At one point some Treasury bonds were callable Consider the

At one point, some Treasury bonds were callable. Consider the prices on the following three Treasury issues as of May 15, 2016:

The bond in the middle is callable in February 2017. What is the implied value of the call feature? Assume a par value of $1,000. (Hint: Is there a way to combine the two noncallable issues to create an issue that has the same coupon as the callable bond?) (Do not round intermediate calculations. Round your answer to 2 decimal places, e.g., 32.16.)

Call value $

7.45		May	20	n		126.56250		126.62500		?	.62500		5.47
9.20		May	20			123.68750		123.75000		?	.12500		5.43
12.95		May	20			153.84375		154.03125		?	.68750		5.51

Solution

ANSWER TO THIS QUESTION

At one point, some Treasury bonds were callable. Consider the prices on the following three Treasury issues as of May 15, 2016:

7.45

May

20

n

126.56250

126.62500

?

.62500

5.47

9.20

May

20

123.68750

123.75000

?

.12500

5.43

12.95

May

20

153.84375

154.03125

?

.68750

5.51

The bond in the middle is callable in February 2017. What is the implied value of the call feature? Assume a par value of $1,000. (Hint: Is there a way to combine the two noncallable issues to create an issue that has the same coupon as the callable bond?) (Do not round intermediate calculations. Round your answer to 2 decimal places, e.g., 32.16.)

Answer:

Step 1: Find the Weighted Average Equal to Bond 2\'s Coupon Rate

As the hint implies, we will first look at the coupon rates. Imagine that we have money to invest, and we want a 9.20% return, but we cannot invest in Bond 2. The other option is to invest part of our money into Bond 1 and part into Bond 3, so that the average rate of return is 9.20%. The formula to do this is:

Rate 2 = Rate 1 * (X) + Rate 3 * (1-X)

It does not matter where the X or the 1-X goes, just as long as we are consistent. The basic idea is that we have 100% money and we are putting X in one investment and what is left over (1 – X) in the other investment.

Placing the coupon rates into our formula, we have:

9.20 = 7.45X + 12.95 (1 – X)

9.20 = 7.45X + 12.95 – 12.95X

9.20 – 12.95 = 7.45X – 12.95X

–3.75 = –5.5X

0.68181 = X

Step 2: Solve for Bond 2\'s Expected Price

So, we invest about 68% in of our money in bond 1, and about 32% in bond 3. This combination of bonds should have the same value as the callable bond, excluding the price of the call, so:

Price 2 = (Price 1 * 0.68181) + (Price 3 * 0.31819)

Price 2 = (126.62500* 0.68181 ) + (154.03125 *0.31819)

Price 2 = (86.33419) + (49.01120)

Price 2 = 135.34539

The price of Bond 2 should be 135.34539 (or 135.34539% of the face value) – this makes sense since the price of Bond 1 is 126.62500 and the price of Bond 3 is 154.03125. However, the actual price of Bond 2 is 123.75000. The difference between what the price of Bond 2 should be (135.34539) and the actual price of Bond 2 (123.75000) is the implied value of the call feature.

135.34539 – 123.75000 = 11.59539

Remember that the prices are actually percentages of the face value, so 11.59539 is actually 11.59539% of the bond\'s face. Typically we assume bond values to be $1000, so the dollar value of the call feature is $11.59539.

Answer: $115.9539

7.45		May	20	n		126.56250		126.62500		?	.62500		5.47
9.20		May	20			123.68750		123.75000		?	.12500		5.43
12.95		May	20			153.84375		154.03125		?	.68750		5.51