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Chapter 25 Factors Affecting Bond Yields Fabozzi:Investment Management Graphics by

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Learning Objectives You will understand why the yield on a Treasury security is the base interest rate. You will study the factors that affect the yield spread between two bonds. You will be able to describe what a yield curve is. You will learn what a spot rate and a spot rate curve mean. You will calculate the theoretical spot rates from the Treasury yield curve.

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Learning Objectives You will explore what the term structure of interest rates is. You will study why the price of a Treasury bond should be based on spot rates. You will learn what is meant by a forward rate and be able to calculate a forward rate. You will explore how long-term rates are related to the current short-term rate and short-term forward rates. You will understand the different theories about the determinants of the shape of the term structure: pure expectations theory, the liquidity theory, the preferred habitat theory, and the market segmentation theory.

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Introduction Bond yields depend on a variety of factors, including the type of issuer, the characteristics of the bond issue, and the state of economy. In this chapter we look at the following factors Investor’s minimum interest rate or yield on U.S. Treasury securities The difference in yields between a non- U.S. and a U.S. Treasury security Bond maturity

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The base interest rate U.S. Treasury securities which are backed by the full faith and credit of the U.S. government, are known throughout the financial world as risk-free securities. Because of this, the interest rate on Treasuries is the key interest rate in world markets and the Treasury market is the most liquid in the world. Benchmark interest rate or base interest rate Minimum interest rate that investors will accept The yield on the most recently issued (on-the-run) Treasury security with a comparable maturity to the investor’s potential bond purchase.

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Risk premium Risk premium – the spread between the rate on non-Treasury securities and a particular on-the-run Treasury security. This is the difference between the risk-free rate and the additional risk inherent in other securities. Or, base interest rate + risk premium Factors affecting the spread 1.type of issuer 2.issuer’s perceived creditworthiness 3.term or maturity of the instrument 4.provisions that grant either issuer or investors the option to do something 5.taxability of the interest received 6.expected liquidity of the security

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Types of issuers Market sectors (classifications of issuers) -U.S. government and its agencies -Municipal governments -Corporations (foreign and domestic) -Foreign governments

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Types of issuers Each sector has different risk/reward ratios as each issuer (except the Treasury) has differing abilities to meet their contractural obligations Intermarket sector spread- difference in interest rates offered by different sectors on security with same maturity Intramarket sector spread – spread between two issues within a market sector

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Perceived creditworthiness of the issuer Default or credit risk The risk that the issuer will be unable to make timely interest or principal payments. This risk is assessed by commercial rating companies Quality spread or credit spread Spread between Treasuries and non-Treasuries that are identical except for quality

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Inclusion of options Embedded options Call provision – issuer can retire debt, in full or partially, prior to maturity Effect on spread Investors require a larger spread on bonds with embedded options comparable to Treasuries if the option benefits the issuer Investors require a smaller spread (or a negative spread!) on bonds with embedded options comparable to Treasuries if the option benefits the investor.

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Taxability of interest Interest income is taxable at the federal level, as well as possible taxes at the state and local level. The interest from municipal bonds is exempt, which makes their yields lower than similar Treasuries. The benchmark for calculating spreads on tax-exempt bonds is a generic AAA G.O. bond with specified maturity.

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Taxability of interest After-tax yield for taxable bond issues After tax-yield = pretax yield x (1- marginal tax rate) Equivalent taxable yield – taxable issue yield = after-tax yield on tax- exempt issue Equivalent taxable yield = tax exempt yield (1- marginal tax rate) Municipal bonds General obligations Revenue Housing Power Hospitals Insured

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Expected liquidity of an issue and term to maturity Expected liquidity of an issue The greater the expected liquidity, the lower required yield. Term to maturity The longer the term to maturity, the greater the bond price volatility Short-term bonds – maturity between 1 and 5 years Intermediate-term bonds – maturity between 5 and 12 years Long-term bonds – maturity > 12 years Maturity spread – spread between two maturity sectors Term structure of interest rates – the relationship between yields on identical bonds with different maturities

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Yield curve Insert Figure 25-1 The Treasury yield curve is a benchmark for pricing bonds and setting yields in other areas of the debt market. While this method has been used for a long time due to the liquidity and creditworthiness of the Treasury market, it has been inexact. That is, securities with the same maturity may have different yields. How then should we price bonds?

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Why the yield curve should not be used to price a bond Recall that the price of a bond = PV of its cash flows and that one interest rate (derived from a Treasury security) is used to discount all cash flows. Problems Different cash flow patterns require different discount rates Each bond should be viewed as package of zero-coupon instruments, with each coupon date being its own maturity. Finding the spot rate - Take the yield on a zero-coupon Treasury for each maturity Spot rate curve – graphical description of the rate and its maturity All zero-coupon Treasuries maturities < 1 year, so we find the theoretical spot rate curve.

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Constructing the theoretical spot rate curve for Treasuries This curve is constructed from the yield on the following Treasury securities: 1.on-the-run Treasury issues 2.on-the-run Treasury issues and selected off-the-run Treasury issues 3.all Treasury coupon securities and bills 4.Treasury coupon strips The methodology depends on which securities are used. Bootstrapping occurs when on-the-run Treasury issues with or without selected off-the-run Treasury issues are used.

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On-the-run Treasury issues These include the most recently auctioned issues of a given maturity with each having an observed yield. Estimated yield is used in the analysis when the issue is not trading at par. The resulting yield curve is the par coupon curve. Here we construct a theoretical spot rate curve with 60 semiannual spot rates – 6 month rate to 30 year rate. Seven maturity points are available, with the rest extrapolated. This yield is found by Yield at higher maturity – yield at lower maturity Number of semiannual periods between the two maturity points

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On-the-run Treasury issues Then, the yield for intermediate points is found by adding to the yield at the lower maturity the amount from the computation above. Problems with this method are: Large gap between maturity points resulting in misleading yields for those maturities estimated Yields for on-the-run issues may be misleading since the true yield tends to be > quoted yield.

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Estimating the theoretical spot rate curve Insert Table 25-6 Use the 6 month Treasury bill with an annualized yield of 5.25% = spot rate and 1 year Treasury with yield of 5.5% as the 1 year spot rate. Coupon rate for a 1.5 year Treasury is 5.75%. Compute the spot rate for a theoretical 1.5 year zero-coupon Treasury. $100 = par Cash flows are: 0.5 year0.575 x $100 x 0.5= $ 2.875 1.0 year0.075 x $100 x 0.5= $ 2.875 1.5 years0.575 x $100 x 0.5 + 100=$102.875 and given: z 1 = one-half the annualized 6 month theoretical spot rate z 2 = one-half the 1 year theoretical spot rate z 3 = one-half the annual value of the 1.5 year theoretical spot rate

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Estimating the theoretical spot rate curve We know that z 1 = 0.02625 (0.0525/2) z 2 = 0.0275 (0.055/2) therefore, 100 = 2.801461 + 2.723166 + 102.875 (1 + z 3 ) 3 94.47537 = 102.875 (1 + z 3 ) 3 (1 + z 3 ) 3 = 1.028798, z 3 = 0.028798 We double this yield to find the bond-equivalent yield of 5.76%. This procedure can be used to find the theoretical spot rates for all maturities. See Table 25-7. Insert Table 25-7

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On-the-run Treasury issues and selected off-the-run Treasury issues To lessen the problem of gaps between maturities, some dealers use selected off-the-run Treasury issues (i.e. 20 year and 25 year issues). The linear extrapolation method is used to fill in the gaps and bootstrapping is used to construct the theoretical spot rate curve. Insert Table 25-8

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All Treasury coupon securities and bills Even when using only on-the-run issues with a few off-the- run issues, information is embodied in Treasury prices is lost to the analysis. It is more appropriae to used all Treasury coupon securities and bills to construct the theoretical spot rate curve.

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Using the theoretical spot rate curve Insert Table 25-9 What forces a Treasury to be price based on the spot rates? Arbitrage. By viewing the bond as a package of zero-coupon instruments and applying the appropriate interest rates, we avoid underpricing the bond relative to its theoretical value, creating an arbitrage opportunity. Base interest rate = theoretical Treasury spot rate for that maturity

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Forward rates Alternative 1: An investor buying a one-year instrument will realize the on-year spot rate, a certain amount. Alternative 2: An investor buying a six-month instrument and then replacing it with another at maturity, will realize an unknown amount. The logic is that in six months rates may be higher. The value of f denotes some rate on a six month instrument six months from now that will make the investor indifferent between alternatives 1 and 2. Insert Figure 25-2 Double this to give the bond-equivalent yield for the 6 month rate, 6 months from now.

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Forward rates: an example Given: 6 month spot rate= 0.0525, z 1 = 0.02625 1 year spot rate = 0.0550, z 2 = 0.02750 That is, the annual rate for f on a bond equivalent basis = 5.75% If the 6 month rate 6 months from now < 5.75%, then the investor should go with alternative one since the total value will be higher. It is important to understand that expectations of future rates are built into the rates offered on investments with different maturities. Forward rate – future interest rate calculated from spot rates or the yield curve

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Relationship between six-month forward rates and spot rates Given a t-period spot rate, the current spot rate and six month forward rates (f t ), the formula for the relationship is z t = [(1 + z 1 )(1 + f 1 )(1 + f 2 )(1+ f 3 )…(1 + f t-1 )] 1/t -1

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Other forward rates We can calculate the forward rates for any time in the future for any investment horizon by using spot rates. But do forward rates do a good job at predicting future interest rates? While studies have concluded that forward rates do not predict rates well, forward rates show how an investor’s expectations must differ from the market consensus to choose the correct alternative.

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Determinants of the shape of the term structure Four typical term structure shapes Insert Figure 25-3 Two theories have evolved to explain these structures: 1. Expectations theories -Pure expectations -Liquidity -Preferred habitat 2.Market segmentation theory

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The pure expectations theory Forward rates exclusively represent the expected future rates. The entire term structure reflects the market’s expectations of future short-term rates. Therefore, a normal curve indicates a rising trend in short-term rates for the near future, a flat term structure indicates constant rates, and a falling structure indicates declining rates. Looking at the normal structure, the following scenario could exist: 1.Long-term horizon investors would shy away from long-term bonds since an increase in rates would lower bond prices. They would buy short-term bonds. 2.Speculators would sell long-term bonds before prices fell and short- sell others. They would reinvest proceeds in short-term debt instruments. 3.Borrowers would tend to borrow now, rather than wait, as lending rates are expected to increase.

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The pure expectations theory Long-term bonds would be sold and demand for short- term debt instruments would rise, making long-term yields rise. That is, the behavior initiated by expectations would affect the predicted result.

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The pure expectations theory: a problem This theory ignores the inherent risks of bonds brought on by the uncertainty of future interest rates. More specifically, the risks are: 1.Price risk Uncertainty about the price of a bond at the end of an investment horizon due to dependence on future interest rates; it may be lower than expected. 2.Reinvestment risk Uncertainty about the rate at which proceeds can be reinvested

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The pure expectations theory: interpretations 1.Investors expect the return for any investment horizon to be the same regardless of maturity strategy. However, due to the price risk associated with investing in bonds with a maturity greater than the investment horizon, expected returns will differ significantly. 2.Local expectations theory states that the returns on bonds of different maturities will be the same over a short-term investment horizon. This interpretation can be sustained given equilibrium. 3.The return-to-maturity expectations interpretation states that the return an investor will realize by rolling over short-term bonds to some investment horizon will be the same as holding zero-coupon bond with a maturity equal to that investment horizon. There is much doubt about this interpretation.

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The liquidity theory Investors will hold longer-term maturities if they are offered a forward rate that reflects both interest rate expectations and a liquidity (or risk) premium. According to this liquidity theory of the term structure, implied forward rates will not be an unbiased estimate of expectations of future interest rates because there is a liquidity premium built in. It also assumes that the risk premium rises uniformly with maturity.

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The preferred habitat theory Given mismatched demand and supply for funds in a maturity range, lenders and borrowers will need to shift to maturities showing the opposite imbalances. They will require a risk premium reflecting the extent of aversion to price or reinvestment risk. The shape of the yield curve, under this theory, is determined by both expectations of future interest rates and the risk premium that will coax investors to shift out of their preferred habitat. Determination of shape Upward and downward sloping as well as humped are possible.

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Market segmentation theory This theory states that the shape of the yield curve is affected by the asset/liability constraints and/or creditors (borrowers) restricting their lending (financing) to specific maturity sectors. Assumes neither investors nor borrowers are willing to shift maturity sectors to take advantage of forward rate or expectations opportunities. Determination of yield curve shape Supply and demand for securities within each maturity sector

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