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Overview

Presenting the most advanced thinking on the topic, this book covers the latest valuation models and techniques. It addresses essential topics such as the subtleties of fixed-income mathematics, new approaches to modeling term structures, and the applications of fixed-income valuation on credit risk, mortgages, munis, and indexed bonds.

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Product Details

  • ISBN-13: 9780471254195
  • Publisher: Wiley
  • Publication date: 12/28/1999
  • Series: Frontiers in Finance Series , #61
  • Edition number: 1
  • Pages: 414
  • Product dimensions: 6.34 (w) x 9.51 (h) x 1.31 (d)

Meet the Author

NARASIMHAN JEGADEESH, PhD, is the Harry A. Brandt Distinguished Professor of Finance at the University of Illinois at Urbana-Champaign. He was formerly a member of the faculty at the University of California at Los Angeles and he received his PhD in finance from Columbia University. Professor Jegadeesh has been published extensively in the Journal of Finance, the Journal of Financial Economics, and other leading financial journals. He serves on the editorial board of the Journal of Securities Market. He is also an investment consultant for the hedge funds managed by Arbitrade Holdings LLC.

BRUCE TUCKMAN, PhD, is Managing Director and Global Head of Relative Value Modeling at Credit Suisse First Boston. After receiving his doctorate in economics from MIT, he became a professor of finance at New York University's Stern School of Business and a visiting professor at UCLA's Anderson School. He began his Wall Street career at Salomon Brothers' Fixed Income Proprietary Trading Group.

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Table of Contents

ADVANCED FIXED-INCOME MATHEMATICS.

Fixed-Income Subtleties and the Pricing of Long Bonds (N. Pearson).

Convexity Bias and the Yield Curve (A. Ilmanen).

Futures vs. Forward Prices: Implications for Swap Pricing and Derivatives Valuation (M. Grinblatt & N. Jegadeesh).

TERM STRUCTURE MODELING.

Discrete-Time Models of Bond Pricing (D. Backus, et al.).

Stochastic Mean Models of the Term Structure of Interest Rates (P. Balduzzi, et al.).

Interest Rate Modeling with Jump-Diffusion Processes (S. Das).

OTHER RISK FACTORS.

Some Elements of Rating-Based Credit Risk Modeling (D. Lando).

Anatomy of Prepayments: The Salomon Brothers Prepayment Model (L. Hayre & A. Rajan).

The Pricing and Hedging of Mortgage-Backed Securities: A Multivariate Density Estimation Approach (J. Boudoukh, et al.).

The Muni Puzzle: Explanations and Implications for Investors (J. Chalmers).

Models of Currency Option Pricing (G. Bakshi & Z. Chen).

NUMERICAL VALUATION TECHNIQUES.

Exploring the Relation between Discrete-Time Jump Processes and the Finite Difference Method (S. Heston & G. Zhou).

Monte Carlo Methods for the Valuation of Interest Rate Securities (L. Andersen & P. Boyle).

Index.

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First Chapter

Note:The Figures and/or Tables mentioned in this sample chapter do not appear on the Web.
Advanced Fixed-Income Mathematics

Part I of this book is about convexity and interest rate risk premia. Familiarity with these concepts is crucial to a deep understanding of fixed income mathematics and fixed income markets.

Prices of bonds without embedded options are convex functions of interest rates. In other words, the fall in a bond's price due to a 1 basis point increase in rates is less than the increase in price due to a 1 basis point decline in rates. Given this property, consider what would happen if there were a 50% chance that rates instantaneously increase by 1 basis point and a 50% chance that they drop by 1 basis point. Since the price rise due to the rate decrease exceeds the price fall due to the rate increase, the expected price change is positive even though the expected rate change is zero. Hence, investors should be willing to pay for convexity.

Now assume that rates are expected to fluctuate by only 0.1 basis points. It is still true that the increase in a bond's price due to a 0.1 basis point fall exceeds the decrease in a bond's price due to a 0.1 basis point rise. However, the difference between these price changes is small compared to the differences resulting from a 1 basis point fluctuation. Therefore, the expected bond price change for a 0.1 basis point fluctuation is positive, but small. Conversely, if rate volatility is 10 basis points, the difference between the price increase due to a rate fall and the price decrease due to a rate rise is larger than in the original example and, consequently, the expected price change is larger than in that example. In short, the return advantages from a bond's convexity increase with interest rate volatility.

Turning to risk premia, it is generally accepted that investors are risk averse. In particular, investors would prefer a short-term bond offering a risk-free return of 5% to the return on a long-term bond of 10% or 0% with equal probability. Alternatively, investors will be attracted to the long-term bond only if its expected return is greater than 5%. The difference between the expected return required by investors from the long-term bond and the risk-free return of 5% is the risk premium of that bond. If a second long-term bond exhibited greater interest rate risk, 1 it would presumably command a greater risk premium. At the same time, if a third bond exhibited less interest rate risk, it would presumably command a smaller risk premium. In fact, it can be shown that a security's risk premium will be proportional to its interest rate risk (e. g., Ingersoll (1987), p. 381). Therefore, any security's risk premium, normalized by that security's interest rate risk, will equal some common interest rate risk premium.

To explore convexity and risk premia in a simple mathematical setting, assume a 1-factor model of interest rates. It then follows that the price of a bond, P, may be written as a function of its yield, y, and time, t: P( y, t). Using Taylor's approximation, the change in the price of the bond, DP, over some small time interval, Dt, is given by the following equation:

Dividing both sides by P,

D DP P P y P Dt P P Dy = ++ 1 11 2 1 2 2 2 ¶P ¶y ¶P ¶t ¶ ¶y ()
D DD P y t P Dy = ++ ¶P ¶y ¶P ¶t ¶ ¶y 1 2 2 2 2 ()

Now, by definition, the modified duration of a bond, D, equals while the convexity of a bond, C, equals

Furthermore, the return on a bond from the passage of time alone, plus its continuously paid coupon rate, c, is equal to its yield. Mathematically,

Hence,

The left-hand side of this equation is the total return from the bond. The right-hand side breaks this return down into three components. First, the contribution of changing yields to the total return is minus the duration times the change in yield. Second, the contribution of the passage of time is the yield times how much time has passed. Third, the contribution of convexity is one-half of the convexity times the change in yield squared.

Taking the expectation of both sides of the equation gives the expected total return of the bond. Note that, over small time intervals, E[( Dy) 2 ] is the variance of yields over time interval Dt, denoted s 2 Dt:

If there are no arbitrage opportunities in bond markets, then the expected return from any bond must equal the short-term rate, r, E P t y y t t DP cD DDD é ë ê ù û ú + ++ =-DE[] 1 2 2 Cs D DDD DP P c t D y y t y + =-++ 1 2 2 C() P c y ¶P ¶t + = 1 2 2 P P ¶ ¶y -1 P ¶P ¶y plus a risk premium that is proportional to the interest rate risk of that particular bond. Denoting that risk premium by l, 2 the no-arbitrage condition states that

E P c t r D DP D Dt Dt é ë ê ù û ú + =-l
Or,
E P c t r P DP D Dt é ë ê ù û ú + =+ 1 ¶P ¶y lDt
Note that, for the risk premium to increase the returns of bonds with higher durations (i. e., with more interest rate risk, l must be negative).

Substituting this no-arbitrage condition into the expected total return equation and rearranging terms gives this final result:

y rDEy = + --] {[D l} Cs 1 2 2

So, the yield on a bond can be broken down into the following components: the short-term rate, representing the reward for holding the bond over time; the duration times the expected change in yields, representing the reward or penalty for expected interest rate changes; minus the duration times l, representing the risk premium for bearing interest rate risk; and a subtraction from yield to offset the convexity benefits of holding that bond. Note that, as discussed above, this convexity benefit increases with the volatility of rates.

Chapter 1 shows how convexity, interest rate risk premia, and other subtleties of fixed income mathematics impact the pricing of very long-term bonds. This application is pedagogically useful because many effects that are "subtle" for shorter term securities become economically significant for longer term securities. Furthermore, there had recently been a spate of issuance in 50-and 100-year bonds. Many market participants believed that these were not issued at prices consistent with the principles described in Chapter 1, and, therefore, they offered relatively attractive investment opportunities.

Chapter 2 describes the effects of convexity on the shape of the yield curve and on expected returns across maturities. The chapter also presents a great deal of empirical evidence on the magnitude of these effects.

Another extremely important manifestation of convexity in fixed income markets is the difference between futures and forward rates. Chapter 3 explains the reason for this difference in the context of Eurodollar futures rates. The chapter then uses a particular term structure model to illustrate the theoretical magnitude of the futures-forward effect and compares the results with historical evidence on empirical futures-forward effects.

1
Fixed-Income Subtleties and the Pricing of Long Bonds
Neil D. Pearson

Suppose that a 40-year noncallable bond is offered at par and has a coupon (and therefore a yield) of 8% per year. A 100-year non-callable bond from the same issuer is also offered at par and has a coupon (and yield) of 8 1 /4 % per year. Which should an investor buy? At first glance, the answer might seem ambiguous. After all, it is common for the yield on the 30-year U. S. Treasury bond to be 1 /4 % higher than the yield on the 10-year note. A difference of 1 /4 % might not seem like a very large term premium for extending the maturity from 40 to 100 years.

Contrary to this first impression, however, the answer is clear. This chapter follows that of Dybvig and Marshall (1996) in arguing that, unless an investor has extreme views about future interest rates or the market risk premium, he should prefer the 100-year bond yielding 8 1 /4 %.

You can reach this conclusion in four steps. First, because long-term forward rates have a very small impact on bond prices and yields, forward rates from years 40 to 100 must be very large to generate a 1 /4 % difference in yield between 40-and 100-year bonds. In fact, that seemingly small yield difference requires that annual forward rates at specific terms exceed 31%.

Second, because interest rates are volatile and because bond prices are convex in interest rates, forward rates (as well as spot rates and yields) will be below expected future rates under the risk-neutral probability measure. Furthermore, since convexity increases with maturity, long-term forward rates will be very much below these expected future rates. Therefore, for forward rates to exceed 31%, expected future rates must be much larger than 31%.

Third, expected future rates under the risk-neutral measure are a combination of investors' expectations of future rates and of the market premium for interest rate risk. So, generating the extremely large risk-neutral expectations necessary to produce a term premium of 1/4 % between 40 and 100 years requires unrealistically high expectations about future rates or an unrealistically high risk premium.

Fourth, uncertainty that investors have with respect to their interest rate models or model parameters also pushes forward rates below expected future rates. Hence, model uncertainty also leads one to question a large-term premium on yields between 40 and 100 years.

As you will see next, the existence of significant term premia in the yields of very long-term bonds implies that distant forward rates are very high. This conclusion depends only on arbitrage arguments and is therefore very robust. We then show how the convexity of bond prices in interest rates lowers forward rates (as well as spot rates and yields) below risk-neutral expected rates and does so in a way that increases with maturity. We then turn to the relationships among expected rates under the risk-neutral probability measure, investors' expectations of forward rates, and the market's interest rate risk premium. Finally, we consider the effect on prices of uncertainty about the parameters of a term structure model.

FORWARD RATES IMPLICIT IN THE YIELDS OF LONG BONDS

To illustrate the effect of long-term forward rates on bond prices and yields, it is useful to derive the price of a coupon bond as a function of spot and forward rates. For ease of exposition, it is assumed that coupons are paid continuously and interest rates are continuously compounded, but the conclusions would be the same without this assumption.

If continuously compounded spot interest rates are flat at r, the value of $1 to be received at time s is given by

e -rs

So the value of a coupon payment of c at time s is given by

ce -rs

If the coupon is paid continuously from time 0 to time T, the value of the stream of coupon payments is

Finally, then, the value of a coupon bond paying a continuous coupon at a rate of c on a face value of 100 until time T and paying that face value at time T is

Evaluating this expression gives the value of

Now assume that spot interest rates are flat at r until time t and then forward rates are flat at f. The value of $1 to be received at time s > t is

e -rt -f (s -t)

The value of a coupon bond maturing at time T is

0 100ce ds 100ce ds 100e rs t rt f s t rt T t f t T -------ò ò + + () ()

100c r e 100e rT () 1-+ --rT

0 100ce ds 100e rs T --rT ò +

ce ds -rs ò 0

Performing the integrations gives a value of

Apply these equations to a specific example: Suppose that spot rates are flat at 8% for 40 years and that forward rates are flat at f thereafter. Then, the price of a 40-year bond that has a face value of $100 and pays a continuous coupon at the rate of 8% per year is equal to its face value, that is,

while the price of a 100-year 8 1 /4 % bond is (1.1)

P ee f e ee f f = -+ æ è ç ö o ÷ -+ -' -' --'-' 8 25 0 08 1 8 25 1 100 0 08 40 0 08 40 60 0 08 40 60 . . () . () .. . 100 8 0 08 1 100 0 08 40 0 08 40 = -+ -' -' . () .. e e 100 1 100 1 100 c r e f e e e rt rt f T t rt f T t () ( ) () () -+ -+ -------

If P = 100, then the yield to maturity of the 100-year bond is 8 1 /4 % and the forward rate f must be greater than 8 1 /4 %. In fact, in order for Equation (1.1) to hold, it must be the case that f = 31.175%. The forward rate must be so high because long-dated forward rates have little impact on the price of a coupon bond. In the expression above, this fact appears in the form of the coefficient e -0.08'40 = 0.0476 that multiplies the terms involving the forward rate. The small size of this coefficient implies that changes in the forward rate f have a small impact on the bond price.

Figure 1.1 shows how the forward rate varies with the term premium h. It shows the pairs (h, f ) that satisfy the equation

100 8 100 0 08 1 8 100 1 0 08 40 0 08 40 60 = + -+ + æ è ç ö o ÷ --' -' -h e e h f e f . () () .. (1.2)

Strikingly, a term premium of only 12.5 basis points implies that the implied forward rate f is 12.85%, and a term premium of 37.5 basis points is impossible because when 100h = .375 there is no forward rate that satisfies Equation (1.2). 1 The figure indicates that, when the interest rate for the first 40 years is about 8%, beyond 40 years the par yield curve must be virtually flat.

FORWARDS, YIELDS, AND RISK-NEUTRAL EXPECTATIONS

The instantaneous forward rate at some time T is less than the expected instantaneous spot rate at time T where expectations are computed using "risk-neutral" probabilities. Also, the yield on a zero coupon bond maturing at time T is less than the average expected instantaneous spot rate from now to time T where, again, expectations are computed using risk-neutral probabilities. (The next section will connect true expectations and the price of interest rate risk with these risk-neutral expectations.)

Let B( t, T) denote the price at time t of a zero-coupon bond paying $1 at time T, and define the instantaneous forward rates through the relation

From the definition, f(t, T), the forward rate applying to time T as observed at time t, is

Forward rates are related to the expected, instantaneous spot interest rates because the price of the zero-coupon bond can also be characterized as

(1.3)

where E Q denotes expectation under the risk-neutral probability, conditional on the information available at time t. Performing the differentiation on the right-hand side,

where the last equality follows from Equation (1.3).

f E rT rsds E rsds T Q T (, tT) (tT , ) (, tT) tT) ( )exp ( ) exp ( ) = - = - -æ è ç o é ë ê ê öù û ÷ú ú -æ è ç o é ë ê ê öù û ÷ú ú ¶lnB ¶T ¶B(, ¶T B E rsds Q t (tT , ) exp ( ) = -æ è ç o é ë ê ê öù û ÷ú ú ò f (, tT) (tT , ) = -¶lnB ¶T B ftsds t (, tT) exp (, ) = -é ë ê ù û ú ò

This result says that the forward rate is a weighted average of the spot rates r( T) that might be observed at time T, where the weights

depend on the interest rates between t and T. For sample paths for which

is large, the interest rate r( T) receives little weight, while when

is small, r( T) receives more weight. An economic interpretation is that the right to receive a cash flow is less valuable when interest rates are high, and more valuable when they are low. Since interest rates tend to move together so that r( T) is large when

is small and r( T) is small when

is large, the forward rate will be less than the expected spot rate.

) } exp{-ò r (sds t ) } exp{-ò r (sds t r sds t () ò r sds t () ò exp ) exp ( ) -æ è ç ö o ÷ -æ è ç o é ë ê ê öù û ÷ú ú r (sds E rsds Q T

Thus,

f(t, T) £ E Q [r( T)] (1.4)

where the inequality is strict if interest rates are not deterministic.

Since forward rates are less than expected spot rates under risk-neutral probabilities, a forward rate larger than 31%, as required in the 100-year bond example, would imply an expected spot rate even greater than that. Without any intuition about how risk-neutral expectations relate to actual expectations, however, one cannot conclude that 31% is too high a number to be reasonable. The next section relates those two expectations and shows that a risk-neutral expectation of 31% is indeed too high.

Focusing on forward rates makes it very clear that pricing the 100-year bond at a yield of 8 1 /4% raises some serious questions. But similar issues arise by focusing on yields. This is not surprising since the yields on zero-coupon bonds are averages of forward rates. An investor may assume that an 8 1/4% yield implies that risk-neutral expected rates are, on average over the life of the bond, 8 1/4%. But, this would be a mistake: Just as forward rates are below risk-neutral expected rates, so are yields below average risk-neutral expected rates. In other words, for the 100-year bond to be priced at a yield of 8 1/4%, the average of future expected risk-neutral rates over the life of the bond would have to exceed 8 1 /4%.

To illustrate this point, consider the case of a zero-coupon bond. The yield at time t on a zero-coupon bond maturing at time T, R( t, T), is defined such that B( t, T) = e -R( t, T)( T -t)

Therefore,

R B T t rsds T t T (, tT) ln (tT , ) lnE exp ( ) () = -- = - -æ è ç o é ë ê ê öù û ÷ú ú

where the inequality follows from Jensen's inequality. It is strict unless is known at time t, i. e. unless there is no interest rate volatility.

ACTUAL AND RISK-NEUTRAL EXPECTATIONS

The local expectations hypothesis asserts that all fixed-income securities have the same short-term expected return. Since the universe of fixed-income securities includes very short-term instruments, the hypothesis implies that the expected rate of return on all fixed-income securities is equal to the short-term interest rate.

A justification for the local expectations hypothesis would be along the following lines. If the expected return from holding security A were greater than that from holding security B, investors would sell their holdings of security B to buy security A. In the process they would push down the price of security B and push up the price of security A, i. e. push up the expected return of security B and push down the expected return of security A. These portfolio adjustments would continue until the expected returns of the two securities were equal.

While the local expectations hypothesis is logically possible, it seems unreasonable as a description of bond market equilibrium because it implies that investors do not require compensation for bearing interest rate risk in the form of extra expected return. In other words, it seems more reasonable to assume that if security A is r sds t () ò -æ è ç ö o æ è ç o ÷÷ é ê-ê öù ú ú è ç o ÷ E r sds T t E rsds T t Q T Q T ln exp () () () () riskier than security B, its expected return would be commensurately higher. Any expected return above (or below) the short-term rate is, therefore, called a risk premium.

In some markets, it is easy to argue that bearing certain risks should earn positive risk premia. For example, the stock market is often thought to be a good proxy for aggregate wealth. Since aggregate wealth fluctuates up and down, some set of individuals must, in equilibrium, bear stock market risk. But, risk averse as they are, individuals would not expose themselves to this risk unless they were paid to do so in the form of excess expected returns. The stan-dard conclusion, therefore, is that stocks with exposure to the risks of aggregate wealth, namely positive beta stocks, earn an expected return above the short-term interest rate or a positive risk premium. Similarly, negative beta stocks (to the extent they exist) would earn an expected return below the short-term interest rate or a negative risk premium.

The theory of risk premia in the fixed-income markets is not as straightforward. Does aggregate wealth increase or decrease with interest rates? In other words, do investors need to earn positive risk premia in order to hold assets that increase in value with the level of interest rates or in order to hold assets that decrease in value with the level of rates? While this point can be debated, the historical evidence is quite clear. Securities that fall in value when interest rates rise, like coupon bonds, earn a positive risk premium. Furthermore, the more sensitive a bond is to interest rate changes, the greater its risk premium.

Given the existence of risk premia in fixed-income markets, bonds can be priced in one of two ways. First, one could compute the expected present value of a bond's cash flows using the actual probabilities of future rates and then penalize that value in accordance with the interest rate risk of that particular bond. Or, second, one could construct risk neutral probabilities that account for risk premia and compute the expected present value of a bond's cash flows using those probabilities. A standard result (see, e. g., Duffie (1996), Chapter 6) is that, in the absence of arbitrage opportunities, there is a risk-neutral probability such that the expected rate of return on each asset is equal to the short-term rate of interest. Or, equivalently, there is a risk-neutral probability such that the prices of fixed-income securities are given by their expected discounted values.

To illustrate the risk-neutral pricing procedure and the relationship between actual and risk-neutral probabilities, consider a simple model that has reasonable qualitative properties (e. g., Vasicek, 1978). In this model, under the actual probability, the short rate follows the process

dr( t) = k(q -r( t)) dt + sdW( t)

and the expected value of the interest rate at time s, given its value at time t, is

E[ r( s)| r( t)] = r( t) e -k( s -t) + q( 1 -e -k( s -t) ) (1.5)

Letting s get large it can be easily seen that q is the long-run or steady-state mean of the interest rate.

The risk-neutral process for the short-term rate in this model is

dr( t) = [k(q -r( t)) -ls] dt + sdW( t)

where l is the risk premium for bearing interest rate risk per unit of interest rate volatility. Under this probability,

E Q [r( s)| r( t)] = r( t) e -k( s -t) + (q -ls/ k)( 1 -e -k( s -t) ) (1.6)

Here, the long-run mean of the interest rate is q -ls/ k instead of q.

Bond prices in this model can be computed using Equation (1.3). Note that the risk-neutral process has to be used in computing the relevant expectation. The resulting price is

B( t, T) = e a( T -t) + b( T -t) r( t)

where

a T t e T t e b T t e T T () () ] () -=--æ è ç o ÷ --ë ê ù û ú - -=- --t --t --t q ls k s k s k() k() k() 2k öé1- 4k [1- 1-

Finally, the instantaneous forward rates can be obtained through the relation

To understand the role of the risk-neutral process, verify by Ito's Lemma that

while

where B r is the partial derivative of the bond price with respect to the short rate. The first of these expectations is true by construc-tion: The risk-neutral process is determined such that the expected return of each bond, under the risk-neutral process, equals the short-term rate. However, the actual expected return of each bond (i. e., that computed using the actual probabilities) equals the short-term rate plus a risk premium. The interest rate risk of a bond, per dollar invested, is B r /B. The market risk premium per unit of inter-est rate risk is ls, which may be further divided into the price of interest rate risk per unit of interest rate volatility, l, and the amount of interest rate volatility, s. In total, the expected return of a bond above the short-term rate is ls(B r /B). Note that since B r < 0, l must be negative for the risk premium to be positive.

The above discussion reveals that the risk-neutral process transforms the risk premium into probabilities so that the expected return on any bond equals the short-term rate. To see how this is done, compare the expected rates under the risk-neutral process to those under the actual process. From the Equations (1.5) and (1.6),

E rsrt Ersrt e Q s [()|()] [()|()] ) -=----t ls k k() (1 E dB B rdt B B r æ è ç ö o ÷ = +ls E dB B rdt Q æ è ç ö o ÷ = f (, tT) (tT , ) = -¶lnB ¶T

So, if l < 0, as the empirical evidence suggests, the expectation of rates under the risk-neutral measure is above the expectation under the actual probabilities.

For this model, plausible parameter values are r( 0) = .06, q =0.06, k =0.2, and s =0.02. These parameter choices imply that the long-run mean of the short rate is 6%, the first-order autocorrelation of monthly interest rates is exp( -k/ 12) = 0.983, and the standard deviation of the steady-state distribution of the short-rate is . While these choices of parameters are only "ball-park" estimates, they have little impact on the conclusions. The choice of l is more important.

In constructing Figure 1.2, l is set to -0.125, corresponding to a risk premium for holding a 30-year zero-coupon bond over the next s k 2 2 0 0316 /( ) . = instant of 1.247 or 124.7 basis points per year. This value is consistent with recent empirical work, e. g. Dhillon and Lasser (1998).

Figure 1.2 shows the expected short rate under both the original and risk-neutral probabilities and the forward rate for maturities of up to 40 years. Because r( 0) = q, the expected interest rate under the original probability is simply equal to q, regardless of maturity. The expected interest rate under the risk-neutral probability increases with maturity because r( t) is less than the long-run mean q -ls/ k. Consistent with Equation (1.4), the forward rate is less than E Q [r( T)], the expected spot rate under the risk-neutral probability. However, the forward rate is greater than E[ r( T)].

To the extent that the model and parameters used to construct Figure 1.2 are reasonable, investors who use the traditional form of the expectations hypothesis will tend to overvalue long-term bonds. Put another way, an investor who prices bonds by discounting at ex-pected interest rates will arrive at prices that are too high. Similarly, although it is almost certainly a rare error, investors who correctly account for the risk premium but ignore the convexity effects discussed in the previous section will tend to discount at rates that are too high and, hence, undervalue long-term bonds.

This argument could also have been made in terms of yield. Since

and

it follows that

The zero-coupon yield curve is an average of the forward curve. This fact, in combination with the finding that, for reasonable estimates of the risk premium, forward rates exceed expected spot rates under

R tT T t f tsds t (, ) (, ) = -ò 1 B tT e R tT Tt (, ) (, )( ) = -- B tT ftsds t (, ) exp (, ) = -é ë ê ù û ú ò

the actual probabilities, implies that yields also exceed these expected spot rates. Hence, an investor who sets yields equal to average expected spot rates will tend to overvalue long-term bonds. Similarly, an investor who sets yields equal to average expected spot rates under risk-neutral probabilities, thus ignoring the lessons of the previous section, will tend to undervalue long-term bonds.

Figure 1.3 illustrates the relationship between actual expectations, risk-neutral expectations, and yields in this example. In Figure 1.4 the interest rate volatility has been set to s =0.06, which implies that the standard deviation of the steady-state distribution of the short-rate is 0.0949. While this is an unreasonably large value, the figure shows that even with a risk premium the forward rate may fall below the expected short rate at the longer maturities.

PARAMETER UNCERTAINTY

In the results and figures we assumed perfect knowledge of the model and the parameters. Of course, actual market participants don't know the model and parameters. What implications does this have for the yields on long bonds? Could this justify high yields on long bonds? In particular, could this indicate that yields on long bonds should be greater than expected future interest rates?

Strikingly, Dybvig, and Marshall (1996) prove a result which suggests that uncertainty about the model and/ or its parameters implies that yields on long bonds should be less than the expected future long-run average interest rate. The argument is based on the fact that the price of a long-term zero-coupon bond is dominated by interest rate paths with the lowest average interest rate. To understand the intuition behind this result, consider the following simple example.

Suppose that all uncertainty about future values of the short rate will be resolved in the next instant. Specifically, assume that in the next instant the short interest rate will become either x or y, where y > x, and remain at that value forever. Also assume for simplicity that investors demand no risk premium for this interest rate risk. Then, the value of a zero-coupon bond paying one dollar at time T is

where p is the probability that the interest rate turns out to be x. Because y > x, as T becomes large, the first term becomes dominant. In particular, the limiting ratio of the first to the second term is

From this, it follows that the long bond yield

So, since y > x, the yield on a long bond, x, is less than the expected rate, px + (1 -p) y. Thus, uncertainty about the model and its parameters cannot explain high yields on long-term bonds. See Dybvig and Marshall (1996) for a more general proof.

CONCLUSION

This chapter showed why, if a 40-year bond has a yield of 8%, it is extremely unlikely that a 100-year bond should sell at a yield of 8 1 /4%.

lim lim ln lim ln R (T) B (T) T = - = -(exp[-xT]) lim ) ] p p p p exp[(x yT Tr¥ exp[-xT] -)exp[-yT] = -=¥ (1 1- B (T) xT] p)exp[ yT] = pexp[-+--(1

This seemingly small term premium would require that forward rates be remarkably high. But, it is unreasonable to assume that expected rates are anywhere near as high as these forward rates. Also, any reasonable value of the interest rate risk premium would not generate forward rates of the required magnitude. Furthermore, interest rate volatility and the convexity of bond prices tend to lower the yield of the 100-year bond relative to the yield of the 40-year bond. Finally, model uncertainty also tends to lower the yields of long bonds. In short, term premia of very long bonds cannot be as high as some investors, at first glance, might believe.

REFERENCES

Dhillon, U. S. and D. J. Lasser. (1998). "Term Premium Estimates from Zero Coupon Bonds: New Evidence on the Expectations Hypothesis," The Journal of Fixed Income ( June), 52- 58.

Duffie, D. (1996). Dynamic Asset Pricing Theory, Princeton, NJ: Princeton University Press.

Dybvig, P. H. and W. J. Marshall. (1996). "Pricing Long Bonds: Pitfalls and Opportunities,"Financial Analysts Journal (January- February), 32-39.

Vasicek, O. (1978). "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, 5, 177-188.

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