A model of the euro-area yield curve with discrete policy rates

Jean-Paul Renne

doi:10.1515/snde-2016-0043

Abstract

This paper presents a no-arbitrage yield-curve model that explicitly incorporates the central-bank policy rate. This model is consistent with the existence of a lower bound for nominal interest rates, which makes it particularly relevant in the current context of extremely low interest rates. Changes in the policy rates depend on the monetary-policy phase, that can be either in an easing, status quo or tightening mode. The estimation of the model, based on daily euro-area yield data, reveals the strong influence of the monetary-policy phases on the shape of the yield curve. This relationship can, in turn, be exploited to estimate the probabilities of being in the different monetary-policy phases. The model is also used to compute term premiums, that are the parts of the yields reflecting the aversion of investors to interest rate risk. The results point to the existence of statistically significant premiums for many dates, even for short horizons.

Keywords: affine term-structure models; zero lower bound; regime switching models

JEL Classification: E43; E44; E47; E52; G12

Acknowledgments

I am deeply grateful to Alain Monfort for extremely valuable suggestions and comments. I have benefited from discussions with Simon Dubecq, Jean-Sébastien Fontaine, René Garcia, Rodrigo Guimaraes, Imen Ghattassi, Wolfgang Lemke, Andrew Meldrum, Jean-Stéphane Mésonnier, Emanuel Moench, Benoît Mojon, Fulvio Pegoraro, Marcel Priebsch, Francisco Rivadeynera Sanchez, Michael Rockinger, Thomas Sargent, Andrew Siegel, Min Wei and Paul Whelan. I thank participants at the Fed of San Francisco workshop on term structure modelling at the ZLB (2013), at SoFiE annual meeting (2013), at Canadian Economic Association annual meeting (2012), at AFSE annual meeting (2012), at ESEM annual meeting (2012), at the ECB workshop “Excess liquidity and money-market functioning” (2012), at AFFI Paris finance meeting (2012) and at seminars at Banque de France, Bank of England, HEC Lausanne, EDHEC Business School and Université Paris-Dauphine. I thank Béatrice Saes-Escorbiac and Aurélie Touchais for excellent research assistance. A substantial part of this work was completed when I was at the Banque de France. This paper however expresses my views only; they do not necessarily reflect those of the Banque de France.

Appendix

A Laplace transform of a Markov-switching process

In the following, I consider a n-state Markov process z_t, valued in {e₁, …, e_n}, the set of columns of I_n, the identity matrix of dimension n×n. I assume that the matrix of transition probabilities is deterministic and denoted by P_t (the columns sum to one). I have: ℙ(zt+1=ei|zt)=e′iPt+1zt. It can be shown that:

(18)Et(exp[α′1zt+1+ … +α′hzt+h])=[1 ⋯ 1][D(expαh)Pt+h]× …… ×[D(expα1)Pt+1]zt.

where exp α is the vector whose entries are of the form exp(α_i)’s and where D(x) is a diagonal matrix whose diagonal entries are the elements of the vector x. Indeed, for h=1:

Et(exp(α′zt+1))=∑i=1nexp(αi)e′iPt+1zt=(∑i=1Mexp(αi)e′i)Pt+1zt=[1 ⋯ 1]D(expα)Pt+1zt.

Now, for h=2, the law of iterated expectations leads to:

Et(exp[α′1zt+1+α′2zt+2])=Et(Et[exp[α′1zt+1+α′2zt+2]|zt+1])=Et(exp[α′1zt+1]Et[exp[α′2zt+2]|zt+1])

Then, using the previous case:

Et(exp[α′1zt+1+α′2zt+2])=Et(exp[α′1zt+1][1 ⋯ 1]D(expα2)Pt+2zt+1)=Et([1 ⋯ 1]D(expα2)Pt+2zt+1exp[α′1zt+1])=Et([1 ⋯ 1]D(expα2)Pt+2zt+1z′t+1D(expα1)[1 ⋯ 1]′).

Using the facts that zt+1z′t+1 commutes with any matrix and that zt+1z′t+1[1 ⋯ 1]′=zt+1, I get:

Et(exp[α′1zt+1+α′2zt+2])=Et([1 ⋯ 1]D(expα2)Pt+2D(expα1)zt+1)=[1 ⋯ 1][D(expα2)Pt+2][D(expα1)Pt+1]zt.

B Pricing formulas

In this Appendix, I briefly explain how to compute the three terms P_z(t,h), P_ξ(t,h) and P_s(t,h) whose product is the bond price P(t,h) [Equation (10)].

B.1 Computation of P_z(t,h)

The targets r̅_t are the only stochastic variables involved in the computation of P₁,(t,h). The previous Appendix shows that the expectation of an exponential-affine combination of a variable that follows a Markov-switching process is available in closed form. This leads to the following formula:

(19)Pz(t,h)=Etℚ(exp[−∑i=0h−1r¯t+i])=Gz(t,h)zt

with Gz(t,h)=[1 ⋯ 1][∏i=h−11D(exp[−Δm])Πt+i∗]D(exp[−Δm])

and where the product operator Π works in a backward direction: if X₁ and X₂ are some square matrices, ∏i=21Xi=X2X1. Defining the vector G(t,h) as G(t,h)=–(1/h)log{G_z(t,h)}, I get P_z(t,h)=exp(–hG(t,h)z_t).

B.2 Computation of P_ξ(t,h)

Using the independence assumption of the ξ_ts, I obtain:

Etℚ(exp[−∑i=0h−1ξt+i])=exp(−ξt)[Etℚ(exp[−ξt+1])]h−1.

According to the definition of the L distribution (see Subsection 2.1.2), the expectation Etℚ(exp[−ξt+1]) is given by:

exp(−w){p×ℱ(αP, βP, −1)+(1−p)×ℱ(αN, βN, 1)}

where ℱ(α, β, v) is given by: ℱ(α, β, v)=1+∑k=0∞vkk!(∏i=0k−1α+iα+β+i) (using the Laplace transform of the Beta distribution). Note that exp(−ξt)/Etℚ(exp[−ξt+1]) is very close to one (especially at the daily frequency). With virtually no impact on the result, I therefore use the approximation:

Pξ(t,h)=Etℚ(exp[−∑i=0h−1ξt+i])≈[Etℚ(exp[−ξt+1])]h=:exp(−hAh(ξ)) (say).

B.3 Computation of P_s(t,h)

We have

Ps(t,h)=Etℚe−∑i=0h−1st+i=Etℚe−∑i=0h−1s1, t+i+s2, t+iwhere [s1, ts2, t]=Φ∗[s1, t−1s2, t−1]+Σεt∗, εt∗∼i.i.d. 𝒩 ℚ(0, I).

The previous equation can be solved using the recursive algorithm proposed by Ang and Piazzesi (2003). A faster computation can be obtained by using the algorithm described in Borgy et al. (2011). The resulting price is of the form:

(20)Ps(t,h)=exp(−hAh(s)−hBhst).

C Computation of the likelihood

Multiplying both sides of Equation (13) by (Λ′sΛs)−1Λ′s, I obtain:

(Λ′sΛs)−1Λ′sRt=(Λ′sΛs)−1Λ′sΛzzt+st+(Λ′sΛs)−1Λ′sηt.

The assumption according to which st=(Λ′sΛs)−1Λ′s(Rt−Λzzt), which reflects the fact the linear combination of yields (Λ′sΛs)−1Λ′sRt is priced without error, implies that the last term in the previous equation is null, that is: Λ′sηt=0. This notably implies that the elements of η_t can then not be independent. However, the elements of a subset of M–1 of them can be independent. Without loss of generality, I assume that the last M–1 elements of η_t are i.i.d. 𝒩(0, σR2).

For a given regime vector z_t, there is the same information in R_t as in {st, R˜t}, where R˜t is any subvector of R_t containing M–1 yields of distinct maturities. My intermediary objective is to compute the likelihood associated to {st, R˜t}t=1, …, T. Without loss of generality, I assume that R˜t=[y(t,h2), …, y(t,hM)]′. Denoting by Γ the (M–1)×M matrix that selects the last M–1 entries of an M×1 vector, I have R˜t=ΓRt. Substituting for s_t in Equation (14) and rearranging the terms in this equation leads to:

ηt=(Id−Λ)Rt−(Id−Λ)Λm, tzm, t,

where Λ=Λs(Λ′sΛs)−1Λ′s. Multiplying both sides of the previous equation by Γ gives:

(21)η˜t=Γ(Id−Λ)Rt−Γ(Id−Λ)Λm,tzm,t,

where η˜t≡Γηt is the vector containing the last M–1 (i.i.d.) elements of η_t.

Besides, let me rewrite the second equation of System (6) after substituting for s_t (≡s_2,t) and s_t–1 (≡s_2,t–1):

(22)ε2,t=1σs(Λ′sΛs)−1Λ′s[(Rt−ρ2Rt−1)−Λz(zt−ρ2zt−1)],

where ε_2,t~𝒩(0, 1) (this is the second element of ε_t). The system made of Equations (21) and (22) involves a set of independent Gaussian shocks {ε_2,t, η_2,t, …, η_M,t} as well as the (unobserved) regime variable z_m,t. The computation of the log-likelihood associated with the previous system of equations is obtained by applying the Kitagawa-Hamilton filter (see e.g. Hamilton (1994), Chapter 22). However, this likelihood is the one associated with the vector {st, R˜t}t=1, …, T, while I need to maximize the one associated with actually observed data {R_t}_{t=1, …, T}. The latter likelihood is obtained by multiplying the former by the determinant of the Jocobian resulting from this change in variables, that is |∂[st, R˜′t]/∂Rt|=1Λ′sΛsΛs,1 where Λ_s,1 is the first entry of Λ_s.

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Supplemental Material:

The online version of this article (DOI: 10.1515/snde-2016-0043) offers supplementary material, available to authorized users.

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Supplemental_Data_and_Code

A model of the euro-area yield curve with discrete policy rates

Abstract

Acknowledgments

Appendix

A Laplace transform of a Markov-switching process

B Pricing formulas

B.1 Computation of P_z(t,h)

B.2 Computation of P_ξ(t,h)

B.3 Computation of P_s(t,h)

C Computation of the likelihood

References

Supplemental Material:

Articles in the same Issue

Articles in the same Issue

A model of the euro-area yield curve with discrete policy rates

Article

Abstract

Acknowledgments

Appendix

A Laplace transform of a Markov-switching process

B Pricing formulas

B.1 Computation of Pz(t,h)

B.2 Computation of Pξ(t,h)

B.3 Computation of Ps(t,h)

C Computation of the likelihood

References

Supplemental Material:

Supplementary Material

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

B.1 Computation of P_z(t,h)

B.2 Computation of P_ξ(t,h)

B.3 Computation of P_s(t,h)