Since the price of a bond fluctuates with market interest rates, the risk that an investor faces is that the price of a bond held in a portfolio will decline if market interest rates rise. This risk is referred to as interest rate risk and is the major risk faced by investors in the bond market.

What can the investor do? The investor cannot force the issuer to change the coupon rate to 6.5%. Nor can the investor force the issuer to shorten the maturity of the bond to a point where a new investor might be willing to accept a 6% coupon rate. The only thing that the investor can do is adjust the price of the bond to a new price where a buyer would realize a yield of 6.5%. This means that the price would have to be adjusted down to a price below par. It turns out, the new price must be 94.4479.¹⁰ While we assumed in our illustration an initial price of par value, the principle holds for any purchase price. Regardless of the price that an investor pays for a bond, an instantaneous increase in market interest rates will result in a decline in a bond’s price.

Suppose that instead of a rise in market interest rates to 6.5%, interest rates decline to 5.5%. Investors would be more than happy to purchase the 6% coupon 20-year bond at par. However, investor X realizes that the market is only offering investors the opportunity to buy a similar bond at par with a coupon rate of 5.5%. Consequently, investor X will increase the price of the bond until it offers a yield of 5.5%. That price turns out to be 106.0195.

Let’s summarize the important relationships suggested by our example.

For now, to understand why, let’s decompose the price of a callable bond into two components, as shown below:

Now, when interest rates decline, the price of an option-free bond increases. However, the price of the embedded call option in a callable bond also increases because the call option becomes more valuable to the issuer. So, when interest rates decline both price components increase in value, but the change in the price of the callable bond depends on the relative price change between the two components. Typically, a decline in interest rates will result in an increase in the price of the callable bond but not by as much as the price change of an otherwise comparable option-free bond.

Similarly, when interest rates rise, the price of a callable bond will not fall as much as an otherwise option-free bond. The reason is that the price of the embedded call option declines. So, when interest rates rise, the price of the option-free bond declines but this is partially offset by the decrease in the price of the embedded call option component.

To see this, we compare a 6% 20-year bond initially selling at a yield of 6%, and a 6% 20-year bond initially selling at a yield of 10%. The former is initially at a price of 100, and the latter 65.68. Now, if the yield for both bonds increases by 100 basis points, the first bond trades down by 10.68 points (10.68%) to a price of 89.32. The second bond will trade down to a price of 59.88, for a price decline of only 5.80 points (or 8.83%). Thus, we see that the bond that trades at a lower yield is more volatile in both percentage price change and absolute price change, as long as the other bond characteristics are the same. An implication of this is that, for a given change in interest rates, price sensitivity is lower when the level of interest rates in the market is high, and price sensitivity is higher when the level of interest rates is low.

First, the longer the time to the next coupon reset date, the greater the potential price fluctuation.¹³ For example, consider a floating-rate security whose coupon resets every six months and suppose the coupon formula is the 6-month Treasury rate plus 20 basis points. Suppose that on the coupon reset date the 6-month Treasury rate is 5.8%. If on the day after the coupon reset date, the 6-month Treasury rate rises to 6.1%, this security is paying a 6-month coupon rate that is less than the prevailing 6-month rate for the next six months. The price of the security must decline to reflect this lower coupon rate. Suppose instead that the coupon resets every month at the 1-month Treasury rate and that this rate rises immediately after the coupon rate is reset. In this case, while the investor would be realizing a sub-market 1-month coupon rate, it is only for one month. The one month coupon bond’s price decline will be less than the six month coupon bond’s price decline.

The second reason why a floating-rate security’s price will fluctuate is that the required margin that investors demand in the market changes. For example, consider once again the security whose coupon formula is the 6-month Treasury rate plus 20 basis points. If market conditions change such that investors want a margin of 30 basis points rather than 20 basis points, this security would be offering a coupon rate that is 10 basis points below the market rate. As a result, the security’s price will decline.

Finally, a floating-rate security will typically have a cap. Once the coupon rate as specified by the coupon reset formula rises above the cap rate, the coupon will be set at the cap rate and the security will then offer a below-market coupon rate and its price will decline. In fact, once the cap is reached, the security’s price will react much the same way to changes in market interest rates as that of a fixed-rate coupon security. This risk for a floating-rate security is called cap risk.

What we are interested in is a first approximation of how a bond’s price will change when interest rates change. We can look at the price change in terms of (1) the percentage price change from the initial price or (2) the dollar price change from the initial price.

For now, we will assume that the valuation model tells us that the price of bond ABC will be 88 if the yield is 6.25%. This means that the price will decline by 2 points or 2.22% of the initial price of 90. If we divide the 2.22% by 25 basis points, the resulting number tells us that the price will decline by 0.0889% per 1 basis point change in yield.

Now suppose that the valuation model tells us that if yields decline from 6% to 5.75%, the price will increase to 92.7. This means that the price increases by 2.7 points or 3.00% of the initial price of 90. Dividing the 3.00% by 25 basis points indicates that the price will change by 0.1200% per 1 basis point change in yield.

We can average the two percentage price changes for a 1 basis point change in yield up and down. The average percentage price change is 0.1044% [= (0.0889% + 0.1200%)/2].

A formula for estimating the approximate percentage price change for a 100 basis point change in yield is:

There is a special name given to this estimate of the percentage price change for a 100 basis point change in yield. It is called duration. As can be seen, duration is a measure of the price sensitivity of a bond to a change in yield. So, for example, if the duration of a bond is 10.44, this means that the approximate percentage price change if yields change by 100 basis points is 10.44%. For a 50 basis point change in yields, the approximate percentage price change is 5.22% (10.44% divided by 2). For a 25 basis point change in yield, the approximate percentage price change is 2.61% (10.44% divided by 4).

Notice that the approximate percentage is assumed to be the same for a rise and decline in yield. When we discuss the properties of the price volatility of a bond to changes in yield in Chapter 7, we will see that the percentage price change is not symmetric and we will discuss the implication for using duration as a measure of interest rate risk. It is important to note that the computed duration of a bond is only as good as the valuation model used to get the prices when the yield is shocked up and down. If the valuation model is unreliable, then the duration is a poor measure of the bond’s price sensitivity to changes in yield.

The approximate dollar price change for a 100 basis point change in yield is sometimes referred to as the dollar duration.

As you will see in Chapter 4, there is not one interest rate or yield in the economy. There is a structure of interest rates. One important structure is the relationship between yield and maturity. The graphical depiction of this relationship is called the yield curve. As we will see in Chapter 4, when interest rates change, they typically do not change by an equal number of basis points for all maturities.

For example, suppose that a $65 million portfolio contains the four bonds shown in Exhibit 1. All bonds are trading at a price equal to par value.

If we want to know how much the value of the portfolio changes if interest rates change, typically it is assumed that all yields change by the same number of basis points. Thus, if we wanted to know how sensitive the portfolio’s value is to a 25 basis point change in yields, we would increase the yield of the four bond issues by 25 basis points, determine the new price of each bond, the market value of each bond, and the new value of the portfolio. Panel (a) of Exhibit 2 illustrates the 25 basis point increase in yield. For our hypothetical portfolio, the value of each bond issue changes as shown in panel (a) of Exhibit 1. The portfolio’s value decreases by $1,759,003 from $65 million to $63,240,997.

Suppose that, instead of an equal basis point change in the yield for all maturities, the 20-year yield changes by 25 basis points, but the yields for the other maturities changes as follows: (1) 2-year maturity changes by 10 basis points (from 5% to 5.1%), (2) 5-year maturity changes by 20 basis points (from 5.25% to 5.45%), and (3) 30-year maturity changes by 45 basis points (from 5.75% to 6.2%). Panel (b) of Exhibit 2 illustrates these yield changes. We will see in later chapters that this type of movement (or shift) in the yield curve is referred to as a “steepening of the yield curve.” For this type of yield curve shift, the portfolio’s value is shown in panel (b) of Exhibit 1. The decline in the portfolio’s value is $2,514,375 (from $65 million to $62,485,625).

Suppose, instead, that if the 20-year yield changes by 25 basis points, the yields for the other three maturities change as follows: (1) 2-year maturity changes by 5 basis points (from 5% to 5.05%), (2) 5-year maturity changes by 15 basis points (from 5.25% to 5.40%), and (3) 30-year maturity changes by 35 basis points (from 5.75% to 6.1%). Panel (c) of Exhibit 2 illustrates this shift in yields. The new value for the portfolio based on this yield curve shift is shown in panel (c) of Exhibit 1. The decline in the portfolio’s value is $2,096,926 (from $65 million to $62,903,074). The yield curve shift in the third illustration does not steepen as much as in the second, when the yield curve steepens considerably.

The point here is that portfolios have different exposures to how the yield curve shifts. This risk exposure is called yield curve risk. The implication is that any measure of interest rate risk that assumes that the interest rates changes by an equal number of basis points for all maturities (referred to as a “parallel yield curve shift”) is only an approximation.

This applies to the duration concept that we discussed above. We stated that the duration for an individual bond is the approximate percentage change in price for a 100 basis point change in yield. A duration for a portfolio has the same meaning: it is the approximate percentage change in the portfolio’s value for a 100 basis point change in the yield for all maturities.

Because of the importance of yield curve risk, a good number of measures have been formulated to try to estimate the exposure of a portfolio to a non-parallel shift in the yield curve. We defer a discussion of these measures until Chapter 7. However, we introduce one basic but popular approach here. In the next chapter, we will see that the yield curve is a series of yields, one for each maturity. It is possible to determine the percentage change in the value of a portfolio if only one maturity’s yield changes while the yield for all other maturities is unchanged. This is a form of duration called rate duration, where the word “rate” means the interest rate of a particular maturity. So, for example, suppose a portfolio consists of 40 bonds with different maturities. A “5-year rate duration” of 2 would mean that the portfolio’s value will change by approximately 2% for a l00 basis point change in the 5-year yield, assuming all other rates do not change.

Consequently, in theory, there is not one rate duration but a rate duration for each maturity. In practice, a rate duration is not computed for all maturities. Instead, the rate duration is computed for several key maturities on the yield curve and this is referred to as key rate duration. Key rate duration is therefore simply the rate duration with respect to a change in a “key” maturity sector. Vendors of analytical systems report key rate durations for the maturities that in their view are the key maturity sectors. Key rate duration will be discussed further later.

Because of these three disadvantages faced by the investor, a callable bond is said to expose the investor to call risk. The same disadvantages apply to mortgage-backed and asset-backed securities where the borrower can prepay principal prior to scheduled principal payment dates. This risk is referred to as prepayment risk.

Reinvestment risk also occurs when an investor purchases a bond and relies on the yield of that bond as a measure of return. We have not yet explained how to compute the “yield” for a bond. When we do, it will be demonstrated that for the yield computed at the time of purchase to be realized, the investor must be able to reinvest any coupon payments at the computed yield. So, for example, if an investor purchases a 20-year bond with a yield of 6%, to realize the yield of 6%, every time a coupon interest payment is made, it is necessary to reinvest the payment at an interest rate of at 6% until maturity. So, it is assumed that the first coupon payment can be reinvested for the next 19.5 years at 6%; the second coupon payment can be reinvested for the next 19 years at 6%, and so on. The risk that the coupon payments will be reinvested at less than 6% is also reinvestment risk.

When dealing with amortizing securities (i.e., securities that repay principal periodically), reinvestment risk is even greater. Typically, amortizing securities pay interest and principal monthly and permit the borrower to prepay principal prior to schedule payment dates. Now the investor is more concerned with reinvestment risk due to principal prepayments usually resulting from a decline in interest rates, just as in the case of a callable bond. However, since payments are monthly, the investor has to make sure that the interest and principal can be reinvested at no less than the computed yield every month as opposed to semiannually.

This reinvestment risk for an amortizing security is important to understand. Too often it is said by some market participants that securities that pay both interest and principal monthly are advantageous because the investor has the opportunity to reinvest more frequently and to reinvest a larger amount (because principal is received) relative to a bond that pays only semiannual coupon payments. This is not the case in a declining interest rate environment, which will cause borrowers to accelerate their principal prepayments and force the investor to reinvest at lower interest rates.

With an understanding of reinvestment risk, we can now appreciate why zero-coupon bonds may be attractive to certain investors. Because there are no coupon payments to reinvest, there is no reinvestment risk. That is, zero-coupon bonds eliminate reinvestment risk. Elimination of reinvestment risk is important to some investors. That’s the plus side of the risk equation. The minus side is that, as explained in Section II, the lower the coupon rate the greater the interest rate risk for two bonds with the same maturity. Thus, zero-coupon bonds of a given maturity expose investors to the greatest interest rate risk.

Once we cover our basic analytical tools in later chapters, we will see how to quantify a bond issue’s reinvestment risk.

Studies have examined the probability of issuers defaulting. The percentage of a population of bonds that is expected to default is called the default rate. If a default occurs, this does not mean the investor loses the entire amount invested. An investor can expect to recover a certain percentage of the investment. This is called the recovery rate. Given the default rate and the recovery rate, the estimated expected loss due to a default can be computed. We will explain the findings of studies on default rates and recovery rates in Chapter 3.

As we will see in Chapter 3, the yield on a bond is made up of two components: (1) the yield on a similar default-free bond issue and (2) a premium above the yield on a default-free bond issue necessary to compensate for the risks associated with the bond. The risk premium is referred to as a yield spread. In the United States, Treasury issues are the benchmark yields because they are believed to be default free, they are highly liquid, and they are not callable (with the exception of some old issues). The part of the risk premium or yield spread attributable to default risk is called the credit spread.

The price performance of a non-Treasury bond issue and the return over some time period will depend on how the credit spread changes. If the credit spread increases, investors say that the spread has “widened” and the market price of the bond issue will decline (assuming U.S. Treasury rates have not changed). The risk that an issuer’s debt obligation will decline due to an increase in the credit spread is called credit spread risk.

This risk exists for an individual issue, for issues in a particular industry or economic sector, and for all non-Treasury issues in the economy. For example, in general during economic recessions, investors are concerned that issuers will face a decline in cash flows that would be used to service their bond obligations. As a result, the credit spread tends to widen for U.S. non-Treasury issuers and the prices of all such issues throughout the economy will decline.

One tool investors use to gauge the default risk of an issue is the credit ratings assigned to issues by rating companies, popularly referred to as rating agencies. There are three rating agencies in the United States: Moody’s Investors Service, Inc., Standard & Poor’s Corporation, and Fitch Ratings.

A credit rating is an indicator of the potential default risk associated with a particular bond issue or issuer. It represents in a simplistic way the credit rating agency’s assessment of an issuer’s ability to meet the payment of principal and interest in accordance with the terms of the indenture. Credit rating symbols or characters are uncomplicated representations of more complex ideas. In effect, they are summary opinions. Exhibit 3 identifies the ratings assigned by Moody’s, S&P, and Fitch for bonds and the meaning of each rating.

In all systems, the term high grade means low credit risk, or conversely, a high probability of receiving future payments is promised by the issuer. The highest-grade bonds are designated by Moody’s by the symbol Aaa, and by S&P and Fitch by the symbol AAA. The next highest grade is denoted by the symbol Aa (Moody’s) or AA (S&P and Fitch); for the third grade, all three rating companies use A. The next three grades are Baa or BBB, Ba or BB, and B, respectively. There are also C grades. Moody’s uses 1, 2, or 3 to provide a narrower credit quality breakdown within each class, and S&P and Fitch use plus and minus signs for the same purpose.

Bonds rated triple A (AAA or Aaa) are said to be prime grade; double A (AA or Aa) are of high quality grade; single A issues are called upper medium grade, and triple B are lower medium grade. Lower-rated bonds are said to have speculative grade elements or to be distinctly speculative grade.

Bond issues that are assigned a rating in the top four categories (that is, AAA, AA, A, and BBB) are referred to as investment-grade bonds. Issues that carry a rating below the top four categories are referred to as non-investment-grade bonds or speculative bonds, or more popularly as high yield bonds or junk bonds. Thus, the bond market can be divided into two sectors: the investment grade and non-investment grade markets as summarized below:

Once a credit rating is assigned to a debt obligation, a rating agency monitors the credit quality of the issuer and can reassign a different credit rating. An improvement in the credit quality of an issue or issuer is rewarded with a better credit rating, referred to as an upgrade; a deterioration in the credit rating of an issue or issuer is penalized by the assignment of an inferior credit rating, referred to as a downgrade. An unanticipated downgrading of an issue or issuer increases the credit spread and results in a decline in the price of the issue or the issuer’s bonds. This risk is referred to as downgrade risk and is closely related to credit spread risk.

As we have explained, the credit rating is a measure of potential default risk. An analyst must be aware of how rating agencies gauge default risk for purposes of assigning ratings in order to understand the other aspects of credit risk. The agencies’ assessment of potential default drives downgrade risk, and in turn, both default potential and credit rating changes drive credit spread risk.

A popular tool used by managers to gauge the prospects of an issue being downgraded or upgraded is a rating transition matrix. This is simply a table constructed by the rating agencies that shows the percentage of issues that were downgraded or upgraded in a given time period. So, the table can be used to approximate downgrade risk and default risk.

Exhibit 4 shows a hypothetical rating transition matrix for a 1-year period. The first column shows the ratings at the start of the year and the top row shows the rating at the end of the year. Let’s interpret one of the numbers. Look at the cell where the rating at the beginning of the year is AA and the rating at the end of the year is AA. This cell represents the percentage of issues rated AA at the beginning of the year that did not change their rating over the year. That is, there were no downgrades or upgrades. As can be seen, 92.75% of the issues rated AA at the start of the year were rated AA at the end of the year. Now look at the cell where the rating at the beginning of the year is AA and at the end of the year is A. This shows the percentage of issues rated AA at the beginning of the year that were downgraded to A by the end of the year. In our hypothetical 1-year rating transition matrix, this percentage is 5.07%. One can view these percentages as probabilities. There is a probability that an issue rated AA will be downgraded to A by the end of the year and it is 5.07%. One can estimate total downgrade risk as well. Look at the row that shows issues rated AA at the beginning of the year. The cells in the columns A, BBB, BB, B, CCC, and D all represent downgrades from AA. Thus, if we add all of these columns in this row (5.07%, 0.36%, 0.11%, 0.07%, 0.03%, and 0.01%), we get 5.65% which is an estimate of the probability of an issue being downgraded from AA in one year. Thus, 5.65% can be viewed as an estimate of downgrade risk.

A rating transition matrix also shows the potential for upgrades. Again, using Exhibit 4 look at the row that shows issues rated AA at the beginning of the year. Looking at the cell shown in the column AAA rating at the end of the year, one finds 1.60%. This is the percentage of issues rated AA at the beginning of the year that were upgraded to AAA by the end of the year.

Finally, look at the D rating category. These are issues that go into default. We can use the information in the column with the D rating at the end of the year to estimate the probability that an issue with a particular rating will go into default at the end of the year. Hence, this would be an estimate of default risk. So, for example, the probability that an issue rated AA at the beginning of the year will go into default by the end of the year is 0.01%. In contrast, the probability of an issue rated CCC at the beginning of the year will go into default by the end of the year is 25.9%.

Liquidity risk is the risk that the investor will have to sell a bond below its indicated value, where the indication is revealed by a recent transaction. The primary measure of liquidity is the size of the spread between the bid price (the price at which a dealer is willing to buy a security) and the ask price (the price at which a dealer is willing to sell a security). The wider the bid-ask spread, the greater the liquidity risk.

A liquid market can generally be defined by “small bid-ask spreads which do not materially increase for large transactions.”¹⁴ How to define the bid-ask spread in a multiple dealer market is subject to interpretation. For example, consider the bid-ask prices for four dealers. Each quote is for $92 plus the number of 32nds shown in Exhibit 5. The bid-ask spread shown in the exhibit is measured relative to a specific dealer. The best bid-ask spread is for Dealers 2 and 3.

From the perspective of the overall market, the bid-ask spread can be computed by looking at the best bid price (high price at which a broker/dealer is willing to buy a security) and the lowest ask price (lowest offer price at which a broker/dealer is willing to sell the same security). This liquidity measure is called the market bid-ask spread. For the four dealers, the highest bid price is 92 and the lowest ask price is 92 . Thus, the market bid-ask spread is 3.

Where are the prices obtained to mark a position to market? Typically, a portfolio manager will solicit bids from several broker/dealers and then use some process to determine the bid price used to mark (i.e., value) the position. The less liquid the issue, the greater the variation there will be in the bid prices obtained from broker/dealers. With an issue that has little liquidity, the price may have to be determined from a pricing service (i.e., a service company that employs models to determine the fair value of a security) rather than from dealer bid prices.

In Chapter 1 we discussed the use of repurchase agreements as a form of borrowing funds to purchase bonds. The bonds purchased are used as collateral. The bonds purchased are marked-to-market periodically in order to determine whether or not the collateral provides adequate protection to the lender for funds borrowed (i.e., the dealer providing the financing). When liquidity in the market declines, a portfolio manager who has borrowed funds must rely solely on the bid prices determined by the dealer lending the funds.

Here is another example of where market liquidity may change. While there are opportunities for those who invest in a new type of bond structure, there are typically few dealers making a market when the structure is so new. If subsequently the new structure becomes popular, more dealers will enter the market and liquidity improves. In contrast, if the new bond structure turns out to be unappealing, the initial buyers face a market with less liquidity because some dealers exit the market and others offer bids that are unattractive because they do not want to hold the bonds for a potential new purchaser.

Thus, we see that the liquidity risk of an issue changes over time. An actual example of a change in market liquidity occurred during the Spring of 1994. One sector of the mortgage-backed securities market, called the derivative mortgage market, saw the collapse of an important investor (a hedge fund) and the resulting exit from the market of several dealers. As a result, liquidity in the market substantially declined and bid-ask spreads widened dramatically.

As another example, consider a portfolio manager in the United Kingdom. This manager’s domestic currency is the pound. If that manager purchases a U.S. dollar denominated bond, then the manager is concerned that the U.S. dollar will depreciate relative to the British pound when the issuer makes a payment. If the U.S. dollar does depreciate, then fewer British pounds will be received on the foreign exchange market.

The risk of receiving less of the domestic currency when investing in a bond issue that makes payments in a currency other than the manager’s domestic currency is called exchange rate risk or currency risk.

For all but inflation protection bonds, an investor is exposed to inflation risk because the interest rate the issuer promises to make is fixed for the life of the issue.

While we discuss these other factors later, we can get an appreciation of one important factor from a general understanding of option pricing. A major factor affecting the value of an option is “expected volatility.” In the case of an option on common stock, expected volatility refers to “expected price volatility.” The relationship is as follows: the greater the expected price volatility, the greater the value of the option. The same relationship holds for options on bonds. However, instead of expected price volatility, for bonds it is the “expected yield volatility.” The greater the expected yield volatility, the greater the value (price) of an option. The interpretation of yield volatility and how it is estimated are explained at in Chapter 8.

Now let us tie this into the pricing of a callable bond. We repeat the formula for the components of a callable bond below:

To see how a change in expected yield volatility affects the price of a putable bond, we can write the price of a putable bond as follows:

This risk that the price of a bond with an embedded option will decline when expected yield volatility changes is called volatility risk. Below is a summary of the effect of changes in expected yield volatility on the price of callable and putable bonds:

The second type of event risk also results in a downgrade and can also impact other issuers. An excellent example occurred in the fall of 1988 with the leveraged buyout (LBO) of RJR Nabisco, Inc. The entire industrial sector of the bond market suffered as bond market participants withdrew from the market, new issues were postponed, and secondary market activity came to a standstill as a result of the initial LBO bid announcement. The yield that investors wanted on Nabisco’s bonds increased by about 250 basis points. Moreover, because the RJR LBO demonstrated that size was not an obstacle for an LBO, other large industrial firms that market participants previously thought were unlikely candidates for an LBO were fair game. The spillover effect to other industrial companies of the RJR LBO resulted in required yields’ increasing dramatically.

Changes in regulations may require a regulated entity to divest itself from certain types of investments. A flood of the divested securities on the market will adversely impact the price of similar securities.

Sovereign risk consists of two parts. First is the unwillingness of a foreign government to pay. A foreign government may simply repudiate its debt. The second is the inability to pay due to unfavorable economic conditions in the country. Historically, most foreign government defaults have been due to a government’s inability to pay rather than unwillingness to pay.