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    In this chapter we look at the factors that affect the yield offered in the bond market. We begin with the minimum interest rate that an investor wants from investing in a bond, the yield on U.S. Treasury securities. Then we describe why the yield on a non-U.S. Treasury security will differ from that of a U.S. Treasury security. Finally, we focus on one particular factor that affects the yield offered on a security: maturity. The pattern of interest rates on securities of the same issuer but with different maturities is called the term structure of interest rates.


    The securities issued by the U.S. Department of the Treasury are backed by the full faith and credit of the U.S. government. As such, interest rates on Treasury securities are the key interest rates in the U.S. economy as well as in international capital.

The minimum interest rate that investors want is referred to as the base interest rate or

    benchmark interest rate that investors will demand for investing in a non-Treasury security. This rate is the yield to maturity (hereafter referred to as simply yield) offered on a comparable maturity

    Treasury security that was most recently issued (―on the run‖).


The difference between the yields of any two bonds is called a yield spread. For example,

    consider two bonds, bond A and bond B. The yield spread is then

    yield spread = yield on bond A yield on bond B

    The normal way that yield spreads are quoted is in terms of basis points. The yield spread reflects the difference in the risks associated with the two bonds.

    When bond B is a benchmark bond and bond A is a non-benchmark bond, the yield spread is referred to as a benchmark spread; that is,

    benchmark spread = yield on non-benchmark bond yield on benchmark bond

    The benchmark spread reflects the compensation that the market is offering for bearing the risks associated with the non-benchmark bond that do not exist for the benchmark bond. Thus, the benchmark spread can be thought of as a risk premium.

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    Some market participants measure the risk premium on a relative basis by taking the ratio of the yield spread to the yield level. This measure, called a relative yield spread, is computed as


    relative yield spread = (yield on bond A yield on bond B) / yield on bond B

The yield ratio is the quotient of two bond yields:

    yield ratio = yield on bond A / yield on bond B

    The factors that affect the yield spread include the type of issuer, the issuer’s perceived credit worthiness, the term or maturity of the instrument, provisions that grant either the issuer or the investor the option to do something, the taxability of the interest received by investors, the expected liquidity of the security.

Types of Issuers

    The bond market is classified by the type of issuer, including the U.S. government, U.S government agencies, municipal governments, credit (domestic and foreign corporations), and foreign governments. These classifications are referred to as market sectors. Different sectors are

    generally perceived to represent different risks and rewards.

    Some market sectors are further subdivided into categories intended to reflect common economic characteristics. For example, within the credit market sector, issuers are classified as follows: industrial, utility, finance, and non-corporate. The spread between the interest rate offered in two sectors of the bond market with the same maturity is referred to as an intermarket sector spread.

    The spread between two issues within a market sector is called an intramarket sector spread.

Perceived Credit Worthiness of Issuer

    Default risk or credit risk refers to the risk that the issuer of a bond may be unable to make timely principal and/or interest payments. Most market participants rely primarily on commercial rating companies to assess the default risk of an issuer. The spread between Treasury securities and non-Treasury securities that are identical in all respects except for quality is referred to as a credit


Inclusion of Options

    It is not uncommon for a bond issue to include a provision that gives either the bondholder and/or the issuer an option to take some action against the other party. The presence of an embedded option has an effect on the spread of an issue relative to a Treasury security and the spread relative to otherwise comparable issues that do not have an embedded option.

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Taxability of Interest

    Because of the tax-exempt feature of municipal bonds, the yield on municipal bonds is less than that on Treasuries with the same maturity. The yield on a taxable bond issue after federal income taxes are paid is called the after-tax yield:

    after-tax yield = pretax yield x (1 marginal tax rate)

    Alternatively, we can determine the yield that must be offered on a taxable bond issue to give the same after-tax yield as a tax-exempt issue. This yield, called the equivalent taxable yield:

    equivalent taxable yield = tax-exempt yield / (1 marginal tax rate).

    The municipal bond market is divided into two major sectors: general obligations and revenue bonds. State and local governments may tax interest income on bond issues that are exempt from federal income taxes. Some municipalities’ exempt interest income from all municipal issues from taxation; others do not. Some states exempt interest income from bonds issued by municipalities within the state but tax the interest income from bonds issued by municipalities outside the state.

    Municipalities are not permitted to tax the interest income from securities issued by the U.S. Treasury. Thus part of the spread between Treasury securities and taxable non-Treasury securities of the same maturity reflects the value of the exemption from state and local taxes.

Expected Liquidity of an Issue

    Bonds trade with different degrees of liquidity. Bonds with greater expected liquidity will have lower yields that investors would require. The lower yield offered on Treasury securities relative to non-Treasury securities reflects the difference in liquidity.

Financeability of an Issue

    A portfolio manager can use an issue as collateral for borrowing funds. By borrowing funds, a portfolio manager can create leverage. The typical market used by portfolio managers to borrow funds using a security as collateral for a loan is the repurchase agreement market or ―repo‖ market.

    When a portfolio manager wants to borrow funds via a repo agreement, a dealer provides the funds. The interest rate charged by the dealer is called the repo rate. There is not one repo rate but a structure of rates depending on the maturity of the loan and the specific issue being financed. With respect to the latter, there are times when dealers are in need of particular issues to cover a short position. When a dealer needs a particular issue, that dealer will be willing to offer to lend funds at a lower repo rate than the general repo rate in the market.

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Term to Maturity

The time remaining on a bond’s life is referred to as its term to maturity or simply maturity. The

    volatility of a bond’s price is dependent on its term to maturity. More specifically, with all other factors constant, the longer the term to maturity of a bond, the greater the price volatility resulting from a change in market yields. Generally, bonds are classified into three maturity sectors: Bonds

    with a term to maturity of between 1 to 5 years are considered short term; bonds with a term to

    maturity between 5 and 12 years are viewed as intermediate term; and long-term bonds are those

    with a term to maturity greater than 12 years. The spread between any two maturity sectors of the market is called a maturity spread. The relationship between the yields on otherwise comparable

    securities with different maturities is called the term structure of interest rates.


    The term structure of interest rates plays a key role in the valuation of bonds.

Yield Curve

    The graphical depiction of the relationship between the yield on bonds of the same credit quality but different maturities is known as the yield curve. In the past, most investors have constructed

    yield curves from observations of prices and yields in the Treasury market. Two factors account for this tendency. First, Treasury securities are free of default risk, and differences in credit worthiness do not affect yields. Therefore, these instruments are directly comparable. Second, as the largest and most active bond market, the Treasury market offers the fewest problems of illiquidity or infrequent trading. The disadvantage, as noted previously, is that the yields may be biased downward because they reflect favorable financing opportunities.

    Market participants are coming to realize that the traditionally constructed Treasury yield curve is an unsatisfactory measure of the relation between required yield and maturity. The key reason is that securities with the same maturity may carry different yields. This phenomenon reflects the role and impact of differences in the bonds’ coupon rates.

Why the Yield Curve Should Not Be Used to Price a Bond

    The price of a bond is the present value of its cash flow. The bond pricing formula assumes that one interest rate should be used to discount all the bond’s cash flows. Because of the different cash

    flow patterns, it is not appropriate to use the same interest rate to discount all cash flows. Instead, each cash flow should be discounted at a unique interest rate appropriate for the time period in which the cash flow will be received.

    The correct way to think about bonds is that they are packages of zero-coupon instruments. Each zero-coupon instrument in the package has a maturity equal to its coupon payment date or, in the case of the principal, the maturity date. The value of the bond should equal the value of all the component zero-coupon instruments. If this does not hold, it is possible for a market participant to generate riskless profits by stripping off the coupon payments and creating stripped securities.

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    To determine the value of each zero-coupon instrument, it is necessary to know the yield on a zero-coupon Treasury with that same maturity. This yield is called the spot rate, and the graphical

    depiction of the relationship between the spot rate and maturity is called the spot rate curve.

    Because there are no zero-coupon Treasury debt issues with a maturity greater than one year, it is not possible to construct such a curve solely from observations of market activity on Treasury securities. Rather, it is necessary to derive this curve from theoretical considerations as applied to the yields of the actually traded Treasury debt securities. Such a curve is called a theoretical spot

    rate curve and is the graphical depiction of the term structure of interest rate.

Constructing the Theoretical Spot Rate Curve for Treasuries

    A default-free theoretical spot rate curve can be constructed from the yield on Treasury securities. The Treasury issues that are candidates for inclusion are (i) on-the-run Treasury issues, (ii) on-the-run Treasury issues and selected off-the-run Treasury issues, (iii) all Treasury coupon securities, and bills, and (iv) Treasury coupon strips

    After the securities that are to be included in the construction of the theoretical spot rate curve are selected, the methodology for constructing the curve must be determined. If Treasury coupon strips are used, the procedure is simple, because the observed yields are the spot rates. If the on-the-run Treasury issues with or without selected off-the-run Treasury issues are used, a methodology called bootstrapping is used.

On-the-Run Treasury Issues

The on-the-run Treasury issues are the most recently auctioned issue of a given maturity. These

    issues include the 3-month, 6-month, and 1-year Treasury bills; the 2-year, 5-year, and 10-year Treasury notes; and the 30-year Treasury bond. Treasury bills are zero-coupon instruments; the notes and the bond are coupon securities.

    There is an observed yield for each of the on-the-run issues. For the coupon issues, these yields are not the yields used in the analysis when the issue is not trading at par. Instead, for each on-the-run coupon issue, the estimated yield necessary to make the issue trade at par is used. The resulting on-the-run yield curve is called the par coupon curve.

    The goal is to construct a theoretical spot rate curve with 60 semiannual spot rates: 6 month rate to 30-year rate. Excluding the three-month bill, there are only six maturity points available when only on-the-run issues are used. The 54 missing maturity points are extrapolated from the surrounding maturity points on the par yield curve. The simplest interpolation method, and the one most commonly used, is linear extrapolation. Specifically, given the yield on the par coupon curve at two maturity points, the following is calculated:

    yield at higher maturity yield at lower maturity. number of semiannual periods between the twoaturity points 1;

    Then, the yield for all intermediate semiannual maturity points is found by adding to the yield at the lower maturity the amount computed here.

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    There are two problems with using just the on-the-run issues. First, there is a large gap between some of the maturities points, which may result in misleading yields for those maturity points when estimated using the linear interpolation method. Specifically, the concern is with the large gap between the five-year and 10-year maturity points and the 10-year and 30-year maturity points. The second problem is that the yields for the on-the-run issues themselves may be misleading because most offer the favorable financing opportunity in the repo market mentioned earlier. This means that the true yield is greater than the quoted (observed) yield.

    To overcome these problems, we convert the par yield curve into the theoretical spot rate curve using bootstrapping. To explain the process of estimating the theoretical spot rate curve from observed yields on Treasury securities, consider (i) a six-month Treasury bill where its annualized yield is the six-month spot rate and (ii) a one-year Treasury where its annualized yield is the one year spot rate. Given these two spot rates, we can compute the spot rate for a theoretical 1.5-year zero-coupon Treasury. The price of a theoretical 1.5-year zero-coupon Treasury should equal the present value of three cash flows from an actual 1.5-year coupon Treasury, where the yield used for discounting is the spot rate corresponding to the cash flow. We can solve for the theoretical 1.5-year spot rate. Doubling this rate, we can obtain the bond-equivalent yield, which is the theoretical 1.5-year spot rate. This rate is the rate that the market would apply to a 1.5-year zero-coupon Treasury security if, in fact, such a security existed. Given the theoretical 1.5-year spot rate, we can obtain the theoretical 2-year spot rate and so forth until we derive theoretical spot rates for the remaining 15 half-yearly rates. The spot rates using this process represent the term structure of interest rates.

    It would seem logical that the observed yield on strips could be used to construct an actual spot rate curve rather than go through the tedious computation procedure to get yields. There are three problems with using the observed rates on strips. First, the liquidity of the strips market is not as great as that of the Treasury coupon market. Thus, the observed rates on strips reflect a premium for liquidity. Second, the tax treatment of strips is different from that of Treasury coupon securities. Specifically, the accrued interest on strips is taxed even though no cash is received by the investor. Finally, there are maturity sectors in which non-U.S. investors find it advantageous to trade off yield for tax advantages associated with a strip.

On-the-Run Treasury Issues and Selected Off-the-Run Treasury Issues

    One of the problems with using just the on-the-run issues is the large gaps between maturities, particularly after five years. To mitigate this problem, some dealers and vendors use selected off-the-run Treasury issues.

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All Treasury Coupon Securities and Bills

    Using only on-the-run issues, even when extended to include a few off-the-run issues, fails to recognize the information embodied in Treasury prices that are not included in the analysis. Thus, it is argued that it is more appropriate to use all Treasury coupon securities and bills to construct the theoretical spot rate curve.

Treasury Coupon Strips

    Treasury coupon strips are zero-coupon Treasury securities. It would seem logical that the observed yield on strips could be used to construct an actual spot rate curve. There are three problems with using the observed rates on strips. First, the liquidity of the strips market is not as great as that of the Treasury coupon market. Second, the tax treatment of strips is different from that of Treasury coupon securities. Finally, there are maturity sectors in which nonU.S. investors

    find it advantageous to trade off yield for tax advantages associated with a strip.

Using the Theoretical Spot Rate Curve

    Arbitrage forces a Treasury to be priced based on spot rates and not the yield curve. The ability of dealers to purchase securities and create value by stripping forces Treasury securities to be priced based on the theoretical spot rates.

Spot Rates and the Base Interest Rate

    The base interest rate for a given maturity should not be seen as simply the yield on the on-the-run Treasury security for that maturity, but the theoretical Treasury spot rate for that maturity. To value a non-Treasury security, we should add a risk premium to the theoretical Treasury spot rates.

Forward Rates

    From the yield curve we can extrapolate the theoretical spot rates. In addition, we can extrapolate what some market participants refer to as the market’s consensus of future interest rates. To illustrate, buying either a one-year instrument or a six-month instrument and when it matures in six months, buy another six-month instrument. Given the one-year spot rate, there is some rate on a six-month instrument six months from now that will make the investor indifferent between the two alternatives. We denote that rate by f which can be readily determined given the theoretical

    one-year spot rate and the six-month spot rate. Doubling f gives the bond-equivalent yield for the

    six-month rate six months from now in which we are interested.

    The market prices its expectations of future interest rates into the rates offered on investments with different maturities. This is why knowing the market’s consensus of future interest rates is critical. The rate that we determined for f is the market’s consensus for the six-month rate six months from

    now. A future interest rate calculated from either the spot rates or the yield curve is called a forward rate.

Relationship Between Six-Month Forward Rates and Spot Rates

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    ), the current six-month spot rate (z), In general, the relationship between a t-period spot rate (zt11/tand the six month forward rates is z = [(1 + z) (1 + f) (1 + f) ??? (1 + f1)] 1 where f is the t112t t

    six-month forward rate beginning t six-month periods from now.

Other Forward Rates

    It is not necessary to limit ourselves to six-month forward rates. The spot rates can be used to calculate the forward rate for any time in the future for any investment horizon.

Forward Rate as a Hedgeable Rate

    A natural question about forward rates is how well they do at predicting future interest rates. The forward rate may never be realized but is important in what it tells investors about his expectation relative to what the market consensus expects. Some market participants prefer not to talk about forward rates as being market consensus rates. Instead, they refer to forward rates as being hedgeable rates. For example, by buying the one-year security, the investor can hedge the six-month rate six months from now.

Determinants of the Shape of the Term Structure

    If we plot the term structurethe yield to maturity, or the spot rate, at successive maturities against maturitywe find three typically shapes: an upward-sloping yield curve; a

    downward-sloping or inverted yield curve, or a flat yield curve.

    Two major theories have evolved to account for these observed shapes of the yield curve: expectations theories and market segmentation theory.

There are several forms of the expectations theory: pure expectations theory, liquidity theory,

    and preferred habitat theory. Expectations theories share a hypothesis about the behavior of

    short-term forward rates and also assume that the forward rates in current long-term bonds are closely related to the market’s expectations about future short-term rates. These three theories

    differ, however, as to whether other factors also affect forward rates, and how. The pure expectations theory postulates that no systematic factors other than expected future short-term rates affect forward rates; the liquidity theory and the preferred habitat theory assert that there are other factors. Accordingly, the last two forms of the expectations theory are sometimes referred to as biased expectations theories.

Pure Expectations Theory

    According to the pure expectations theory, the forward rates exclusively represent the expected future rates. Thus, the entire term structure at a given time reflects the market’s current expectations of the family of future short-term rates.

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Liquidity Theory

    The pure expectations theory states that investors will hold longer-term maturities if they are offered a long-term rate higher than the average of expected future rates by a risk premium that is positively related to the term to maturity. According to this theory, which is called the liquidity

    theory of the term structure, the implied forward rates will not be an unbiased estimate of the market’s expectations of future interest rates because they embody a liquidity premium.

Preferred Habitat Theory

    The preferred habitat theory also adopts the view that the term structure reflects the expectation of the future path of interest rates as well as a risk premium. However, the preferred habitat theory rejects the assertion that the risk premium must rise uniformly with maturity.

Market Segmentation Theory

    The market segmentation theory also recognizes that investors have preferred habitats dictated by the nature of their liabilities. However, the market segmentation theory differs from the preferred habitat theory in that it assumes that neither investors nor borrowers are willing to shift from one maturity sector to another to take advantage of opportunities arising from differences between expectations and forward rates.

The Main Influences of the Shape of the Yield Curve

    Empirical evidence suggests that the three main influences on the shape of the Treasury yield curve are (1) the market’s expectations of future rate changes, (2) bond risk premiums, and (3) convexity bias. The convexity bias influence is the least well known of the three influences. The longer the maturity, the more convexity the security has. That is, longer-term Treasury securities have a more attractive feature due to convexity than shorter-term Treasury securities. As a result, investors are willing to pay more for longer-term Treasury securities and therefore accept lower returns. This influence on the shape of the Treasury yield curve is what is referred to as the convexity bias.


    The basic elements of an interest rate swap are important for us to understand because it is a commonly used interest rate benchmark. In fact, the interest rate swap market in most countries is increasingly used as an interest rate benchmark despite the existence of a liquid government bond market.

    In a generic interest rate swap, the parties exchange interest rate payments on specified dates: one party pays a fixed rate and the other party pays a floating rate over the life of the swap. In a typical swap, the floating rate is based on a reference rate, and the reference rate is typically the London Interbank Offered Rate (LIBOR). LIBOR is the interest rate at which prime banks in London pay other prime banks on U.S. dollar certificates of deposits.

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The fixed interest rate that is paid by the fixed rate counterparty is called the swap rate. Dealers in

    the swap market quote swap rates for different maturities. The relationship between the swap rate and maturity of a swap is called the swap rate yield curve or, more commonly, the swap curve.

    Because the reference rate is typically LIBOR, the swap curve is also called the LIBOR curve.

    There is a swap curve for most countries. For Euro interest rate swaps, the reference rate is the Euro Interbank Offered Rate (Euribor), which is the rate at which bank deposits in countries that have adopted the euro currency and are member states of the European Union are offered by one prime bank to another prime bank.

    The swap curve is used as a benchmark in many countries outside the United States. Unlike a country’s government bond yield curve, however, the swap curve is not a default-free yield curve.

    Instead, it reflects the credit risk of the counterparty to an interest rate swap.

     market would One would expect that if a country has a government bond market, the yields in thatbe the best benchmark. That is not necessarily the case. There are several advantages of using a

    swap curve over a country’s government securities yield curve. First, there may be technical

    reasons why within a government bond market some of the interest rates may not be representative

    of the true interest rate but instead be biased by some technical or regulatory factor unique to that

    market. Second, to create a representative government bond yield curve, a large number of

    maturities must be available. Finally, the ability to compare government yields across countries is difficult because there are differences in the credit risk for every country.

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