Term Structure Models

1. Zero-coupon bond prices and yields 2. Vasicek model 3. Cox-Ingersoll-Ross model 4. Multifactor Cox-Ingersoll-Ross models 5. Ane models 6. Completely ane models 7. Bond risk premia 8. Inflation and nominal asset prices

1

Readings and References Back, chapter 17. Due, chapter 7. Vasicek, O., 1977, An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188. Cox, J., J. Ingersoll, and S. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385-407. Longsta↵, F., and E. Schwartz, 1992, Interest rate volatility and the term structure: A two-factor general equilibrium model, Journal of Finance 47, 1259-1282. Due, D., and R. Kan, 1996, A yield-factor model of interest rates, Mathematical Finance 6, 379-406. Du↵ee, G., 2002, Term premia and interest rate forecasts in ane models, Journal of Finance 57, 405-443. Drechsler, I., A. Savov, and P. Schnabl, A Model of Monetary Policy and Risk Premia, Journal of Finance forthcoming.

2 Summary of the Continuous-Time Financial Market

dS0,t dSk,t I Security prices satisfy = rt dt and =(µk,t k,t) dt + k,tdBt. S0,t Sk,t I Given tight tr. strat. ⇡t and consumption ct, portfolio value Xt satisfies the

WEE: dXt = rtXt dt + ⇡t(µt rt1) dt + ⇡tt dBt ct dt. •

No arbitrage if ⇡tt =0then ⇡t(µt rt1) = 0 ✓t s.t. t✓t = µt rt1 I ) )9 dXt = rtXt dt + ⇡tt(✓t dt + dBt) ct dt. ) t 1 t 2 d ✓ dBs ✓s ds ⇤ = = 0 s0 2 0 | | I Under emm ⇤ given by dP ZT where Zt e , P P t R R B = Bt + ✓s ds is Brownian motion. • t⇤ 0 t rsRds Let t = e 0 and sdf process Mt = tZt.ThentheWEEcanalsobewritten: R WEE*: dtXt + tct dt = t⇡tt dB • t⇤ WEE-M: dMtXt + Mtct dt = Mt[⇡tt ✓tXt] dBt • T u T T Mu MT I So Xt = Et⇤ t cu du + XT = Et t cu du + XT if ⇡ is mtgale-gen. { t t } { Mt Mt } R R I If is nonsingular, every c.plan (c, XT ) can be generated by a mtgale-gen. tr.strat.

3

Zero-Coupon Bond Prices and Yields

I Now let’s go through some classic models of the prices of default-free bonds.

T T I Let Pt be the time t price of the zero maturing at time T and yt be its yield. I Suppose we’re in a complete market. We can represent zero prices in various ways:

T T yT (T t) ru du MT P e t = E⇤ e t = Et . (1) t ⌘ t { } { M } R t

I For the purpose of modeling bond prices and yields, we can work entirely under the risk-neutral measure . P⇤ I In order to characterize expected bond returns, we need to work under the true measure . P

4 The Vasicek Model

I In the Vasicek (1977) model, the instantaneous riskless rate rt is modeled as a constant-volatility mean-reverting Ornstein-Uhlenbeck process,

drt = (¯r rt) dt + dB⇤ , (2)

where r¯ is the “long-run mean” of rt and  > 0 is its “speed of mean-reversion.” I The Ornstein-Uhlenbeck process is the continuous-time analogue of the discrete- time AR(1) process. The solution of (2)foranyu t 0 is I u (u t) (u t) (u s) ru = e rt +(1 e )¯r + e dBs⇤, (3) Zt

so ru is normally distributed with conditional mean and variance

(u t) (u t) E⇤ ru = e rt +(1 e )¯r and (4) t { } u 2 2 2(u s) 2(u t) vart⇤ ru = e ds = (1 e ) . (5) { } Zt 2 I Normality of ru means negative interest rates are possible in this model.

5

T T yT (T t) ru du It follows from P e t = E e t that zero yields are “ane” in I t ⌘ t⇤{ } rt,andthusalsonormal: R

T y = a(T t)+b(T t)rt, where (6) t 2 2 r¯2 1 e ⌧ 2 1 e 2⌧ a(⌧)=¯r +( )( ) ( ) and (7) 22 3 ⌧ 43 ⌧ 1 1 e ⌧ b(⌧)= ( ) . (8)  ⌧

I The derivation of this result from Back (2010) is as follows:

T T T u (u t) (u s) ru du =(T t)¯r (¯r rt) e du + e dBs⇤ du t t t t Z Z Z TZ u 1 (T t) (u s) =(T t)¯r (¯r rt)(1 e )+ e dBs⇤ du .  Zt Zt I Switching the order of integration in the last integral gives

T u T T T (u s) (u s) (T s) e dBs⇤ du = e du dBs⇤ = (1 e ) dBs⇤ . Zt Zt Zt Zs  Zt

6 I The integrand of this stochastic integral is nonrandom, so the integral is normal with mean zero and variance

2 T 2 2 2 (T s) 2 2 (T t) 2(T t) (1 e ) ds =(T t) (1 e )+ (1 e ) . 2 2 3 3  Zt   2 T Thus ru du is normal with I t R T 1 (T t) Et⇤ ru du = (T t)¯r + (¯r rt)(1 e ) and (9) { Zt }  T 2 2 2 2 (T t) 2(T t) var⇤ r du =(T t) (1 e )+ (1 e ) , t u 2 3 3 { Zt }   2 (10)

I and it follows from the usual rule for expectations of exponentials of normals that

T T ru du 1 (T t) log P = log E⇤ e t = (T t)¯r + (¯r rt)(1 e ) t t { }  R 2 2 2 1 2 (T t) 2(T t) + [(T t) (1 e )) + (1 e ))] . (11) 2 2 3 23

T I Homework: Verify that Pt satisfies the fundamental PDE for derivative prices.

7

The Cox-Ingersoll-Ross Model

I In the Cox-Ingersoll-Ross (1985) model, the instantaneous riskless rate rt is modeled as a mean-reverting square-root process,

drt = (¯r rt) dt + prt dB⇤ , (12)

where r>¯ 0 is the long-run mean of rt,  > 0 is its speed of mean-reversion, and the solution to (12) is nonnegative for all t, so it is possible to take the square 2 root. If r¯ /2,thenrt is a.s. strictly positive for all t. I The exact solution to (12) is more complicated than the solution to the O-U process in Vasicek, so we derive zero prices by solving their fundamental PDE. I Since the square-root process (12) is Markov, zero prices are functions of rt:

T T ru du T P = E⇤ e t = p (rt,t) . (13) t t { } R I Assuming the price function pT is smooth enough for an application of Itˆo’s lemma,

1 @P T (¯r r)pT + 2rpT + = rpT s.t. pT (r, T )=1. (14) r 2 rr @t

8 I The solution to (14)hasyieldsane in rt:

T ↵(T t) (T t)rt T p (rt,t)=e and y = a(T t)+b(T t)rt . (15) t where a(⌧)=↵(⌧)/⌧ and b(⌧)=(⌧)/⌧ and

2r¯ ( + )⌧ 2 2(e⌧ 1) ↵(⌧)= [ + log ] and (⌧)= , (16) 2 2 c(⌧) c(⌧)

where c(⌧)=2 +( + )(e⌧ 1) and = p2 +22. 2 To verify this solution, note that (15)impliespr = p, prr = p,and I pt =(↵0 + r0)p, so the fundamental PDE holds if and only if

1 2 2 (¯r r) + r + ↵0 + 0r = r. (17) 2

This holds for all values of r if and only if

1 2 2  + + 0 =1 and ↵0 = r¯ . (18) 2

The ODE for above is said to be of the Riccati type.

9

I The terminal condition pT (r, T )=1is equivalent to ↵(0) = (0) = 0. I Di↵erentiating in (16) shows it is a solution to (18).

I Given , integrating ↵0 in (18) gives ↵ in (16).

10 The Multifactor CIR Model

We can get a multifactor version of the CIR model by taking r to be the sum of square-root processes: rt = X1t + X2t where

dXit = i(X¯i Xit) dt + ipXit dB⇤ . (19) i

I Independence of the Xi implies

T T T T ru du X1 du X2 du p = E⇤ e t = E⇤ e t u E⇤ e t u (20) t t { } t { } t { } ↵ (T R t) ↵ (T t) (T t)RX (T t)X R = e 1 2 1 1t 2 2t (21)

where the ↵i and i are defined as in the single-factor CIR model.

T = T + T T = ↵i(⌧) + i(⌧) I This implies yt y1t y2t where yit ⌧ ⌧ Xit. Other Multifactor Models Similarly, we could construct other multifactor interest rate processes as sums of independent processes, and get product pricing formulas if price formulas exist for each independent process.

11

Ane Term Structure Models

In an n-factor ane term structure model, the short rate r is ane in n Markov state variables X whose drift and instantaneous covariance matrix are also ane in X:

rt = 0 + 0Xt and dXt =( + KXt) dt + (Xt) dB⇤, where (22)

B is a vector of n independent Brownian motions under , I ⇤ P⇤ 0 is a constant, and are a constant n-vectors, K is a constant n n matrix, I ⇥ is an n n matrix-valued function s.t. each element of the covariance matrix I ⇥ (x)(x)0 is ane in x. Parametric assumptions are made to ensure the diagonal elements of (x)(x)0 are nonnegative and uniqueness of the solution for X.

I Both the Vasicek and CIR models are examples of ane models.

I If is constant, then the model is Gaussian, in the sense that conditional on Xt, (ru,Xu) is multivariate normal for all u t. I It can be shown that in any two-factor Gaussian model, the two factors can be taken to be the short rate and its drift.

12 I In an ane model, zero prices are exponential-ane in X and yields are ane:

n n T ↵(T t) i(T t)xi T p (x, t)=e i=1 and y = a(T t)+ bi(T t)xi , (23) t P iX=1

as shown by verifying the fundamental PDE for p with BC ↵(0) = i(0) = 0.

I The i solve a multi-dimensional ODE, for which no closed-form is available in general, and ↵ can be solved by integrating the i.

I A useful feature of ane models is that one can take a vector of zero yields to be the factors. For the vector Y of n yields for fixed times to maturity ⌧1,...,⌧n,let be the vector of the a(⌧i) and the matrix of the bj(⌧i).Then A B

1 Yt = + Xt Xt = (Yt ) , (24) A B ) B A

1 1 provided is invertible. Substituting Xt = (Yt ) and dX = dYt B B A B produces an ane model with the n yields as factors.

Similarly, one could use n 1 zero yields and the short rate as ane factors. I

13

T Y To get the fundamental PDE, write P = e t where Y = ↵(T t)+(T t) Xt I t > I From Itˆo’s lemma

dP 1 2 = dY + (dY ) = ↵0(T t) dt + 0(T t)>Xdt (T t)> dX P 2 1 + (T t)>(dX)(dX)>(T t) dt (25) 2 = ↵0(T t) dt + 0(T t)>Xdt (T t)>[( + KX) dt + (X) dB⇤] 1 + (T t)>(X)(X)>(T t) dt . (26) 2

I Setting the appreciation rate of P equal to the short rate r = 0 + 0X gives:

0 + 0X = ↵0(T t)+0(T t)>X (T t)>( + KX) (27) 1 + (T t)>(X)(X)>(T t) . (28) 2

I Equating the coecients of the Xi on each side gives a system of n ODEs in the functions i,calledRiccati equations because they are ane in the i0 and quadratic in the i. I Matching constant terms on each side, given the i,determines↵0,whichcanbe integrated from ↵(0) = 0 to give ↵.

14 Completely Ane Models

In a completely ane model, factor dynamics are ane under both the risk-neutral measure and the true measure . This is set up as follows. P⇤ P I Assume (X) in the ane model (23) is of the form S(X) where is a constant n n matrix and S is a diagonal n n matrix-valued function with each ⇥ ⇥ squared diagonal element being an ane function of X. Then each element of 2 the covariance matrix (x)(x)0 = S(x) 0,isanane function of x. d ⇤ = ( )= I Assume the market price of risk ✓ in the Radon-Nikodym derivative dP Z T T 1 t 2 P ✓u dBu ✓u du e 0 2 0 | | is of the form ✓t = S(Xt)✓ for a constant vector ✓. R R Then dB = dBt + S(X)✓ dt and the model is also ane under : I t⇤ P

dXt =( + KXt) dt + S(Xt) dB⇤ (29) 2 =[ + KXt + S(Xt) ✓] dt + S(Xt) dBt (30)

=(ˆ + KXˆ ) dt + S(Xt) dBt (31)

for some constant vector ˆ and matrix Kˆ,becauseS(X)2 is diagonal with ane functions of X on the diagonal.

15

Bond Risk Premia

In a completely ane model, bond risk premia are also ane in X.

I To see this, note from (26) that the volatility vector of the zero price P T is (T t) S(Xt). 0 Thus, the risk premium on the (T t)-year zero is I

T 2 µ rt = (T t)0S(Xt) ✓ , (32) t

which is ane in Xt.

I In a Gaussian completely ane model, S(X) is a constant matrix, so the risk premium of a zero depends only on its time to maturity.

16 Inflation and Nominal Asset Prices

I Until now, we have been working with real prices Sk,t, in units of the consumption good, and the sdf Mt for these real prices. When we go to the data, we can in some contexts ignore inflation, but it’s an issue when pricing nominal bonds, i.e., claims to a dollar rather than a unit of consumption.

I Let qt be the price level, e.g., dollars per unit of consumption, sometimes measured by CPI.

I Then nominal asset prices are qtSk,t.

q Mt Therefore, the sdf for nominal asset prices is M ,sinceMtSt = Et MuSu I t qt q q ⌘ { } implies M qtSt = Et M quSu . t { u } t t 1 t 2 dqt ◆s ds+ q,s dBs q,s ds Suppose = ◆t + q,t dBt,orqt = e 0 0 2 0 | | . I qt R R R dqt Interpret ◆t as expected inflation over the period and as realized inflation. I qt

17

Locally Riskless Inflation

I If q,t =0, i.e., inflation is locally riskless or “known at the beginning of the period,” then nominal asset prices qtSk,t follow

dqtSk,t =(µk,t + ◆t k,t) dt + k,t dBt , (33) qtSk,t

t t 1 t 2 q M (rs+◆s) ds ✓ dBs ✓s ds the sdf for nominal prices is M = t = e 0 0 s0 2 0 | | ,and I t qt I the nominal riskless rate, i.e., the nominalR return onR a locallyR riskless nominal money market account is rt + ◆t.

In this case, nominal excess expected returns (µ + ◆t) (rt + ◆t) are equal to I k,t real excess returns µ rt, so theory developed for real excess returns may be k,t able to be applied directly to empirical work with nominal excess returns. I The price of a “nominal bond” paying a dollar at time T is

q T q,T MT (rs+◆s ds P = Et 1 = E⇤ e t } . (34) t {M q } t { } t R

If both rt and ◆t are ane in a set of ane state variables, then bond prices will n q,T ↵(T t) (T t)x again be exponentially ane, p (x, t)=e i=1 i i. P 18 Inflation with Shocks

If q,t =0, i.e., if realized inflation over the period contains both its expectation I 6 ◆t dt and a shock q,t dBt, then nominal asset prices qtSk,t follow

dq S dS dq dq dS t k,t = k,t + t +( )( k ) (35) qtSk,t Sk,t qt q Sk

=(µ + ◆t + 0 ) dt +( + q,t) dBt , (36) k,t k,t q,t k,t k,t

I the sdf for nominal prices is

t t 1 t 2 1 t 2 q Mt (rs+◆s) ds (✓ +q,s) dBs ✓s ds+ q,s ds M = = e 0 0 s0 2 0 | | 2 0 | | , (37) t q t R R R R t 2 t 1 t 2 (rs q,s✓s+◆s q,s ) ds (✓ +q,s) dBs ✓s+ ds = e 0 | | 0 s0 2 0 | q,s0 | , (38) R R R q I and it follows from the form of Mt that the nominal riskless rate, the rate on a 2 locally riskless nominal money market account, is rt q,t✓t + ◆t q,t . | |

19

The Nominal Riskless Rate and the Inflation Risk Premium

I To get to this more intuitively, note that the nominal money market account, which delivers a locally riskless nominal return and zero shocks, must have shocks to its real returns that are exactly opposite the shocks to the price level. In other

words, if the nominal money market is security k⇤,withrealpriceSk⇤,itsreal return shocks must by equation (36) have loadings identically equal to q. k⇤

Thus the nominal money market’s real risk premium µ rt must be ✓ = I k⇤,t k⇤ q✓, which it must pay because its real return in units of consumption is risky. I As we will see shortly, this q✓ gives rise to the so-called “inflation risk premium” in the term structure of nominal bond yields.

I As we will see later, in equilibrium q✓ is the instantaneous covariance between shocks to inflation and shocks to aggregate consumption, times the relative risk aversion of the representative agent. Estimates suggest that this has been positive historically, though perhaps not since the crisis.

20 Now to get the nominal riskless rate, note that by equation (36)with = q, I k⇤ the nominal money market account’s nominal price follows

dq S t k⇤,t =( + + ) +( + ) µk⇤,t ◆t k⇤,tq,t0 dt k⇤,t q,t dBt (39) qtSk⇤,t 2 =(rt q,t✓t + ◆t q,t ) dt . (40) | |

2 Therefore, the nominal riskless rate is rt q,t✓t + ◆t q,t . I | | I To gain economic intuition for all this, let’s redefine inflation as the rate of decline 1 in the consumption price of dollars q , rather than the rate of increase in the dollar price of consumption q or CPI, that is more typically quoted in practice. Viewing currency (e.g., dollars) as an asset priced in units of consumption will clarify the economics of its real return and the real returns of nominal assets more generally.

I By Itˆo’s lemma,

1 1 1 2 1 1 2 d = 2 dqt + 3 (dq)= (◆t dt + q,t dBt)+ q,t dt (41) qt qt 2qt qt qt| |

21

1/qt 2 2 I so d =( ◆t + q,t ) dt q,t dBt = it dt+1/q,t dBt,whereit = ◆t q,t 1/qt | | | | is the rate of decline in the real price of dollars, our alternative measure of inflation,

and = q,t is the volatility of the real price of dollars. 1/q,t I Then we can see the nominal riskless rate as rt + it + 1/q,t✓t,therealriskless rate on rate rt plus compensation for the decline in the real price of dollars it plus

compensation for the risk of the real price of dollars 1/q,t✓t.

I If shocks to inflation and aggregate consumption are positively correlated, i.e., q✓ > 0, then the nominal money market is a hedge against negative consumpion shocks, because it pays dividends in dollars, which are worth more in real terms, when inflation and consumption are down. In that case, it makes sense that the

real risk premium on the nominal money market is negative, ✓ = q✓ < 0. 1/q

22 Nominal Bond Prices

I The nominal price of the nominal zero paying a dollar at time T is

q M T 2 T 1 T 2 q,T T (rs q,s✓s+◆s q,s ) ds (✓s0 +q,s) dBs ✓s+q,s0 ds P = Et 1 = Et e t | | t 2 t | | t {M q } { } t R R R (42) T 2 q (rs q,s✓s+◆s q,s ) ds = E e t | | (43) t { } R under the risk-neutral measure for nominal prices q given by P q t 1 t 2 d (✓ +q,s) dBs ✓s+ ds P = e 0 s0 2 0 | q,s0 | , (44) d P R R q t under which B Bt + (✓ + q,s) ds is a standard Brownian motion. t ⌘ 0 s0 I Therefore, if rt and ◆t areR ane in a set state variables Xt that are ane under q ,andifq,t = qS(Xt) and ✓t = S(Xt)✓,whereq and ✓ constant vectors P and S(X) a diagonal matrix as in the completely ane model, then the nominal 2 riskless rate rt q,t✓t + ◆t q,t will be ane in the state variables and nominal n | | q,T ↵(T t) (T t)x zero prices will again be exponentially ane, p (x, t)=e i=1 i i. P

23

Nominal Risk Premia

I Note that under q, nominal expected returns on all assets are equal to the P q nominal riskless rate, as can be seen by substituting dB (✓t + q,t) dt for dBt t in equation (36):

dqtSk,t =(µk,t + ◆t + k,tq,t0 k,t) dt +(k,t + q,t) dBt (45) qtSk,t 2 q =(rt q,t✓t + ◆t q,t ) dt +( + q,t) dB . (46) | | k,t k,t t This justifies calling q a risk-neutral measure for nominal returns. P I Nominal risk premia, expected nominal returns minus the nominal riskless rate, are

2 (µ + ◆t + 0 ) (rt q,t✓t + ◆t q,t ) (47) k,t k,t q,t | | 2 = µk,t rt + q,t✓t + q,t + k,tq,t0 (48) q | | = k,t(✓t + q,t0 ) (49)

q where + q,t is the vector of security k’s nominal return risk loadings, k,t ⌘ k,t and ✓ + q0 is the mpr paid by nominal returns. Now the nominal risk premia di↵er from the real risk premia µ rt. I k,t

24