In fact, -m,+i has the same structure as asset returns did in the example of Chapter 7, Section 7.1.4. As in that example, it is important to realize that -im,+i is not a univariate process even though its conditional expectation x, is univariate. Thus the univariate autocorrelations of - ) do not tell us all we need to know for asset pricing; different sets of parameter values, with different implications for asset pricing, could he consistent with the same set of univariate autocorrelations for - wi,+ . For example, these autocorrelations could all be zero because a1 = 0, which would make interest rates constant, but they could also be zero for а2 ф 0 if fi lakes on a particular value, and in ibis case interest rates would vary over time.
Wc can determine die price of a one-period bond by noting that when it = 1, n i.,+i = /Ai.i+i = 0, so die terms involving -i.i+i in equation -j(11.1.1) drop out. Substituting (11.1.5) and (11.1.3) into (11.1.1), wc have
j / , = E,[ /,+ 1] + (1/2) Var, [/ ,+ ,] = -x, + fiV/2. (11.1.6)
I The one-period bond yield уц = -fi\ so
= x, - f}2o2/2. (11.1.7)
The short rate equals the state variable less a constant lerm, so il inherits the AR(1) dynamics of the stale variable. Indeed, wc can think of the short rale as measuring the state of the economy in this model. Nole that there is nothing in equation (11.1.7) lhat rules out a negative short rate.
Wc now guess that the form of the price function for an n-pcriod bond
-p , = A + B xt. (11.1.8)
Since the n-pcriod bond yield yu, = -/ / и, we are guessing that the yield on a bond of any maturity is linear or a/fine in the slate variable x, (Ikown and Schaefer ). We already know that bond prices for и = 0 and n = 1 satisfy equation (11.1.8), with Л = Д, = 0, A, = -f)2o2/2, and 7i, = 1. We [proceed to verify our guess by showing that it is consistent with the pricing relation (11.1.1). Al die same lime we can derive recursive formulas for the coefficients A and Ii .
Our guess for die price function (1 1.1.8) implies that the two terms on the right-hand side of (11.1.1) arc
E/[ /+i 4- /i -i./+i ] = -x, - Л - H -iO - ф)ц - /Vis* X,. Var,[i ,+ I 4- / ,.,+ ,] = (/J + . ,)V. (II.1.9)
Substituting (11.1.8) and (11.1.9) into (11.1.1), we get,
A + B x, - x, - A i - /> -i(l - ф)ц - В -\ф x,
4- (/J + /J ,)V-72 = 0. (I 1.1.10)
1 his must hold for any x so the coefficients on .v, must sum lo zero and the remaining coefficients must also sum lo zero. This implies
II = 1 +фВ -\ = (I -</> )/< 1 -</>).
A -A i = (1 -0)/tK . - (/) + /s )VJ/2. (II.I.II)
We have now verified the guess (I 1.1.8), since with the coefficients in (11.1.11) the price function (11.1.8) satisfies die asset pricing equation (11.1.1) and its assumption that bond returns are conditionally lognormal.
Imputations of the Homoskedaslic Model
The homoskedaslic bond pricing model has several interesting implications. First, the coefficient B measures the fall in the log price of an n-period bond when there is an increase in the stale variable x, or equivalently in the one-period interest rate ylh Ii therefore measures the sensitivity of the /.-period bond relurn lo the one-period interest rate. Equation (11.1.11) shows that the coefficient B follows a simple univariate linear difference equation in n, with solution (1 - $ )/( - ф). As n increases, ll approaches a limit В - 1/(1 - ф). Thus bond prices fall when short rates rise, and the sensitivity of bond returns to short rates increases with maturity.
Nole thai B is different from duration, defined in Section 10.1.2 of Chapter 10. Duration measures the sensitivity of the /(-period bond return to die n-period bond yield, and for zero-coupon bonds duration equals maturity. B measures ihe sensitivity of the n-period bond return to die one-period interest rate; it is always less than maturity because the n-pcriod bond yield moves less than one-for-one with the one-period interest rate.
A second implication of the model is that the expected log excess return on an /(-period bond over a one-period bond, ЕД r J+\ ] - yu = E,[ ii ] - p, + pn, is given by
E,[/ .,+ i] -ум = - Gov, [/ .,+ ., wi,+l] - Var, / .,H ]/2
= B i Gov,I дг/+ j, m,+1 I - II2 i Var, [.v,+, ]/2
= -В фо2 - /i;; 0-2/2. (11.1.12)
The first equality in (11.1.12) is a general result, discussed in Chapter 8, that holds for the excess log return on any asset over the riskfree interest rate. It can be obtained by taking logs of the fundamental relation I = E,[(l + /( ,+ i)(W,+i] for the n-pcriod bond and die short interest rale, and then taking the difference between the two equations. It says lhat the expected excess log return is the sum of a risk premium term and a Jensens Inequality term in the own variance which appears because we are working in logs.
I /. term-Strut tun- Лнн/гЛ
The second equality in (11.1.12) uses the fad dial I lie uiu-xpt-i led com-poiicnt ol die log return on an -period bond is jusi -H \ times I lie- in-iiiivaliini in the Mali- variable. The third equality in (I 1.1.12) uses the Iacl dial the conditional vat iance ol.v , and ilsconditional covariance with /,+ , are constants lo show dial die expected log excess return on any bond is constant over lime, so that the log expectations hypothesis-but not die log pure expeclalions hypothesis-holds.
- It i is the coefficient Irom a regression ol /(-period log bond relurns on stale variable hinovalioiis, so we can interpret -Л as the bonds loading on the single source ol risk and fin as die reward lor bearing a unil of risk. Allernalivclv. following; Vask ck (H)77) and others, we might calculate the price of risk as the ratio of die expected excess log relurn on a bond, plus one half ils own variance lo adjust for Jensens Inequality, to lite standard deviation ol the exi ess log return on the bond. Defined this way, the price of risk is just --/(a in this model.
Ihe homoskedastic bond pricing model also has implications for (Inpatient of lorward rales, and hence lot the shape of the vield curve. To derive these implications, we nole lhat in any tci m-slitit lure model die -period-ahead forward lale / , satisfies
11,1 - l> i l> ii./
= -/и i (K/l/vml - / п., + /и) - П.,!/ . 11 - / /)
VI/ f (l/l i I.m I I - Vl,) - (K,f/V/tl I - / /) (I I.1.13)
In Ibis model l./l/ ./11 \-l> , = -/( K, A.v(+1], and E,[/ +,. , -у is given by (I 1.1.12). .Substituting into (I 1.1.13) and using ll = (1 - ф )/(\ - ф), we gel
- + ф (х,-ц)
The lirsl equality in ( I 1.1.1 I) shows dial the change in the -period forward rale is < times die change in ,v, Thus movements in the forward rale die out geomeli ii allv ai rale (/>. This can be undcrsiood bv noting thai the log expeclalions livpoihesis holds in this model, so forward-rate movements relict I movements in die expected future short rate which are given bv ф limes movements in the current short rate.
t 1+ fl<i-ФУ
ILL Affine-Yield Modeb
As maturity n increases, the forward rate approaches M-(/) + l/(l-0))V/2,
a constant that does not depend on the current value of the state variablej x,. Equation (11.1.7) implies that the average short rate is ц - /fV/2. Thus the difference between the limiting forward rate and the average short rate
-(i/(i -0))V-72-(/уи ~ф)у. j
This is the same as the limiting expected log excess return on a long-tejrm bond. Because of the Jensens Inequality effect, the log forward-rate curve tends to slope downwards towards its limit unless в is sufficiendy negative, В < -1/2(1 -0). >
As x, varies, the forward-rate curve may take on different shapes. The second equality in (11.1.14) shows that the forward-rate curve can be written as the sum of a component that does not vary with n, a component that dies out with n at rate </>, and a component that dies out with n at rate </>2. The third component has a constant coefficient with a negative sign; thus there is always a steeply rising component of the forward-rate curve. The second component has a coefficient that varies with x so this component may be slowly rising, slowly falling, or flat. Hence the forward-rate curve may be rising throughout, falling throughout (inverted), or maybe rising at first and then falling (hump-shaped) if the third component initially dominates and then is dominated by the second component further out along the curve. These are the most common shapes for nominal forward-rate curves. Thus, if one is willing to apply the model to nominal interest rates, disregarding the fact that it allows interest rates to go negative, one can fit most observed nominal term structures. However the model cannot generate a forward-rate curve which is falling at first and then rising (inverted hump-shaped), as occasionally seen in the data.
Il is worth noting that when ф - 1, the one-period interest rate follows a random walk. In this case the coefficients A and B never converge as n increases. We have B = n and A - A i = -(в + n- l)zo2/2. The forward rate becomes / , = x, - (3 + п)2ст2/2, which may increase with maturity at first if В is negative but eventually decreases with maturity forever. Thus the homoskedastic bond pricing model does not allow the limiting forward rate to be both finite and time-varying; either ф < 1, in which case the limiting forward rate is constant over time, or ф = 1, in which case there is no finite limiting forward rate. This restriction may seem rather counterintuitive; in fact it follows from the very general result-derived by Dybvig, Ingersoll, and Ross (1996)-that the limiting forward rate, if it exists, can never fall. In the homoskedastic model with ф < 1 the limiting forward rate never falls because it is constant; in the homoskedastic model with ф = 1 the limiting forward rate does not exist. j
The cliscrclc-timc model developed in this section is closely related to the continuous-lime model of Vasicek (1977). Vasicek specilies a continuous-lime AR(1) or Ornstcin-Uhlenbeck process for the short interest rate r, given by the following stochastic differential equation:
j dr = к (в ~r)dt + o dli, (11.1.15)
jtvltcre к, в, and a are constants.2 Also, Vasicek assumes that the price of interest rate risk-the ratio of the expected excess return on a bond lo the standard deviation of the excess return on the bond-is a constant that docs , pot depend on the level of the short interest rate. The model of this section derives an AR( I) process for the short rale and a constant price of risk from primitive assumptions on the stochastic discount factor.
equilibrium Interpretation of the Model
<pur analysis has shown that the sign of the coefficient fi determines the ign of all bond risk premia. To understand this, consider the effects of a positive shock which increases die state variable i and lowers all bond prices. When /3 is positive the shock also drives down m,+), so bond returns are positively correlated with the stochastic discount factor. This correlation has hedge value, so risk premia on bonds are negative. When fi is negative, on the other hand, bond returns are negatively correlated with the stochastic discount factor, and risk premia arc positive, j We can get more intuition by considering the case where the stochastic discount factor reflects the power utility function of a representative agent, as in Chapter 8. In this case M,+ \ = &(C,+ \f C,)~Y, where 5 is the discount factor and у is the risk-aversion coefficient of the representative agent. Taking logs, we have
w,+i = log(o) - уДе.+ь (11.1.16)
Il follows that xt н= E,[-w/+] = - log(c5) 4- yE,[Ac,+ i], and i = - ml+\ - E,[ - m,+ ] = y(Ac,+ \ - Е,[Д(Г,+ ]). x, is a linear function of expected consumption growth, and i is proportional lo the innovation in consumption growth. The term-structure model of this section then implies that expected consumption growth is an AR(1) process, so that realized consumption growth is an ARMA(1,1) process. The coefficient fi governs the covariance between consumption innovations and revisions in expected future consumption growth. If fi is positive, ihcn a positive consumption shock today drives up expected future consumption growth and increases interest rates; die resulting fall in bond prices makes bonds covary negatively with consumption and gives them negative risk premia. If
2 As in Chapter *.l, tilt in (11.1.tf>) denotes ihe increment to a Brownian motion; it should not lie confused wilh Ihe bond price coefficients Jl of this section.
fi is negative, a positive shock lo consumption lowers interest tales so bonds have posilive risk premia.
Campbell (1986) explores the relation between bond risk premia and the time-series properties of consumption in a related model. Campbells model is similar lo the one here in that consumption and asset returns are conditionally lognormal and homoskedastic. It is more restrictive than die model here because it makes consumption growth (rather than expected consumption growth) a univariate stochastic process, but it is more general in that il does not require expected consumption growth to follow an AR( 1) process. Campbell shows thai die sign of the risk premium for an n-period bond depends on whether a consumption innovation raises or lowers consumption growth expected over (n - 1) periods. Backus and /.in (1994) explore this model in greater detail. Backus, Gregory, and /in (1989) also relate bond risk premia to the time-scries properties of consumption growth and interest rales.
Cox, Ingersoll, and Ross (1985a) show how to derive a continuous-lime term-structure model like the one in this section from an underlying production model. Sun (1992) and Backus (1993) restate their results in discrete time. Assume that there is a representative agent with discount factor & and time-separable log utility. Suppose lhat the agent faces a budget constraint of the form
A,+ , = (K, - OX, V,+ l, (11.1.17)
where A, is capital at the start of the period, (A, - (.,) is invested capital, and X, V,+j is the return on capital. This budget constraint has constant returns lo scale because the return on capilal does not depend on the level of capital. X, is the anticipated component of the return and V,+ is an unanticipated technology shock. With log utility it is well-known that the agent chooses C,/A, = (1 - 5). Substituting this into (11.1.17) and taking logs we find thai
Д<-,+ 1 = log(<5) + x, + vn. (11.1.18)
where u,+. = log( V,+ i), and- m,+i = - Iog(6) +Ai,ti = x, + ). This derivation allows X/ to follow any process, including the AR(1) assumed by the term-structure model.
11.1.2 Л Square-Root Sinnfe-hactor Model
The homoskedaslic model of die previous section is appealing because of its simplicity, but it has several unattractive features. First, it assumes that interest rate changes have constant variance. Second, the model allows interest rates to go negative. This makes it applicable to real interest rates, but less appropriate for nominal interest rales. Third, it implies lhat risk premia
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