- Email: [email protected]
Financial shocks and the maturity of the monetary policy rate
Economics Letters 107 (2010) 333–337
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Financial shocks and the maturity of the monetary policy rate Petra Gerlach-Kristen a,⁎, Barbara Rudolf b a b
Bank for International Settlements, Centralbahnplatz 2, 4002 Basel, Switzerland Swiss National Bank, Börsenstrasse 15, 8022 Zürich, Switzerland
a r t i c l e
i n f o
Article history: Received 5 November 2008 Received in revised form 14 January 2010 Accepted 25 February 2010 Available online 10 March 2010
a b s t r a c t Monetary policy is typically formulated with a very short-term interest rate, while longer rates matter in the transmission mechanism. We show that financial market shocks impact less on the macroeconomy if policy is set with a longer rate. © 2010 Elsevier B.V. All rights reserved.
Keywords: Monetary policy framework Maturity of the policy interest rate Financial shocks Three-month libor JEL classification: E43 E52 E58
1. Introduction The paper is motivated by the effects of the US subprime crisis on international money markets. As Fig. 1 shows, both the Federal Reserve and the Swiss National Bank succeeded in keeping their monetary policy rate (federal funds rate and three-month CHF libor, respectively) close to the announced target level during the first year of the crisis.1 It appears that the Swiss National Bank's practice of setting monetary policy in terms of a range for three-month libor has enabled it to absorb shocks to that rate, whereas USD three-month libor displayed large variations. To the extent that longer rates matter more for the economy than shorter rates, this may have alleviated the macroeconomic impact of the financial crisis. In this paper we examine the impact of financial shocks on the output gap and inflation under two monetary policy frameworks which differ with respect to the interest rate targeted by the central bank. We add to the literature in three ways. First, while a number of authors present macroeconomic models with two interest rates (e.g. Eijffinger et al., 2009; Fendel, 2009; Lansing and Trehan, 2003; and Svensson, 2000), they assume that the economically relevant longer rate obeys the pure expectations hypothesis. Empirically, however,
⁎ Corresponding author. Tel.: + 41 61 280 8757; fax: + 41 61 280 9100. E-mail address: [email protected] (P. Gerlach-Kristen). 1 The Swiss National Bank usually announces (i) a hundred basis points target range for three-month libor and (ii) the point within the target range it is aiming at.
0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.02.012
financial risk premia matter, too, and we allow for this. Second, we assume, realistically, that this premium varies at a higher frequency than the macro data. In practice, financial data vary continuously, whereas data on GDP are available on a quarterly basis. Third, we explore both the standard policy framework in which monetary policy is formulated in terms of a one-week rate, and an alternative framework where policy is set with the three-month rate.2 The paper is structured as follows. Section 2 presents a model that assumes that policy is implemented using a one-week rate, but that the three-month rate, which depends on the path of current and expected one-week rates and a risk premium, matters in the IS curve. Central banks may formulate policy with either of these rates and keep the rate in question constant between the regular quarterly policy meetings. We refer to these two regimes as the one-week rate framework and the three-month rate framework. Section 3 studies the macroeconomic effects of movements in the risk premium under the two frameworks. The model simulations suggest that inflation and the output gap are more stable under the three-month rate framework since it allows for the adjustment of the one-week rate between policy meetings in response to financial shocks. Section 4 concludes.
2 Of course, the federal funds rate is an overnight rather than a one-week rate. Policy in the US has traditionally been implemented in terms of mainly overnight and twoweek repos. Thus, the 1w rate framework is only a rough proxy for the Federal Reserve's policy regime. Kulish (2007) and McGough et al. (2004) also consider frameworks where policy is set with a long rate but do not allow for risk premia.
334
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337
Fig. 1. Interest rate data (in percent).
2. The model
2.2. The term structure
2.1. The macroeconomy
Typically, the literature assumes that the central bank sets It⁎ . In practice, however, the central bank only sets a very short-term rate, it, which we think of as a 1w rate. We assume that there are twelve weeks to a quarter and that policy decisions are taken every three months in week 0 of quarter t. We denote week j in quarter t by t, j. The weekly value of the 3m market rate depends on the current and expected 1w rates through the expectations hypothesis
We consider a small New Keynesian model (see e.g. Fuhrer and Moore, 1995; and Rotemberg and Woodford, 1997). The IS curve (in its log-linearized form) is given by r y yt = 1−ρy Et yt+1 + ρy yt−1 −γ It −Et πt+1 −I + et ;
ð1Þ
where yt denotes the output gap, πt the rate of inflation, It the oneperiod nominal interest rate and Ir the steady state level of It. New estimates of the output gap are available quarterly in practice, so we think of the time passing between t and t + 1 in terms of quarters. It thus is a 3m interest rate. The demand shock eyt follows y et
=
y ρey et−1
where
+
vyt ∼ N(0,
y vt ;
σ2y ).
ð2Þ The Phillips curve is given by π
πt = ð1−ρπ ÞEt πt+1 + ρπ πt−1 + δyt + et
ð3Þ
with π et
=
⁎
It = I
+
π vt :
ð4Þ
⁎;r
⁎ + πt + 0:5 πt −π + 0:5yt ;
ð5Þ
where I⁎,r denotes the steady state level of the real Taylor-rule rate.3 Alternatively, we allow for interest-rate smoothing and let I⁎t follow =I
⁎;r
+ 1−ρI⁎
h
i ⁎ ⁎ πt + 0:5 πt −π + 0:5yt + ρI⁎ It−1 ;
ð6Þ
with ρ⁎I the smoothing parameter. Such a policy rule is optimal if the squared change in It⁎ enters the central bank loss function.4 3
11 1 Et;j ∑ it;j 12 n=0
+ n
+ θt;j ;
The results are qualitatively unchanged if we assume that the central bank determines It⁎ optimally. 4 For a discussion of the smoothing literature, see Gerlach-Kristen (2004). If a smoothing term enters the implementation objective function of the two policy frameworks (Eqs. (10) and (13) below), the qualitative results are not altered.
ð7Þ
where θt,j is a premium that we assume for simplicity as exogenous and that evolves according to θ
θt;j = θ0 + ρθ θt;j−1 + et;j
ð8Þ
with eθt,j ∼ N(0, σ2θ ).5 The constant θ0 captures a fixed term premium, whereas the variable part of θt,j reflects counterparty and liquidity risk.6 We assume that policymakers observe θt,j imperfectly. Formally, we have that ^ θt;j = θt;j + ut;j ;
π ρeπ et−1
where vπt ∼ N(0, σ2π). As a baseline model, we assume that monetary policy is set according to the Taylor rule
⁎ It
It;j =
ð9Þ
where ut,j ∼ N(0, σ2u). Since θt,j is autocorrelated, policymakers use the Kalman filter to assess the risk premium as θ˜ t;j = κ^ θt;j + ð1−κÞρθ θ˜ t;j−1 ; with κ the gain scalar, which is a function of ρθ, σ2θ , and σ2u (for details see Hamilton, 1994). 2.3. The policy decision At each meeting, policymakers determine It⁎.7 Policy is implemented with the 1w rate and may be communicated either with this or the 3m rate. Under the 1w rate framework the central bank formulates policy in terms of the 1w rate. Therefore, this rate is held constant between 5 Svensson (2000) also presents a model that allows for a premium for the longer rate. However, he assumes that it is constant. 6 See Hess and Kamara (2005) for a detailed discussion. 7 Since new data on inflation and the output gap are assumed to become available once a quarter, this rate stays constant between policy meetings. We write It,0⁎ = It,j⁎ = I⁎t for compactness.
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337
335
Table 1 Timing of events. Time
Information flow
1w rate framework
3m rate framework
t, 0
New data on πt, yt and θt,0: Taylor rule rate It⁎ is determined
Policy is announced and implemented in terms of 1w rate it,0
t, 1
New data on θt,1
… t, 11
1w rate is kept constant at it,1 = it,0; 3m market It,1 reacts to θt,1
Policy is announced in terms of 3m market rate It,0 and implemented with 1w rate it,0 1w rate it,1 is adjusted so that 3m market rate It,1 is expected constant at It,0
New data on θt,11
1w rate is kept constant at it,11 = it,0; 3m market It,11 reacts to θt,11 Policy is announced and implemented in terms of 1w rate it + 1,0
New data on πt + 1, yt + 1 and θt + 1,0: Taylor rule rate It⁎+ 1 is determined
t+1, 0
policy decisions and only adjusted every twelve weeks. Policymakers choose it,0 at the time of the policy meeting such as to minimize the expected deviation of the 3m market rate from its Taylor rule level. Formally,
Vt;0 = min Et;0 it;0
11 j ⁎ 2 ∑ β It;j −It ;
j=0
ð10Þ
where β is the real discount factor. The 1w rate is thus chosen not only to minimize the squared deviation of the 3m market rate from I⁎t at the time of the policy decision, but for all twelve weeks up to the next meeting. The 3m rate at the time of the policy meeting equals It;0 = it;0 + θt;0 ;
1w rate it,11 is adjusted so that 3m market rate It,11 is expected constant at It,0 Policy is announced in terms of 3m market rate it + 1,0 and implemented with 1w rate it + 1,0
and the 3m rate j weeks after the meeting is given by ! j−1 11 1 E It;j = ∑ i + ∑ it + 1;0 + θt;j 12 t;j n = j t;0 n=0 h i 1 E ð12−jÞit;0 + jit + 1;0 + θt;j ; = 12 t;j
ð11Þ
where the two summations capture the fact that the 1w rate is being held constant until, and after, the next policy meeting, respectively. Substituting Eq. (11) into Eq. (10), differentiating with respect to it,0, and solving yields 11 j ⁎ it;0 = Et;0 ∑ β ð12−jÞ It −jit j=0
+ 1;0 −θt;j
11
j
2
= ∑ β ð12−jÞ : j=0
Fig. 2. Simulations. Note: ρ⁎I = 0, σθ = 0.05 and σu = 0.0015. Solid red lines show the simulation for the 1w rate framework, dotted blue lines that for the 3m rate framework.
336
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337
Since policymakers observe the risk premium imperfectly, they announce and implement policy as 11 j ⁎ i˜t;0 = Et;0 ∑ β ð12−jÞ It −jit j=0
11 j 2 ˜ + 1;0 −θt;j = ∑ β ð12−jÞ : j=0
Baseline Taylor rule (Eq. (5))
ð12Þ
Thereafter, the 1w rate is held constant until the next policy meeting. Table 1 clarifies the timing. Under the 3m rate framework the central bank formulates policy with the 3m rate. To keep the 3m market rate as close as possible to It⁎, policymakers adjust it weekly to minimize the squared deviation of It,j from It⁎, Vt;j
⁎ 2 = min Et;j It;j −It :
Substituting Eq. (7) into Eq. (13), differentiating, solving for the 1w rate and accounting for uncertainty yields 11
n=1
var(πt) var(yt) var(It) var(it)
+ n:
ð14Þ
This implementation rate is adjusted weekly to keep the 3m market rate as close as possible to the Taylor rule rate. Thereby, the 1w rate absorbs financial shocks. 3. Simulations The model is next simulated to assess how the maturity of the policy rate matters for the macroeconomic impact of financial shocks. We set the inflation target π⁎ = 2% and the equilibrium 3m real interest rate I⁎,r = 1.5%, implying a real discount factor β = (1 + I⁎,r/ 400)− 1 = 0.996. The impact of the nominal 3m rate on the output gap in the IS curve is set to γ = 1 and the impact of the output gap on inflation in the Phillips curve to δ = 0.2, with ρy = ρπ = 0.5. The smoothing coefficient ρI⁎ is either zero or 0.5 and the term premium is set as θ0 = 0.1. For the shock processes, we assume that the autoregressive coefficients equal ρey = 0.75 and ρeπ = ρθ = 0.5, and the variances are fixed as σy = σπ = 0.05. The model is solved using the rational expectations methods proposed by Sims (2002). We first assume no interest-rate smoothing. We set the variance of the financial shock as σθ = 0.05 and σu = 0.0015, which suggests that policymakers' observations of this shock lie with 95% probability within a range of ±0.3 basis points around its true value. The upper row of Fig. 2 shows the paths of the output gap, inflation, and the 3m Taylor rule rate the central bank would like to see in the market, It⁎. The second row shows the 1w rate implemented by the central bank, it,j, the 3m market rate, It,j, and the deviation of this rate from It⁎. Any such deviation causes additional volatility of the output gap and inflation. The simulation shows that under the 3m rate framework, It,j is close to It⁎ since it,j is adjusted continuously. Under the 1w rate framework, it,j is set only every twelve weeks, resulting in larger variations in It,j and thus in the macroeconomy. Table 2 reports in the upper panel the variances of πt, yt, It, and it under the two frameworks in the baseline model. Results under interest-rate smoothing are reported in the lower panel. The left part of the table assumes σθ = 0.05 and σu = 0.0015, while the right part increases financial volatility by doubling σθ and σu. We find that for all specifications, yt and It are less volatile, and that it varies more, under the 3m rate framework than under the 1w framework. Inflation is more volatile under the 3m rate framework in all cases except for high financial shocks under interest-rate smoothing. For all model specifications, macroeconomic volatility – given by the sum of the inflation and the output gap variance – is higher under the 1w rate framework. Comparing the cases of financial calm and turbulence, we find that under the 1w rate framework all variances rise. Under the 3m
Low financial volatility: σθ = 0.05 and σu = 0.0015
High financial volatility: σθ = 0.1 and σu = 0.003
1w rate framework
3m rate framework
1w rate framework
3m rate framework
0.015 0.029 0.038 0.036
0.019 0.021 0.030 0.093
0.016 0.033 0.057 0.048
0.018 0.021 0.030 0.310
Taylor rule with smoothing (Eq. (6)), ρ⁎I = 0.5
ð13Þ
it;j
⁎ i˜t;j = 12It −12θ˜ t;j −Et;j ∑ it;j
Table 2 Variances.
var(πt) var(yt) var(It) var(it)
Low financial volatility: σθ = 0.05 and σu = 0.0015
High financial volatility: σθ = 0.1 and σu = 0.003
1w rate framework
3m rate framework
1w rate framework
3m rate framework
0.026 0.029 0.073 0.074
0.028 0.016 0.019 0.077
0.033 0.035 0.105 0.099
0.015 0.028 0.019 0.285
Note: Simulated variances for 50,000 draws.
rate framework, the variances of inflation, the output gap, and the 3m interest rate are essentially unchanged, but that of the 1w rate more than triples. Thus, macroeconomic stability under the 3m rate framework is paid for with higher volatility at the short end of the yield curve. 4. Conclusions The financial crisis of 2007/08 has shown that the market interest rate used the central bank to formulate monetary policy displays comparatively little variation in response to financial shocks. In this paper we have extended the standard monetary policy model to allow for interest rates of different maturity: a 1w rate, which the central bank uses for the implementation of monetary policy, and a 3m rate, which matters economically and which depends on the expected path of the 1w rate and a financial risk premium. Monetary policy is set every quarter, while financial shocks occur weekly. We consider two central bank frameworks that formulate policy with respect to either the 1w or the 3m rate. Under the 1w rate framework, financial shocks impact on the 3m market rate and thereby increase the volatility of the macroeconomy. Under the 3m rate framework, financial shocks are essentially absorbed by the 1w rate. One issue not addressed here is that policy under the 1w rate framework may be adjusted between scheduled meetings if there is a large financial shock. This would improve the relative performance of this monetary policy framework. Acknowledgements The views expressed in this paper are the authors' and do not necessarily reflect those of the Bank for International Settlement or of the Swiss National Bank. We thank an anonymous referee, Stefan Gerlach, Sébastien Kraenzlin, Michel Peytrignet, Marcel Savioz, Martin Schlegel, Paul Söderlind and Mathias Zurlinden for useful comments and discussions. References Eijffinger, S., Schaling, E., Verhagen, W., 2009. The term structure of interest rate and inflation forecast targeting. South African Journal of Economic and Management Sciences 12 (2), 162–179. Fendel, R., 2009. Note on Taylor rules and the term structure. Applied Economics Letters 16 (11), 1–5. Fuhrer, J.C., Moore, G., 1995. Inflation persistence. Quarterly Journal of Economics 110 (1), 127–159.
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337 Gerlach-Kristen, P., 2004. Interest Rate Smoothing: Monetary Policy Inertia or Unobserved Variables? B.E. Journals in Macroeconomics (Contributions to Macroeconomics) 4 (1) Article 3. Hamilton, J.D., 1994. Time Series Analysis. Princeton University Press, Princeton. Hess, A.C., Kamara, A., 2005. Conditional time-varying interest rate risk premium: evidence from the treasury bill futures market. Journal of Money, Credit, and Banking 37 (4), 679–698. Kulish, M., 2007. Should Monetary Policy Use Long-Term Rates? B.E. Journal of Macroeconomics, Advances 7 (1) Article 15. Lansing, K.J., Trehan, B., 2003. Forward-looking behavior and optimal discretionary monetary policy. Economics Letters 81 (2), 249–256.
337
McGough, B., Rudebusch, G., Williams, J.C., 2004. Using a Long-Term Interest Rate as the Monetary Policy Instrument: Federal Reserve Bank of San Francisco Working Paper, vol. 2004-22. Rotemberg, J.J., Woodford, M.J., 1997. An optimization-based econometric framework for the evaluation of monetary policy. NBER Macroeconomics Annual vol. 12. MIT Press, Cambridge MA, pp. 297–346. Sims, C.A., 2002. Solving linear rational expectations models. Computational Economics 20 (1–2), 1–20. Svensson, L.E.O., 2000. Open-economy inflation targeting. Journal of International Economics 50, 155–183.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e t
Financial shocks and the maturity of the monetary policy rate Petra Gerlach-Kristen a,⁎, Barbara Rudolf b a b
Bank for International Settlements, Centralbahnplatz 2, 4002 Basel, Switzerland Swiss National Bank, Börsenstrasse 15, 8022 Zürich, Switzerland
a r t i c l e
i n f o
Article history: Received 5 November 2008 Received in revised form 14 January 2010 Accepted 25 February 2010 Available online 10 March 2010
a b s t r a c t Monetary policy is typically formulated with a very short-term interest rate, while longer rates matter in the transmission mechanism. We show that financial market shocks impact less on the macroeconomy if policy is set with a longer rate. © 2010 Elsevier B.V. All rights reserved.
Keywords: Monetary policy framework Maturity of the policy interest rate Financial shocks Three-month libor JEL classification: E43 E52 E58
1. Introduction The paper is motivated by the effects of the US subprime crisis on international money markets. As Fig. 1 shows, both the Federal Reserve and the Swiss National Bank succeeded in keeping their monetary policy rate (federal funds rate and three-month CHF libor, respectively) close to the announced target level during the first year of the crisis.1 It appears that the Swiss National Bank's practice of setting monetary policy in terms of a range for three-month libor has enabled it to absorb shocks to that rate, whereas USD three-month libor displayed large variations. To the extent that longer rates matter more for the economy than shorter rates, this may have alleviated the macroeconomic impact of the financial crisis. In this paper we examine the impact of financial shocks on the output gap and inflation under two monetary policy frameworks which differ with respect to the interest rate targeted by the central bank. We add to the literature in three ways. First, while a number of authors present macroeconomic models with two interest rates (e.g. Eijffinger et al., 2009; Fendel, 2009; Lansing and Trehan, 2003; and Svensson, 2000), they assume that the economically relevant longer rate obeys the pure expectations hypothesis. Empirically, however,
⁎ Corresponding author. Tel.: + 41 61 280 8757; fax: + 41 61 280 9100. E-mail address: [email protected] (P. Gerlach-Kristen). 1 The Swiss National Bank usually announces (i) a hundred basis points target range for three-month libor and (ii) the point within the target range it is aiming at.
0165-1765/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2010.02.012
financial risk premia matter, too, and we allow for this. Second, we assume, realistically, that this premium varies at a higher frequency than the macro data. In practice, financial data vary continuously, whereas data on GDP are available on a quarterly basis. Third, we explore both the standard policy framework in which monetary policy is formulated in terms of a one-week rate, and an alternative framework where policy is set with the three-month rate.2 The paper is structured as follows. Section 2 presents a model that assumes that policy is implemented using a one-week rate, but that the three-month rate, which depends on the path of current and expected one-week rates and a risk premium, matters in the IS curve. Central banks may formulate policy with either of these rates and keep the rate in question constant between the regular quarterly policy meetings. We refer to these two regimes as the one-week rate framework and the three-month rate framework. Section 3 studies the macroeconomic effects of movements in the risk premium under the two frameworks. The model simulations suggest that inflation and the output gap are more stable under the three-month rate framework since it allows for the adjustment of the one-week rate between policy meetings in response to financial shocks. Section 4 concludes.
2 Of course, the federal funds rate is an overnight rather than a one-week rate. Policy in the US has traditionally been implemented in terms of mainly overnight and twoweek repos. Thus, the 1w rate framework is only a rough proxy for the Federal Reserve's policy regime. Kulish (2007) and McGough et al. (2004) also consider frameworks where policy is set with a long rate but do not allow for risk premia.
334
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337
Fig. 1. Interest rate data (in percent).
2. The model
2.2. The term structure
2.1. The macroeconomy
Typically, the literature assumes that the central bank sets It⁎ . In practice, however, the central bank only sets a very short-term rate, it, which we think of as a 1w rate. We assume that there are twelve weeks to a quarter and that policy decisions are taken every three months in week 0 of quarter t. We denote week j in quarter t by t, j. The weekly value of the 3m market rate depends on the current and expected 1w rates through the expectations hypothesis
We consider a small New Keynesian model (see e.g. Fuhrer and Moore, 1995; and Rotemberg and Woodford, 1997). The IS curve (in its log-linearized form) is given by r y yt = 1−ρy Et yt+1 + ρy yt−1 −γ It −Et πt+1 −I + et ;
ð1Þ
where yt denotes the output gap, πt the rate of inflation, It the oneperiod nominal interest rate and Ir the steady state level of It. New estimates of the output gap are available quarterly in practice, so we think of the time passing between t and t + 1 in terms of quarters. It thus is a 3m interest rate. The demand shock eyt follows y et
=
y ρey et−1
where
+
vyt ∼ N(0,
y vt ;
σ2y ).
ð2Þ The Phillips curve is given by π
πt = ð1−ρπ ÞEt πt+1 + ρπ πt−1 + δyt + et
ð3Þ
with π et
=
⁎
It = I
+
π vt :
ð4Þ
⁎;r
⁎ + πt + 0:5 πt −π + 0:5yt ;
ð5Þ
where I⁎,r denotes the steady state level of the real Taylor-rule rate.3 Alternatively, we allow for interest-rate smoothing and let I⁎t follow =I
⁎;r
+ 1−ρI⁎
h
i ⁎ ⁎ πt + 0:5 πt −π + 0:5yt + ρI⁎ It−1 ;
ð6Þ
with ρ⁎I the smoothing parameter. Such a policy rule is optimal if the squared change in It⁎ enters the central bank loss function.4 3
11 1 Et;j ∑ it;j 12 n=0
+ n
+ θt;j ;
The results are qualitatively unchanged if we assume that the central bank determines It⁎ optimally. 4 For a discussion of the smoothing literature, see Gerlach-Kristen (2004). If a smoothing term enters the implementation objective function of the two policy frameworks (Eqs. (10) and (13) below), the qualitative results are not altered.
ð7Þ
where θt,j is a premium that we assume for simplicity as exogenous and that evolves according to θ
θt;j = θ0 + ρθ θt;j−1 + et;j
ð8Þ
with eθt,j ∼ N(0, σ2θ ).5 The constant θ0 captures a fixed term premium, whereas the variable part of θt,j reflects counterparty and liquidity risk.6 We assume that policymakers observe θt,j imperfectly. Formally, we have that ^ θt;j = θt;j + ut;j ;
π ρeπ et−1
where vπt ∼ N(0, σ2π). As a baseline model, we assume that monetary policy is set according to the Taylor rule
⁎ It
It;j =
ð9Þ
where ut,j ∼ N(0, σ2u). Since θt,j is autocorrelated, policymakers use the Kalman filter to assess the risk premium as θ˜ t;j = κ^ θt;j + ð1−κÞρθ θ˜ t;j−1 ; with κ the gain scalar, which is a function of ρθ, σ2θ , and σ2u (for details see Hamilton, 1994). 2.3. The policy decision At each meeting, policymakers determine It⁎.7 Policy is implemented with the 1w rate and may be communicated either with this or the 3m rate. Under the 1w rate framework the central bank formulates policy in terms of the 1w rate. Therefore, this rate is held constant between 5 Svensson (2000) also presents a model that allows for a premium for the longer rate. However, he assumes that it is constant. 6 See Hess and Kamara (2005) for a detailed discussion. 7 Since new data on inflation and the output gap are assumed to become available once a quarter, this rate stays constant between policy meetings. We write It,0⁎ = It,j⁎ = I⁎t for compactness.
P. Gerlach-Kristen, B. Rudolf / Economics Letters 107 (2010) 333–337
335
Table 1 Timing of events. Time
Information flow
1w rate framework
3m rate framework
t, 0
New data on πt, yt and θt,0: Taylor rule rate It⁎ is determined
Policy is announced and implemented in terms of 1w rate it,0
t, 1
New data on θt,1
… t, 11
1w rate is kept constant at it,1 = it,0; 3m market It,1 reacts to θt,1
Policy is announced in terms of 3m market rate It,0 and implemented with 1w rate it,0 1w rate it,1 is adjusted so that 3m market rate It,1 is expected constant at It,0
New data on θt,11
1w rate is kept constant at it,11 = it,0; 3m market It,11 reacts to θt,11 Policy is announced and implemented in terms of 1w rate it + 1,0
New data on πt + 1, yt + 1 and θt + 1,0: Taylor rule rate It⁎+ 1 is determined
t+1, 0
policy decisions and only adjusted every twelve weeks. Policymakers choose it,0 at the time of the policy meeting such as to minimize the expected deviation of the 3m market rate from its Taylor rule level. Formally,
Vt;0 = min Et;0 it;0
11 j ⁎ 2 ∑ β It;j −It ;
j=0
ð10Þ
where β is the real discount factor. The 1w rate is thus chosen not only to minimize the squared deviation of the 3m market rate from I⁎t at the time of the policy decision, but for all twelve weeks up to the next meeting. The 3m rate at the time of the policy meeting equals It;0 = it;0 + θt;0 ;
1w rate it,11 is adjusted so that 3m market rate It,11 is expected constant at It,0 Policy is announced in terms of 3m market rate it + 1,0 and implemented with 1w rate it + 1,0
and the 3m rate j weeks after the meeting is given by ! j−1 11 1 E It;j = ∑ i + ∑ it + 1;0 + θt;j 12 t;j n = j t;0 n=0 h i 1 E ð12−jÞit;0 + jit + 1;0 + θt;j ; = 12 t;j
ð11Þ
where the two summations capture the fact that the 1w rate is being held constant until, and after, the next policy meeting, respectively. Substituting Eq. (11) into Eq. (10), differentiating with respect to it,0, and solving yields 11 j ⁎ it;0 = Et;0 ∑ β ð12−jÞ It −jit j=0
+ 1;0 −θt;j
11
j
2
= ∑ β ð12−jÞ : j=0
Fig. 2. Simulations. Note: ρ⁎I = 0, σθ = 0.05 and σu = 0.0015. Solid red lines show the simulation for the 1w rate framework, dotted blue lines that for the 3m rate framework.
336
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Since policymakers observe the risk premium imperfectly, they announce and implement policy as 11 j ⁎ i˜t;0 = Et;0 ∑ β ð12−jÞ It −jit j=0
11 j 2 ˜ + 1;0 −θt;j = ∑ β ð12−jÞ : j=0
Baseline Taylor rule (Eq. (5))
ð12Þ
Thereafter, the 1w rate is held constant until the next policy meeting. Table 1 clarifies the timing. Under the 3m rate framework the central bank formulates policy with the 3m rate. To keep the 3m market rate as close as possible to It⁎, policymakers adjust it weekly to minimize the squared deviation of It,j from It⁎, Vt;j
⁎ 2 = min Et;j It;j −It :
Substituting Eq. (7) into Eq. (13), differentiating, solving for the 1w rate and accounting for uncertainty yields 11
n=1
var(πt) var(yt) var(It) var(it)
+ n:
ð14Þ
This implementation rate is adjusted weekly to keep the 3m market rate as close as possible to the Taylor rule rate. Thereby, the 1w rate absorbs financial shocks. 3. Simulations The model is next simulated to assess how the maturity of the policy rate matters for the macroeconomic impact of financial shocks. We set the inflation target π⁎ = 2% and the equilibrium 3m real interest rate I⁎,r = 1.5%, implying a real discount factor β = (1 + I⁎,r/ 400)− 1 = 0.996. The impact of the nominal 3m rate on the output gap in the IS curve is set to γ = 1 and the impact of the output gap on inflation in the Phillips curve to δ = 0.2, with ρy = ρπ = 0.5. The smoothing coefficient ρI⁎ is either zero or 0.5 and the term premium is set as θ0 = 0.1. For the shock processes, we assume that the autoregressive coefficients equal ρey = 0.75 and ρeπ = ρθ = 0.5, and the variances are fixed as σy = σπ = 0.05. The model is solved using the rational expectations methods proposed by Sims (2002). We first assume no interest-rate smoothing. We set the variance of the financial shock as σθ = 0.05 and σu = 0.0015, which suggests that policymakers' observations of this shock lie with 95% probability within a range of ±0.3 basis points around its true value. The upper row of Fig. 2 shows the paths of the output gap, inflation, and the 3m Taylor rule rate the central bank would like to see in the market, It⁎. The second row shows the 1w rate implemented by the central bank, it,j, the 3m market rate, It,j, and the deviation of this rate from It⁎. Any such deviation causes additional volatility of the output gap and inflation. The simulation shows that under the 3m rate framework, It,j is close to It⁎ since it,j is adjusted continuously. Under the 1w rate framework, it,j is set only every twelve weeks, resulting in larger variations in It,j and thus in the macroeconomy. Table 2 reports in the upper panel the variances of πt, yt, It, and it under the two frameworks in the baseline model. Results under interest-rate smoothing are reported in the lower panel. The left part of the table assumes σθ = 0.05 and σu = 0.0015, while the right part increases financial volatility by doubling σθ and σu. We find that for all specifications, yt and It are less volatile, and that it varies more, under the 3m rate framework than under the 1w framework. Inflation is more volatile under the 3m rate framework in all cases except for high financial shocks under interest-rate smoothing. For all model specifications, macroeconomic volatility – given by the sum of the inflation and the output gap variance – is higher under the 1w rate framework. Comparing the cases of financial calm and turbulence, we find that under the 1w rate framework all variances rise. Under the 3m
Low financial volatility: σθ = 0.05 and σu = 0.0015
High financial volatility: σθ = 0.1 and σu = 0.003
1w rate framework
3m rate framework
1w rate framework
3m rate framework
0.015 0.029 0.038 0.036
0.019 0.021 0.030 0.093
0.016 0.033 0.057 0.048
0.018 0.021 0.030 0.310
Taylor rule with smoothing (Eq. (6)), ρ⁎I = 0.5
ð13Þ
it;j
⁎ i˜t;j = 12It −12θ˜ t;j −Et;j ∑ it;j
Table 2 Variances.
var(πt) var(yt) var(It) var(it)
Low financial volatility: σθ = 0.05 and σu = 0.0015
High financial volatility: σθ = 0.1 and σu = 0.003
1w rate framework
3m rate framework
1w rate framework
3m rate framework
0.026 0.029 0.073 0.074
0.028 0.016 0.019 0.077
0.033 0.035 0.105 0.099
0.015 0.028 0.019 0.285
Note: Simulated variances for 50,000 draws.
rate framework, the variances of inflation, the output gap, and the 3m interest rate are essentially unchanged, but that of the 1w rate more than triples. Thus, macroeconomic stability under the 3m rate framework is paid for with higher volatility at the short end of the yield curve. 4. Conclusions The financial crisis of 2007/08 has shown that the market interest rate used the central bank to formulate monetary policy displays comparatively little variation in response to financial shocks. In this paper we have extended the standard monetary policy model to allow for interest rates of different maturity: a 1w rate, which the central bank uses for the implementation of monetary policy, and a 3m rate, which matters economically and which depends on the expected path of the 1w rate and a financial risk premium. Monetary policy is set every quarter, while financial shocks occur weekly. We consider two central bank frameworks that formulate policy with respect to either the 1w or the 3m rate. Under the 1w rate framework, financial shocks impact on the 3m market rate and thereby increase the volatility of the macroeconomy. Under the 3m rate framework, financial shocks are essentially absorbed by the 1w rate. One issue not addressed here is that policy under the 1w rate framework may be adjusted between scheduled meetings if there is a large financial shock. This would improve the relative performance of this monetary policy framework. Acknowledgements The views expressed in this paper are the authors' and do not necessarily reflect those of the Bank for International Settlement or of the Swiss National Bank. We thank an anonymous referee, Stefan Gerlach, Sébastien Kraenzlin, Michel Peytrignet, Marcel Savioz, Martin Schlegel, Paul Söderlind and Mathias Zurlinden for useful comments and discussions. References Eijffinger, S., Schaling, E., Verhagen, W., 2009. The term structure of interest rate and inflation forecast targeting. South African Journal of Economic and Management Sciences 12 (2), 162–179. Fendel, R., 2009. Note on Taylor rules and the term structure. Applied Economics Letters 16 (11), 1–5. Fuhrer, J.C., Moore, G., 1995. Inflation persistence. Quarterly Journal of Economics 110 (1), 127–159.
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