Models of the TS. Carlo A Favero. February Carlo A Favero () Models of the TS February / 47

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1 Models of the TS Carlo A Favero February 201 Carlo A Favero () Models of the TS February / 4

2 Asset Pricing with Time-Varying Expected Returns Consider a situation in which in each period k state of nature can occur and each state has a probability π(k), in the absence of arbitrage opportunities the price of an asset i at time t can be written as follows: P i,t = k s=1 π t+1 (s)m t+1 (s) X i,t+1 (s) where m t+1 (s) is the discounting weight attributed to future pay-o s, which (as the probability π) is independent from the asset i, X i,t+1 (s) are the payo s of the assets ( in case of stocks we have X i,t+1 = P t+1 + D t+1, in case of zero coupon bonds, X i,t+1 = P t+1 ), and therefore returns on assets are de ned as 1 + R s,t+1 = X i,t+1 P i,t. Carlo A Favero () Models of the TS February / 4

3 Asset Pricing with Time-Varying Expected Returns For the safe asset, whose payo s do not depend on the state of nature, we have: P s,t = X i,t+1 k s=1π t+1 (s)m t+1 (s) 1 + R s,t+1 = 1 + R s,t+1 = 1 m π t+1 (s)m t+1 (s) j=1 1 E t (m t+1 ) Carlo A Favero () Models of the TS February / 4

4 Asset Pricing with Time-Varying Expected Returns consider now a risky asset : E t (m t+1 (1 + R i,t+1 )) = 1 Cov (m t+1 R i,t+1 ) = 1 E t (m t+1 ) E t (1 + R i,t+1 ) E t (1 + R i,t+1 ) = Cov (m t+1 R i,t+1 ) + (1 + R s,t+1 ) E t (m t+1 ) Turning now to excess returns we can write: E t (R i,t+1 R s,t+1 ) = (1 + R s,t+1 ) cov (m t+1 R i,t+1 ) Assets whose returns are low when the stochastic discount factor is high (i.e. when agents values payo s more) require an higher risk premium, i.e. an higher excess return on the risk-free rate. Carlo A Favero () Models of the TS February / 4

5 Bond Returns Cash- ows from di erent type of bonds: t + 1 t + 2 t T general CF t+1 CF t+2 CF t+3... CF T coupon bond C C C C 1-period zero period zero (T t) -period zero Carlo A Favero () Models of the TS February / 4

6 ZC Bonds De ne the relationship between price and yield to maturity of a zero-coupon bond as follows: P t,t = 1 (1 + Y t,t ) T t, (1) where P t,t is the price at time t of a bond maturing at time T, and Y t,t is yield to maturity. Taking logs we have the following relationship: p t,t = (T t) y t,t, (2) which clearly illustrates that the elasticity of the yield to maturity to the price of a zero-coupon bond is the maturity of the security. Carlo A Favero () Models of the TS February 201 / 4

7 ZC Bonds Price and YTM of zero-coupon bonds Maturity P t,t Y t,t p t,t y t,t Carlo A Favero () Models of the TS February 201 / 4

8 ZC Bonds The one-period uncertain holding-period return on a bond maturing at time T, rt,t+1 T, is then de ned as follows: r T t,t+1 p t+1,t p t,t = (T t 1) y t+1,t + (T t) y t,t (3) = y t,t (T t 1) (y t+1,t y t,t ), = (T t) y t,t (T t 1) y t+1,t, (4) which means that yields and returns di er by the a scaled measure of the change between the yield at time t + 1, y t+1,t, and the yield at time t, y t,t. Think of a situation in which the one-year YTM stands at 4.1 per cent while the 30-year YTM stands at per cent. If the YTM of the thirty year bonds goes up to.1 per cent in the following period, then the period returns from the two bonds is the same. Carlo A Favero () Models of the TS February / 4

9 A simple model of the Term Structure Apply the no arbitrage condition to a one-period bond (the safe asset) and a T-period bond: E t r t,t+1 T rt,t+1 1 E t r t,t+1 T = E t r T t,t+1 y t,t+1 = φ T t,t+1 = y t,t+1 + φ T t,t+1 Solving forward the di erence equation p t,t = p t+1,t rt,t+1 T, we have : y t,t = = 1 (T t) 1 (T t) n 1 i=0 n 1 i=0 E t r T t+i,t+i+1 E t y t+i,t+i+1 + φ T t+i,t+i+1 Carlo A Favero () Models of the TS February / 4

10 Forward Rates Forward rates are returns on an investment at time t, made in the future at time t 0 with maturity at time T. The return on this strategy is equivalent to the return on a strategy that buys at time t zero coupon with maturity T and sells at time t the same amount of bonds with maturity t 0. The price of the investment strategy is ( (T t) y t,t + (t 0 t) y t,t 0)and using the usual formula that links prices to returns we have : f t,t 0,T = (T t) y t,t (t 0 t) y t,t 0 T t 0 (5) Carlo A Favero () Models of the TS February / 4

11 Forward Rates Applying the general formula to speci c maturities we have : f t,t+1,t+2 = 2y t,t+2 y t,t+1 () f t,t+2,t+3 = 3y t,t+3 2y t,t+2 () f t,t+3,t+4 = 4y t,t+4 3y t,t+3 (8) f t,t+n 1,t+n = ny t,t+n (n 1)y t,t+n 1 (9) Carlo A Favero () Models of the TS February / 4

12 Forward Rates Using all these equations we have: y t,t+n = 1 n (y t,t+1 + f t,t+1,t+2 + f t,t+2,t f t,t+n 1,t+n ) (10) y t,t+n = 1 n n 1 i=0 E t y t+i,t+i+1 + φ T t+i,t+i+1 f t,t+i,t+i+1 = E t y t+i,t+i+1 + φ T t+i,t+i+1 Carlo A Favero () Models of the TS February / 4

13 Forward Rates Think of using forward rates to assess the impact of monetary policy. Let us analyze a potential movement of spot and forward rates around a shift in the central bank target rate. Before CB intervention 1-year spot and forward rates maturity i=1 i=2 i=3 i=4 i=5 y t,t+i f t,t+i,t+i Carlo A Favero () Models of the TS February / 4

14 Forward Rates After CB intervention : 1-year spot and forward rates maturità i=1 i=2 i=3 i=4 i=5 y t,t+i f t,t+i,t+i Please remember that the interpretion of future forward as expected rates requires some assumption on the risk premium. Carlo A Favero () Models of the TS February / 4

15 Instantaneous Forward Rates De ne the instanteous froward as the forwad rate on the contract with in nitesimal maturity: f t,t 0 = lim T!t 0 f t,t 0,T (11) given the sequence of forward rates you can de ne forward rate at any settlement date as follows : f t,t 0,T = R T τ=t 0 f τt dτ (T t 0 ) Carlo A Favero () Models of the TS February / 4

16 Instantaneous Forward Rates As a consequence the relationship between spot and forward rate is written as: R T τ=t y t,t = f τtdτ (T t) and therefore f t,t = y t,t + (T t) y t,t (12) T so instantaneous forward rates and spot rates coincide at the very short and very long-end of the term structure, forward rates are above spot rates when the yield curve slopes positevely and forward rates are below spot rates when the yield curve slopes negatively. Carlo A Favero () Models of the TS February / 4

17 Factor Models of the Term Structure Gurkanyak et al. estimate the following interpolant at each point in time, by non-linear least squares, on the cross-section of yields: 1 exp k τ1 y t,t+k = L t + SL t +C 2 t k τ 1 1 exp k τ 2 k τ 2 + C 1 t exp 1 exp k τ 1 k τ 2 k τ 1 1 A exp k τ 1 1 A( which is an extension originally proposed by Svensson(1994) on the original parameterization adopted by Nelson and Siegel (198) that sets C 2 t = 0. Carlo A Favero () Models of the TS February / 4

18 Factor Models of the Term Structure Forward rates are easily derived as f tk = L t + SL t exp k τ 1 + C 1 t k exp τ 1 k τ 1 + C 2 t k exp τ 2 (14) When maturity k goes to zero forward and spot rates coincide at L t + SL t, and when maturity goes to in nite forward and spot coincide at L t. Terms in C 1 t and C 2 t describes two humps starting at zero at di erent starting points and ending at zero. k τ 2 Carlo A Favero () Models of the TS February / 4

19 Factor Models of the Term Structure L t, SL t, C 1 t, C 2 t, which are estimated as parameters in a cross-section of yields, can be interpreted as latent factors. L t has a loading that does not decay to zero in the limit, while the loading on all the other parameters do so, therefore this parameter can be interpreted as the long-term factor, the level of the term-structure. The loading on SL t is a function that starts at 1 and decays monotonically towards zero; it may be viewed a short-term factor, the slope of the term structure. In fact, rt rf = L t + SL t is the limit when k goes to zero of the spot and the forward interpolant. We naturally interpret rt rf as the risk-free rate. C t are medium term factor, in the sense that their loading start at zero, increase and then decay to zero (at di erent speed). Such factors capture the curvature of the yield curve. Carlo A Favero () Models of the TS February / 4

20 A general state-space representation To generalize the NS approach we can put the dynamics of the term structure in a state-space framework: y t,t+n = 1 n A n + B 0 nx t + εt,t+n ε t i.i.d.n(0, σ 2 I ) (15) X t = µ + ΦX t 1 + v t v t i.i.d.n(0, Ω) (1) In the case of original NS we have Bn 0 = n, 1 e λn λ, 1 e λn λ ne λn and A n = 0 Carlo A Favero () Models of the TS February / 4

21 The object to our interest We consider two approaches to assessing the importance of the risk premium: single equation based-evidence and VAR based evidence. Carlo A Favero () Models of the TS February / 4

22 Single Equation based evidence To assess the importance of RP, three di erent implications of the Expectations theory can be brought to the data: y t,t (T t 1) E t (y t+1,t y t,t ) = y t,t+1 + φ t,t. f t,t+i,t+i+1 = E t y t+i,t+i+1 + φ T t+i,t+i+1 y t,t+n y t,t+1 = 1 E t y t+i,t+i+1 n 1 i=1 + 1 n 1 n n 1 φ T t+i,t+i+1 i=1 Carlo A Favero () Models of the TS February / 4

23 Single Equation based evidence (a) Estimate the following model : y t+1,t y t,t = β 0 + β 1 1 T t 1 (y t,t y t,t+1 ) + u t+1 to test β 0 = 0, β 1 = 1. Carlo A Favero () Models of the TS February / 4

24 Single Equation based evidence (b) Estimate the following model : to test β 0 = 0, β 1 = 1. S t,n = β 0 + β 1 S t,n + u t S t,n = y t,t+n y t,t+1 1 S t,n = n 1 i=1 1 n y t+i,t+i+1 Carlo A Favero () Models of the TS February / 4

25 Single Equation based evidence (c) Estimate the following model : (y t+i,t+i+1 y t,t+1 ) = β 0 + β 1 (f t,t+i,t+i+1 y t,t+1 ) + u t+i+1 to test β 0 = 0, β 1 = 1. Carlo A Favero () Models of the TS February / 4

26 Single Equation based evidence (d) Estimate the following model : y t,t (T t 1) (y t+1,t y t,t ) y t,t+1 = β 0 + β 1 (f t,t+i,t+i+1 y t,t+1 ) + to test β 1 = 0. Carlo A Favero () Models of the TS February / 4

27 Single Equation based evidence The empirical evidence shows that: i) high yields spreads fare poorly in predicting increases in long rates(see Campbell, 1995) ii) the change in yields does not move one-to-one with the forward spot spread (see Fama and Bliss,198) iii) period excess returns on long-term bond are predictable using the information in the forward-spot spread (see Cochrane,1999,Cochrane-Piazzesi 2005) Carlo A Favero () Models of the TS February / 4

28 The VAR based evidence The VAR based evidence is due to Campbell-Shiller(198) and considers the following version of the ET S t,t = S t,t = T t 1 j=1 γ j E [ r t+j j I t ] (1) (1) shows that a necessary condition for the ET to hold puts constraints on the long-run dynamics of the spread. In fact, the spread should be stationary being a weighted sum of stationary variables. Obviously, stationarity of the spread implies that, if yields are non-stationary, they should be cointegrated with a cointegrating vector (1,-1). However, the necessary and su cient conditions for the validity of the ET impose restrictions both on the long-run and the short run dynamics. Carlo A Favero () Models of the TS February / 4

29 The VAR based evidence Assuming that R t,t and r t are cointegrated with a cointegrating vector (1,-1), CS construct a bivariate stationary VAR in the rst di erence of the short-term rate and the spread : r t = a(l) r t 1 + b(l)s t 1 + u 1t S t = c(l) r t 1 + d(l)s t 1 + u 2t Carlo A Favero () Models of the TS February / 4

30 The VAR based evidence Stack the VAR as: 2 4 r t.. r t p+1 S t.. S t p = 2 4 a 1.. a p b 1.. b p c 1.. c p d 1.. d p r t 1.. r t p S t 1.. S t p u 1t.. 0 u 2t (18) This can be written more succinctly as: z t = Az t 1 + v t Carlo A Favero () Models of the TS February / 4

31 The VAR based evidence The ET null puts a set of restrictions which can be written as : g 0 z t = T 1 j=1 γ j h 0 A j0 z t (19) where g0 and h0 are selector vectors for S and r correspondingly ( i.e. row vectors with 2p elements, all of which are zero except for the p+1st element of g0 and the rst element of h0 which are unity). Since the above expression has to hold for general z t, and, for large T, the sum converges under the null of the validity of the ET, it must be the case that: g 0 = h 0 γa(i γa) 1 (20) Carlo A Favero () Models of the TS February / 4

32 The VAR based evidence which implies: g 0 (I γa) = h 0 γa (21) and we have the following constraints on the individual coe cients of VAR(??): fc i = a i, 8ig, fd 1 = b 1 + 1/γg, fd i = b i, 8i = 1g (22) The above restrictions are testable with a Wald test. Carlo A Favero () Models of the TS February / 4

33 The VAR based evidence By doing so using US data between the fties and the eighties Campbell and Shiller (198) rejected the null of the ET. However, when CS construct a theoretical spread St,T, by imposing the (rejected) ET restrictions on the VAR they nd that, despite the statistical rejection of the ET, St,T and S t,t are strongly correlated. Carlo A Favero () Models of the TS February / 4

34 The VAR based evidence Carlo A Favero () Models of the TS February / 4

35 No-Arbitrage A ne Factor Models First building block is the dynamics of the factors determining risk premium: X t = µ + ΦX t 1 + Σɛ t interpret this as a companion form representation. Second is a speci cation for the one-period rate r t which is assumed to be a linear function of the factors: r t = δ 0 + δ 0 1X t Carlo A Favero () Models of the TS February / 4

36 No-Arbitrage A ne Factor Models The third is a pricing kernel. The assumption of no-arbitrage guarantees the existence of a risk-neutral measure Q such that the price of any asset V t that does not pay any dividends at time t+1 satis es the following relation: V t = E Q t (exp ( r t ) V t+1 ) the Radon-Nikodym derivative (which converts the risk neutral measure to the data-generating measure) is denoted by ξ t+1. Carlo A Favero () Models of the TS February / 4

37 No-Arbitrage A ne Factor Models So for any random variable Z t+1 we have E Q t (Z t+1 ) = E t (ξ t+1 Z t+1 ) /ξ t The assumption of no-arbitrage allows us to price any nominal bond in the economy. Assume that ξ t+1 follows the log-normal process: 1 ξ t+1 = ξ t exp 2 λ0 t λ t λt 0 ε t+1 where λ t are the time-varying market prices of risk associated with the sources of uncertainty ɛ t. Carlo A Favero () Models of the TS February / 4

38 No-Arbitrage A ne Factor Models Parameterize λ t as an a ne process: de ne the pricing kernel m t+1 as λ t = λ 0 + λ 1 X t m t+1 = exp ( r t ) ξ t+1 /ξ t substituting from the processes for the short-rate and ξ t+1 we have: m t+1 = exp δ 0 δ1x 0 1 t 2 λ0 t λ t λt 0 ε t+1 Carlo A Favero () Models of the TS February / 4

39 No-Arbitrage A ne Factor Models Now the total one-period gross return of any nominal asset satis es: E t (m t+1 R t+1 ) = 1 If pt n represents the price of an n-period zero coupon bond, then we can use this equation to compute recursively bond prices as: p n+1 t = E t (m t+1 p n t+1) Carlo A Favero () Models of the TS February / 4

40 No-Arbitrage A ne Factor Models Guess that the log of bond prices are linear functions of the state variable we have: p n t = exp A n + B 0 nx t This guess is easily veri ed for the one-period bond, in which case we have: p 1 t = E t (m t+1 ) = exp ( r t ) = exp δ 0 + δ 0 1X t Carlo A Favero () Models of the TS February / 4

41 No-Arbitrage A ne Factor Models But it also applies to n-period bonds, in which case we have: p n+1 t = E t (m t+1 pt+1) n = E t exp δ 0 δ1x 0 1 t 2 λ0 t λ t λt 0 ε t+1 + A n + BnX 0 t+1 = exp δ 0 δ1x 0 1 t 2 λ0 t λ t + A n E t exp λ 0 t ε t+1 + BnX 0 t+1 = exp δ 0 δ1x 0 1 t 2 λ0 t λ t + A n E t exp λ 0 t ε t+1 + Bn 0 (µ + ΦX + Σɛ t+1 ) = exp δ 0 + BnΦ 0 δ1 0 1 Xt 2 λ0 t λ t + A n + Bnµ 0 E t exp λ 0 t ε t+1 + BnΣɛ 0 t+1 Carlo A Favero () Models of the TS February / 4

42 No-Arbitrage A ne Factor Models = exp δ 0 δ1x 0 1 t 2 λ0 t λ t + A n E t exp λ 0 t ε t+1 + Bn 0 (µ + ΦX + Σɛ t+1 ) = exp δ 0 + BnΦ 0 δ1 0 1 Xt 2 λ0 t λ t + A n + Bnµ 0 E t exp λ 0 t ε t+1 + BnΣɛ 0 t+1 δ0 + B = exp nφ 0 δ1 0 Xt + A n + Bn 0 (µ Σλ 0 ) B0 nσσ 0 B n BnΣλ 0 1 X t here the last step uses log-normality and the fact that λ 0 t λ t = λ 0 tvar (ε t+1 ) λ t. Carlo A Favero () Models of the TS February / 4

43 No-Arbitrage A ne Factor Models By matching coe cients we now have: A n+1 = δ 0 + A n + B 0 n (µ Σλ 0 ) B0 nσσ 0 B n B 0 n+1 = B 0 n (Φ Σλ 1 ) δ 0 1 Carlo A Favero () Models of the TS February / 4

44 No-Arbitrage A ne Factor Models To sum up, we can characterize a traditional A ne TS model as follows: y t,t+n = 1 n A n + B 0 nx t + εt,t+n ε t i.i.d.n(0, σ 2 I ) (23) a n+1 = a 1 1 X t = µ + ΦX t 1 + v t v t i.i.d.n(0, Ω) (24) n b n+1 = 1 (n+1) n (n+1) i=0 i=0 Φ 0 λ1ω 0 i b 1 B (i), where B (i) = Bi 0 (µ Ωλ 0 ) B0 i ΩB i. Carlo A Favero () Models of the TS February / 4

45 Risk Premium Given the knowledge of the model parameters the risk premium can be derived naturally: RPt n 1 = y t,t+n n E t n 1 r t+j j=0 E t r t+j = δ 0 + δ1e 0 t X t+j = δ 0 + δ1 0 _ µ + Φ j X t _ µ where X t = µ + ΦX t 1 + Σɛ t _ µ = (I Φ) 1 µ Carlo A Favero () Models of the TS February / 4

46 Risk Premium and, in absence of measurement error, we have: y t,t+n = 1 n A n + B 0 nx t = _ A n + _ B 0 nx t RP n t = _ A n δ 0 n + _ B 0 n δ1 0 1 n δ 0 1 n 1 j=0 I 1 n Φ j! X t n 1 j=0 Φ j! _ µ + but B 0 n+1 = B0 n (Φ Σλ 1 ) δ 0 1,so when Σ = 0 or λ 1 = 0 the term multiplying X t in the last expression vanishes and the risk premium becomes constant. Carlo A Favero () Models of the TS February / 4

47 Extensions Recently the no-arbitrage approach has been extended to include some observable macroeconomic factors in the state vector and to explicit allow for a Taylor-rule type of speci cation for the risk-free one period rate. This extension of the small information set using macroeconomic variable can be further expanded by moving to large-information set. In this case rather than including in the state vector some speci c macroeconomic variables, common factors can be extracted from a large panel of macroeconomic variables using static principal components, as suggested by Stock and Watson (2002). Carlo A Favero () Models of the TS February / 4