Portfolio Substitution Effect of Quantitative Easing∗ Nikola Mirkov†

Barbara Sutter‡

First Draft: November 2011 This Version: June 2013 Abstract If money is an imperfect substitute for other financial assets, a large increase in the money supply raises prices and reduces yields on alternative, non-money assets. We refer to this effect as the portfolio substitution effect in the sense of Bernanke and Reinhart (2004) and estimate that a $100 billion increase in Non-borrowed reserves at the Federal Reserve reduces the 10-year Treasury yield by 5 basis points on average. We use the case of the Swiss National Bank to show that the existence of the portfolio substitution channel is independent of whether the expansion of the Central Bank’s balance sheet comes from purchases of Treasury securities, as in the Fed’s case, or foreign currency, as in the case of the SNB. Keywords: Interest rates, LSAPs, substitution effect, Bayesian MCMC JEL Classifications: E43, E52, C11, G12

∗

We are greatful to Paul Söderlind, Eric Swanson and Glenn Rudebusch for important suggestions at the earlieast stage of the project. A great thanks to Nina Larsson, Ragna Alstadheim, Lars Svensson, Anders Vredin, Francesco Audrino, Ragnar Nymoen, Jens Christensen, Michael Bauer, Sylvain Leduc and seminar and conference participants at the Norges Bank, University of Oslo, University of St.Gallen, Federal Reserve Bank of San Francisco and the 14th INFER Annual Conference. The views expressed in the paper do not necessarily reflect those of the Swiss National Bank. † Nikola Mirkov (corresponding author), Swiss National Bank, Borsenstrasse 15, 8022 Zurich, Switzerland, E-mail: [email protected], Tel: +41(0)76.22.98.176 ‡ Barbara Sutter, Swiss National Bank, Borsenstrasse 15, 8022 Zurich, Switzerland, E-mail: [email protected], Tel: +41(0)44.63.13.736

1

1

Introduction

In his seminal paper from 1969, James Tobin argues that a change in the supply of any asset alters the structure of rates of return on all assets in the economy in a way that makes the public willing to hold the new supply. When the asset’s own rate of return can change, the adjustment primarily happens through the increase or fall in the price of that asset. Yet, if the rate of return on the asset is fixed (e.g. money), the whole adjustment takes place through increases in prices and reduction in yields of other assets. The effect of a large increase in the money supply on long-term interest rates is discussed in Bernanke and Reinhart (2004) in the context of monetary policy near the zero lower bound (ZLB) and the authors refer to it as the “portfolio substitution” effect. This paper assess the empirical relevance of the Tobin’s portfolio substitution effect at the ZLB. In particular, we ask whether the increase in reserve balances at the Federal Reserve (Fed) given by the large scale asset purchases (LSAPs) has effected the longterm Treasury yields beyond producing the well-established supply effect. The supply effect arises when the Fed purchases an asset, reduces the supply of it and bids-up its price.1 One possible explanation for the existence of the supply effect in the Treasury market is offered by the preferred habitat model of interest rates, which assumes imperfect substitutability between securities with similar maturities.2 In addition to the supply effect, an outright purchase of the asset by the Fed raises the amount of credit in the form of reserves available in the aggregate. Assuming an unchanged amount of government debt, i.e. Treasury bonds, prices of these bonds should, all else equal, increase as there is more money to buy the same amount of assets. Since the short-term government bonds constitute a close substitute for money when the Fed funds rate is at the ZLB, some of the additional credit might have found its way into longer-term Treasury bonds and thereby brought down the long-term yields. To the best of our knowledge, this is the first study to empirically measure such substitution effect between the reserve money at the Fed and the Treasuries. Oda and Ueda (2007) test the “balance sheet expansion” hypothesis of Bernanke and Reinhart (2004) on Japanese data. The closest to our paper is the study by 1

See for example Hamilton and Wu (2010a), Vayanos and Vila (2009), Greenwood and Vayanos

(2010), Neely (2010), Kuttner (2006) and D’Amico and King (2010). 2 See Li and Wei (2012).

2

Krogstrup, Reynard and Sutter (2012) that estimates the “liquidity effect” of asset purchases on the Treasury yield curve. We estimate that an increase in non-borrowed reserves (NBR) at the Fed by $100 billion caused a fall in the 10-Year Treasury yield by 5 basis points on average, across different modeling assumptions. The effect is estimated by using an affine term structure model (ATSM) in which most of the variation in yields is driven by the information extracted from the yield curve itself and unrelated to the changes in central bank reserves. In such a way, we try to control for various supply effects of the LSAPs found in the previous studies, such as the announcement effect,3 the stock effect,4 and the effect of change in the maturity structure of the Treasury debt,5 because the reduced supply of long-term Treasury securities should be instantly priced in yields. Therefore, the NBR factor should only explain a marginal variation in yields given by the higher amount of reserves money after the purchases. Most importantly, the information in the yield curve should capture changes in expectations regarding the future path of policy that LSAPs might have affected, because any increase in reserve balances above the level necessary to keep the Fed funds rate at zero implicitly signals a continuation of an accommodative monetary policy stance.6 Finally, we show that the portfolio substitution effect exists independently of how the increase in central bank reserves occurs. To this end, we apply our estimation strategy to the Swiss data, as the reserves at the Swiss National Bank (SNB) increased considerably since the policy rate has been essentially set to zero and the increase came exclusively from the purchases of foreign currency. We estimate a 16 basis points fall in the 10-year government bond yield associated with an average increase in reserves at the SNB of 100 billion Swiss francs (SFr.). The rest of the paper is organized as follows. Section 2 illustrates the dataset. In Section 3, we introduce the term-structure model we use to estimate the portfolio substitution effect and Section 4 provides the details on the model estimation. Finally, Section 5 presents the results. 3

See for example Gagnon, Raskin, Remache and Sack (2010). See D’Amico and King (2010) and Li and Wei (2012). 5 See Hamilton and Wu (2010b) and Greenwood and Vayanos (2010). 6 See Bauer and Rudebusch (2011) for the discussion of possible signaling effects that LSAPs 4

could have had.

3

2

Data

2.1

United States

We use the NBR held at the Fed to measure the effect of an increase in the Fed’s balance sheet i.e. the effect of an increase in the supply of reserves on the yield curve. NBR can be considered a narrow monetary aggregate7 and hence less affected by money demand shocks.8 The NBR are calculated by subtracting the borrowed reserves, equal to the sum of credit extended through the Fed’s regular discount window programs and other liquidity facilities, from the amount of total reserves in the system. The data on aggregate reserves held at the Fed are published every Wednesday in the H.3 release by the Federal Reserve Board. The interest rate data stem from off-the-run US Treasury bills and bonds and their closing prices on Wednesdays. We use 3-month and 6-month secondary market Tbill rates9 and 1- to 10-year off-the-run constant maturity yields from Gurkaynak, Sack and Wright (2007).10 The yields are continuously compounded, whereas a quarterly compounding is assumed for the 3-month yield and the semi-annual compounding for other maturities.11 The data sample starts with the second week of December 2008, when the FOMC set the target range for the Fed funds rate to 0 - 25 basis points. We do not use the data before the Fed funds rate reached the ZLB in order to empirically assess the effects of “increasing the size of the central bank’s balance sheet beyond the level needed to set the short-term policy rate at zero” discussed in Bernanke and Reinhart (2004). The upper panel of Figure 1 illustrates the US data. The NBR amounted to roughly $540 billion at the time the Fed funds rate reached the ZLB in December 2008 and continued to grow to roughly $1,500 billion by December 2012. 7

See Pagan and Robertson (1995). See e.g. Bernanke and Blinder (1992), Sims (1992), or Christiano and Eichenbaum (1995). 9 Obtained from the Federal Reserve Economic Data base (FRED), under DTB3 and DTB6. The 8

two are not constant-maturity yields which is acceptable due to their short maturity. 10 The data can be downloaded from http://www.federalreserve.gov/econresdata/

researchdata.htm. We use off-the-run Treasuries to avoid the treatment of “repo-specialness” implicit in the on-the-run Treasuries, see Duffie (1996) and Jordan and Jordan (1997). 11 See Hull (2008).

4

FIGURE 1 ABOUT HERE

2.2

Switzerland

We use the data on bank’s sight deposit accounts held at the SNB by both domestic and foreign banks as the measure of supply of reserve balances. Interest rate data are the 3-month and the 6-month Swiss franc Libor rates and 1- to 10year constant-maturity zero-coupon yields on the Swiss Confederation bonds. The sample starts with the second week of December 2008, when the SNB decreased its targeted level for the 3-month Libor from 1% to 0.5%, and ends on the 26th December 2012. The lower panel of Figure 1 illustrates the Swiss dataset.

3

The Model

This Section defines the term structure model to estimate the portfolio substitution effect. We start from the general asset pricing equation, define the pricing kernel and specify a transition equation for the underlying state factors. Then, we specify the one-period interest rate and formulate the bond prices across the maturity spectrum.

3.1

General Setting and State Dynamics

The general asset pricing equation12 under the physical probability measure P reads

£ ¤ P n,t = E t M t+1 P n−1,t+1 | I t ,

(1)

where P n,t is the price of an n-periods to maturity zero-coupon bond at time t, M t+1 denotes the stochastic discount factor, and I t represents the agents’ current information set. In a risk-neutral world, where investors request no risk compensation, the price of the bond P n,t equals 12

See Campbell, Lo and MacKinlay (1997)

5

¤ Q£ P n,t = E t exp(− y1,t )P n−1,t+1 | I t ,

(2)

where Q is the risk-neutral probability measure and y1,t is the short-term interest rate. The no-arbitrage argument assures that the two prices in (1) and (2) are equal. There exists an equivalent martingale measure Q according to which (2) holds13 with the stochastic discount factor taking the form

exp(− y1,t ) = E t [ M t+1 | I t ] = exp(− y1,t )E t [( d Q/ d P) t+1 | I t ] .

d Q/ d P is the Radon-Nykodim derivative14 which follows a log-normal process, so it reads

1 ( d Q/ d P) t+1 = exp − (λ t )0 λ t − (λ t )0 ε t+1 2 µ

¶

(3)

where λ t is the market price of risk associated with the sources of uncertainty ε t 15 . Following Duffee (2002), the market price of risk is an “essentially affine” function of the state variables X t , so it can be written as

λ t = λ0 + λ1 X t .

(4)

Equations (3) to (4) jointly define the pricing kernel of the model, where the essentially affine market price of risk constitutes the first fundamental building block of the Gaussian term structure model. Another fundamental building block of the Gaussian term structure model is the multivariate state variable X t . It follows a discrete version of the constant volatility Ornstein-Uhlenbeck process16 . Under the physical probability measure P, the 13

See Harrison and Kreps (1979). See Dai, Singleton and Yang (2007). 15 See Ang and Bekaert (2002) and Ang and Piazzesi (2003). 16 See Phillips (1972). 14

6

process is

X t+1 = ( I − Ψ)µ + Ψ X t + Σε t+1 .

(5)

The first term on the right-hand side of equation (5) is a vector of the factors’ means. Ψ is the VAR matrix, Σ is the covariance matrix that normalizes the residuals ε t which are assumed to be standard normal i.i.d. shocks.

3.2

Short Rate and Bond Prices

Following Duffie and Kan (1996), the one-period interest rate is an affine function of risk factors X t as

y1,t = A 1 + B1 X t where the coefficient A 1 corresponds to the average one-period rate in the sample and B1 is a vector of loadings of the risk factors on y1,t . The risk factors are 

l 1,t



   Xt =   

l 2,t

     

l 3,t R eserves t

(6)

The first three factors l ti , i = 1, 2, 3 denote the latent factors backed out from yields. As commonly in the term structure literature, these factors can be interpreted as a level, a slope, and a curvature factor.17 In order to estimate the portfolio substitution effect on the longer-end of the yield curve, this paper adds central bank reserves, as the fourth factor to X t .18 As it can be noticed, the model does not assume that the short rate can reach the ZLB. Yet, as shown in the results section, the model fits the data reasonably well and thus can be considered as a good approximation. 17 18

See Section 4 for details on how yield-only factors are “backed-out” from yields. Similarly, Ang and Piazzesi (2003), for instance, are among many studies that use macro-

economic variables as explicit factors.

7

Assuming joint log-normality of bond prices and the pricing kernel in equation (1), the n-periods to maturity nominal bond price is an affine function of the state variables19 and thus takes the form:

p n,t = − A n − B n X t ,

(7)

with:

¡ ¢ 1 A n = A n−1 + B n−1 ( I − Ψ)µ − Σλ0 + B n−1 ΣΣ0 B0n−1 + A 1 2 B n = B n−1 (Ψ − Σλ1 ) + B1

(8)

4

Estimation

In this section we derive the likelihood function used to construct the joint posterior of parameters and data. The model is estimated with a simple version the Bayesian Markov-Chain Monte-Carlo (MCMC) method and the section provides the rationale for using MCMC and the description of the algorithm.

4.1

Likelihood Function

Following Chen and Scott (1993), the 6-month, the 5-year and the 10-year yields are set to be observable, while the rest is measured with error. Let yo,t be a vector of observed yields i.e. yields perfectly priced by the model

"

yo,t R eserves t

19

#

" =

Ao 01×1

# " +

Bo

03×1

01×3

1

See for example: Cochrane and Piazzesi (2009).

8

#"

X o,t R eserves t

#

where A o is a 3 × 1 vector and B o a 3 × 3 matrix of factor loadings. X o,t are the three latent factors obtained by inverting the above equation as

"

#

X o,t R eserves t

" =

Bo

03×1

01×3

1

#−1 Ã"

yo,t R eserves t

# " −

Ao

#!

01×1

The first part of the likelihood function refers to the evolution of the state variables

X t as given in equation (5). The assumption that ε t is multivariate-Gaussian implies that the conditional probability density function of X t is ³ ¡ 0 ¢−1 ´ 1 0 exp − ε εt ¡ ¢ 2 t ΣΣ . f X t | yo,t−1 , R eserves t−1 = p (2π)T | Σ0 Σ |

(9)

Let the yu,t be a vector of remaining N − 3 yields priced by the model with an error

yu,t = A u + B u X t + ξ t The second part of the likelihood function refers to the pricing errors ξ t . They ¡ ¢ are assumed to be distributed as i.i.d. N 0, ω2 I , with the same variance ω and zero-correlations across yields, where I is a unity matrix. The conditional density of yu,t is thus given by µ ³¡ ´−1 ¶ ¢ 1 0 2 0 2 exp − 2 ξ t ω I ω I ξt ¡ ¢ f yu,t | X t = r ¯¡ ¯ ¢0 ¯ ¯ (2π)T ¯ ω2 I ω2 I ¯

(10)

The log-likelihood function is just the sum of logarithms of the “time-series part” in equation (9) and the “cross-sectional part” in equation (10) and thus takes the form

¡ ¢ ¡ ¢ ln L (·) = log( f X t | yo,t−1 , R eserves t−1 ) + log( f yu,t | X t )

9

(11)

4.2

Econometric Identification

Solid identification of parameters is an essential part of dynamic term structure model estimation. The proposed identification scheme stems mainly from Dai and Singleton (2000) and Hamilton and Wu (2010a). To begin with, the upper-left block of the VAR matrix Ψ that drives the dynamics of the yields-only factors in X o,t is set to be a power law structure,20 with zero non-diagonal elements and the following power relation on the diagonal

ψ zz = ψ11 α z−1

where ψ11 is the largest eigenvalue and the AR coefficient of the first latent factor, α is a scaling parameter controlling the distance between the eigenvalues, and z = 2, ...Z , where Z is the number of latent factors. Preliminary estimation showed that the ψ11 parameter is near one. In line with the near co-integration assumption from previous studies21 , we simply set ψ11 to 1,22 and estimate ψ zz , where z = {2, 3}, together with the AR(1) coefficient of the reserves dynamics. The off-diagonal elements of the Ψ are set to zero, as well as vector the µ vector and the off-diagonal elements of matrix Σ in the transition equation (5). We impose the usual boundary condition A 0 = B0 = 0 on the parameters of the pricing equation given in (7). A 1 is normalized to average the one-period interest rate in the sample23 while B1 is normalized to [1 1 1 b R eserves ]0 .24 Alternatively, one could set the covariance matrix of the transition equation (5) to a unity matrix and estimate all the elements of B1 .25 Finally, the market price of risk dynamics is restricted to 20

See also Calvet, Fisher and Wu (2010) and Bauer and de los Rios (2011). See for instance Giese (2008) and Jardet, Monfort and Pegoraro (2011). 22 As in Diebold and Li (2006), Söderlind (2010) and Bauer (2011). 23 Following Favero, Niu and Sala (2007). 24 As in Ang, Bekaert and Wei (2008). 25 See for instance Ang and Piazzesi (2003). 21

10



λ0,1

  λ  0,2 λst =   0  0

 

0 0 0 λ1,14

    0 0 0 λ 1,24   +   0 0 0 0   0 0 0 0

     X t,  

(12)

so that both “level” and “slope” risks are priced in yields.26 This is the main market price of risk specification of the study. The results are also reported for the riskneutral Q measure, where λ0 = λ1 = 0, and for the case where only the “level” shock is being a compensated risk, i.e. where λ0,2 and λ1,24 are set to 0.27 It is indeed a restricted set of models, yet previous studies show that many restrictions on the market price of risk are supported by the data.28 In addition, we estimated an unrestricted model with λ1,i j 6= 0 for i = {1, 4}, j = {1, 4} and the main result holds.29

4.3

Bayesian Inference

The yield curve implied by the model is a complicated non-linear function of the underlying parameters. As this non-linearity tend to produce a multi-modal likelihood function30 , fitting a yield curve model with a standard maximum likelihood estimation is a daunting task. Bayesian Markov Chain Monte Carlo (MCMC) method seem to be a powerful alternative, providing both efficiency and tractability.

4.3.1

Setting

Let Θ be a vector of length K collecting all the parameters of the model to be estimated

© ª Θ = α, ψR eserves , Σ, λ0 , λ1 , ω, b R eserves 26

As in Duffee (2010) and Joslin, Priebsch and Singleton (2010). We follow exactly this specification from Cochrane and Piazzesi (2009) and add the slope factor. 28 See for example Joslin et al. (2010) and Bauer (2011) 29 We do not report the output from the unrestricted model estimation, but the results are avail27

able upon request. 30 See Chib and Ergashev (2009).

11

The key idea behind Bayesian estimation is to consider the vector as a multivariate random variable, and use the Bayes’ rule to “learn” about the variable given the data

p (Θ | data) =

p (data | Θ) p (Θ) p (data)

(13)

where p (Θ | data) is the posterior density of Θ, p (data | Θ) is the likelihood function and p (Θ) denotes the prior density of the parameters. The term p (data) is known as "normalising constant" and it is independent of Θ.31 Consequently, the rule in (13) can be re-written as

ln p (Θ | data) ∝ lnL + ln p (Θ) where lnL is the logarithm of the likelihood function defined in (9) and (10):

lnL

= +

TX −1 t=0 TX −1

¡ ¢ lnpdf X t | yo,t−1 , r t−1 ¡ ¢ lnpdf yu,t | X t , yo,t−1 , r t−1

t=0

(14)

4.3.2

Priors

In the estimation, the priors p (Θ) are set to be non-informative or “flat”, so that the posterior density of the model parameters is drawn with equal probability from the pre-defined support interval. Alternatively, we could derive the prior distributions for parameters Ψ and Σ, given the normality assumption of the state VAR process,32 and for ω given the assumption of the Gaussian measurement error.33 Chib (2001) propose a scaled beta distribution as an alternative to the uniform distribution. Nevertheless, we choose not to impose lower (or higher) probability R In particular: p (data) = p(data | Θ) p(Θ) d Θ. See Koop (2003). 32 For instance, see Ang, Dong and Piazzesi (2007). 33 See for example Mikkelsen (2001). 31

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areas from which the candidate values of parameters are drawn. In such a way, the estimation is almost completely data-driven and proves to be computationally efficient. The parameters’ suport intervals are specified by following the no-arbitrage condition and previous studies. In particular, the eigenvalues of the VAR matrix are set to be positive and less than one and the volatility parameters on the diagonal of Σ are set to be non-negative. The parameter b NBR is constrained to be inside the unit-circle. The lower bound of the parameters in λ0 vector and λ1 matrix are set as in Chib and Ergashev (2009).

4.3.3

Markov Chain Monte Carlo

We use a simple version of the “Metropolis within Gibbs” algorithm34 to draw the parameters from their posterior densities.³ The´parameter candidates are drawn from continuous uniform distributions U Θ, Θ where the lower and the upper boundaries Θ and Θ for each parameter in Θ are specified in Table 1. The algorithm can be described in five steps. Step 1: Set the initial values of parameters Θ0 . Two Markov chains with different starting values are set up. The initial values for the VAR parameters in first chain are obtained from OLS and the data descriptive statistics. The starting values of the market price of risk parameters and of the second chain are chosen arbitrarily.35 ¡ ¢ Step 2: Draw a candidate log-posterior density ln p Θ∗ | Θmc−1 , data conditional 34

The “Metropolis within Gibbs” is a simple method and therefore often used in the literature,

as for example in Gilks (1996), Koop (2003) and Lynch (2007). In particular, this algorithm features two favorable characteristics. First, in standard Metropolis new proposals for the parameter values are drawn all at once, whereas the Gibbs sampler draws only one proposal for only one parameter value at a time. Therefore, the Gibbs sampler results much more efficient, i.e. the estimator converges quicker. Second, the tuning of the estimator is simpler, where tuning refers to setting the scale factor in step 2 to different values. In standard Metropolis, one scale factor must “fit all”. The Gibbs sampler is more flexible as it allows for a separate set of values for each parameter. 35 For example, the volatility parameters’ starting values are set to be 3 times larger in the second chain, the Ψ matrix parameters are set to 0.8 and the B1 parameter is set to 0.5 and -0.5 in the first and the second Markov chain, respectively.

13

on previously drawn parameters’ values Θmc−1 . The number mc denotes current iteration. The draws are performed separately for every parameter in Θ. For instance, a proposal for the first element in the vector Θ is generated by the following Markov chain:

θ1∗ = θ1mc−1 + ν1U1

where νk is a scaling factor and Uk is an uniformly distributed random number from interval [-1,1]. We initialise νk for parameters α and ψ NBR to 0.01, for diagonal elements of Σ matrix and the market price of risk parameters to 0.1, and for the ω parameter to 0.0000136 . The scaling factor is then automatically updated after every 5,000 sweeps37 to obtain the acceptance ratio in step 4 of approximately 0.5. Step 3: For every parameter in Θ∗ , calculate the difference between the posterior density with the candidate value and the posterior density with the previously drawn parameter value, keeping the other parameter values unchanged. Using again the first element in Θ as an example, the difference reads:

¢ ¡ mc−1 } | Θmc−1 , data δ = ln p {θ1∗ , θ2mc−1 , ..., θK ¡ ¢ mc−1 − ln p {θ1mc−1 , θ2mc−1 , ..., θK } | Θmc−1 , data

(15)

Step 4: Draw a random number u ∼ U (0, 1) and accept the single parameter candidate from Step 2, whenever the following holds for the difference in Step 3:

min(0, δ) > log( u) 36

Proposed in Ang et al. (2007) so that it roughly corresponds to a 30 basis points bid-ask spread

on the Treasuries. An average spread on the OTC plain vanilla swap market might be similar. See also Skarr (2010). 37 The algorithm is ran for 100,000 times. The scaling factor is updated starting from the 10,000th iteration to the 40,000th iteration.

14

Step 5: Repeat the Steps 2 to 5 until the joint posterior density of parameters converge in distribution. The algorithm is ran 100,000 times and the first 40,000 are discarded as the burnin period. The two Markov chains with different starting values for both joint posterior and the single parameters’ posteriors converge to literally the same posterior distributions. Before estimating the entire model, the proposed parametrization is used to estimate the risk neutral specification. The model under Q converges even quicker and thus the algorithm is ran for 50,000 times and the first 20,000 are discarded as burn-in.38

5

Results

Before turning to the main result, we must be assured that the estimated model produces reasonable parameter estimates and fits the observed interest rates well. We also show that most of the variation in yields is driven by the factors extracted from the yield curve itself and that the central bank reserves, as a fourth factor in the term structure model, explain only a marginal variation in fitted yields. Therefore we arguably control for different supply effects reported in previous studies.

5.1 5.1.1

Model Performance Parameters

Table 1 and 2 report the estimated parameters for the US and Switzerland, respectively. Each table illustrates the parameter’s modes and numerical standard errors (in brackets). Figure 2 presents the posterior distributions of the parameter estimates for the US. TABLES 1 AND 2 ABOUT HERE The first two columns in the tables provide the support intervals, and the last 38

The scaling factor is also automatically adapted until the 20,000th iteration.

15

two the average acceptance ratios39 and inefficiency factors (IF)40 for the two subsamples. The coefficient estimates are depicted in the third and the fourth column. FIGURE 2 ABOUT HERE As it can be noticed, all the parameters converge to a static mode. The estimated factor loadings of the reserves’ factors in the two countries on the short rate is negative, yet insignificant, which means that any increase in reserves on the longterm yields affects the term premia and not the monetary policy expectations embedded in the long-term yields.41 Put differently, the factor loading of the NBR factor on the one-period interest rate is basically zero and thus has no effect on the expected future one-period rate. The estimated time-varying prices of risk coefficients, λ14 and λ24 , are statistically significant in the two samples. λ14 is associated with the effect of the NBR at the Fed (or the sight deposits at the SNB) on the market price of “level” risk. The significantly negative λ14 suggests that during the ZLB period, the discount factor decreases with an expansion in the NBR (or the sight deposits), and so the term premia increases in the level. This could be an evidence of the Fisher effect: expansionary monetary policy eventually leads to higher inflation rates which, anticipated in the form of increasing inflation expectations and premia, drives up the nominal interest rates. Evidence of the substitution effect is found in the market price of risk associated with the slope of the yield curve. λ24 turns significantly positive at the ZLB, which suggests that an increase in the NBR (or the sight deposits) leads to a flattening of the yield curve. Since short-term interest rates are close to zero, the substitution effect drives the long-end of the yield curve down.42 39

The acceptance ratio is the number of accepted parameters’ proposals divided by the number of

iterations after the burn-in. The rate between 0.25 and 0.75 is often acceptable, see Lynch (2007) and Koop (2003). P 40 The Inefficiency Factor measures how well the sampler “mixes” is computed as 1 + 2 Ll=1 ρ ( l ), where ρ ( l ) is the autocorrelation at lag l in the Markov chain sequence of a parameter, and L is the lag at which the autocorrelation function goes to zero. See Chib (2001) for details. 41 The result is in line with the previous literature showing that the LSAPs lowered the risk premia on long-term Tresuries. 42 The parameter λ24 is statistically significant and positive even when we estimate an unrestricted version of the model, where λ1,i j 6= 0 for i = {1, 4}, j = {1, 4}.

16

5.1.2

Pricing

By construction, the 3-month, the 5-year and the 10-year yields are explained perfectly by the model. Table 3 reports the cross-sectional fit for both countries. Allowing for the time-varying level and slope risks improves the fit above the Q model,43 yet, on the whole, the pricing performance is relatively similar across modeling assumptions. TABLE 3 ABOUT HERE

5.2

Variance Decomposition and Latent Factor Dynamics

Table 4 reports the variance decomposition for selected yields and shows that most of the variance in yields is explained by the three latent factors.44 The contribution of the NBR factor (and the sight deposits factor for the SNB) does not exceed 3.7% (0.4%). TABLE 4 ABOUT HERE Figure 3 shows the estimated latent factors for the US and the Swiss yield curve. The factor l 1 is by definition a unit-root process45 and thus drives the level of both yield curves, as it can be noticed by comparing the evolution of the latent factors with the 5- and 10-year yields in the two countries depicted in Figure 1. FIGURE 3 ABOUT HERE

5.3 5.3.1

The Factor Loadings The Non-Borrowed Reserves at the Fed

We report our key results in Table 5. The table shows the factor loadings of the NBR factor on selected yields under different market price of risk specifications. 43

Our preliminary analysis also showed that the reserves factor contribution to pricing perfor-

mance was minimal. We do not report this result. 44 The result is broadly in line with Litterman and Scheinkman (1991), Ang and Piazzesi (2003) and Joslin, Singleton and Zhu (2011), who all show that almost entire variation in the cross-section of yields can be explained by some three latent factors (or the first three principal components of the yields matrix). 45 See Section 4.2.

17

A factor loading indicates by how much a particular yield changes in response to a $100 billion change in the NBR. TABLES 5 ABOUT HERE We find a statistically and economically significant portfolio substitution effect at the ZLB of an increase in the NBR on the long-term yields. When the NBR rise by $100 billion, the 10-year yield falls by 5 basis points on average across modeling assumptions. With the US nominal GDP of roughly $15.1 trillion, the estimates suggest that an increase in the NBR of the size of one percent of GDP lowers longterm rates by up to 8 basis points. The model under the Q measure suggests a smaller effect at long maturities.

5.3.2

Non Borrowed Reserves and Treasury Balances

When the Fed purchases an asset, e.g. a Treasury note, the asset side of the Fed’s balance sheet marks an increase in assets that corresponds to the amount of purchased Treasury notes. Contemporaneously, the liability side of the balance sheet increases by the same amount as the Fed credits the deposit account of the assetseller depository bank with reserve money. This section addresses the balance sheet identity and asks whether our main result is driven by the supply effect, i.e. the fall in yields of purchased Treasury securities given by the fall in their supply available to the public and thus by the asset-side increase in the Fed’s balance sheet. In particular, we orthogonalize the NBR factor on the amount of purchased Treasury bonds with maturities from 5 to 10 years as

NBR t = α + βTB t + ε t

(16)

and then use the residual from the regression ε t as the fourth explicit factor in the estimation. Figure 4 illustrates the orthogonalised NBR factor and Table 6 reports the factor loadings for the three modeling alternatives. FIGURE 4 ABOUT HERE The estimated effect on the 10-year yield is essentially unchanged. The Q model still measures the smallest effect and the average effect across modeling assump18

tions is a 5 basis points drop in the yield in response to a $100 billion increase in the NBR orthogonal to the increase in the holdings of 5 to 10-year Treasury securities. TABLES 6 ABOUT HERE

5.3.3

Sight Deposit Accounts at the Swiss National Bank

NBR at the Fed increased because the Fed was purchasing long-term Treasury and Mortgage-Backed securities. In order to confirm that the portfolio substitution effect of Tobin (1969) is empirically relevant independently of why the Central Bank reserves increase, we use the case of the SNB who has never been engaged in asset purchases. Table 7 reports the reserves factor loadings across different market price of risk settings and selected yields. TABLE 7 ABOUT HERE We find a negative and significant effect of an increase in reserves held at the SNB on the long term yields according to all the model specifications. The estimates suggest that the increase in reserves of SFr.100 billion reduced the 10-year yield by 16 basis points on average, 11 basis points according to the Q model, 27 basis points according to the model in which the level factor is priced, and 11 basis points according to the model in which both level and slope risks are priced in yields. In terms of Swiss GDP, which amounts to approximately SFr.600 billion, an increase of reserves at the SNB of one percent of nominal GDP lowers rates by roughly 1 basis point.

5.4 5.4.1

Cumulative Effects United States

Figure 5 plots the estimated overall effect of an increase in the NBR during the ZLB period corresponding to different modeling assumptions. As the 90% confidence bands indicate, the effect is statistically significant only for the yields of long maturity bonds.

19

FIGURE 5 ABOUT HERE The total increase in NBR of roughly $1 trillion is associated with a fall in the 10-year yield of 40 basis points on average across the modeling specifications. The model under the Q measure estimates a total decrease of roughly 20 basis points for both mid- and long-term yields. The models accounting for the time-varying level risk and the time-varying level and slope risks estimate a fall in the mid- and long-term yields of approximately 50 basis points as of December 2012. Figure 7 illustrates that most of the drop in yields can be explained by a fall in risk premium. FIGURE 7 ABOUT HERE

5.4.2

Switzerland

According to our estimates, the cumulative effect of the expansion in sight deposits at the SNB from approximately SFr.26 billion francs in December 2008 to SFr.370 billion December 2012 has led to an estimated fall in the 10-year yield of 60 basis points on average across market price of risk settings. Figure 6 shows the estimated cumulative effect as well as the corresponding 90%- credible interval. The Q and the L&S models estimate a fall of roughly 40 basis points in the yield during the period, whereas the L model points to a drop of around 80 basis points. FIGURE 6 ABOUT HERE

6

Conclusion

This study provides evidence of Tobin’s portfolio substitution effect at the longer end of the yield curve, when the short-term interest rates are near the ZLB. Increases in central bank reserves at the Fed and the SNB seem to have had a statistically and economically significant effect on the long-term government bond yields in the two countries. Due to the sizes of the respective secondary markets, the average effect in the Swiss case is estimated to be higher relative to the one found in the US data. The presented analysis might be relevant in the discussion of exit strategies from the expansionary monetary policy measures, implemented in the two countries. 20

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7

Appendix - Tables and Figures

Table 1: Parameter Estimates for the US. The table reports the estimated posterior modes of the parameters for the US together with numerical standard errors (in brackets). The first two columns provide the support intervals, and the last two the average acceptance (AccRatio) ratios and inefficiency factors (IF) for the two subsamples. The acceptance ratio is the number of accepted parameters’ proposals divided by the number of iterations after burn-in. Inefficiency Factor is computed P as 1 + 2 Ll=1 ρ ( l ), where ρ ( l ) is the autocorrelation at lag l in the Markov chain sequence of a parameter, and L is the lag at which the autocorrelation function goes to zero. α

aΘ 0.00

bΘ 0.99

ψR

0.00

0.99

σ11

0.00

15.0

σ22

0.00

15.0

σ33

0.00

15.0

σ NBR

0.00

15.0

λ01

-100

100

λ02

-100

100

λ14

-100

100

λ24

-100

100

ω

0.00

10.0

b NBR

-1.00

1.00

Θ 0.93 (0.00) 0.99 (0.00) 0.14 (0.00) 0.31 (0.01) 0.23 (0.01) 0.07 (0.00) 4.89 (0.22) -2.31 (0.22) -0.13 (0.02) 0.21 (0.04) 0.21 (0.01) -0.03 (0.18)

26

AccRatio 0.80

IF 1894

0.82

69

0.23

1493

0.55

200

0.23

15

0.58

21

0.67

2157

0.32

947

0.83

328

0.64

503

0.19

49

0.36

544

Table 2: Parameter Estimates for Switzerland. The table reports the estimated posterior modes of the parameters for the US together with numerical standard errors (in brackets). The first two columns provide the support intervals, and the last two the average acceptance (AccRatio) ratios and inefficiency factors (IF) for the two subsamples. The acceptance ratio is the number of accepted parameters’ proposals divided by the number of iterations after burn-in. Inefficiency Factor is P computed as 1 + 2 Ll=1 ρ ( l ), where ρ ( l ) is the autocorrelation at lag l in the Markov chain sequence of a parameter, and L is the lag at which the autocorrelation function goes to zero. α

aΘ 0.00

bΘ 0.99

ψSi ghtD e po

0.00

1.00

σ11

0.00

15.0

σ22

0.00

15.0

σ33

0.00

15.0

σSi ghtD e po

0.00

16.0

λ01

-100

100

λ02

-100

100

λ14

-100

100

λ24

-100

100

ω

0.00

10.0

b Si ghtD e po

-1.00

1.00

27

Θ 0.92 (0.00) 1.00 (0.00) 0.08 (0.00) 0.18 (0.01) 0.17 (0.01) 0.17 (0.01) 1.95 (0.21) -1.06 (0.20) -0.04 (0.02) 0.10 (0.05) 0.20 (0.01) 0.03 (0.08)

AccRatio 0.70

IF 294

0.71

40

0.23

645

0.39

83

0.21

35

0.31

9

0.64

693

0.27

442

0.79

308

0.69

433

0.21

16

0.28

348

Table 3: Pricing Errors. Mean absolute pricing errors in basis points across different modelling assumptions are reported for selected yields in the US (upper panel) and Switzerland (lower panel), and for the two subsamples.

Q l l&s

Q l l&s

6M 7.8 8.5 7.7

1Y 14.4 16.4 15.0

USA 3Y 10.7 7.8 6.6

6Y 5.7 4.2 4.1

8Y 10.0 8.9 9.0

9Y 7.0 6.7 6.8

6M 9.5 9.7 9.4

CH 1Y 3Y 12.9 8.2 13.1 6.0 12.5 6.7

6Y 4.1 3.3 3.7

8Y 6.2 5.2 5.7

9Y 4.1 3.4 3.8

Table 4: Variance Decomposition. The table reports variance decomposition of selected yields in % for the US (upper panel) and Switzerland (lower panel). The variance is decomposed by dividing each single state variable shock j to an nj periods yield: MSE n = B0n Σ j B n + B0n ΨΣ j ΨB n , where Σ j is a K × K matrix with zeros and a non-zero j j element corresponding to the volatility of state variable j ; with the overall Mean Squared Error of forecasting the states 1 period ahead: MSE n = B0n ΣB n + B0n ΨΣΨB n .

l1 l2 l3 NBR

6M 23.2 47.1 28.8 0.9

l1 l2 l3 Si ghtD e po

6M 43.8 41.8 14.0 0.3

US 1Y 3Y 27.6 43.7 48.4 44.1 22.8 9.7 1.2 2.5 CH 1Y 3Y 49.8 67.9 40.0 28.9 10.0 3.2 0.2 0.0

28

6Y 62.0 30.5 3.9 3.6

8Y 70.7 23.1 2.5 3.7

9Y 74.2 20.1 2.1 3.6

6Y 82.9 15.8 1.1 0.3

8Y 88.2 10.7 0.6 0.4

9Y 90.1 9.0 0.5 0.4

Table 5: Factor Loadings for the US. The table reports the values of the NBRfactor loadings for selected yields, together with the Z-score from the Z-test (in parenthesis). The levels of significance of 0.1, 0.05 and 0.01 are denoted with *, ** and ***, respectively. Q

B Z-score

3M -0.01** -(1.92)

1Y -0.01* -(1.92)

3Y -0.01* -(1.92)

6Y -0.01* -(1.92)

8Y -0.01* -(1.92)

10Y -0.01* -(1.92)

8Y -0.08*** -(4.39)

10Y -0.07*** -(3.92)

8Y -0.08*** -(2.89)

10Y -0.07*** -(2.94)

L B Z-score

3M -0.12*** -(5.45)

1Y -0.11*** -(5.39)

B Z-score

3M -0.01 -(0.22)

1Y -0.03 -(0.60)

3Y 6Y -0.1*** -0.09*** -(5.19) -(4.77) L&S 3Y 6Y -0.06 -0.08** -(1.55) -(2.57)

Table 6: Factor Loadings for the US with the orthogonal NBR-factor. The table reports the values of the NBR-factor loadings for selected yields, together with the Z-score from the Z-test (in parenthesis). The levels of significance of 0.1, 0.05 and 0.01 are denoted with *, ** and ***, respectively. Q

B Z-score

3M -0.04*** -(2.60)

1Y -0.04*** -(2.65)

3Y -0.03*** -(2.74)

6Y -0.02*** -(2.77)

8Y -0.02*** -(2.76)

10Y -0.01*** -(2.74)

3Y 6Y -0.06* -0.05** -(1.69) -(2.01) L&S 3Y 6Y -0.1** -0.1*** -(2.29) -(2.71)

8Y -0.05** -(2.17)

10Y -0.05** -(2.26)

8Y -0.09*** -(2.84)

10Y -0.08*** -(2.91)

L B Z-score

3M -0.07 -(1.35)

1Y -0.07 -(1.44)

B Z-score

3M -0.09 -(1.51)

1Y -0.1* -(1.77)

29

Table 7: Factor Loadings for Switzerland. The table reports the values of the reserves factor loadings for selected yields, together with the Z-score from the Ztest (in parenthesis). The levels of significance of 0.1, 0.05 and 0.01 are denoted with *, ** and ***, respectively. Q

B Z-score

3M -0.11*** -(4.33)

1Y -0.11*** -(4.34)

B Z-score

3M -0.45*** -(11.14)

1Y -0.43*** -(11.04)

B Z-score

3M 0.3*** (3.54)

1Y 0.2*** (2.68)

3Y -0.11*** -(4.39)

6Y -0.11*** -(4.45)

L 3Y 6Y -0.4*** -0.34*** -(10.70) -(10.00) L&S 3Y 6Y 0.03 -0.08 (0.56) -(1.60)

30

8Y -0.11*** -(4.48)

10Y -0.11*** -(4.52)

8Y -0.3*** -(9.36)

10Y -0.27*** -(8.59)

8Y -0.11** -(2.24)

10Y -0.11** -(2.36)

Figure 1: Interest Rates and Central Bank Reserves. The figure reports selected interest rates (left scale) and the central bank reserves (right scale) in the US (upper panel) and Switzerland (lower panel). A. The US

B. Switzerland

31

Figure 2: Estimated Posteriors for the US. The Figure illustrates the estimated posterior densities for all the parameters of the model where both level and slope risks are priced in the yield curve (subsequently denoted as “L&S model”).

32

Figure 3: Latent Factors. The figure plots the estimated latent factors for the US yield curve (upper panel) and the Swiss yield curve (lower panel). The reported are l 1 (solid blue), l 2 (solid green), l 3 (solid red) factors. A. The US

B. Switzerland

33

Figure 4: Orthogonalized NBR-Factor. The figure plots the residuals from the equation (16).

34

Figure 5: Cumulative Effects for the US. The figure plots a cumulative effect of increase in NBR on the 3-month (left-hand side panels) and 10-year (right-hand side panels) interest rates according to the Q-model (top row), the L-model (middle row) and L&S -model (bottom row plots) together with a 90 % credible interval (left panels). The credible interval is calculated using the posterior parameters’ distributions, i.e. every 1000th set of parameters along the first Markov chain and after burn-in.

35

Figure 6: Cumulative Effects for Switzerland. The figure plots a cumulative effect of increase total sight deposits at the SNB on the 3-month (left-hand side panels) and 10-year (right-hand side panels) interest rates according to the Q-model (top row), the L-model (middle row) and L&S -model (bottom row plots) together with a 90 % credible interval (left panels). The credible interval is calculated using the posterior parameters’ distributions, specifically every 1000th set of parameters along the first Markov chain and after burn-in.

36

Figure 7: Cumulative Effects Due to Fall in Risk Premia. The figure plots the cumulative effects (dashed black line) of increase in NBR at the Fed and sight deposits at the SNB on 10-year yields in the US (panel A) and Switzerland (panel B), respectively. Also, it illustrates the effect of increase in central bank reserves on risk premia (solid red line) together with a 90 % credible interval. Estimated effects are obtained from the L&S -model output. A. The US

B. Switzerland

37