Term-structure modelling at the zero lower bound: Implications for estimating the forward term premium

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Term-structure modelling at the zero lower bound: Implications for estimating the forward term premium Tsz-Kin Chung a, Cho-Hoi Hui b, Ka-Fai Li b,∗ a b

Graduate School of Social Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa Hachioji, 192-0397 Tokyo, Japan Research Department, Hong Kong Monetary Authority, 55/F, 2 IFC, Central, Hong Kong

a r t i c l e

i n f o

Article history: Received 31 May 2016 Revised 2 December 2016 Accepted 6 December 2016 Available online xxx JEL classification: C11 C32 E43 E44 G12

a b s t r a c t Although the affine Gaussian term-structure model has been a workhorse model in termstructure modelling, it remains doubtful whether it is an appropriate model in a low interest rate environment. This paper uses an alternative quadratic Gaussian-term structure model which is well known to be as tractable as the affine model and yet is suitable for interest rates close to zero. Compared with the quadratic model under the zero lower bound, we illustrate how the forward term premium can be biased upward under the affine model both theoretically and empirically. © 2016 Elsevier Inc. All rights reserved.

Keywords: Forward term premium Zero lower bound Quadratic Gaussian term-structure model Bayesian MCMC

1. Introduction The risk premium component of long-term interest rates is commonly referred as the forward term premium (FTP). To obtain a timeliness measure of FTP, economists typically prefer model-based estimates over infrequently sampled surveybased measures.1 One popular model is the affine Gaussian term-structure model (affine model).2 However, it is well known that the affine model is not suitable when interest rates are subjected to the zero lower bound (ZLB). Instead, Longstaff (1989), Leippold and Wu (2002), Ahn et al. (2002) and Realdon (2006) extend to the quadratic Gaussian term-structure model (quadratic model) which is as tractable as its affine counterpart with closed-form bond pricing formula available. Kim and Singleton (2012) test the quadratic model in Japan and conclude it is a useful model under the ZLB. By using a simple and transparent setup through which the affine and quadratic models can be readily compared, this paper compares the efficacy of the models for generating a realistic FTP.

∗

Corresponding author. E-mail addresses: [email protected] (T.-K. Chung), [email protected] (C.-H. Hui), kfl[email protected] (K.-F. Li). 1 Standard surveys such as the Blue Chip Survey of forecaster asks respondents for their long-term forecasts of the short-term interest rates only twice per year. 2 See Piazzesi (2010) for a survey of the affine model. http://dx.doi.org/10.1016/j.frl.2016.12.001 1544-6123/© 2016 Elsevier Inc. All rights reserved.

Please cite this article as: T.-K. Chung et al., Term-structure modelling at the zero lower bound: Implications for estimating the forward term premium, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.12.001

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Under the ZLB, we show that the FTP estimated from the affine model (affine FTP) is upward biased both theoretically and empirically. Specifically, when interest rates are persistently near zero, the expectation of future interest rates is downward biased in affine model since the model predicts that it is equally likely future interest rates to rise or fall due to its Gaussian assumption. In the decomposition of long-term interest rates, a downward bias in the expected future interest rates is equivalent to an upward bias in FTP, other things being equal. Since risk premia are countercyclical in nature - high during recessions and low during expansions - an upward biased FTP may overstate the severity of a recession. On the contrary, the quadratic model naturally retains the asymmetric interest rate response under the ZLB and avoids the downward bias in expected future interest rates. Thus, FTP estimated by the quadratic model (quadratic FTP) is more realistic under the ZLB. The ZLB is an important issue since it not only forms the basis that justifies the use of the unconventional monetary policies (see Bernanke (2012)), but it also affects how the term structure models are formulated.3 In turn, these term structure models are used to measure the effectiveness of the unconventional monetary policies (see Bernanke et al. (2004), Hamilton and Wu (2012), Jarrow and Li (2011), Joyce et al. (2011), and Li and Wei (2013)). In view of the need to guard against misspecification, this paper is related to the recent studies of term-structure modelling under the ZLB. The shadow rate term–structure model proposed originally by Black (1995) is also a popular candidate under the ZLB (see Krippner (2013), Bauer and Rudebusch (2016), Ichiue and Ueno (2013), Christensen and Rudebusch (2015) and Wu and Xia (2016)).4 However, these papers typically focus exclusively on when the exit of the ZLB would occur and neglect the implication on FTP under the ZLB. We contribute to the literature by showing theoretically and empirically on how affine FTP is severely biased under the ZLB.5 . Moreover, we also find that the longer horizon affine FTP is more prone to the bias. The paper is organised as follows. Section 2 presents a simple model showing that affine FTP is biased under the ZLB. Section 3 presents the data and estimation method. In Section 4, we quantitatively illustrate the discrepancy of short-term and long-term FTP under the affine and quadratic models. The final section concludes. 2. A simple model For any maturity pair m and n with n > m, FTP is defined as:

F T Ptm,n = ftm,n −

n−m +1

EtP [rt+m+i ]

(1)

i=0

where ftm,n are forward interest rates for a (n-m)-period bond to be commenced at m-periods ahead and EtP [rt+m+i ] denotes the time-t expected future short-term interest rates where the expectation is taken with respect to the physical measure P. Joslin et al. (2014) call F T Ptm,n as “in-n-years-for-m-years” FTP. In Eq. (1), ftm,n can be rewritten as ftm,n = (n − m )−1 (logPtm − logPtn ) where Ptm and Ptn denote the price of a bond maturing in m and n periods respectively. For any term structure models with a reasonable good fit of the bond yield, the discrepancy in forward interest rates should be negligible. Meanwhile, the expectation EtP [rt+m+i ] depends on the assumptions made on the interest rate process rt . As we shown below, affine model will lead to an overestimation of FTP. We start with a continuous time example where the state variable (xt ) follow a mean-reverting process:

dxt = (α − β xt )dt + σ dWt

(2)

where β is the mean reversion speed to the unconditional mean level α /β and σ is the volatility of the Brownian motion term dWt .6 Solving Eq. (2) forward,

xt = x0 e−β t +

 α 1 − e −β t + β



0

t

σ e−β (t−s) dWs , t ≥ 0

(3)

From Eq. (3), we know xt is a Gaussian random variable distributed as xt ∼ N (μ ¯ , σ¯ 2 ) with μ ¯ = E[xt ] = x0 e−β t + 2 σ¯ 2 = σ2β (1 − e−2β t ). To model the persistent low interest rate environment, one possible parameter com-

α −β t ) and β (1 − e

bination is x0 → 0 and β → 0 such that μ ¯ → 0. Hence, Eq. (2) will generate a sequence of xt which will stay around zero. Without loss of generality, we assume observed interest rates (rt ) has a simple relationship with the factor xt . In the affine model, we have rtAG = δ xt with δ > 0 is a constant. The probability density function (pdf) for rtAG is

f



rtAG





=φ

rtAG /δ − μ ¯ σ¯



1

 

= √ exp − 2π σ¯ 2

2

rtAG /δ − μ ¯ 2σ¯ 2

(4)

3 Meanwhile, Jarrow (2013) argues that ZLB is just a myth, rather than a reality as he shows that negative interest rates are totally consistent with no arbitrage using the Heath-Jarrow-Morton (Heath et al. (1992)) model. 4 It is noteworthy that the estimated shadow rates are very sensitive to the model assumptions due to the singularity at the truncated lower bound. Using the Japanese government bond market from 20 0 0 to 2006 as an example, Ueno et al. (2006) estimate a one-factor model and find that the shadow rate can be as low as −15% On the contrary, Kim and Singleton (2012) estimate a two-factor model and find that the shadow rate is only about −1%. 5 Andreasen and Meldrum (2015) compare the performance of affine, quadratic and shadow rate models for the US. Regarding the bias in affine FTP, they focus on the bias arising due to the persistence of interest rates in the VAR under physical measure. This paper focuses on the bias due to the expectation of future interest rates under the ZLB. 6 For illustration purposes, we ignore the distinction of physical and risk neutral measures.

Please cite this article as: T.-K. Chung et al., Term-structure modelling at the zero lower bound: Implications for estimating the forward term premium, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.12.001

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In Eq. (4), the probability of negative interest rates depends on the level α /β and the mean-reverting parameter β . We can also derive





E rtAG = δ E[xt ] = δ x0 e−β t + δ

α ( 1 − e −β t ) β

(5)

Under the assumed condition x0 → 0 and β → 0, E[rtAG ] will coverage to zero. For the quadratic model, we have rtQG = a(xt + b)2 with a > 0 and b are constant. The pdf is







f rtQG =



1 2π σ¯ 2 rtQG /a

exp −

¯ + b) rtQG /a + (μ 2σ¯ 2

2





cosh



μ¯ + b QG r /a t σ¯ 2

(6)

which is a non-central chi-square distribution. Kim and Singleton (2012) argue this pdf can precludes negative interest rates and is positively skewed with the shape dependent on the ratio (μ ¯ + b)/σ¯ . The expected interest rate is





 



E rtQG = a E xt2 + 2bE[xt ] + b2 = a σ¯ 2 + a(μ ¯ + b)

2

(7)

which is always positive. Hence, if x0 → 0 and β → 0, the expected interest rate converges to a positive level E[rtQG ] = a( σ¯ 2 + b2 ) > 0. As a result, the expected interest rate for the quadratic model should be higher than under an affine model under the ZLB. From Eq. (1), lower expected interest rates in the affine model would manifest into a higher FTP, other things being equal. 3. Quantitative analysis In the quantitative analysis, it is more convenient to adopt a discrete time latent factor term structure model.7 The latent state vector Xt = (x1t , x2t , x3t ) follows a VAR(1) process:

Xt+1 = μQ + Q Xt + εt+1 with ɛt ∼ N(0, I3 × 3 ),

μQ

is a 3 × 1 vector and

(8)

Q

is a 3 × 3 matrix. The latent factor Xt is related to observed factors as

Xˆt = Xt + ωX,t

(9)

where Xˆt = (xˆ1t , xˆ2t , xˆ3t ) is the observed state variables and ωX, t are i.i.d. normals. The observed factors are constructed as follows: 1. Level (xˆ1t ) = yt1 , i.e., the three-month Treasury yield; 2. Slope (xˆ2t ) = yt40 − yt4 , i.e., the one- to ten-year term spread; 3. Curvature (xˆ3t ) = yt40 − 2yt20 + yt4 , i.e., the one- to five- to ten-year butterfly spread).8 Diebold et al. (2006), Bikbov and Chernov (2010) and Hamilton and Wu (2012) employ similar proxies in their termstructure models. The notation Q in Eq. (8) denotes the risk-neutral probability measure. We specify the market price of risk as λt = λ0 + λ1 Xt with λ0 is a 3 × 1 vector and λ1 is a 3 × 3 matrix. Hence, real-world dynamics of the state vector are given by:

Xt+1 = μP + P Xt + εt+1

(10)

with μQ = μP − λ0 and Q = P − λ1 . These assumptions imply the existence of a stochastic discount factor (Duffee (2002)). For any given short-rate function linking short-term interest rates with the state vector rt = ρ (Xt ), the price of a n-period zero-coupon bond is



Ptn

=

EtQ



exp −

n−1 



rt+i

(11)

i=0

For the affine model, the short-rate function is given by rtAG = δ0 + δ1T Xt where δ 0 is a scalar and δ 1 is a 3 × 1 vector. Hence, the short-rate is a linear function of the state variables. We can solve the expectation in Eq. (11) as Ptn = exp(An + BTn Xt ), where An is a scalar and Bn is a 3 × 1 vector which solves a system of two equations recursively. (see Duffee (2002)) The bond yield is also linear in Xt with ytn = − 1n logPtn = an + bTn Xt by taking an = −An /n and bn = −Bn /n as the factor loadings. For the quadratic model, the short-rate function is given by rtQG = α0 + β0T Xt + XtT 0 Xt where α 0 is a scalar and β 0 is a 3 × 1 vector and  0 is a 3 × 3 matrix. The n-period zero coupon bond price is Ptn = exp(An + BTn Xt + XnT Cn Xt ) where An is a scalar and Bn is a 3 × 1 vector and Cn is a 3 × 3 matrix which solves a system of three equations recursively. (see Ang et al., 2011) Bond yields are ytn = an + bTn Xt + XnT cn Xt with an = −An /n and bn = −Bn /n and cn = −Cn /n as the factor loadings 7 8

The difference in between continuous time and discrete time Gaussian term structure models is empirically negligible. The superscript on the bond yield denotes its maturity in quarters.

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Fig. 1. Graphical illustration of the short-rate functions. This figure plots the short-rate functions (i.e., rt = ρ (x1t ) for the affine and quadratic models. The short-rate functions for the affine and quadratic models are rtAG = x1t (45-degree line) and rtQG1 = ηx21t (quadratic curve) respectively. We set the parameter η such that both models generate identical short-rates when x1t = 2%. Table 1 Summary statistics of the US Treasury yields and the factors in the term-structure model.

y1 (Level) y2 y4 y8 y12 y16 y20 y24 y28 y32 y36 y40 Slope Curvature

Min

Max

Mean

Standard deviation

0.01% 0.04% 0.12% 0.22% 0.32% 0.47% 0.65% 0.86% 1.07% 1.28% 1.48% 1.68% −3.79% −0.70%

7.90% 7.85% 8.32% 8.48% 8.54% 8.56% 8.57% 8.57% 8.63% 8.72% 8.80% 8.86% 0.23% 1.61%

3.06% 3.17% 3.45% 3.72% 3.96% 4.19% 4.40% 4.59% 4.76% 4.91% 5.05% 5.16% −1.71% 0.18%

2.23% 2.23% 2.35% 2.32% 2.24% 2.15% 2.06% 1.98% 1.91% 1.85% 1.80% 1.75% 1.19% 0.51%

Note: The quarter-end yield data is from the first quarter of 1990 to the second quarter of 2014. The superscript on the bond yield denotes its maturity in quarters. Level factor is the three-month yield (y1 ). Slope factor is the one-to-ten year term spread (y40 -y4 ). Curvature factor is the the one- to five- to ten-year butterfly spread (y40 − 2y20 + y4 ).

To facilitate a transparent comparison between the affine and quadratic models, we assume the short-term interest rates depend on the level factor in the model, i.e., rt = ρ (x1t ). This setup is similar to that used in previous studies such as Bernanke et al. (2004) and Ang et al. (2011).9 For the affine model, we set rtAG = x1t . For the quadratic model, we set rtQG = ηx21t where η is a parameter used to ensure both models to produce identical short-rate at a specific level. The 45degree line and the quadratic curve in Fig. 1 represent the short-rate function for the affine and quadratic models respectively. If we want both models to generate the same short-rate when x1t = xu = 2%, we can take η = 50 such that ηx2u = 0.02.10 4. Data and the estimation method The quarterly (end-of-quarter) dataset of zero-coupon US Treasury yields from January 1990 to June 2014, with yield maturities of one-, two-, four-, eight-, and up to forty-quarter (total 12 maturities) are obtained from Gurkaynak et al. (2007). A brief summary statistics of the data is provided in Table 1. We re-cast both models as follows:

ytn = an + bTn Xt + XnT cn Xt + ωn,t

9 10

(12)

None of our quantitative results are sensitive to this assumption. We ignore the negative root in the estimation.

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Table 2 Summary statistics of the estimated forward term premium.

F T Pt5,10

Affine Quadratic F T Pt5,10 Affine F T Pt1,2 Quadratic F T Pt1,2

Min

Max

Mean

Standard deviation

2.74% 1.80% 0.43% 0.06%

6.91% 5.88% 2.50% 2.79%

4.74% 3.84% 1.35% 1.21%

1.13% 0.92% 0.48% 0.68%

Note: F T Pt5,10 and F T Pt1,2 denote the “in-ten-years-for-five-years” and the “in-two-years-for-one-year” forward term premium respectively.

Fig. 2. In-ten-years-for-five-years forward term premium (FTPt 5,10 ). This figure plots the estimated “in-ten-years-for-five-years” forward term premium of different models and the federal funds target rate.

with n = 1, 2, ….. N and ωn, t are measurement errors which are i.i.d. normals.

Xt+1 = μP + P Xt + εt+1

(13)

where Xˆt = Xt + ωX,t . Eqs. (12) and (13) are the measurement and state equation respectively in a non-linear state space model. The model is estimated using the Bayesian MCMC method.11 We choose the number of iterations of the Gibbs sampler to be 20,0 0 0 and discard the first 10,0 0 0 burn-in samples. We conduct statistical inference based on the sample of these remaining draws. 5. Empirical results In Table 2, we report the summary statistics of the “in-ten-years-for-five-years” and the “in-two-years-for-one-year” FTP ( F T Pt5,10 and F T Pt1,2 ) estimated from both models. We can see the long-horizon affine FTP almost exceed the quadratic FTP by almost a percentage point. The over-estimation of short-horizon affine FTP is also notable, albeit it is less severe than the long-horizon counterparts. We can also illustrate the over-estimation graphically. Fig. 2 shows the long-horizon FTP estimated from both models. FTP estimates exhibit a countercyclical pattern, rising notably during recessions. Although the two FTP move in tandem during most of the sample period, it is noteworthy that there are two episodes where the affine FTP is tangibly higher than the quadratic FTP. The first episode occurs around 2002 and lasts for about two years. The second episode starts shortly after the global financial crisis in 2008 and has persisted since then. The disparity is due to the downward bias of expected future interest rates in the affine model under the ZLB as we argued in Section 3. Indeed, when the US Fed decided to raise the policy rate in late 2004, the end of ZLB helped to narrow the disparity because the probability of negative interest rates in affine model dwindled significantly. The shorter-horizon FTP also depicts a similar pattern to its longer-horizon counterpart. Fig. 3 shows the “in-two-yearsfor-one-year” forward term premium (F T Pt1,2 ). When compared with Fig. 2, the disparity is still pronounced from 2008 onwards, but the disparity in the longer-horizon term premium during 20 02-20 04 is not present. The reason for this is

11

We follow Ang et al. (2011) on the MCMC algorithm, prior distributions and initial conditions.

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Fig. 3. In-two-years-for-one-year forward term premium (FTPt 1,2 ). This figure plots the estimated “in-two-years-for-one-years” forward term premium of different models and the federal funds target rate.

probably related to the level of interest rates and the forecasting horizon to compute the expectation term in Eq. (1). Specifically, when the federal funds rate was still one percentage point above the ZLB in 20 02-20 04, the probability of negative interest rates in one-year’s time is lower than the corresponding probability in five-year’s time since the uncertainty is increasing with the forecasting horizon. On the contrary, since the financial crisis, the affine model is more likely to generate negative interest rates as the federal funds rate has been in the range of 0-25 basis points. 6. Conclusion Although the affine model has been the workhorse model in term-structure modelling, its inability to generate an asymmetric interest rate response under the ZLB undermines its usefulness in the low interest rate environment. Specifically, we show that the affine FTP is biased upwards and may lead to an inaccurate assessment of the effectiveness of QE. On the contrary, the quadratic model shares the analytical tractability with unbiased expected interest rates, and therefore produces plausible term premium estimates under the ZLB. References Ang, A., Boivin, J., Dong, S., Rudy, L., 2011. Monetary policy shifts and the term structure. Rev. Econ. Stud. 78 (2), 429–457. Ahn, D., Dittmar, R.F., Gallant, R.A., Latane, H.A., 2002. Quadratic term structure models: theory and evidence. Rev. Financ. Stud. 15 (1), 243–288. Andreasen, M.M., Meldrum, A., 2015. Dynamic Term Structure Models: The Best Way to Enforce the Zero Lower Bound. Bank of England working paper No.550. Bauer, M.D., Rudebusch, G.D., 2016. Monetary policy expectations at the zero lower bound. J. Money Credit Bank. 48 (7), 1439–1465. Bernanke, B.S., 2012. Monetary Policy since the Onset of the Crisis. In: Speech at the Federal Reserve Bank of Kansas City Economic Symposium August 31, 2012. Bernanke, B.S., Reinhart, V.R., Sack, B.P., 2004. Monetary policy alternatives at the zero bound: an empirical assessment. Brookings Papers Econ. Activity 2, 1–100. Bikbov, R., Chernov, M., 2010. No-arbitrage macroeconomic determinants of the yield curve. J. Econ. 159 (1), 166–182. Black, F., 1995. Interest rates as options. J. Finance 50 (5), 1371–1376. Christensen, J.H.E., Rudebusch, G.D., 2015. Estimating shadow-rate term structure models with near-zero yields. J. Financ. Econ. 13 (2), 226–259. Diebold, F.X., Rudebusch, G.D., Aruoba, S.B., 2006. The macroeconomy and the yield curve: a dynamic latent factor approach. J. Econ. 131 (1-2), 309–338. Duffee, G.R., 2002. Term premia and interest rate forecasts in affine models. J. Finance 57, 405–443. Gurkaynak, R.S., Sack, B.P., Wright, J.H., 2007. The U.S. treasury yield curve: 1961 to the present. J. Monet. Econ. 54 (8), 2291–2304. Hamilton, J.D., Wu, J.C., 2012. The effectiveness of alternative monetary policy tools in a zero lower bound environment. J. Money Credit Bank. 44 (1), 3–46. Heath, D., Jarrow, R., Morton, A., 1992. Bond pricing and the term structure of interest rates: a new methodology for contingent clams valuation. Econometrica 60, 77–105. Ichiue, H., Ueno, Y., 2013. Estimating Term Premia at the Zero Bound: An Analysis of Japanese, US, and UK Yields. Bank of Japan Working Paper Series 13-E-8. Jarrow, R., 2013. The zero-lower bound interest rates: myth or reality? Finance Res. Lett. 10, 151–156. Jarrow, R., Li., H., 2011. The Impact of Quantitative Easing on the US Term Structure of Interest Rates. Cornell University working paper. Joyce, M., Lasaosa, A., Stevens, I., Tong, M., 2011. The financial market impact of quantitative easing in the United Kingdom. Int. J. Central Bank. 7 (3), 113–161. Joslin, S., Priebsch, M., Singleton, K.J., 2014. Risk premiums in dynamic term structure models with unspanned macro risks. J. Finance 69 (3), 1197–1233. Kim, D., Singleton, KJ., 2012. Term structure models and the zero bound: an empirical investigation of Japanese yields. J. Econ. 170 (1), 32–49. Krippner, L., 2013. A Tractable Framework for Zero Lower Bound Gaussian Term Structure Models. Reserve Bank of New Zealand Discussion Paper Series DP2013/02, Reserve Bank of New Zealand. Leippold, M., Wu, L., 2002. Asset Pricing under the quadratic class. J. Financ. Quant. Anal. 37 (2), 271–295.

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