[Solved] What is the implied value of the call feature?

[Solved] What is the implied value of the call feature?

Problem: The bond in the middle is callable. What is the implied value of the call feature? (Hint: Is there a way to combine the two noncallable issues to create an issue that has the same coupon as the callable bond?)

Difficulty Level: Hard

If you have never seen a problem like this one, then figuring out how to solve it can be very difficult. However, as you will see, the solution makes a lot of sense once you understand the thought process behind it. The key words provided in the hint are "combine" and "the same coupon." Here is an example data set of bond issues.

Assume it is the year 2040 and the middle bond (Bond 2) is callable May 2044 in 4 years. Notice that both the bid and ask prices for Bond 2 are less than those for Bond 1, even though Bond 1 also has a lower coupon rate. This lower price on Bond 2 is due to the call feature on Bond 2, which can be called 4 years earlier than the other two bonds. To solve this problem, we do not need to know the amount of time until maturity or the yield (hence, no yield column). We only need to look at the coupon rates and the current ask prices.

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 5.9% 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 5.9%. The formula to do this is:

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

  • Rate 1, 2, and 3 are the coupon rates for Bonds 1, 2, and 3.
  • X represents our percentage of money in Bond 1 and 1-X represents our percentage of money in Bond 3.

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:

5.9 = 4.2X + 7.3 (1 – X)

Note: It is alright to use the percentage form of the coupon rate instead of decimal form (meaning using 5.9 instead of 0.059) because it does not affect this portion of the problem. However, in the next step (after we solve for this formula), it will matter that the percentages are used as the appropriate decimals. More on this subject in Step 2.

After a little algebraic distribution of the 7.3 * (1 – X), we get 7.3 – 7.3X:

5.9 = 4.2X + 7.3 – 7.3X

We can move the 7.3 to the left side by subtracting it from both sides, and then combine terms.

5.9 – 7.3 = 4.2X – 7.3X

We will end up with negatives on both sides of the equation, but these will ultimately cancel out.

–1.4 = –3.1X

Now we simply get X by itself by dividing by –3.1 on both sides:

0.4516 = X

So 45.16% is the amount of money we need to invest in Bond 1 in order to get an average rate of return of 5.9% like we could get from Bond 2. Solving for the simple 1 – X shows us that 0.5484 (54.84%) needs to be invested in Bond 3. If you want, you can plug these percentages back into the formula to make sure that (4.2 * 0.4516) + (7.3 * 0.5484) = 5.9. Now we move on to the next step.

Step 2: Solve for Bond 2's Expected Price

Now this is where it gets interesting. We used the coupon rates to estimate what percentage of our money should be invested in Bond 1 and Bond 3 in order to equal the rate of Bond 2. We were able to do this because we knew all three coupon rates. However, the basis of this question implies that the current price of Bond 2 is not what it should be, due to the "callable feature." And in fact, if you look at the price of Bond 2, you could probable deduce that something was affecting the price since it is lower than both Bond 1 and Bond 3 despite having a coupon rate in the middle of the two other bonds.

So since we now know what percentage of our investments in Bond 1 and Bond 3 will equal Bond 2's coupon rate, we can extrapolate this information and apply it to the bond price in order to estimate what the price of Bond 2 should be without the callable feature. Essentially, we are wanting to "pay" for Bond 2's price without buying Bond 2, which we can do by purchasing a portion of the cheaper Bond 1 and a portion of the more expensive Bond 3 (we already solved for these portions using the coupon rates).

Again, our basic assumption is that the prices of Bond 1 and Bond 3 are correct, but that the price of Bond 2 is not what it should be because it is affected by the callable feature. The formula we use looks basically the same as the previous formula:

Price 2 = (Price 1 * 0.4516) + (Price 3 * 0.5484)

Note: As mentioned previously, it matters that we are using decimals for our "weighing" percentages because we are using them to "weigh" the prices. You will get an error if you try to use 45.16 instead of 0.4516. Technically, our prices are in percentage form too, but because we are not using the price to weigh anything, it does not affect our answer to use the percentage version instead of decimals. So we can use 112.78 instead of the appropriate decimal of 1.1278 (112.78%).

Now, simply plug in the prices we know are unaffected by a call feature, which is the prices for Bonds 1 and 3, and then solve for what the price of Bond 2 should be. Always use the Ask prices when using the price of the bond. The Ask price is always the price an investor will pay to purchase the bond (it is the dealer's asking price).

Price 2 = (112.78 * 0.4516) + (114.61 * 0.4516)

Price 2 = (50.93) + (62.85)

Price 2 = 113.78

The price of Bond 2 should be 113.78 (or 113.78% of the face value) – this makes sense since the price of Bond 1 is 112.78 and the price of Bond 3 is 114.61. However, the actual price of Bond 2 is 109.65. The difference between what the price of Bond 2 should be (113.78) and the actual price of Bond 2 (109.65) is the implied value of the call feature.

113.78 – 109.65 = 4.13

Remember that the prices are actually percentages of the face value, so 4.13 is actually 4.13% of the bond's face. Typically we assume bond values to be $1000, so the dollar value of the call feature is $41.30.

Answer: $41.30


Congratulations! You now know how to solve this difficult problem!


Deriving the Formula for Future Value with an Annuity

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