupon payments and the par value at maturity.)

Bond Price Year 1 Year 2 Year 3

1

2

3

99.50

101.25

100.25

105

6

7

0

106

7

0

0

107

Let Pi be the price of bond i. Let ci denote the dollar coupon associated with bond i, and y1 as a one-year spot rate of interest. Then, we can denote the price of the first bond as

P1=(100+c1)/((1+y1))

Then, using known data, solve for y1.

Finally, we can solve for the one-year implied zero price, z1, as follows:

z1=1/((1+y1) )=P1/(100+c1)

Note: z1 is also referred to as the “discount factor” or the present value of a $1 to be received at time t in the future. The two terms to the right of the z1 equation are equivalent ways of finding the answer. They should give the same answer.

Following the same procedure, we can solve iteratively for y2 and z2, etc ….

Note that, after solving for yi in one step it becomes a known value in the next step. For example, when solving for y2 and z2, P1, P2 and y1 are known values that you can just plug in the formula for P2 to find y2 and then solve for z2.

Required task: Use the above procedure, called “Bootstrapping” to complete Table 2 below for maturities 2 and 3.

(Note: I provided the answers for maturity 1, for illustration. However, show your work, i.e., show the steps you followed to reach the answers, including your work to verify the answers already provided).

Table 2: Implied Zeroes and Spot Rates. (Answers for maturity 1 are already given to you for illustration. This will not count in your grade.)

Maturity (Years) Implied Zero (zi) Spot Rate (yi)

1

z1 = 0.9476

y1 = 5.53%

2

z2 = ?

y2 = ?

3

z3 = ?

Y3 = ?