possible in a model with any number of factors. Longstaff and Schwartz (1992) present their two-factor model as one in which the volatility of the short rate and the level of the short rate are the factors; the model could be written equally well in terms of any two bond yields of fixed maturities. Third, affine-yield models with К state variables imply that the term structure of interest rates can be summarized by the levels of К bond yields at each 1 point in time and the constant coefficients relating other bond yields to the К basis yields. In this sense affine-yield models are linear; their nonlinearily is confined to the process governing the intertemporal evolution of the К basis yields and the relation between the cross-sectional coefficients and the underlying parameters of the model.

Affine-yield models also have some disadvantages. The linear relations among bond yields mean that the covariance matrix of bond returns has rank К-equivalcntly, wc can perfectly fit the return on any bond using a regression on К other contemporaneous bond returns. This implication will always be rejected by a data sel containing more than К bonds, unless we add extra error terms to the model. Affine-yield models also limit the way in which interest rate volatility can change with the level of interest rates; for example a model in which volatility is proportional lo the square of the interest rate is not affine. Finally, as Constantinidcs (1992) emphasizes, single-factor affine-yield models imply that risk premia on long-term bonds always have the same sign.

If we move outside the affine-yield class of models, wc can no longer work with equation (11.1.1) but must return lo the underlying nonlinear difference equation (11.0.1) or its n-period representation (11.0.2). In general these equations must be solved numerically. One common method is to sel up a binomial tree for die short-term interest rale. Black, Dermau, and Toy (1990) and Black and Karasinski (1991), for example, assume that the simple one-period yield Y\, is conditionally lognormal (as opposed to the assumption of affine-yield models lhat (1 4- Yu) is conditionally lognormal). They use a binomial tree to solve their models for the implied term structure of interest rates. Constantinidcs (1992), however, presents a model that can be solved in closed form. His model makes the log stochastic discount factor a sum of noncentral chi-squared random variables rather than a normal random variable, and Consianiinidcs is then able to calculate the expeclalions in (11.0.2) analytically.

J 11.2 Fitting Term-Structure Models to the Data

, 11,2.1 Real Bonds, Nominal Bonds, and Inflation

The term-structure models described so far apply to bonds whose payoffs are liskless in real terms. Almost all actual bonds instead have payolfs that arc

riskless in nominal terms/ We now discuss how the models can be adapted lo deal with this fact.

To study nominal bonds we need lo introduce some new notation. We write the nominal price index at lime ( as Qj, and the gross rate of inflation from ; to I + 1 as П,+ , = Qj+l/Qj. We have already defined t , lo be the real price of an n-pcriod real bond which pays one goods unit at time / 4- n; we now define I*, to be the nominal price of an n-pcriod nominal bond which pays $1 at lime t + n. From these definitions it follows that the nominal price of an n-pcriod real bond is / , (>, and the real price of an N-period nominal bond is /*J,/Q,. We do not adopt any special notation for these last two concepts.

If we now apply the general asset pricing condition,

1 = + ,.,+ , )M,+ I],

to the real relurn on an n-period nominal bond, we find that

1 ni

= E,

(11.2.1)

Multiplying through by Qj, we have

t = E,

Civ,

= E,

и-1.1+1

П,+ .

(11.2.2)

where M($+ s M,+ i/n,+ i can be thought of as a nominal stochastic discount factor that prices nominal returns.

The empirical literature on nominal bonds uses this result in one of two ways. The first approach is to take the primitive assumptions that we made about M,+ in Section 11.1 and to apply them instead to fhe real

term-structure models of the last section are (hen reinterpreted as nominal term-structure models. Brown and Oybvig (1980), for example, do this when

Some governments, notably those ot Canada, Israel, and (he UK. have issued Inintls whose nominal payoffs are linked to a nominal price index. In HMHi the U.S Ticasury is considering issuing similar securities. These index-linked bonds approximate real ImiikIs but are rarely exactly equivalent to real bonds, lirown and Schacler (HI.M) give a lucid discussion ol the imperfections in the UK indexing system, and apply the Cox, Ingersoll, and Ross (IJHfta) model to UK index-linked bonds. See also ban and Campbell (W.)li) and Campbell and Shiller (1!)%).

11. 7hm-Struclurc Models

they apply the Cox, Ingcrsoll, and Ross (НЖГ>а) square-root model directly to data on US nominal bond prices. The square-root model restricts interest rates to he positive, and in this respect it is more appropriate for nominal interest rates than lor real interest tales.

The second approach is to assume that the two components of the nominal stochastic discount factor, Л/ - and 1/П/+, are independent of each other. To see how this assumption facilitates empirical work, take logs of the nominal stochastic discount factor to get

(11.2.3)

When the components tit and jr are independent, we can price nominal bonds by using the insights of Cox, Ingersoll, and Ross (1985a) and Dybvig (1989). Recall from Section I 1.1.3 their result that the log bond price in a model with two independent components of the stochastic discount factor is the sum of the log bond prices implied by each component. We can, for example, apply the l.ongstaff and Schwartz (1992) model to nominal bonds by assuming that шм is described by a square-root single-factor model, - hi = v + x\leLn i, and that ттп1 is known at / and equal to a state variable .v.,. We then get ш* = - ш , + л,+ - = x + .v[fi + *>/. and the I .ongsiaff-Schwarl/ model describes nominal bonds.

More generally, the assumption lhat Mni and 1/П,+ 1 are independent implies that prices of nominal bonds are just prices of real bonds multiplied by the expectation of die future real value of money, and that expected real returns on nominal bonds are the same as expected real returns on real bonds. To see this, consider equation (11.2.2) with maturity n = 1, arid note thai the independence of Лand 1/П,+ 1 allows us to replace the expectation of their product by the product of their expectations:

>l = 1.<1л/,5и1 = К, 1Л/М,1 к,

= IuQjV;

(11.2.4)

since У-, = K,(a/m i I and 1/П,+ = Ci/Ci+t- Thus the nominal price of a bond which pays $1 tomorrow is the nominal price of a bond which pavs one unit of goods tomorrow, times the expectation of the real value of $1 tomorrow.

We now guess thai a similar relationship holds for all maturities n, and we prove this by induction. If lite (n- 1 J-period relationship holds, Pj , , = / -i.f Ci I-*rl 1/Qf i11. Hum

l.M I rt,i l --

lit i

Ill l.M I (1 I I I

L<Ar .

4н J

11.2. Filling Term-Structure Models to tlie Data

Q,E,

P i +i M,+1 E,+

= ftitJE,

where the last equality uses both the independence of real variables frohi the price level (which enables us to replace the expectation of a product by the product of expectations), and the fact that P , = E,[P + M,+i1. Equation (11.2.5) is the desired result that the nominal price of a bond which pays $1 at time t + n is the nominal price of a bond which pays one unit of goods at time I + n, times the expected real value of $1 at time I + n. Dividing (11.2.5) by Qj, we can see that the same relationship holds between the real prices of nominal bonds and the real prices of real bonds. Further, (11.2.5) implies that the expected real return on a nominal bond equals the expected real return on a real bond:

(11.2.5!)

n-I.M-l Qj

= E,

= E,

E(+i[i/(2/+ ]Pn-i.(+i Qj+i Ui/Q,+\]Pn,Qj

Qjul

(11.2.6)

Gibbons and Ramaswatrty (1993) use these results to test the implications of real term-structure models for econometric forecasts of real returns on nominal bonds.

Although it is extremely convenient to assume that inflation is independent of the real stochastic discount factor, this assumption may be unrealistic. Rarr and Campbell (1995), Campbell and Ammer (1993), and Pen-nacchi (1991), using respectively UK data on indexed and nominal bonds, rational-expectations methodology applied to US data, and survey data, all find that innovations to expected inflation are negatively correlated in the short run with innovations to expected future real interest rates. More directly, Campbell and Shiller (1996) find that inflation innovations are correlated with stock returns and real consumption growth, proxies for the stochastic discount factor suggested by the traditional CAPM of Chapter 5 and the consumption CAPM of Chapter 8.

11.2.2 Empirical Evidence on Affine- Yield Models 1

All the models we have discussed so far need additional error terms if they! are to fit the data. To see why, consider a model in which the real stochastic! discount factor is driven by a single state variable. In such a model, returnsj on all real bonds are perfectly correlated because the model has only a single shock. Similarly, returns on all nominal bonds are perfectly correlated in any

model where a single stale variable drives (he nominal stochastic discount laclor. In reality (here are no deterministic linear relationships among relurns on different bonds, so these implications are bound lo be rejected by the dala. Adding extra state variables increases the rank of the variance-covariance matrix of bond returns from one to K, where Л is the number of state variables, but whenever there arc more than A bonds the matrix remains singular-cquivalently, there are deterministic linear relationships among bond returns. So these models, loo, are trivially rejected by the data.

To handle this problem empirical researchers allow additional error terms to affect bond prices. These errors may be thought of as measurement errors in bond prices, errors in calculating implied zero-coupon prices from an observed coupon-bearing term structure, or model specification errors arising from tax effects or transactions costs. Alternatively, if one uses a model for the real stochastic discount factor and tests il on nominal bonds in the manner of Gibbons and Ramaswamy (1993), the errors may arise from unexpected inflation.

Whatever the source of the additional errors, auxiliary assumptions about their behavior are needed to keep the model testable. One common assumption is that bond-price errors are serially uncorrelated, although they may be correlated across bonds. This assumption makes it easy to examine ihe time-series implications of term-structure models. Other authors assume that bond-price errors arc uncorrelated across bonds, although they may be correlated over time.

Affine-Yield Models as Latent-Variable Models

Stambaugh (1988) and Hcston (1992) show thai under fairly weak assunip-tioiQ> about the additional bond price errors, an affine-yield model implies a la tent-variabk structure for bond returns. Variables that forecast bond lett rns can do so only as proxies for the underlying slate variables of ihe model; if there are fewer state variables than forecasting variables, this puus testable restrictions on forecasting equations for bond returns.

A general affine-yield model with A. stale variables takes the form

p i = A + IS , xtl 1-----h В/,- xKl,

(11.2.7)

whe rc хм, к = 1 ... A, are the stale variables, and A and Zfo, к = 1 ... A, are constants. The model also implies that expected excess returns on long bonds over the short interest rate can be written as

(11.2.8)

x A* and BJ , к = 1 ... К, are constants. The model puts cross-scclijonal restrictions on these constants which are related to the lime-series process driving the stale variables, but we ignore this aspect of the model here.

Now suppose that we do nol observe die true excess relurns on long bonds, but instead observe a noisy measure

,../+1 = .i i ~ Уи + ./! i. (11.2.9)

where )/ ./+! is an error term. We assume that i is orthogonal lo a vector h, containing / instruments /( , j = 1.../:

К1 ./.ц I h,l = <>. (11.2.10)

The vector h, might contain lagged variables, for example, if the return error i) .)+i is serially uncorrelated. We further assume thai for each stale variable л* It = 1 ... A, die expectation of the slale variable conditional on the instruments is linear in the instruments:

E[x*, I h,J = £iV/ (11.2.11)

for some constant coefficients tV*y.

These assumptions imply that the expectation of e J+l conditional on the instruments, which from (11.2.10) is the same as the expectation of the li ue excess return r ,+ i - yu conditional on die instruments, is linear in the instruments:

I-lVi+i I h,J = E [/ .,+1 -yu I h,J =

*=i *=i j=\

If we define c to be die vector [e\ rll ... r,v.H i ] for assets n = I ... N, then (11.2.12) can be rewritten in vector form as

c,+ i = A* -f-Ch, + t/,h, (11.2.13)

where A* is a vector whose nth element is A* and С is a matrix of coefficients whose (n, j) element is

C i = Yl b l<r (П.2.14)

Equations (11.2.13) and (11.2.14) define a lalenl-variable model Unexpected excess bond returns with A latent variables. Kquation (11.2.14) says that the (N x J) matrix of coefficients of N assets on J instruments lias rank at most A, where A is the number of state variables in the underlying term-structure model. The instruments forecast excess bond returns only through their ability to proxy for the stale variables (measured by (he 0kl