Answer questions on bonds: Yield, Interest Rate, Coupon

I will try my best to answer the question someone had posted on the bond topic.  Since the question was very broad, I am not sure to which extent I should cover this subject. As a result, I will cover the basics with some intuitive explanation, as I believe people can remember better when they truly understand the basic concept. It is definitively a good question and it is very understandable that it may be a bit confusing.

The most important bond pricing relationship to understand is the inverse relationship between bond prices and interest rates (or bond yields) — as interest rates rise, bond prices fall and as interest rates fall, bond prices rise. So if the yield of the bond has fallen below the coupon rate, the price must have increased above par.

But before we go there, first let's look at the difference between the coupon rate and the Yield?

Coupon rate refers to the interest payments the bond issuer makes to the investor.

Yield refers to the return on a bond and includes both the interest income paid each year as well as the theoretical gain or loss each year based on the difference between the investor's purchase price and the eventual maturity price of the bond.

When interest rates increase, the price of the bond fall and yields increase. Using the approximate yield to maturity to better understand how yield is impacted by both interest payments and price, let’s use the following example: assume you want to purchase a bond that has a coupon rate of 5% and the general level of interest rates is currently 5% and the price of the bond is trading at 100. Assume the bond matures in 4 years.
Approximate yield to maturity = ($5 + $0)/ ([100 +100]/2) = 5%

What happens if interest rates increase and the bond price decrease to 96? If you buy the bond today and pay 96, and it matures in 4 years at a price of 100, you have a theoretical gain of $1 per year for the next 4 years. When you combine this gain with the interest income, here is what happens to the yield on the bond: ($5 + $1) / ([96 + 100] / 2) = 6.12%
This example shows how the interest income and the price difference combine to create a yield.

Now we have some basics in mind, let’s look what causes the inverse relationship between interest rates and bond prices?

The fact that the bond price is lower is what causes a higher yield in the first place. Using the approximate yield to maturity formula, look at the following 3 scenarios that should help explain the relationship:

Scenario 1: Assume a bond that pays 5% per year in interest, matures in 4 years and trades at 100.
According to the approximate yield to maturity formula, the yield of this bond would be:
= (Interest Income + annual price change)/ [(100 + bond price)/2]
= ($5 + $0) / [(100 + 100)/2] = 5%

Scenario 2: What happens if interest rates increase in the market? The existing bond will not be as valuable because new issuers will issue bonds with a coupon rate that provides more interest income in order to keep pace with the higher market interest rate.
So assume the price of the bond above was now at 96 instead of 100.
The approximate yield to maturity would be:
= ($5 + $1) / [(100 + 96)/2]
= 6.12%
Why is the yield higher if the bond price is lower? Yield includes not only of the interest income earned each year, but also the theoretical gain or loss per year based on the difference between the purchase price and maturity price. The bond is now at 96, so if an investor were to buy that bond today, he or she would enjoy not only $5 in interest income each year, but also would gain $1 per year in value since he or she paid 96 but will get back 100 in 4 years (= (100 – 96 )/ 4))

Scenario 3: What if, rather than increasing, market interest rates had dropped. This would result into a higher bond price because the bond in existence would pay more interest than what new issuers are offering. The existing bond would be more attractive to buyers.
Let's say the bond price increased to 104. What would the approximate yield to maturity be?
= ($5 - $1) / [(100 + 104)/2]
= 3.92%
Since the investor today would pay 104 for something that will only pay back 100 in 4 years, the investor is theoretically losing $1 per year. So even though the investor gains $5 from interest income, the investor also loses $1 per year. As you can see, the reason why yield moves in the opposite direction to price, as mentioned earlier, is that yield consists of both interest income and gain or loss.