Lecture 7: Term Structure Models Simon Gilchrist Boston Univerity - - PowerPoint PPT Presentation

Lecture 7: Term Structure Models

Simon Gilchrist Boston Univerity and NBER

EC 745

Fall, 2013

Bond basics

A zero-coupon n period bond is a claim to a sure payoff of 1 at time t + n. The price is denoted P (n)

t

and it satisfies the recursion: P (n)

t

= Et

• Mt+1P (n−1)

t+1

• ,

P (0)

t

= 1. We define the yield of a bond with maturity n at time t through the equation P (n)

t

= 1

• 1 + Y (n)

t

n , i.e. it is the per period (e.g. per year) return that you get if you buy a bond today and hold it until it matures. The holding period return is the return if you buy a bond of maturity n at time t and sell it back at time t + 1 : R(n)

t+1 = P (n−1) t+1

P (n)

t

.

Yield curve with iid stochastic discount factor

Suppose that {Mt+1} is iid. Then the yield curve is constant and flat, i.e. y(n)

t

= yt = y for all t, n.

Proof: Let p = E(M). Start with n = 2 and show P (n)

t

= Et

• Mt+1P (n−1)

t+1

• = Et
• Mt+1pn−1

= p.pn−1 = pn. If P (n)

t

= pn, then y(n)

t

= y with 1 + y = 1

P .

Implication: if the short-rate is constant, all yields are constant, and hence bonds of all maturities have risk-free returns (Whereas in general, only the one-period bond has a return that is risk-free.)

Nominal discount factors

Arbitrage: Pt qt = Et

• Mt+1

Dnom

t

qt+1

• ,

where qt =price index (CPI). Hence, inflation is πt+1 = qt+1/qt. Define a nominal SDF Mnom

t+1 = Mt+1

πt+1 , The bond pricing recursion holds for nominal bonds using this nominal SDF: P (n)

t

= Et

• Mnom

t+1 P (n−1) t+1

• .

Comovement

All yields (long and short) are highly correlated - they tend to move up and down together a lot; more precisely, one can do principal components to find the factors which move yields. The first, by far most important factor is the “level”: all yields move up and down together; second, there is a “slope” effect i.e. long term yields and short term yields move in opposite direction; last, there is a “curvature” effect i.e. the concavity of the yield curve changes somewhat.

Monthly Nominal Treasury Yields (1985:2013)

5 10 15 1985m1 1990m1 1995m1 2000m1 2005m1 2010m1 2015m1 date treas03m treas02y treas05y treas10y

Daily Nominal Treasury Yields (1996:2013)

2 4 6 8 01jan1996 01jan1998 01jan2000 01jan2002 01jan2004 01jan2006 01jan2008 01jan2010 01jan2012 01jan2014 date treas03m treas02y treas05y treas10y treas15y treas30y

Daily Forward Rates (1996:2013)

2 4 6 8 01jan1996 01jan1998 01jan2000 01jan2002 01jan2004 01jan2006 01jan2008 01jan2010 01jan2012 01jan2014 date ftreas02y1 ftreas04y1 ftreas09y1

Term Spread

On average the yield curve is somewhat upward sloping; i.e. the yield on long-term bonds is larger than on short-term bonds. The slope of the yield curve, i.e. the term spread measured as the difference between a long rate (≥ 5 years) and a short rate (≤ 1 year) is correlated with the business cycle: an inverted yield curve predicts a recession, and at the trough of the recession, the yield curve is steeply upward sloping.

Monthly Term Spreads (1985:2013)

• 1

1 2 3 4 1985m1 1990m1 1995m1 2000m1 2005m1 2010m1 2015m1 date TS_10y_2y TS_10y_3m

Volatility

Long-term bond prices are fairly volatile; the std dev of the 10y return is about 8% per year, i.e. almost half that of stocks. In terms of yields, the std dev of yields as a function of maturity is hump-shaped. n (years) E(hpr) s.e. σ(hpr) 1 5.83 .42 2.83 2 6.15 .54 3.65 3 6.40 .69 4.66 4 6.40 .85 5.71 5 6.36 .98 6.58

Table1

Expectation hypothesis (EH)

The EH states that the expected log (holding period) returns on all bonds is the same: Ethpr(n)

t+1 = Ethpr(1) t+1 = y(1) t , alln ≥ 0.

The N-period (log) yield is the average of expected future

• ne-period (log) yields:

y(N)

t

= 1 N Et

• y(1)

t

+ y(1)

t+1 + ... + y(1) t+N−1

• .

The forward rate equals the expected future spot rate (in logs): fN→N+1

t

= Et

• y(1)

t+N

• .

Weaker version of EH

A weaker form of the EH is that these relations hold “up to a constant”, i.e. Ethpr(n)

t+1

= y(1)

t+1 + constant,

y(N)

t

= 1 N Et

• y(1)

t

+ y(1)

t+1 + ... + y(1) t+N−1

• + constant,

fN→N+1

t

= Et

• y(1)

t+N

• + constant,

where the constant may depend on maturity n but not on time t. Key point: the EH is almost assuming risk-neutrality - expected returns should be the same on all assets. (It is not quite that, because it is in logs instead of levels.)

Implication and test of EH:

Under the EH, if the long-term yield is high today relative to the short yield, it must be that the short yield will rise in the future, so that if you invest in short rate only every period you will end up getting the same return at the end. In the data, the expectation hypothesis does not work very well: the expected return on bonds is forecastable, Ethpr(n)

t+1 − y(1) t+1 = α + βXt

i.e. there are times when investing in long-term bonds brings excess returns.

Intuition

One way to summarize the results is to go back to the example explaining the EH – in the data, on average, when short < long, the short yield does not increase enough in the future, so there is a positive excess return to borrowing short term and buying long-term bonds. More precisely, the variable Xt that researchers use to predict returns on long-term bonds is usually based on current yields or forward rates. Fama and Bliss use the difference between the forward rate at time t + n and the current short rate, to forecast the maturity n bond excess return. Cochrane and Piazzesi find that a particular combination of forward rates forecast all maturities of excess bonds returns. (Refs: Fama and Bliss (1988), Campbell and Shiller (1991), Cochrane and Piazzesi (2005)).

Testing the EH

The expectation hypothesis is often tested through to the following equation: yn−1

t+1 − yn t = α + βn

yn

t − y1 t

n − 1

• + εt+1.

The expectation hypothesis implies that βn = 1. In the data, βn < 1, often negative, and decreasing with the horizon n.

Table: Expectation Hypothesis Tests

n = 2 n = 3 n = 4 n = 5 Slope Coefficients - 1961-1979 −1.03 −1.52 −1.55 −1.43 [0.65] [0.71] [0.83] [0.96] Slope Coefficients - 1988-2006 0.61 −0.13 −0.19 −0.21 [0.89] [0.90] [0.97] [1.02]

Summing up

Just like for stocks, we need a model which explains:

1

the mean return on long-term bonds, relative to short-term bonds,

2

the volatility of long-term bond returns

3

the variation over time in the expected returns.

A simple affine model:

Assume conditional log-normality of Mt+1 and P n

t then for

n > 1 pnt = Et (mt+1 + pn−1,t+1) + 1 2vart (mt+1 + pn−1,t+1) and p1t = Etmt+1 + 1 2vart (mt+1) Suppose mt+1 = −xt + et+1 where xt+1 = (1 − φ) µ + φxt + vt+1 Also assume et+1 = βvt+1 + ηt+1 where vt+1 and ηt+1 are iid uncorrelated.

Simplified version:

For simplicity set ηt+1 = 0 so that mt+1 = −xt + βvt+1 (it only affects the level of the term structure). In this case mt+1 is an ARMA(1,1) (sum of AR1 and white noise). Note if β = 1/φ we have mt+1 = −xt+1 φ + (1 − φ) φ µ which may be interpreted as renormalized consumption growth, hence a form of CARRA.

One period yield:

For the one period yield y1t = −p1t so y1t = xt − β2 σ2 2 Since short-rate inherits AR1 dynamics we can think of xt as the process for the short-rate process itself.

N period yield

Guess a solution of the form −pnt = An + Bnxt Since the yield on an n-period bond is ynt = −pnt n we are guessing that the yield on any maturity is an affine function of xt.

Implications

With this guess we have pnt = Et [mt+1 + pn−1,t+1] = Et [−xt + βvt+1 − An−1 − Bn−1xt+1] and vart [mt+1 + pn−1,t+1] = [β + Bn−1]2 σ2 So −An − Bntxt = −xt − An−1 − Bn−1 (1 − φ) µ − Bn−1φxt + [β + Bn−1]2 σ2 2

Solution:

For n = 0, 1 A0 = B0 = 0 A1 = −β2 σ2 2 , B1 = 0 Equating coefficients we have: Bn = 1 + φBn−1 = 1 − φn 1 − φ An − An−1 = (1 − φ) µBn−1 − (β + Bn−1)2 σ2/2 Interpretation:

The coefficient Bn measures the sensitivity of the bond price to an increase in the short-rate. Bond prices fall when short rates rise. The higher the maturity, the greater the response.

Expected excess holding period return:

The expected holding period return is Etrn,t+1 = Etpn−1,t+1 − pn,t =

• − (1 − φ) µBn−1 + (β + Bn−1)2 σ2/2
• + xt

so all expected holding period returns increase one for one with the short-rate. The excess holding period return is defined as Etrn,t+1 − y1t + vart (rn,t+1) 2 = −covt (rn,t+1, mt+1) Etrn,t+1 − y1t + vart (xt+1) 2 = Bn−1covt (xt+1, mt+1) This implies Etrn,t+1 − y1t + B2

n−1σ2

2 = −Bnβσ2 We can interpret Bnσ as the amount of risk for an n period holding period return and βσ as the price of risk.

Interpretation:

Assume that mt+1 = log δ − γ∆ct+1 then γ (∆ct+1 − Et∆ct+1) = βvt+1 If β > 0 a shock to vt+1 represents positive news about future consumption growth which then follows a persistent AR1

• process. (i.e. β governs the covariance between consumption

innovations and expected future consumption growth). So if β > 0 a positive shock signals that future consumption growth and hence interest rates will be high. A rise in expected future interest rates implies a reduction in current bond prices. So bond returns covary negatively with current consumption growth – they are therefore a good hedge and carry a negative risk premium.

Forward rates:

The forward rate satisfies fnt = pnt − pn+1,t = −p1t + (Etpn,t+1 − pn+1,t + p1t) − (Etpn,t+1 − pnt) = y1t + Etrn+t,t+1 − y1t − (Etpn,t+1 − pnt) where pn,t+1 − pnt = −BnEt∆xt+1 This implies fnt = µ −

• β + 1 − φn

1 − φ 2 σ2 2 + φn (xt − u) With this specification we can get rising, humped-shaped or inverted yield curves depending on xt. If xt = µ the yield curve is initially rising relatively steeply

• wing to the squared term.

Conditional Heteroskedasticity

Cox-Ingersoll-Ross in discrete time: −mt+1 = xt + √xtβvt+1 xt+1 = (1 − φ) µ + φxt + √xtvt+1 We now have p1t = Etmt+1 = −xt

• 1 − β2 σ2

2

• so A1 = 0, B1 =
• 1 − β2 σ2

2

• and the recursion

Bn = 1 + φBn−1 − (β + Bn−1)2 σ2 2 An − An−1 = (1 − φ) µBn−1 In the limit Bn solves a quadratic but is roughly equal to

1 1−φ as

before so it is positive and increasing in n.

Expected excess log bond return:

We now have Etrn,t+1 − y1t + B2

n−1

vart (xt+1) 2 = Bn−1covt (xt+1, mt+1) Etrn,t+1 − y1t + B2

n−1xt

σ2 2 = −Bn−1xtβσ2 We can again interpret Bn−1√xtσ as the amount of risk and β√xtσ as the price of risk. Thus, in this model, the price of risk rises with the (square root)

• f the level of interest rates.

Ten-year term premium vs ten minus 2 year term spread

• 1

1 2 3 4 01jan1996 01jan1998 01jan2000 01jan2002 01jan2004 01jan2006 01jan2008 01jan2010 01jan2012 01jan2014 date term_premium term_spread

Ten-year term premium vs ten-year treasury yield

2 4 6 8 treas10y

• 1

1 2 term_premium 01jan1996 01jan1998 01jan2000 01jan2002 01jan2004 01jan2006 01jan2008 01jan2010 01jan2012 01jan2014 date term_premium treas10y