are constant over time, contrary to the evidence presented in Section 10.2.1 ol Chapter 10. One can alter the model to handle these problems, while retaining much ol the simplicity ol the basic structure, by allowing the state variable .v, to follow a conditionally lognormal but heteroskedastic square-root process. This change is entirely consistent with the equilibrium foundations lot the model given in the previous section.

Ihe square-root model, which is a discrete-time version of the famous Cox, Ingersoll, and Ross (1085a) continuous-time model, replaces (11.1.5) and (1 1.1.3) with

- / i = x, +х/-(1И = x, + xlriBS,+ i, (11.1.10)

.v,tl = (\ -ф)ц+фх1 + хУ-$1+1. (11.1.20)

The new element here is that the shock , is multiplied by x)r~. To understand the importance of this, recall that in the homoskedastic model .v,+1 and /H, are normal conditional on x, for all i > 1. This means lhat one can analyze the homoskedastic model either by taking logs of (11.0.1) to get the recuisive equation (I 1.1.1), or by taking logs of (1 1,0.2) to get an /(-period loglinear equation:

l>,a = I,! in, i i I -( /, t I -К I /2) Var, ( mH.)+ + ,nH. . (11.1.21)

Calculations based on (11.1.21) are more cumbersome than the analvsis presented in (he previous section, but they give the same result. In the square-root model, by contrast, .v,+ i and m,+ ] are normal conditional on x, but x,+l and w,+, are nonnormal conditional on x, for all z > 1. This means that one can only analyze the square-root model using the recursive equation (11.1.1); the /(-period loglinear relation (11.1.21) does not hold in the square-root model.

Proceeding with the recursive analysis as before, we can determine the price of a one-period bond by substituting (11.1.19) into (1 1.1.1) to gel

l>u = l-d /i till 1/2) Vai ,1 /H 11 = -x,(l - /iV/2). (11.1.22)

The one-period bond yield yu = - j,u is now proportional to the stale variable x,. Once again the short rate measures the slate of the economy in the model.

Since the short rale is proportional to the state variable, it inherits the property that ils conditional variance is proportional lo its level. Many authors have noted that interest rale volatility tends to be higher when interest rates are high; in Section I 1.2.2 we discuss the empirical evidence-on this point. This property also makes it hard for the interest rale to go negative, since the upward drift in Ihe slate variable tends to dominate the random shocks as v, declines towards zero. Cox, Ingersoll, and Ross (1985a)

A = 1 +</>A,-1 -№ + B,.,)V/2,

A -AH-i = (l-ci)A,-. (ll.l.p)

Comparing (11.1.23) with (11.1.11), we see lhat the term in a2 has been moved from the equation describing A to the equation describing Bn. This is because the variance is now proportional to the state variable, so it affects the slope coefficient rather than the intercept coefficient for the bond price. The limiting value of B , which we write as B, is now the solution to a quadratic equation, but for realistic parameter values this solution is clse to the limit 1/(1 - ф) from the previous model. Thus B is positive and increasing in n.

The expected excess log bond return in the square-root model is given

Е/[r .,+11 ~yu - -Cov,[r +i, ml+i) - Var,[r ,(+i]/2

= A i Cov,[xl+1, m,+\] - Д2 , Var,[x,+]/2

= (-BnBcr2-B\ xcj2l2)xt. (11.1.24)

The first two equalities here are the same as in the previous model. The third equality is the formula from the previous model, (11.1.12), multiplied by the state variable x,. Thus the expected log excess return is proportional to the state variable x, or, equivalently, to the short interest rate y\,. This is the expected result since the conditional variance of interest rates is proportional to x,. Once again the sign of /3 determines the sign of the risk premium term in (11.1.24). Since the standard deviation of excess bond returns is proportional to the square root of x the price of interest rate risk-the ratio of the expected excess log return on a bond, plus one half its own variance to adjust for Jensens Inequality, to the standard deviation of

Depending on the parameter values, it maybe possible for the interest rale to be zero in the continuous-lime model, l-ongstaff (1°(.)2) discusses alternative ways to model this possibility.

show that negative interest rates are ruled out in the continuous-time version *, of this model, where the instantaneous interest rate follows the process (/r = к(в-r)dt-rarl2dB. Time-variation in volatility also produces time-variation in term premia, so that the log expectations hypothesis no longer holds in this model. ;

We now guess that the price function for an ?i-period bond has the same linear form as before, -p , = A + B x equation (11.1.8). In this model Au - A, = 0, Л = 0, and At = 1 - B2a2/2. It is straightforward to verify the guess and to show lhat A and B obey

the excess log return on the bond-is also proportional to the square root оГх

The forward rate in the square-root model is given by

fnt = >i( + 5 (E/[Ax/+,]-Cov([jC,+ i, ,+,])-Z?*Var([x(+il/2

= (1 -fS2a2/2)x, - в,[(1 -Ф)(х, ~ H) + x,fia2]

-D2,x,a2/2. (11.1.25)

The first equality in (11.1.25) is the same as in the homoskedaslic model, virile the second equality multiplies variance terms by x, where appropri-itc. It can be shown that the square-root model permits the same range of : hapes for the yield curve-upward-sloping, inverted, and humped-.is the lomoskedastic model.

! Pearson and Sun (1994) have shown that the square-root model can be generalized to allow the variance of the state variable to be linear in the level of the state variable, rather than proportional to it. One simply replaces the x,2 terms, multiplying the shocks in (11.1.19) and (11.1.20) with terms of the form (0/04-0/1 x,)>/2. The resulting model is tractable because it remains in the affme-yield class, and it nests both the homoskedaslic model (the caseao = l.a- = 0) and the basic square-root model (the case a = 0, a, = 1).

11.1.3 A Two-Factor Model

So far we have only considered single-factor models. Such models imply that all bond returns arc perfectly correlated. While bond returns do lend to be highly correlated, their correlations arc certainly not one and so it is natural lo ask how this implication can be avoided.

We now present a simple model in which there arc two factors rather than one, so that bond returns arc no longer perfectly correlated.1 The model is a discrete-time version of the model of LongstafT and Schwartz (1992). U replaces (11.1.19) with

- m,+ , = x\, + x>i, + xl/2e,+ l, (11.1.20)

and replaces (11.1.20) with a pair of equations for the state variables:

= (I -v*t)/t +01 x -f-x.fu+t- (11.1.27)

*z..+ i = (I - fa)li-i + Ф-2 *!, + xjf 1. (11.1.28)

Although bond returns are not pit Icrtly correlated in this model, the covarianre matrix of bond returns has rank two and hence is singular whenever we observe more than two bonds. We discuss this point further in Section 11.1.4.

Finally, the relation between the shocks is

(ш = №,+ >. (H-l-29)

and the shocks £.,+ and \ are uncorrelated with each other. We will write crj- for the variance of £ and rr.j- for the variance ol .щ.

In this model, minus the log .stochastic discount factor is forecast by two state variables, хц and x2/. The variance of the innovation to the log stochastic discount factor is proportional to the level of л , as in the square-rool model; and each of the Iwo state variables follows a square-root autoregressive piocess. Finally, the log stochastic discount factor is conditionally correlated with xt but not with x2. This last assumption is requited lo keep Ihe model in the tractable affme-yield class. Note that the two-factor model nests the single-factor square-root model, which can be obtained by selling x = 0, but does not nest the single-factor homoskedaslic model.

Proceeding in the usual way, we find that die price of a one-period bond

f,u = E,[ .,+ i] -r(l/2)Var,[m,+ 1] = -x - .v + x fi2a2/2. (11.1.30)

fhe one-period bond yield yu = -fn, is no longer proportional lo die state variable xi because it depends also on х-ц. The short interest rate is no longer sufficient to measure the stale of the economy in this model. I.ongstalf and Schwartz (1992) point out, however, that the conditional variance til the short rale is a different linear function of the two stale variables:

Var,[y:.,+1] = (1 - fi2o2m2o2 .v + a2 x . (11.1.31)

Thus die short rate and its conditional volatility summarize the state of the economy, and one can always state the model in terms of these two variables.

We guess thai the price function for an n-period bond is linear in the two stale variables: -/>, = A + />, x + lh x-ц. We already know lhat Au = lim = Iho = 0, Ai = 0, = 1 - a2/2. and lin = 1. his straightforward to show lhat A , li\ , ami !!> obey

Ли = 1 +01Й1..-1 - (/ 4- ,. ., )- Г/--li, = \+fabi. -i-liiK-ia;/2.

A - /l i

= (1 -0, ) ,/),. .. - +(1 -ф-,),с,1Ъ. -1- (II.1.32)

/ /. leini-Sli in lure Model*

The difference equation for H\ is (lie same as in ilie single-factor square-root model, (11.1.23), hut the difference equation for 11, includes only a term in the own variance of.*) because .* is uncorrelated with m and docs not allcct the variance ol /. Ihe difference equation for Л is just the stun of two lertns, each of which has the familiar Conn from the single-factor square-root model.

fhe expected excess log bond return in the two-factor model is given

I-il .mi I - vii = - Gov, I r .m. 1,4.1 J - Var,(r 4. 1/2

= /*(...-i Gov,.vi.,.M, 1 - B\ Var,.v.,+ i/2 -/iL-Vai-,l*../+,l/2

- -/J.. -i/iffr-/(b,-i/2]xi,

Illl, ,n.;/2.v.,. (11.1.33)

This is the same as in the square-root model, with the addition of an extra term, arising from Jensens Inequality, in the variance of *> + . The forward rate in the two-factor model is given by

f i = Yi, + i (K,A.v,.,h I - Gov,[.v.,m, ? ,+ ,]) + B, E,\Ax-,.l+i I - /;r V-l-v,.m1]/2- /(.7 Var,[.v,.,+ 1]/2 = (1 - ti<,-/>)xtl + x.,- , (1 -</>)(.v -/<,)

- 0..,)(.v., -Ц->)-П, .х;,ваf

- .v cTf/2 - rr.;/2. (11.1.3-1)

Ibis is the obvious generalization of the square-root model. Importantly, it can generate more complicated shapes for the yield curve, including inverted hump shapes, as the independent movements of both .v, and л\>, a flee i the term structure.

fhe analysis of this model illustrates an important principle. As Cox, Ingersoll, and Ross (1(.ЖГ>а) and Dybvig (1989) have emphasized, under certain circumstances one can construct inultilactor term-structure models simply by adding up single-factor models. Whenever the stochastic discount factor w, н can be written as the sum of two independent processes, then the resulting term structure is the sum of the lenn structures that would exist under each of these processes. In the 1 .ongstalf and Schwartz (1992) model the stochastic discount factor is the sum of - ,vi, - x],2tt£i,n[ and -vj, and these components are independent of each other. Inspection of (11.1.34) shows dial the resulting term structure is just the sum of a general

ill Affine-Yield Models 441

square-root term structure driven by the x\, process and a special square-root? term structure with parameter restriction в - 0 driven by the хц process.

ILL4 Beyond Affine-Yield Models

We have considered a sequence of models, each of which turns out to have the property that log bond yields are linear or affine in the underlying state variables. Brown and Schaefer (1991) and Duffie and Kan (1993) have clarified the primitive assumptions necessary to get an affine-yield model. In the discrete-time framework used here, these conditions are most easily stated by defining a vector x, which contains the log stochastic discount factor m, and the time t values of the state variables relevant for forecasting future m,+i, i = 1... n. If the conditional forecast of x one period ahead, F-i[Xf+i]> is affine in the state variables, and if the conditional distribution of x one period ahead is normal with a variance-covariance matrix Var,[x,+)] which is affine in the state variables, then the resulting term-structure model is an affine-yield model.

To see this, consider the steps we used to derive the implications of each successive term-structure model. We first calculated thelog short-term interest rate; this is affine in the underlying state variables if m,+\ is conditionally normal and E,[m,+i] and Vart[mti.i] are affine in the state variables. We next guessed that log bond yields were affine and proceeded to verify the guess. If yields are affine, and if x is conditionally normal with affine variance-covariance matrix, then the risk premium on any bond is affine. Finally we derived log forward rates; these are affine if the short rate, risk premium, and the expected change in the state variable are all affine. Affine forward rates imply affine yields, verifying that the model is in the affine-yield class.

Brown and Schaefer (1991) and Duffie and Kan (1993) state conditions on the short rate which deliver an affine-yield model in a continuous-tirt)e setting. They show that the risk-adjusted drift in the short rate-the expected change in the short rate less the covariance of the short rate wi(h the stochastic discount factor-and the variance of the short rate must both be affine to get an affine-yield model. The models of Vasicek (1977), Co\, Ingersoll, and Ross (1985a), and Pearson and Sun (1994) satisfy these requirements, but some other continuous-time models such as that of Brennaii and Schwartz (1979) do not.

Affine-yield models have a number of desirable properties which help to explain their appeal. First, log bond yields inherit the conditional normality assumed for the underlying state variables. Second, because log bond yields are linear functions of the state variables we can renormalize the modejl so that the yields themselves are the state variables. This is obvious in a one-factor model where the short rate is the state variable, but it is equally