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10. lixed-lneomeSecurities

Stales. litis finding is reminiscent of the finding discussed in Chapter , lhat stock relurns lend lo be higher when dividend yields arc high al the stall of the holding period. As in die slock market case, it is nol clear whether this result reflects a failure of rationality on die part of investors or the presence of lime-varying risk premia. Froot (1989) has used survey data to argue dial bond market investors have irrational expectations, but there is also much theoretical work, discussed in the next chapter, that explores the impact of time-varying risk on the term structure of interest rates.

This chapter has concentrated on the forecasting power of yield spreads for future movements in nominal interest rales. Yield spreads are also useful in forecasting other variables. For example, one can decompose nominal rates into inflation rales and real interest rates; the evidence is that most of the long-run forecasting power of the term structure is for inflation rather than real interest rates (see Fama 11975, 1990] and Mishkin [ 1990a, 1990!))). We mentioned in Chapter 8 that the slope of the term structure has some ability lo forecast excess returns on slocks as well as bonds. Other recent studies by Chen (1991b) and F.slrella and Hardouvelis (1991) have shown lhat the term structure forecasts real economic activity, since inverted yield curves lend lo precede recessions and steeply upward-sloping yield curves tend to precede expansions.

Problems-Chapter 10

10.1 You are told dial an 8-year nominal zero-coupon bond has a log yield lo maturity of 9.1 %, and a 9-year nominal zero-coupon bond has a log yield of 8.0%.

10.1.1 Can the pure expectations theory of the term structure describe these data?

10.1.2 A year goes by, and die bonds in part (a) still have the same yields to maturity. Can the pure expectations theory of the term structure describe these new dala?

10.1.3 1 low would your answers change if you were told thai the bonds have an 8% coupon rale per year, rather than zero coupons?

10.2 Suppose that the monetary authority controls short-term interest rales by selling

yii = yi.i i + My-i - yu) + with X > 0. Intuitively, die monetary authority tries lo smooth interest rales but raises iheni when die vield curve is sleep. Suppose also thai the two-period bond vield satisfies

y-ji = (yi/ i /./l.vi./, i )/2 + .v,.

Problems 425 >

where x, is a term premium that represents the deviation of the two-period yield from the log pure expectations hypothesis, equation (10.2.10). The variable x, follows an AR( 1) process

X, = фх,-\ + n,.

The error terms e, and n, are serially uncorrelated and uncorrelated with each other.

10.2.1 Show that this model can be solved for an interest-rate procejss of the form

Уи = yi.t-l + YX,-r(,. Express the coefficient у as a function of the other parameters inlthe model.

10.2.2 The expectations hypothesis of the term structure is often tested in the manner of equation (10.2.17) by regressing the scaled change in the short rate onto the yield spread,

<0>i./+i ->w)/2 = a + B(y.2t-yi,)+u,+\,

and testing the hypothesis that the coefficient в = 1. If the m<)del described above holds, what is the population value of the regression coefficient 8}

10.2.3 Now consider a version of the problem involving n-period bojids. The monetary authority sets short-term interest rates as

Уи = >t./-i +Uy t - yu) +e<. and the n-period bond yield is determined by

y i - Уи = (n-l)Ei [> ./+1 -y t] + x

where x, now measures the deviation of the n-period yield from the log pure expectations hypothesis (10.2.14). (This formulation ignores the disiinction between y , and y i,/.) As before, x, follows an AR(1) process. What is the coefficient у in this case? What is the regression coefficient В in a regression of the form (10.2.16),

(j .i+i ~ y t) = a + fi(ynl - yu)/(n - 1) + u,+i ?

10.2.4 Do you find the model you have studied in this problem to be a plausible explanation of empirical findings on the term structure? Why or why not?

Note: This problem is based on McCallum (1994).



Term-Structure Models

Tins CHAPTER EXPLORES the large modern literature on fully specified general-equilibrium models of the term structure of interest rales. Much of ibis literature is set in continuous time, which simplifies some of the theoretical analysis but complicates empirical implementation. Since we focus on the econometric testing of the models and their empirical implications, we adopt a discrete-time approach; however we take care to relate all our results lo their continuous-lime equivalents. We follow the literature by first developing models for real bonds, but we discuss in some detail how these models can be used to price nominal bonds.

All the models in this chapter slarl from the general asset pricing condition introduced as (8.1.3) in Chapter 8: 1 = F.,l(l + /<;.,+ W+t L where /{,.,+ ! is the real return on some assel i and M!+\ is the .stochastic discount factor. As we explained in Section 8.1 of Chapter 8, this condition implies lhat the expected return on any assel is negatively related to its covariance with the stochastic discount factor. In models with utility-maximizing investors, ihe stochastic discount factor measures the marginal utility ol investors. Assets whose returns covary positively with the stochastic discount factor lend to pay off when marginal utility is high-they deliver wealth al limes when wealth is most valuable lo investors. Investors are willing to pay high prices and accept low returns on such assets.

Fixed-income securities are particularly easy lo price using this framework. When cash Hows are random, the stochastic properties of the cash flows help to determine the covariance of an assets relurn with the stochastic discount factor. But a fixed-income security has deterministic cash flows, so it covaries with the stochastic discount factor only because there is time-variation in discount rates. This variation in discount rales is driven by die time-series behavior of the stochastic discount factor, so term-structure models are equivalent lo time-series models for the stochastic discount factor.

From (10.1.4) in Chapter 10, we know that returns on n-pcriod real zero-coupon bonds are related to real bond prices in a particularly simple



. term-Structure Models

way: (I + /i .,h) = ii,-\.h-\Ii\,i- Substituting this into (8.1.3), wo find that the real price ol an н-period real bond, A* satisfies

l , = K,/ -i.,+ iM/+i]. (П.0.1)

fhiseipiation lends itself to a recursive approach. We model / , as a function of those state variables lhat are relevant for forecasting the Ml+\ process. Given that process and the function relating / -1./ to state variables, we can calculate the function relating / , lo slate variables. We start the calculation by noting that / = 1.

Equation (11.0.1) can also be solved forward to express the -period bond price as the expected product of и stochastic discount factors:

>, = Y.,\MHl...Ml+ ]. (11.0.2)

Although we emphasize the recursive approach, in some models il is more convenient to work directly with (11.0.2).

Section I I.I explores a class of simple models in which all relevant variables are conditionally lognormal and log bond yields arc linear in state variables. These affine-yield models include all the most commonly used term-strucliire models. Section I 1.2.shows how these models can be lit lo nominal interest rate data, and reviews their strengths and weaknesses. One of the main uses of term-structure models is in pricing interest-rate derivative securities; we discuss this application in Section 11.3. We show how standard lenn-slrm lure models can be modified so that they lit the current lenn structure exactly. We then use the models to price forwards, futures, and options on fixed-income securities.

11.1 Affine-Yield Models

To keep mailers simple, we assume ihroughoul this section lhat the distribution of the slot haslic discount factor M,+ i is conditionally lognormal. We specify models in which bond prices are jointly lognormal with Ml+\. We can then take logs of (1 1.0.1) to obtain

/ / = E/I i i-A, i.nil + (1/2) Var, [/>/,+ , + /vi.i+il. (11.1.1)

where as usual lowercase letters denote the logs of the corresponding U> percase letters so for example z/z,+ i = log(M,+ i). This is the basic equation we shall use.

We begin wilh two models in which a single slate variable forecasts the stochastic discount factor. Section 1 1.1.1 discusses the first model, in which 1 i is homoskedastic. while Section 11.1.2 discusses the second model, in which the conditional variance of mHi changes over lime. These are

./. Affine-Yield Models

discrete-lime versions of the well-known models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985a), respectively. Section 11.1.3 then considers a more general model with two state variables, a discrete-time version of the model of Longstaff and Schwartz (1992). AH of these models have the property that log bond prices, and hence log bond yields, are linear or affene in the state variables. This ensures the desired joint lognormality of bond prices with the stochastic discount factor. Section 11.1.4 describes the general properties of these affine-yield models, and discusses some alternative modelling approaches.

11.1.1 A Homoskedastic Single-Factor Model

It is convenient to work with the negative of the log stochastic discount factor, -ml+\. Without loss of generality, this can be expressed as the sum of its one-period-ahead conditional expectation x, and an innovation

- m,+ i = x, + e-,+i. (11.1.2)

We assume lhat is normally distributed with constant variance. j

Next we assume that x(+i follows the simplest interesting time-series process, a univariate AR(1) process with mean u. and persistence ф. The shock to is written

x,+1 = (1 -ф + фъ + Цм. (1U.3)

The innovations to m,+i and x,+i may be correlated. To capture this write e,+i as

7+1 = №+1 + П/+1. 01

1.4)

where i and n,+i are normally distributed with constant variances anc are uncorrelated with each other. . i

The presence of the uncorrelated shock / + only affects the avenige level of the term structure and not its average slope or its time-series behavior. To simplify notation, we accordingly drop it and assume that = Equation (11.1.2) can then be rewritten as !

-m,+ i = х/4-pW (11.1.5)

The innovation fj/+i is now the only shock in the system; accordingly we can write its variance simply as a1 without causing confusion.

Equations (11.1.5) and (11.1.3) imply that - ml+\ can be written as an ARMA( 1,1) process since it is the sum of an AR(1) process and white noise.

Our ilisritlc-liine presentation follows .Singleton (lJJfl), Sun (1992), and especially Backus (1993). Sun (1992) explores ilie relation between discrete-time and continuous-time uio/lcls in more detail.



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