What model for the target rate

Bruno Feunou; Jean-Sébastien Fontaine; Jianjian Jin

doi:10.1515/snde-2019-0005

Abstract

The Federal Reserve target rate has a lower bound. Changes to the target rate occur with discrete increments. Using out-of-sample forecasts of the target rate, we evaluate models incorporating these two realistic non-linear features. Incorporating these features mitigates in-sample over-fitting and improves out-of-sample forecast accuracy of the target rate level and volatility. A model with these features performs better relative to the linear models because (i) it produces stronger responses of forecasts to inflation and unemployment and a weaker response to lagged target rate, and because (ii) it yields very different forecast distributions when the target rate is close to the lower bound.

Keywords: financial markets; interest rates

JEL Classification: E43

Acknowledgement

We thank Antonio Diez, Geoffrey Dunbar, Jonathan Witmer, Bruce Mizrach (the editor), and an anonymous referee for comments and suggestions. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Bank of Canada and the British Columbia Investment Management Corporation.

A Appendix

A.1 Moment-Generating Functions for rt+1∗

A.1.1Et[exp⁡(urt+1∗)]

The one-step-ahead conditional characteristic function of rt+1∗ is given by:

(17)ψt(u)≡Et[exp⁡(urt+1∗)]=exp⁡((ωt−1+β⊤Et[Yt+1])u+(σr2+β⊤ΣtΣt⊤β)2u2),

for u a real or complex scalar. Note that the partial derivatives ψt⊤(u) and ψt⊤⊤(u) are given by:

(18)ψt⊤(u)=(ωt−1+β⊤Et[Yt+1]+(σt2+β⊤ΣtΣt⊤β)u)ψt(u)

(19)ψt⊤⊤(u)=[(σr2+β⊤ΣtΣt⊤β)+(ωt−1+β⊤Et[Yt+1]+(σr2+β⊤ΣtΣt⊤β)u)2]ψt(u).

A.1.2Et[exp⁡(art+1∗)1[rt+1∗≤x]]

We are also interested in Et[rt+1∗1[rt+1∗≤x]] and Et[(rt+1∗)21[rt+1∗≤x]] to forecast the level and variance of the target rate one-period ahead in the B-Linear model. Define φt(a;x)≡Et[exp⁡(art+1∗)1[rt+1∗≤x]] the truncated generating function, with a scalar. Then, using result in Duffie, Pan, and Singleton (2000), we have

(20)φt(a;x)=ψt(a)2−1π∫0∞Im(ψt(a+iv)e−ivx)vdv.

The partial derivatives with respect to the first argument are as follows:

φt⊤(a;x)=Et[rt+1∗exp⁡(art+1∗)1[rt+1∗≤x]]φt⊤⊤(a;x)=Et[(rt+1∗)2exp⁡(art+1∗)1[rt+1∗≤x]],

This leads to to following solution:

(21)Et[rt+1∗1[rt+1∗≤x]]=φt⊤(0;x)=ωt−1+β⊤Et[Yt+1]2−1π∫0∞Im(ψt⊤(iv)e−ivx)vdv

(22)Et[(rt+1∗)21[rt+1∗≤x]]=φt⊤⊤(0;x)=σr2+β⊤ΣtΣt⊤β+(ωt−1+β⊤Et[Yt+1])22−1π∫0∞Im(ψt⊤⊤(iv)e−ivx)vdv.

A.1.3Et[exp⁡(umax(rt+1∗,0))]

In the B-Linear model, the conditional moment-generating function Et[exp⁡(umax(rt+1∗,0))] is given by:

Et[exp⁡(umax(rt+1∗,0))]=Et[exp⁡(umax(rt+1,0))1rt+1∗>0]+Et[exp⁡(umax(rt+1,0))1rt+1∗≤0]=Et[exp⁡(urt+1∗)(1−1rt+1≤0)]+Pt(rt+1∗≤0),

leading to the following closed-form solution:

(23)Et[exp⁡(umax(rt+1∗,0))]=ψt(u)−φt(u;0)+Φ(−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β).

In particular, evaluating the partial derivatives at u = 0:

(24)Et[max(rt+1∗,0)]=ψt⊤(0)−φt⊤(0;0)

(25)Et[max(rt+1∗,0)2]=ψt⊤⊤(0)−φt⊤⊤(0;0).

A.2 Density and Probability Distribution

We derive the density ft(rt+1) for the Linear, B-Linear and Square models. In each case, we start with the computation of ft(rt+1|Yt+1) and then derive ft(rt+1). Similarly, we derive the probability distribution function P_t(n) for the Ordered and B-Ordered model.

A.2.1 Linear

In the Linear model,

ft(rt+1|Yt+1)=1σrϕ(rt+1−(ωt−1+β⊤Yt+1)σr),

and

ft(rt+1)=1σ2+β⊤ΣΣ⊤βϕ(rt+1−(ωt−1+β⊤Et[Yt+1])σ2+β⊤ΣΣ⊤β).

A.2.2 B-Linear

In the B-Linear model,

ft(rt+1|Yt+1)=1σϕ(rt+1−(ωt−1+β⊤Yt+1)σ)1[rt+1>0]+Φ(rt+1−(ωt−1+β⊤Yt+1)σ)1[rt+1=0],

and

ft(rt+1|Yt+1)=1σ2+β⊤ΣΣ⊤βϕ(rt+1−(ωt−1+β⊤Et[Yt+1])σ2+β⊤ΣΣ⊤β)1[rt+1>0]+Φ(rt+1−(ωt−1+β⊤Et[Yt+1])σ2+β⊤ΣΣ⊤β)1[rt+1=0].

A.2.3 Square

In the Square model,

ft(rt+1|Yt+1)=12σrt+1[ϕ(rt+1−(ωt−1+β⊤Yt+1)σ)+ϕ(rt+1+(ωt−1+β⊤Yt+1)σ)]1[rt+1>0],

and

ft(rt+1)=12σ2+β⊤ΣΣ⊤βrt+1[ϕ(rt+1−(ωt−1+β⊤Et[Yt+1])σ2+β⊤ΣΣ⊤β)+ϕ(rt+1+(ωt−1+β⊤Et[Yt+1])σ2+β⊤ΣΣ⊤β)]1[rt+1>0].

A.2.4 Ordered and B-Ordered

For the Ordered models, the mapping from the latent rt+1∗ to the observed target rate r_t+1 works via 2. This implies that the conditional probability distribution for r_t+1 collapses to the conditional probability distribution for n:

Pt(n)≡Pt(rt+1=rt+0.25n)=Pt(rt+0.25n<rt+1∗≤rt+0.25(n+1)),

as in 9. First,

Pt(n|Yt+1)={Φ(rt+(\underline{n}+1)c−(ωt−1+β⊤Yt+1)σ)forn=\underline{n}Φ(rt+(n+1)c−(ωt−1+β⊤Yt+1)σ)−Φ(rt+nc−(ωt−1+β⊤Yt+1)σ)for\bn<n<n¯Φ((ωt−1+β⊤Yt+1)−(rt+n¯c)σ)forn=n¯

, which implies

Pt(n)={Φ(rt+(\underline{n}+1)c−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)forn=\bnΦ(rt+(n+1)c−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)−Φ(rt+nc−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)for\bn<n<n¯Φ((ωt−1+β⊤Et[Yt+1])−(rt+n¯c)σr2+β⊤ΣtΣt⊤β)forn=n¯.

A.3 Conditional variance or r_t+1

A.3.1 Linear

In the Linear model, the conditional variance of r_t+1 is given directly by

Vart[rt+1]=β⊤ΣtΣt⊤β+σr2.

A.3.2 B-linear

Then, the conditional variance of r_t+1 can be computed from Var(x)=Ex2−(Ex)2. Using 24 and 25:

Et[rt+1]=ωt−1+β⊤Et[Yt+1]2+1π∫0∞Im(ψt⊤(iv))vdv

Et[rt+12]=Et[(rt+1∗)2]−ψt⊤⊤(0)2+1π∫0∞Im(ψt⊤⊤(iv))vdv=Et[(rt+1∗)2]2+1π∫0∞Im(ψt⊤⊤(iv))vdv=σr2+β⊤ΣtΣt⊤β+(ωt−1+β⊤Et[Yt+1])22+1π∫0∞Im(ψt⊤⊤(iv))vdv.

A.3.3 Square

In the Square model, we use standard results:

Vart[rt+1]=Vart[(rt+1∗)2]=Vart[rt+1∗]2Vart[(rt+1∗−Et[rt+1∗]Vart[rt+1∗]+Et[rt+1∗]Vart[rt+1∗])2]=2(β⊤ΣtΣt⊤β+σr2)2(1+2Et[rt+1∗]2Vart[rt+1∗])=2(β⊤ΣtΣt⊤β+σr2)2(1+2(ωt−1+β⊤Et[Yt+1])2σr2+β⊤ΣtΣt⊤β)=2(β⊤ΣtΣt⊤β+σr2)(σr2+β⊤ΣtΣt⊤β+2(ωt−1+β⊤Et[Yt+1])2).

A.3.4 Ordered

In the Ordered model, the conditional variance can be computed directly from its definition and the solution for P_t(n):

Vart(rt+1)=∑n(rt+0.25n)2Pt(n).

A.3.5 B-Ordered

In the B-Ordered model, the conditional variance can be computed directly from its definition and the solution for P_t(n):

Vart(rt+1)=∑n(max(rt+0.25n,0))2Pt(n).

A.4 Response coefficients

A.4.1 Linear

In the Linear model, the response coefficient is given by:

∂Et[rt+1]∂Yt=β∂Et[Yt+1]∂Yt,

and

∂Et[rt+1|Yt+1]∂rt=ρ.

A.4.2 Black linear

In the Black Linear model, the response coefficient is given by:

∂Et[rt+1]∂Yt=ωt−1+β⊤∂Et[Yt+1]∂Yt2+1π∫0∞Im((ωt−1+β⊤∂Et[Yt+1]∂Yt+(σr2+β⊤ΣtΣt⊤β)iv)ψt(iv))vdv+1π∫0∞Im((ωt−1+β⊤Et[Yt+1]+(σr2+β⊤ΣtΣt⊤β)iv)β⊤∂Et[Yt+1]∂Ytivψt(iv))vdv,

and

∂Et[rt+1|Yt+1]∂rt=[Φ(ωt−1+β⊤Yt+1σr)+2(ωt−1+β⊤Yt+1σr)ϕ(ωt−1+β⊤Yt+1σr)]ρ.

A.4.3 Square

In the Square model, the response coefficient is given by:

∂Et[rt+1]∂Yt=2(ωt−1+β⊤Et[Yt+1])β∂Et[Yt+1]∂Yt

and

∂Et[rt+1|Yt+1]∂rt=2(ωt−1+β⊤Yt+1)ρ.

A.4.4 Ordered

In the Ordered model, the response coefficient is given by:

∂Et[rt+1]∂Yt=−1σr2+β⊤ΣtΣt⊤β(rt+\underline{n}c)ϕ(rt+(\underline{n}+1)c−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)β∂Et[Yt+1]∂Yt−1σr2+β⊤ΣtΣt⊤β∑\underline{n}<n<n¯(rt+nc)[ϕ(rt+(n+1)c−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)−ϕ(rt+nc−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)]β∂Et[Yt+1]∂Yt+1σr2+β⊤ΣtΣt⊤β(rt+n¯c)ϕ((ωt−1+β⊤Et[Yt+1])−(rt+n¯c)σr2+β⊤ΣtΣt⊤β)β∂Et[Yt+1]∂Yt

and

∂Et[rt+1|Yt+1]∂rt=−1σr(rt+\underline{n}c)ϕ(rt+(\underline{n}+1)c−(ωt−1+β⊤Yt+1)σr)ρ−1σr∑\underline{n}<n<n¯(rt+nc)[ϕ(rt+(n+1)c−(ωt−1+β⊤Yt+1)σr)−ϕ(rt+nc−(ωt−1+β⊤Yt+1)σr)]ρ+1σr(rt+n¯c)ϕ((ωt−1+β⊤Yt+1)−(rt+n¯c)σr)ρ.

A.5 Cumulative probability distributions

We derive the cumulative probability distribution in each model. We repeatedly use the fact that

X∼N(X¯,α2)→E[Φ(X)]=Φ(X¯1+α2).

A.5.1 Linear

In the Linear model, for z∈R:

Pt[rt+1≤z]=Et[Pt[rt+1≤z|Yt+1]]=Et[Φ(z−(ωt−1+β⊤Yt+1)σr)]=Φ(z−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β).

A.5.2 B-linear

In the B-Linear model, for z∈R:

Pt[rt+1≤z|Yt+1]=Pt[max(ωt−1+β⊤Yt+1+σrεt+1,0)≤z|Yt+1]=1[z≥0]Φ(−(ωt−1+β⊤Yt+1)σr)+Pt[−(ωt−1+β⊤Yt+1)σr≤εt+1≤z−(ωt−1+β⊤Yt+1)σr|Yt+1]=1[z≥0]Φ(−(ωt−1+β⊤Yt+1)σr)+[Φ(z−(ωt−1+β⊤Yt+1)σr)−Φ(−(ωt−1+β⊤Yt+1)σr)]1[z≥0]=Φ(z−(ωt−1+β⊤Yt+1)σr)1[z≥0],

and therefore,

Pt[rt+1≤z]=Et[Φ(z−(ωt−1+β⊤Yt+1)σr)1[z≥0]]=Φ(z−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)1[z≥0].

A.5.3 Square

In the Square model, for z∈R:

and therefore:

Pt[rt+1≤z]=1[z≥0](Et[Φ(z−(ωt−1+β⊤Yt+1)σr)]−Et[Φ(−z−(ωt−1+β⊤Yt+1)σr)])=1[z≥0](Φ(z−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)−Φ(−z−(ωt−1+β⊤Et[Yt+1])σr2+β⊤ΣtΣt⊤β)).

A.5.4 Ordered and ordered B-linear

In the Ordered model, for z∈N:

Pt[rt+1≤z]=∑n=n_n=zPt(n),

and in the B-Ordered model:

Pt[rt+1≤z]=∑n=n_n=zPt(n)1z≥0.

A.6 Option prices

We can derive option prices using 10. We need a solution for Et[rt+11[rt+1≥z]] for each model. The solution Pt[rt+1≥z] is given in the previous section.

A.6.1 Linear

In the Linear model:

Et[rt+11[rt+1≥z]]=Et[rt+1∗1[rt+1∗≥z]]=Et[rt+1∗]−Et[rt+1∗1[rt+1∗≤z]]=ωt−1+β⊤Et[Yt+1]−Et[rt+1∗1[rt+1∗≤z]]=ωt−1+β⊤Et[Yt+1]−φt⊤(0,z).

A.6.2 B-linear

In the B-Linear model:

Et[rt+11[rt+1≥z]]=Et[max(rt+1∗,0)1[max(rt+1∗,0)≥z]]=Et[rt+1∗1[rt+1∗≥max(z,0)]]=Et[rt+1∗]−Et[rt+1∗1[rt+1∗≤max(z,0)]]=ωt−1+β⊤Et[Yt+1]−Et[rt+1∗1[rt+1∗≤max(z,0)]]=ωt−1+β⊤Et[Yt+1]−φt⊤(0,max(z,0)).

A.6.3 Square

In the Square model:

Et[rt+11[rt+1≥z]]=Et[(rt+1∗)21[(rt+1∗)2≥z]]=Et[(rt+1∗)21[|rt+1∗|≥z]]=Et[(rt+1∗)21[rt+1∗>z]]+Et[(rt+1∗)21[rt+1∗<−z]]=Et[(rt+1∗)2[1−1[rt+1∗<z]]]+Et[(rt+1∗)21[rt+1∗<−z]]=Et[(rt+1∗)2[1−1[rt+1∗<z]]]+Et[(rt+1∗)21[rt+1∗<−z]]=Et[(rt+1∗)2]+Et[(rt+1∗)21[rt+1∗<−z]]−Et[(rt+1∗)21[rt+1∗<z]],

where, from Section A.1.1:

Et[(rt+1∗)2]=σr2+β⊤ΣtΣt⊤β+(ωt−1+β⊤Et[Yt+1])2,

and from Section A.1.2:

Et[rt+1∗1[rt+1∗≤x]]=φt⊤(0;x)Et[(rt+1∗)21[rt+1∗≤x]]=φt⊤⊤(0;x).

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Supplementary Material

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Published Online: 2020-01-23

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Supplementary Material Details

What model for the target rate

Abstract

Acknowledgement

A Appendix

A.1 Moment-Generating Functions for rt+1∗

A.2 Density and Probability Distribution

A.2.1 Linear

A.2.2 B-Linear

A.2.3 Square

A.2.4 Ordered and B-Ordered

A.3 Conditional variance or r_t+1

A.3.1 Linear

A.3.2 B-linear

A.3.3 Square

A.3.4 Ordered

A.3.5 B-Ordered

A.4 Response coefficients

A.4.1 Linear

A.4.2 Black linear

A.4.3 Square

A.4.4 Ordered

A.5 Cumulative probability distributions

A.5.1 Linear

A.5.2 B-linear

A.5.3 Square

A.5.4 Ordered and ordered B-linear

A.6 Option prices

A.6.1 Linear

A.6.2 B-linear

A.6.3 Square

References

Supplementary Material

Articles in the same Issue

Articles in the same Issue

What model for the target rate

Article

Abstract

Acknowledgement

A Appendix

A.1 Moment-Generating Functions for rt+1∗

A.2 Density and Probability Distribution

A.2.1 Linear

A.2.2 B-Linear

A.2.3 Square

A.2.4 Ordered and B-Ordered

A.3 Conditional variance or rt+1

A.3.1 Linear

A.3.2 B-linear

A.3.3 Square

A.3.4 Ordered

A.3.5 B-Ordered

A.4 Response coefficients

A.4.1 Linear

A.4.2 Black linear

A.4.3 Square

A.4.4 Ordered

A.5 Cumulative probability distributions

A.5.1 Linear

A.5.2 B-linear

A.5.3 Square

A.5.4 Ordered and ordered B-linear

A.6 Option prices

A.6.1 Linear

A.6.2 B-linear

A.6.3 Square

References

Supplementary Material

Supplementary Material

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

A.3 Conditional variance or r_t+1