In Part 2 of this series on bond basics I explained the relationship between bond price and bond yield: when one goes up, the other goes down. However, I also stated that saying when interest rates rise, bond prices fall (or vice versa) is not really an accurate statement. This is because there are many different interest rates in our economy, and the price of a bond is only affected by the interest rate, or more precisely the yield, of that particular bond or bonds very similar to it. Below I'll discuss the precise relationship between the price and yield of a particular bond a bit more, then explain why the relationship between interest rates in general and bond prices in general is not so precise.

For a particular bond, price and yield are precisely related by a mathematical formula. Because of this precise relationship, price and yield are really two sides of the same coin--two different ways of expressing the value of a bond. So for a particular bond, it is absolutely true that when yield increases, price decreases, and vice versa.

Although it's not a perfect analogy, this analogy may help. You can express mileage as miles per gallon (mpg) or gallons per mile (gpm): 20 mpg is the same as 1/20 gpm (or 0.05 gpm, since 1/20 = 0.05). We can express this generically as mpg = 1/gpm, or equivalently, gpm = 1/mpg. If the denominator on the right-hand side of either of these equations increases, the value of the left-hand side of the equation decreases, and vice versa. In other words, mpg and gpm are inversely related.

The mathematical formula relating bond price and bond yield is more complex, but the mathematical foundation for the inverse relationship between bond price and yield is similar. In a subsequent post I'll get into more detail about the formula that relates bond price and yield, but for now I'll just note that in the equation that expresses bond price as a function of bond yield, yield is in the denominator of the right-hand side of the equation. As with the mpg vs. gpm example, this is the mathematical foundation for the inverse relationship between bond price and yield.

The yield of a bond is related to the riskiness of the bond: a riskier bond will have a higher yield than a less risky bond. Since bond price is inversely related to bond yield, we can also say that a riskier bond will have a lower price than a less risky bond (assuming both bonds have the same coupon rate). Risk is proportional to the uncertainty that the expected return will be realized. For a bond we can think of the yield as a reasonable measure of the annualized expected return. There are two dominant factors that contribute to the riskiness of a bond; i.e., the uncertainty that an investor will earn the yield of the bond.

The two main risk factors for bonds are credit quality, a measure of default risk, and term to maturity, which affects term risk, commonly referred to as interest-rate risk. Credit quality is a measure of the certainty that the bond issuer will make interest and principal payments (or make them on time). As explained in Part 1, term to maturity is the number of years until a bond matures and repays its principal.

An investor will demand a higher yield for a bond with lower credit quality to compensate for the higher uncertainty that interest and principal payments will be received (or received on time). If a bond issuer defaults on interest or principal payments, the investor will not earn the original yield (yield at the time the bond was purchased).

Investors usually demand higher yields for bonds with longer terms to maturity. Although the investor will receive face value for the bond at maturity, assuming no default, that may not be the case if the bond is sold before maturity. The longer the term to maturity, the longer the investor must bear the uncertainty of having to sell before maturity and receive a price other than face value. Also, the longer the term to maturity, the higher the uncertainty of the impact of unexpected inflation on the purchasing power of the bond's interest and principal payments.

A standard benchmark for bond yields for different terms to maturity are the yields of U.S. Treasuries, published by the U.S. Department of the Treasury. You can find the latest published yields as well as historical yields here: Daily Treasury Yield Curve Rates. If you click on the link, you'll see yields for terms to maturity from 1 month to 30 years, and as of this writing you'll see that yield increases as term to maturity increases, which is typical. For example, yields on the 1-year, 5-year and 10-year Treasuries as of December 29, 2016 are 0.85%, 1.96% and 2.49% respectively.

Incidentally, the U.S. Treasury Department uses the terms

Since U.S. Treasuries generally are considered to have no default risk, the Treasury yield curve gives us insight into the term risk the bond market is assessing for different terms to maturities. The degree to which longer-term Treasuries have higher yields is referred to as the steepness of the yield curve.

With this background, we can now start to understand the imprecision of simply saying that bond prices fall when interest rates rise, or vice versa. First let's look at this just from the perspective of term to maturity, and consider U.S. Treasuries since they have no default risk.

Yields on Treasuries of different maturities change every day, and they change by different amounts. The price of a Treasury of a given term to maturity is directly related to the yield of that Treasury, but not necessarily related to the yield of a Treasury of a different term to maturity. For example, looking at the current Treasury yield curve, we see that between December 28 and 29 of 2016, the yield of the 6-month Treasury did not change at all--it was 0.62% on both days--but the yield on the 5-year Treasury fell by 6 basis points--from 2.02% to 1.96%. So the price of the 5-year Treasury increased due to the decrease in the 5-year yield, but the price of the 6-month Treasury did not change since its yield did not change.

Although over long time periods, yields of all maturities tend to move up and down together, over shorter time periods this is not necessarily the case, since the yield curve can steepen or flatten as yields for different maturities change at different rates and even in different directions. This is an example of why it is imprecise to simply refer to bond prices falling due to interest rates rising, or vice versa.

For non-Treasury bonds, such as corporate and municipal bonds, the bond market's changing assessment of credit risk also can cause changes in yields. For example, the yield on a certain 5-year corporate bond could increase or decrease on a day when the 5-year Treasury yield did not change, resulting in a change in price for the 5-year corporate bond but no change in price for the 5-year Treasury. This is another example of why we can't accurately specify the relationship between bond prices and interest rates (yields) without specifying which bond or bonds we're talking about.

In the example in Part 2 of this series, I used a bond with a 1-year term to maturity to explain the inverse relationship between price and yield for a particular bond. In this part I've explained that prices and yields of bonds with different terms to maturity and different default risks can change by different amounts and even in different directions. So although the relationship between price and yield for a particular bond is mathematically precise, there is no such precise relationship between bond prices and yields (or interest rates) in general.

In the next part of this series I'll start getting into more advanced bond topics, such as the mathematical formula that determines the relationship between bond price and bond yield.

For a particular bond, price and yield are precisely related by a mathematical formula. Because of this precise relationship, price and yield are really two sides of the same coin--two different ways of expressing the value of a bond. So for a particular bond, it is absolutely true that when yield increases, price decreases, and vice versa.

Although it's not a perfect analogy, this analogy may help. You can express mileage as miles per gallon (mpg) or gallons per mile (gpm): 20 mpg is the same as 1/20 gpm (or 0.05 gpm, since 1/20 = 0.05). We can express this generically as mpg = 1/gpm, or equivalently, gpm = 1/mpg. If the denominator on the right-hand side of either of these equations increases, the value of the left-hand side of the equation decreases, and vice versa. In other words, mpg and gpm are inversely related.

The mathematical formula relating bond price and bond yield is more complex, but the mathematical foundation for the inverse relationship between bond price and yield is similar. In a subsequent post I'll get into more detail about the formula that relates bond price and yield, but for now I'll just note that in the equation that expresses bond price as a function of bond yield, yield is in the denominator of the right-hand side of the equation. As with the mpg vs. gpm example, this is the mathematical foundation for the inverse relationship between bond price and yield.

The yield of a bond is related to the riskiness of the bond: a riskier bond will have a higher yield than a less risky bond. Since bond price is inversely related to bond yield, we can also say that a riskier bond will have a lower price than a less risky bond (assuming both bonds have the same coupon rate). Risk is proportional to the uncertainty that the expected return will be realized. For a bond we can think of the yield as a reasonable measure of the annualized expected return. There are two dominant factors that contribute to the riskiness of a bond; i.e., the uncertainty that an investor will earn the yield of the bond.

The two main risk factors for bonds are credit quality, a measure of default risk, and term to maturity, which affects term risk, commonly referred to as interest-rate risk. Credit quality is a measure of the certainty that the bond issuer will make interest and principal payments (or make them on time). As explained in Part 1, term to maturity is the number of years until a bond matures and repays its principal.

An investor will demand a higher yield for a bond with lower credit quality to compensate for the higher uncertainty that interest and principal payments will be received (or received on time). If a bond issuer defaults on interest or principal payments, the investor will not earn the original yield (yield at the time the bond was purchased).

Investors usually demand higher yields for bonds with longer terms to maturity. Although the investor will receive face value for the bond at maturity, assuming no default, that may not be the case if the bond is sold before maturity. The longer the term to maturity, the longer the investor must bear the uncertainty of having to sell before maturity and receive a price other than face value. Also, the longer the term to maturity, the higher the uncertainty of the impact of unexpected inflation on the purchasing power of the bond's interest and principal payments.

A standard benchmark for bond yields for different terms to maturity are the yields of U.S. Treasuries, published by the U.S. Department of the Treasury. You can find the latest published yields as well as historical yields here: Daily Treasury Yield Curve Rates. If you click on the link, you'll see yields for terms to maturity from 1 month to 30 years, and as of this writing you'll see that yield increases as term to maturity increases, which is typical. For example, yields on the 1-year, 5-year and 10-year Treasuries as of December 29, 2016 are 0.85%, 1.96% and 2.49% respectively.

Incidentally, the U.S. Treasury Department uses the terms

*bills*,*notes*and*bonds*to refer to U.S. Treasuries of maturities of up to 1 year, 1-10 years, and more than 10 years respectively. This distinction is not particularly useful for our purposes, since the bond basics being discussed in this series are the same for all of them. It is common to refer to a Treasury security as simply a*Treasury*, whether referencing a bill, note or bond, but I also will use the generic term*bond*to refer to all of them.Since U.S. Treasuries generally are considered to have no default risk, the Treasury yield curve gives us insight into the term risk the bond market is assessing for different terms to maturities. The degree to which longer-term Treasuries have higher yields is referred to as the steepness of the yield curve.

With this background, we can now start to understand the imprecision of simply saying that bond prices fall when interest rates rise, or vice versa. First let's look at this just from the perspective of term to maturity, and consider U.S. Treasuries since they have no default risk.

Yields on Treasuries of different maturities change every day, and they change by different amounts. The price of a Treasury of a given term to maturity is directly related to the yield of that Treasury, but not necessarily related to the yield of a Treasury of a different term to maturity. For example, looking at the current Treasury yield curve, we see that between December 28 and 29 of 2016, the yield of the 6-month Treasury did not change at all--it was 0.62% on both days--but the yield on the 5-year Treasury fell by 6 basis points--from 2.02% to 1.96%. So the price of the 5-year Treasury increased due to the decrease in the 5-year yield, but the price of the 6-month Treasury did not change since its yield did not change.

Although over long time periods, yields of all maturities tend to move up and down together, over shorter time periods this is not necessarily the case, since the yield curve can steepen or flatten as yields for different maturities change at different rates and even in different directions. This is an example of why it is imprecise to simply refer to bond prices falling due to interest rates rising, or vice versa.

For non-Treasury bonds, such as corporate and municipal bonds, the bond market's changing assessment of credit risk also can cause changes in yields. For example, the yield on a certain 5-year corporate bond could increase or decrease on a day when the 5-year Treasury yield did not change, resulting in a change in price for the 5-year corporate bond but no change in price for the 5-year Treasury. This is another example of why we can't accurately specify the relationship between bond prices and interest rates (yields) without specifying which bond or bonds we're talking about.

In the example in Part 2 of this series, I used a bond with a 1-year term to maturity to explain the inverse relationship between price and yield for a particular bond. In this part I've explained that prices and yields of bonds with different terms to maturity and different default risks can change by different amounts and even in different directions. So although the relationship between price and yield for a particular bond is mathematically precise, there is no such precise relationship between bond prices and yields (or interest rates) in general.

In the next part of this series I'll start getting into more advanced bond topics, such as the mathematical formula that determines the relationship between bond price and bond yield.

Yet another excellent primer on bonds. Keep them coming!

ReplyDeleteThanks! Part 4 is in the works.

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