🔎 Yield to Maturity/Call/Worst

Background: Coupon Rate

Coupon Rate = a cash interest payment that you receive every year for owning a bond.

• Coupon Rate is not full rate of return you get from holding a bond - it is just part of the return you get. You may get additional return because you purchase the bond at a discount. For more information, see 🔎 Why are i and c different?

Yield to Maturity (YTM)

Yield to Maturity = the total return you receive for purchasing a bond and holding the bond until it matures.

The total return you get from an investment is always calculated using the Internal Rate of Return (IRR) of purchasing the investment. Therefore:

• YTM is the Internal Rate of Return you would receive if you purchase a bond at a specific price and hold it to maturity.

Yield to Call (YTC)

some bonds are callable, so you may not get to hold the bond until maturity because the issuer calls it. We need to calculate the yield to call, which reflects the return you would get if the issuer calls the bond.

Yield to Call = the total return you receive for holding a bond until it is called (assuming the bond is called at the earliest possible date)

In other words:

• Yield to Call is the IRR of purchasing a bond and then having it be called at the earliest possible date.

Yield To Worst (YTW)

Unfortunately, we now have two interest rates:

• Yield to maturity: the interest rate if it is not called and we hold it to maturity.
• Yield to call: the interest rate if it is called.

How do we know which to use?

Unfortunately, we cannot know which to use. In this situation, it is always important to be prepared for the worst-case scenario. Therefore, investors sometimes wish to calculate the yield to the worst outcome. This is the IRR they will receive in the worst case, which is the minimum of the yield to call and the yield to maturity.

Example: YTM, YTC, YTW

✏️ Consider the following bond:
F = $1000
T = 10
c = 10%
Price = $900
Assume that the bond is callable after 2 years at its face value.

Note that this bond is sold at a $100 discount to its face value. Do you think you’d get a better deal if the bond is called or if you hold it to maturity? In other words, would you typically prefer that this bond be called?

✔ Click here to view answer

We can break the return you get from holding this bond into two components:

1. The 10% coupon you receive for lending your money.
2. The extra return you get by buying a $1,000 bond for $900.

We can think of the coupon rate as the base interest rate you receive from the bond. The extra $100 you get by buying the bond at a discount is the extra interest. You always receive the base coupon rate, but the extra $100 can be received in two years if the bond is called, or in ten years if it is not called and you hold it to maturity.

If you are going to be given $100 for free, your return each year will be higher if you receive that $100 over the course of just two years than if you have to wait ten years to receive it. Therefore, we would expect the yield to call to be higher than the yield to maturity. Because the yield to call is higher than the yield to maturity, the yield to worst would equal the yield to maturity.

We will see how this plays out in the following problem.

For background on this, see 🔎 Why are i and c different?

✏️ Calculate YTM, YTC, and YTW for the bond in the previous example:
F = $1000
T = 10
c = 10%
Price = $900
Assume that the bond is callable after 2 years at its face value.

✔ Click here to view answer

To calculate IRR, write down the cash flows you receive, then apply the annuity formula or use a built-in formula in a spreadsheet or financial calculator. In some cases, you can also do it with algebra.

Let’s calculate the cash flows if the bond is called and if the bond is not called.

If the bond is called:

T	CF
0	-$900 (purchasing the bond)
1	$100 (first coupon payment)
2	$100 (second coupon payment) $1000 (receive face value when bond is called)

We enter this into our spreadsheet as follows:

T	CF
0	-900
1	100
2	1100
YTC	16.2%	=IRR(B2:B4)

Next, let’s calculate the IRR of holding the bond to maturity. When we put this into Excel, we get:

T	CF
0	-900
1	100
2	100
3	100
4	100
5	100
6	100
7	100
8	100
9	100
10	1100
YTM	11.8%	=IRR(B2:B12)
YTW	11.8%	=MIN(16.2%, 11.8%)

As we predicted in the answer to the previous problem, the yield to call was higher than the yield to maturity and the yield to call (because it was lower) was equal to the yield to worst.

Three Yields

Excel Google Sheets

Download: L9 Three Yields.xlsx

How can we make up additional practice problems?

Most bonds are sold at a price close to their face value. To create a realistic bond problem, choose:

• a coupon rate
• a yield to maturity
• a face value
• a bond price

Make sure the bond price is within 10% of the face value, and you will have a reasonable, realistic bond problem.

From here, you already have a good yield-to-maturity problem. You can solve for yield to maturity using a financial calculator or by the usual means.

Once you have done that, you can check your answer by using the yield to maturity to calculate the bond price. The catch is that you need to use a high-precision value for the yield to maturity if you want the result to be close to the bond price you started with. The less precision you use for the yield to maturity you are using as the discount rate, the farther the resulting bond price will be from the bond price you started with.

Let’s use the same problem we worked through earlier and calculate the bond price a couple of times, using different levels of precision for our yield to maturity.

Making Practice Problems

Excel Google Sheets

Download: L9 - Make Own Problems.xlsx