Yield

We first discussed the concept of yield in the preferred stock chapter. The concept is the same for bonds; yield measures the overall return of an investment. With bonds, there are several factors that determine bond yields. These factors include:

• Interest rate (coupon)

• Purchase price

• Length of time until maturity

While the interest rate of a bond and its yield sound like they’re the same thing, they are not (except for the nominal yield, which is discussed below). The interest rate represents the amount of annual interest paid by the issuer to the bondholder. The yield represents the overall rate of return of the bond. The bond’s interest rate factors into the yield, but the interest rate and yield will be different if the bond is bought at a discount or a premium.

We’ll discuss the following yields in this chapter:

• Nominal yield
• Current yield
• Yield to maturity (YTM)
• Yield to call (YTC)

Nominal yield

The nominal yield is another way to refer to the interest rate (coupon). You will likely never do a calculation to find the nominal yield, but you may be asked what the formula is.

For example:

A $1,000 par, 4% bond

$\text{Nominal yield}=\frac{\text{Annual\hspace{0.33em}income}}{\text{Par}}$

$\text{Nominal\hspace{0.33em}yield}=\frac{\text{\$40}}{\text{\$1,000}}$

$\text{Nominal\hspace{0.33em}yield}=4\%$

As you can see, we went through a calculation only to find the 4% that was originally provided to us. The nominal yield is based on two factors, both of which never change over the life of the bond. This bond has a $1,000 par value and pays $40 annually to its bondholder, regardless of what happens in the market.

The nominal yield is a fixed, unaffected rate throughout the life of the bond. Unlike all the other yields we’ll discuss, the bond’s market price is not a factor. If you are given information on a bond, the nominal yield is always the first % referenced.

Current yield

Let’s add some information to the example we used in the last section.

A $1,000 par, 4% bond bought for $800

If this bond is bought at $800 (discount), the bond’s overall rate of return will be higher than the interest rate of the bond. How do we know this? The investor is earning two separate forms of return:

Coupon

• Pays the investor $40 annually

Discount

• Investor earns $200 over the life of the bond

The investor is receiving 4% of par ($1,000) annually, but they’re also benefiting from the difference between the purchase price and the maturity value (par). The extra $200 return adds to their overall return (yield), so we can assume it’s something above 4%.

In the preferred stock chapter, we discussed the current yield, which is also an important yield for bonds. As a reminder, the current yield is found by dividing the annual income by the market price of the security.

A $1,000 par, 4% bond bought for $800. What’s the current yield?

Can you figure it out?

As you can see, the current yield reflects a higher yield than the interest rate of the bond. Therefore, you can always assume the current yield for discount bonds will always be higher than the coupon.

Although the current yield is an important yield that must be known for the exam, it’s not a terribly effective yield to utilize as an investor. Why? Because the current yield does not factor in time.

You’ve probably heard the term “time is money.” This couldn’t be more true with bonds. When time is not a factor in a yield, the yield is not really telling the full story. Yield is an annualized number that represents the annual overall return to the investor.

A large component of this bond’s return is the discount, which gives the investor an additional $200 over the life of the bond. Shouldn’t there be a difference in the annualized return if the bond had a one-year maturity as compared to a 30-year maturity? Yes, there should!

Here’s a video breakdown of a practice question on current yield:

Let’s also discuss how current yield works with premium bonds. As a reminder, premium bonds trade in the market at prices above par ($1,000).

A $1,000 par, 4% bond bought for $1,100

The investor purchasing this bond will receive $40 a year in interest from the issuer. They’ll lose money over the life of the bond by buying it at a premium because bonds mature at par. Paying $1,100 for a bond that will mature at $1,000 results in a $100 loss at maturity. Because of this dynamic, yields of premium bonds will always be lower than the coupon.

A $1,000 par, 4% bond bought for $1,100. What is the current yield?

Can you figure it out?

When the coupon is coupled with the loss of the premium, we find that the investor’s current yield reflects a more accurate rate of return than the coupon. Therefore, you can always assume the current yield for premium bonds will always be lower than the coupon.

The current yield is only an approximate yield that shouldn’t be taken too seriously in the real world but can show up on the Series 7 exam. The next two yields we’ll discuss are more applicable in the investing world.

Yield to maturity (YTM)

Yield to maturity and yield to call formulas are difficult to memorize and typically are not tested. Exam questions are more likely to focus on the relationships of the yields, which is best depicted on the bond see-saw (discussed at the end of this chapter). Additionally, it’s possible a test question focuses on the components of these yield formulas. Don’t spend a significant amount of time focusing on the math related to these yields.

Unlike the current yield, the yield to maturity (YTM) (also known as a bond’s basis) does factor in time. In fact, the YTM assumes the investor buys the bond and holds it until the bond’s maturity. Let’s look at an example:

A 10 year, $1,000 par, 4% bond is trading at $800. What is the yield to maturity (YTM)?

$$\begin{gathered} \begin{array}{rl}\mathrm{ytm}& =\frac{C+\frac{F-P}{n}}{\frac{F+P}{2}}\\ \mathrm{ytm}& =\frac{40+\frac{1000-800}{10}}{\frac{1000+800}{2}}\\ \mathrm{ytm}& =\frac{40+20}{900}\\ \mathrm{ytm}& =6.7\%\end{array} \\ \begin{array}{rl}\text{where:}& \\ C& =\text{coupon\hspace{0.33em}interest payment}\\ F& =\text{face\hspace{0.33em}value (par)}\\ P& =\text{price}\\ n& =\text{years\hspace{0.33em}to maturity}\end{array} \end{gathered}$$

The YTM is not the easiest formula to work through, so let’s break it down. The annual income is $40, based on the 4% coupon. The annualized discount is found by dividing the overall discount ($200) by the number of years until maturity (10). Remember, the discount is equal to the difference between the market price ($800) and par ($1,000). From there, find the average between the market price ($800) and par ($1,000).

We see this yield (6.7%) is higher than the coupon (4%). The trend remains the same; bonds bought at discounts have yields higher than their coupon because the bond’s discount is providing them with an extra return.

Let’s look at how YTM looks with a premium bond.

A 10 year, $1,000 par, 4% bond is trading at $1,100. What is the yield to maturity (YTM)?

$$\begin{gathered} \begin{array}{rl}\mathrm{ytm}& =\frac{C-\frac{P-F}{n}}{\frac{F+P}{2}}\\ \mathrm{ytm}& =\frac{40-\frac{1,100-1,000}{10}}{\frac{1,000+1,100}{2}}\\ \mathrm{ytm}& =\frac{40-10}{1050}\\ \mathrm{ytm}& =2.9\%\end{array} \\ \begin{array}{rl}\text{where:}& \\ C& =\text{coupon\hspace{0.33em}rate}\\ F& =\text{face\hspace{0.33em}value (par)}\\ P& =\text{price}\\ n& =\text{years\hspace{0.33em}to maturity}\end{array} \end{gathered}$$

The annual income of the bond is $40, based on its 4% coupon. The annualized premium is found by dividing the premium ($100) by the number of years to maturity (10 years). Instead of adding, we subtract the annualized premium because the investor is losing money over time. The average value of the bond is found by adding the market value ($1,100) to the par value ($1,000) and dividing by 2.

We see the trend again with premium bonds. When purchased at a market price above par, the YTM is lower than the bond’s coupon.

Yield to call (YTC)

Remember the advice provided above - the math behind the yield to call formula is not important.

Yield to call (YTC) only applies to callable bonds. If a bond is not callable, YTC does not exist. It represents a bond’s overall rate of return if held until the bond is callable. Essentially, we’re assuming the bond will be called as soon as it’s eligible.

The YTC formula is similar to the YTM formula, but there are some differences. Calculating YTC is also less important for the exam than calculating YTM.

A 10 year, $1,000 par, 4% bond is trading at $800. The bond is callable at par after 5 years. What is the yield to call (YTC)?

$$\begin{gathered} \begin{array}{rl}\mathrm{ytc}& =\frac{C+\frac{CP-MP}{t}}{\frac{CP+MP}{2}}\\ \mathrm{ytc}& =\frac{40+\frac{1000-800}{5}}{\frac{1000+800}{2}}\\ \mathrm{ytc}& =\frac{40+40}{900}\\ \mathrm{ytc}& =8.9\%\end{array} \\ \begin{array}{rl}\text{where:}& \\ C& =\text{coupon\hspace{0.33em}rate}\\ CP& =\text{call\hspace{0.33em}price}\\ MP& =\text{market\hspace{0.33em}price}\\ t& =\text{years to\hspace{0.33em}call}\end{array} \end{gathered}$$

Again, we see the yield we calculated (8.9%) is higher than the coupon (4%). Also, the YTC (8.9%) is higher than the YTM (6.7%). Why is that?

Remember, this investor is earning a discount of $200 over the life of the bond. If the bond is held to maturity, it will take the investor 10 years to earn the $200 discount. If the bond is called in 5 years, the customer earns the $200 5 years earlier than expected, which increases their annualized rate of return.

Let’s take a look at YTC with a premium bond:

A 10 year, $1,000 par, 4% bond is trading at $1,100. The bond is callable at par after 5 years. What is the yield to call (YTC)?

$$\begin{gathered} \begin{array}{rl}\mathrm{ytc}& =\frac{C-\frac{MP-CP}{t}}{\frac{CP+MP}{2}}\\ \mathrm{ytc}& =\frac{40-\frac{1,100-1,000}{5}}{\frac{1,000+1,100}{2}}\\ \mathrm{ytc}& =\frac{40-20}{1050}\\ \mathrm{ytc}& =1.9\%\end{array} \\ \begin{array}{rl}\text{where:}& \\ C& =\text{coupon\hspace{0.33em}rate}\\ CP& =\text{call\hspace{0.33em}price}\\ MP& =\text{market\hspace{0.33em}price}\\ t& =\text{years to\hspace{0.33em}call}\end{array} \end{gathered}$$

We see the yield we calculated (1.9%) is lower than the coupon (4%). Also, the YTC (1.9%) is lower than the YTM (2.9%). Why is that? Remember, this customer is losing a premium of $100 over the life of the bond. If the bond is held to maturity, it will take the investor 10 years to lose the $100 premium. If the bond is called in 5 years, the customer loses the $100 5 years earlier than expected, which decreases their annualized rate of return.

Discount bond yield relationships

Let’s look at a summary of our discount bond example from the previous sections.

A 10 year, $1,000 par, 4% bond is trading at $800. The bond is callable at par after 5 years.

• Coupon = 4%

• Current yield = 5%

• YTM = 6.7%

• YTC = 8.9%

The order of these yields is no coincidence. In fact, every discount bond will exhibit the same relationship between the yields. The coupon will be the lowest rate, followed by the current yield, then the YTM, and last the YTC. You could spend time calculating each yield separately, or you can utilize a beloved visual with bonds: the bond see-saw.

As you can see, the bond see-saw is a quick way of determining the relationship between a bond’s price and its yields. Many test-takers commit the bond see-saw to memory and write it out on their scratch paper when they take the exam. By doing this, you can avoid doing most of the calculations we just went through.

NASAA is more concerned that you know the relationship between the yields (which is lower or higher) than your ability to calculate these yields. While you certainly can be required to calculate current yield, the order of the yields and their relationships are more important topics to master. Plus, knowing the order of yields will help you eliminate wrong answers to yield calculation questions.

Premium bond yield relationships

Let’s continue using our premium bond example from the previous sections.

A 10 year, $1,000 par, 4% bond is trading at $1,100. The bond is callable at par after 5 years.

• Coupon = 4%

• Current yield = 3.6%

• YTM = 2.9%

• YTC = 1.9%

Again, the order of these yields is no coincidence. In fact, every premium bond will exhibit the same relationship between the yields. The YTC will be the lowest rate, followed by the YTM, then the current yield, and last the coupon. You could spend time calculating each yield separately, or you can utilize the bond see-saw.

As we witnessed with discount bonds, the bond see-saw gives us a great visual representation of the yields. Now, we have the price side pointed upward, as premium bonds are purchased at prices above par. Remember the specific order of the yields on the see-saw and yield-based questions will be easy to answer!

Yield for par bonds

We’ve gone through how prices affect yields for bonds purchased at discounts and premiums. What if a bond is purchased at par ($1,000)? We won’t need to go into much detail, because this concept is fairly simple. When a bond is purchased at par, all of the yields are equal to the coupon.

The investor isn’t making or losing any money through the purchase price of their bond. If the bond is bought at par ($1,000), it will mature at par ($1,000). The only return the investor is realizing is the coupon. For a bond purchased at its par value, the see-saw looks like this:

As you can see, the coupon is at the same level as the current yield, YTM, and YTC. If you see a question on par bond yields, keep it simple. All of them are the same.

The bond see-saw

Yield is a very important concept on the Series 7 exam, and hopefully, the bond see-saw helps visualize the relationship between bond prices, interest rate changes, and the yields. We’ve looked at the see-saw through the lens of a discount bond, premium bond, and par bond. Let’s take a look at all of them together:

If you’re going to utilize a “dump sheet,” it’s highly recommended that this be a part of it. A dump sheet is a list of visual items that are written out on your scratch paper after you start your exam. Many students memorize concepts like the bond see-saw in order to rewrite them for the exam and rely on it for yield questions.

I’ve also seen a lot of successful test-takers use acronyms, like this:

CYM Call

• CY = Current Yield

• M = yield to Maturity

• Call = yield to Call

Whatever will help you commit these terms to memory is encouraged. It doesn’t matter how you remember it, as long as you remember it!

Yield to worst

Two yields are typically provided when a bond quote is given to investors (and also disclosed on trade confirmations). First, the bond’s nominal yield (coupon) is disclosed, which provides insight into the security’s interest payments. Second, the bond’s yield to worst is furnished, which represents the lower of the YTM or YTC. Providing an investor with the worst of these two yields gives them a “worst-case scenario.” Let’s go through the two general scenarios.

If a callable bond is bought at a discount, the investor is provided the YTM. An investor would make a higher annualized return if a discount bond is redeemed sooner as they would earn the discount faster. For example, let’s assume a 20 year bond that is callable in 10 years at par is purchased for 90 ($900). The investor would earn the $100 discount faster if the bond is called before maturity, resulting in a higher annualized return. The YTM assumes the discount bond is held to maturity, which reflects the discount being gained slower and the lowest possible yield (unless the bond defaults).

If a callable bond is bought at a premium, the investor is provided the YTC. An investor would make a higher annualized return if a premium bond is held to maturity as they would lose the premium slower. For example, let’s assume a 20 year bond that is callable in 10 years at par is purchased for 110 ($1,100). The investor would lose the $100 premium slower if the bond is held to maturity, resulting in a higher annualized return. The YTC assumes the premium bond is called at the first possible date, which reflects the premium being lost faster and the lowest possible yield (unless the bond defaults).

No YTC exists if a bond is not callable. Therefore, the YTM is always provided when quoting a non-callable bond.

Assuming a bond is callable, here’s the “yield to worst” summary:

• Discount bonds = YTM is quoted
• Premium bonds = YTC is quoted

Key points

Nominal yield

• $\text{NY}=\frac{\text{Annual\hspace{0.33em}income}}{\text{Par}}$
• Measures the interest paid annually to investor
• Never changes over the life of the bond
• Also known as:
  • Coupon
  • Interest rate
  • Stated rate

Current yield

• $\text{CY}=\frac{\text{Annual income}}{\text{Market\hspace{0.33em}price}}$
• Measures overall rate of return based on the current market price
• Discount bonds CY > coupon
• Premium bonds CY < coupon

Yield to maturity (YTM)

• Measures overall rate of return if the bond is held to maturity
• Discount bonds YTM > coupon
• Premium bonds YTM < coupon

Yield to call (YTC)

• Measures overall rate of return if the bond is held until called
• Discount bonds YTC > coupon
• Premium bonds YTC < coupon

Discount bond yield relationships

• Current yield, YTM, and YTC are higher than the coupon
• Order of all yields (lowest to highest)
  • Nominal yield (coupon)
  • Current yield
  • Yield to maturity (YTM)
  • Yield to call (YTC)

Premium bond yield relationships

• Current yield, YTM, and YTC are lower than the coupon
• Order of all yields (lowest to highest)
  • Yield to call (YTC)
  • Yield to maturity (YTM)
  • Current yield
  • Nominal yield (coupon)

Par bond yield relationships

• All yields are the same

Yield to worst

• Lower of YTM or YTC provided in callable bond quotes
• Discount bonds = YTM is quoted
• Premium bonds = YTC is quoted