inflation expectations and risk premiums in an arbitrage-free

Inflation Expectations and Risk Premiumsin an Arbitrage-Free Model ofNominal and Real Bond Yields

Jens H. E. ChristensenJose A. Lopez

Glenn D. Rudebusch

Federal Reserve Bank of San Francisco

Term Structure Modeling and the Lower Bound ProblemLecture III.4 (a)

European University InstituteFlorence, September 9, 2015

The views expressed here are solely the responsibility of the authors and should not be interpreted as reflecting the views

of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. 1 / 77

Motivation

Inflation expectations are important because:

They impact the price setting behavior of firms andconsumers.They can be used to gauge the credibility of monetarypolicy.

Approaches to extracting inflation expectations:

Historical inflation data: Inherently backward looking.Survey data: Forward looking, but infrequent and fewtime horizons.Bond yields: Forward looking, available at dailyfrequency and for a wide range of maturities.

This motivates our focus on bond yields for the extraction ofinflation expectations.

2 / 77

The Fisher Equation

The Fisher Equation states that

yNt (τ) = yR

t (τ) + πet (τ),

orBEIt(τ) ≡ yN

t (τ)− yRt (τ) = πe

t (τ).

Thus, the difference between nominal and real yields of thesame maturity, also known as the breakeven inflation rate(BEI), should equal the expected inflation.

Problem: No room for any inflation risk premium.

Thus, we need a model framework to decompose BEI intoexpected inflation and an inflation risk premium.

3 / 77

Market-Implied Price Level Dynamics

Standard assumption in finance: Nominal and real stochasticdiscount factors exist

dMNt /M

Nt = −rN

t dt − Γ′tdW Pt ,

dMRt /M

Rt = −rR

t dt − Γ′tdW Pt ,

such that zero-coupon bond prices are given by

PNt (τ) = EP

t

[MN

t+τ

MNt

]and PR

t (τ) = EPt

[MR

t+τ

MRt

].

Since

MNt denotes NPV in t0 dollars of $1 at time t and

MRt denotes NPV in t0 dollars per goods basket at t ,

the ratio

Πt =MR

t

MNt

defines a price level as the number of dollars attime t per goods basket at time t .

4 / 77

Breakeven Decomposition (1)

Nominal zero-coupon bond prices can then be rewritten as:

PNt (τ) = EP

[MN

t+τ

MNt

]

= EP

[MR

t+τ/Πt+τ

MRt /Πt

]= EP

[MR

t+τ

MRt

Πt

Πt+τ

]

= EP

[MR

t+τ

MRt

]× EP

[Πt

Πt+τ

]×(

1 +cov

[MR

t+τ

MRt, ΠtΠt+τ

]

EP[

MRt+τ

MRt

]× EP

[Πt

Πt+τ

])

Now, we recall that the yield of a zero-coupon bond is

PNt (τ) = e−yN

t (τ)τ ⇐⇒ yNt (τ) = −1

τln PN

t (τ).

Applying this to the equation above, it can be converted intoa yield-to-maturity expression ...

5 / 77


... so that breakeven inflation can be written as

BEIt(τ) ≡ yNt (τ)− yR

t (τ) = πet (τ) + φt(τ),

where the nominal and real zero-coupon yields are

yNt (τ) = −1

τln EP

[MN

t+τ

MNt

]and yR

t (τ) = −1τ

ln EP

[MR

t+τ

MRt

],

the market-implied expected rate of inflation from t to t + τ is

πet (τ) = −1

τln EP

[Πt

Πt+τ

]= −1

τln EP

t

[e−

∫ t+τt (rN

s −rRs )ds

],

and the corresponding inflation risk premium is given by

φt(τ) = −1τ

ln

(1 +

covPt

[MR

t+τ

MRt, ΠtΠt+τ

]

EPt

[MR

t+τ

MRt

]× EP

t

[Πt

Πt+τ

]).

Note: P-dynamics are key in calculating πet (τ) and φt(τ). 6 / 77


Bottom line: With a minimum of assumptions we haveobtained a decomposition of the breakeven inflation


t (τ) = πet (τ) + φt(τ).

Note: The inflation risk premium, φt(τ), is positive if and onlyif

cov[MR

t+τ

MRt

,Πt

Πt+τ

]< 0.

This happens when the real discount factor/marginal utilitytends to be high at the same time as Πt

Πt+τis low, i.e., πe

t (τ) ishigh and purchasing power is low.

Plainly: Downturns should tend to coincide with high inflation.

Goal: A model of rNt and rR

t that fits nominal and real yieldswell, while accurately capturing the P-dynamics of the riskfactors to produce good estimates of πe

t (τ). 7 / 77

The Arbitrage-Free Nelson-Siegel Models

Proposition: If the nominal risk-free rate is defined by

rt = Lt + St

and the Q-dynamics of Xt = (Lt ,St ,Ct) are given by

dLt

dSt

dCt

=

0 0 00 λ −λ0 0 λ

θQ1θQ

2θQ

3

−

Lt

St

Ct

dt +ΣdW Q

t ,

where Σ is a constant matrix, then zero-coupon yields have theNelson-Siegel factor structure:

yt(τ) = Lt +(1 − e−λτ

λτ

)St +

(1 − e−λτ

λτ− e−λτ

)Ct −

A(τ)τ

.

This defines the AFNS model class.

The yield-adjustment term, A(τ)/τ , ensures absence ofarbitrage and is a deterministic function of λ, θQ, and Σ.

Nice analytical pricing formulas to work with. 8 / 77

The Risk Premium Specification

If we combine arbitrage-free Nelson-Siegel models definedunder the Q-measure with the essentially affine risk premiumspecification (see Duffee (2002))

Γt = γ0 + γ1Xt ,

we preserve the affine dynamics under the P-measure

dXt = K P(θP − Xt)dt + ΣdW Pt .

In order to econometrically identify the models we fix θQ at 0,which is without loss of generality.

Furthermore, building on the insights from CDR, we limit ourfocus to models with diagonal Σ.

9 / 77

Nominal Bond Yields

For the nominal bond yields, we use the off-the-runzero-coupon Treasury bond yields made publicly available inGürkaynak, Sack, and Wright (2007).

We choose to work at a weekly frequency (Fridays) with asample covering the period from January 6, 1995 to March28, 2008 and containing 8 different constant maturities (inyears)

{0.25, 0.5, 1, 2, 3, 5, 7, 10}.

What is the relevance of the Nelson-Siegel model formodeling these yields?

10 / 77

Principal Component Analysis of Nominal Yields

Maturity First Second Thirdin years P.C. P.C. P.C.

0.25 -0.42 -0.42 0.520.5 -0.43 -0.35 0.161 -0.43 -0.20 -0.252 -0.39 0.05 -0.473 -0.35 0.21 -0.375 -0.29 0.37 -0.047 -0.24 0.45 0.23

10 -0.19 0.51 0.47% explained 94.17 5.50 0.30

Three factors explain 99.9% of the variation.

Clear pattern of level, slope, and curvature in the style ofthe Nelson-Siegel model.

11 / 77

Real Bond Yields

For the real yields, we use yields on TIPS bonds.

TIPS coupons are fixed at issuance. However, as timepasses, they are adjusted with the change in the headline,non-seasonally adjusted CPI - with about a three-month lag.

Gürkaynak, Sack, and Wright (2010) extract zero-couponreal yield curves from TIPS prices daily since Jan. 1999.

To minimize effects from poor liquidity in the early years ofTIPS trading, our sample runs from 1/3/2003 to 3/28/2008.

To minimize effects from lagged indexation, we focus on thelong end of the TIPS curve and use the following maturities:

{5, 6, 7, 8, 9, 10}.

Again, we may ask what the relevance of the Nelson-Siegelmodel is in terms of modeling TIPS yields?

12 / 77

Principal Component Analysis of Real Yields

Maturity First Second Thirdin years P.C. P.C. P.C.

5 -0.54 0.61 0.516 -0.47 0.22 -0.357 -0.41 -0.06 -0.508 -0.36 -0.27 -0.289 -0.33 -0.43 0.11

10 -0.29 -0.56 0.53% explained 97.66 2.32 0.02

Three factors explain 99.9% of the variation.

Clear pattern of level, slope, and curvature in the style ofthe Nelson-Siegel model.

Note, however, the very limited role of the third factor.13 / 77

Step 1: Separate Estimations

First, we make separate estimations using nominal and realyields only.

The results show that:

For nominal yields, a three-factor model with level, slope,and curvature is well-specified.

For real yields, a two-factor model with a level and aslope factor is sufficient - likely caused by the limitedmaturity range we consider.

Important observations:

The nominal and real level factor are highly correlatedwith a correlation coefficient of ρ = 0.90.

The same holds for the two slope factors with acorrelation of ρ = 0.92.

14 / 77

Nominal and Real Risk Factors

Estimated nominal and real level and slope factors

2003 2004 2005 2006 2007 2008

0.02

0.03

0.04

0.05

0.06

0.07

Est

imat

ed fa

ctor

val

ue

Nominal level factor Real level factor

2003 2004 2005 2006 2007 2008−

0.06

−0.

04−

0.02

0.00

0.02

Est

imat

ed fa

ctor

val

ue

Nominal slope factor Real slope factor

The strong connection between each pair of nominal andreal factors motivates the structure of our joint models fornominal and real yields.

15 / 77

Step 2: A Joint Three-Factor Model

The instantaneous nominal risk-free rate is assumed to equal

rNt = LN

t + SNt .

The instantaneous real risk-free rate combines scaledversions of the nominal level and slope factor

rRt = αR

L LNt + αR

S SNt .

Now, nominal and real zero-coupon yields are given by

yNt (τ) = LN

t +(1 − e−λτ

λτ

)SN

t +(1 − e−λτ

λτ− e−λτ

)CN

t +AN(τ)

τ,

yRt (τ) = αR

L LNt + αR

S

(1 − e−λτ

λτ

)SN

t + αRS

(1 − e−λτ

λτ− e−λτ

)CN

t +AR(τ)

τ.

Problem: Too strict, poor fit of shorter end of both curves.

Consequence: In order to simultaneously fit both yield curvessatisfactorily, we introduce a fourth factor.

16 / 77

Step 3: A Joint Four-Factor Model

We allow for separate level factors in nominal and real yields,but maintain the common slope and curvature factor:

rNt = LN

t + St ,

rRt = LR

t + αRSt .

Now, nominal and real zero-coupon yields are given by

yNt (τ) = LN

t +(1 − e−λτ

λτ

)St +

(1 − e−λτ

λτ− e−λτ

)Ct +

AN(τ)

τ,

yRt (τ) = LR

t + αR(1 − e−λτ

λτ

)St + αR

(1 − e−λτ

λτ− e−λτ

)Ct +

AR(τ)

τ.

Level factors represent permanent components that mayvary across nominal and real yields.

Slope and curvature reflect monetary policy andexpectations about future monetary policy, respectively.For that reason, they impact both nominal and real yields.

17 / 77

Model Fit

Maturity Three-Factor Model Four-Factor Model

Nom. yields Mean RMSE Mean RMSE0.25 -4.59 43.54 -0.26 10.380.5 -4.58 30.46 0.00 0.001 -2.41 15.24 1.76 6.182 0.63 2.74 2.30 4.153 0.80 2.91 0.00 0.005 -0.56 2.33 -2.88 3.817 -0.31 0.99 -0.08 2.8510 0.00 0.00 10.04 11.53

TIPS yields Mean RMSE Mean RMSE5 -24.91 28.34 -4.04 10.196 -15.42 18.25 -2.23 6.537 -8.51 10.56 -0.99 3.198 -3.53 4.63 0.00 0.009 0.00 0.00 0.77 2.9410 2.43 3.61 1.36 5.54

18 / 77

The Estimated P-Dynamics (1)

As noted earlier, the P-dynamics of state variables arecritical to getting accurate estimates of the market-impliedexpected inflation.

To find the appropriate P-dynamics, we use ageneral-to-specific strategy starting from the unrestrictedK P-matrix:

K P =

κP11 κP

12 κP13 κP

14κP

21 κP22 κP

23 κP24

κP31 κP

32 κP33 κP

34κP

41 κP42 κP

43 κP44

.

In each step, we eliminate the parameter with the lowestt-statistic.

We use the Akaike Information Criterion, the BayesianInformation Criterion, and marginal likelihood ratio tests tofind the optimal stopping point.

19 / 77


Alternative Goodness of fit statisticsspecifications log L k p-value AIC BIC(1) Free K P 42,354.0 40 n.a -84,628.1 -84,446.5(2) κP

31 = 0 42,354.0 39 0.89 -84,630.0 -84,453.1(3) κP

32 = 0 42,353.9 38 0.69 -84,631.9 -84,459.4(4) κP

13 = 0 42,353.9 37 0.69 -84,633.7 -84,465.8(5) κP

34 = 0 42,353.6 36 0.46 -84,635.2 -84,471.8(6) κP

12 = 0 42,353.2 35 0.39 -84,636.4 -84,477.6(7) κP

24 = 0 42,352.7 34 0.31 -84,637.4 -84,483.1(8) κP

43 = 0 42,352.5 33 0.50 -84,639.0 -84,489.2(9) κP

41 = 0 42,350.7 32 0.06 -84,637.5 -84,492.3(10) κP

42 = 0 42,345.9 31 < 0.01 -84,629.8 -84,489.1(11) κP

21 = 0 42,335.4 30 < 0.01 -84,610.9 -84,474.7(12) κP

14 = 0 42,326.5 29 < 0.01 -84,595.0 -84,463.4(13) κP

23 = 0 42,303.3 28 < 0.01 -84,550.7 -84,423.6

20 / 77


The preferred specification has P-dynamics given by

dLNt

dSt

dCt

dLRt

=

κP11 0 0 κP

14κP

21 κP22 κP

23 00 0 κP

33 0κP

41 κP42 0 κP

44

θPLN

θPSθP

CθP

LR

−

LNt

St

Ct

LRt

dt+ΣdW P

t .

There are important interactions between the nominaland real level factors through κP

14 and κP41.

The slope factor has feedback effects from the nominallevel factor (κP

21) and generates feedback effects to thereal level factor (κP

42).

The curvature factor is not impacted by any of the otherfactors, but generates a feedback effect to the slopefactor (κP

23).21 / 77

Model-Implied Inflation Expectations

Five-year (left) and ten-year (right) expected inflation.

2003 2004 2005 2006 2007 2008

1.0

1.5

2.0

2.5

3.0

Rat

e in

per

cent

Preferred K matrix Full K matrix Diagonal K matrix Survey−based inflation forecast

2003 2004 2005 2006 2007 2008

1.0

1.5

2.0

2.5

3.0

Rat

e in

per

cent

Preferred K matrix Full K matrix Diagonal K matrix Survey−based inflation forecast

Reasonable model-implied expected inflation from thepreferred K P-specification relative to survey forecasts(Blue Chip at 5-year horizon, SPF at 10-year horizon).

To us, this provides further evidence that we have anappropriate specification of the P-dynamics. 22 / 77

Model-Implied Inflation Risk Premiums

2003 2004 2005 2006 2007 2008

−1.

0−

0.5

0.0

0.5

1.0

Rat

e in

per

cent

Ten−year IRP Five−year IRP

Small, +/- 50 basis points at the 5- and 10-year horizons.Volatile and frequently negative.

Potential explanations:1 Low liquidity of TIPS ⇒ High real yields and reduced BEI

⇒ Negative IRP (given unchanged infl. exp.).2 Headline CPI may not be the inflation index of the

marginal investor ⇒ Premium for exposure to this risk. 23 / 77

Conclusion

We introduce a joint four-factor model for nominal andreal U.S. Treasury yields.

We demonstrate the model’s ability to fit both yieldcurves simultaneously.

We conduct an extended search to find the mostappropriate specification of the model’s P-dynamics.

Based on the preferred model specification, wedecompose breakeven inflation into market-impliedexpected inflation and an inflation risk premium.

The derived inflation expectations align well with thesurvey forecasts at comparable horizons, while theinflation risk premium is found to be relatively small, andeven frequently negative.

24 / 77

Pricing Deflation Riskwith U.S. Treasury Yields

Jens H. E. ChristensenJose A. Lopez

Glenn D. Rudebusch

Federal Reserve Bank of San Francisco

Term Structure Modeling and the Lower Bound ProblemLecture III.4 (b)

European University InstituteFlorence, September 9, 2015

The views expressed here are solely the responsibility of the authors and should not be interpreted as reflecting the views

of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.

25 / 77

Motivation: Deflation Risk in TIPS Bonds

Features of TIPS (or real Treasury) bonds:

Coupon and principal payments indexed to headline CPI.Provide protection against deflation as the principal is notpermitted to fall below par.

The deflation protection option:

Has received limited attention in the literature, CLR(2012) and Grishchenko et al. (2013) are examples.Has been of relatively little value in the generallyinflationary environment in the U.S. since 1997.However, CLR (2012) find that deflationary concernsappeared to be significant during the financial crisis.

Goal of this paper:

Develop an arbitrage-free term structure model that canaccurately price the deflation protection option in TIPS.Demonstrate how to modify and validate AFNS models. 26 / 77

Overview: The CV and SV Models

Our approach:

1 Start from the CLR (2010) model of nominal and realTreasury yields characterized by

Four-factor arbitrage-free term structure model;

Separate nominal and real yield level factors;

Slope and curvature factors are common to both curves;

Assumes constant volatility (hence CV).

2 Then incorporate spanned stochastic volatility (SV) —the two level factors drive their own volatility dynamics.

The two models’ conditional mean dynamics are quite similaras could be expected.

More importantly, the SV model more closely matches theobserved proxy for the deflation protection option.

27 / 77

CLR (2010) Modeling Framework (1)

CLR (2010) introduce a joint, four-factor arbitrage-free model ofnominal and real Treasury yields centered around thearbitrage-free Nelson-Siegel (AFNS) models introduced inChristensen, Diebold, and Rudebusch (2011).

The CLR model has four factors Xt = (LNt ,St ,Ct ,LR

t ).

The instantaneous nominal and real risk-free rates are defined as

rNt = LN

t + St ,

rRt = LR

t + αRSt ,

while the risk-neutral factor dynamics are assumed given by

dLNt

dSt

dCt

dLRt

=

0 0 0 00 λ −λ 00 0 λ 00 0 0 0

θQLN

θQSθQ

CθQ

LR

−

LNt

St

Ct

LRt

dt+ΣdW Q

t .

For identification, and without loss of generality, we fix θQ = 0. 28 / 77

CLR (2010) Modeling Framework (2)

Nominal Treasury zero-coupon bond yields are modeled as

yNt (τ) = LN

t +

(1 − e−λτ

λτ

)St +

(1 − e−λτ

λτ− e−λτ

)Ct −

AN(τ)

τ.

The real TIPS zero-coupon bond yields are modeled as

yRt (τ) = LR

t + αR(

1 − e−λτ

λτ

)St+ αR

(1 − e−λτ

λτ− e−λτ

)Ct−

AR(τ)

τ.

Note: The Nelson-Siegel factor loading structure is preserved forboth yield curves.

These two equations when combined in state-space formconstitute the measurement equation used in the Kalman filter.

We link the risk-neutral and real-world yield dynamics using theessentially affine risk premium specification (Duffee 2002).

This framework is the basis for our four-factor Gaussian model thatwe refer to as the CV model .

29 / 77

CLR Model with Stochastic Volatility (1)

We replace the constant volatility assumption with aspecification of stochastic volatility as per CLR (2014a):

dLNt

dSt

dCt

dLRt

=

κQLN 0 0 0

0 λ −λ 00 0 λ 0

0 0 0 κQLR

θQLN

00θQ

LR

−

LNt

St

Ct

LRt

dt

+

σ11 0 0 00 σ22 0 00 0 σ33 00 0 0 σ44

√LN

t 0 0 0

0√

1 0 00 0

√1 0

0 0 0√

LRt

dW LN ,Qt

dW S,Qt

dW C,Qt

dW LR ,Qt

.

30 / 77


The nominal zero-coupon bond yield function becomes

yNt (τ) = gN

(κQ

LN

)LN

t +

(1 − e−λτ

λτ

)St +

(1 − e−λτ

λτ− e−λτ

)Ct −

AN(τ)

τ.

The real zero-coupon bond yield function becomes

yRt (τ) = gR

(κQ

LR

)LR

t +αR

(1 − e−λτ

λτ

)St+α

R

(1 − e−λτ

λτ−e−λτ

)Ct−

AR(τ)

τ.

The slope and the curvature factors preserve their CVmodel factor loadings exactly.

By fixing κQLN = κQ

LR = 10−7, the two level factorsapproximate their CV model factor loadings closely.

31 / 77


To complete the model, we use the extended affine riskpremium specification of Cheridito et al. (2007). Under theP-dynamics,

dLNt

dSt

dCt

dLRt

=

κP11 0 0 κP

14

κP21 κP

22 κP23 κP

24

κP31 κP

32 κP33 κP

34

κP41 0 0 κP

44

θP1

θP2

θP3

θP4

−

LNt

St

Ct

LRt

dt

+

σ11 0 0 00 σ22 0 00 0 σ33 00 0 0 σ44

√

LNt 0 0 0

0√

1 0 00 0

√

1 00 0 0

√

LRt

dW LN ,Pt

dW S,Pt

dW C,Pt

dW LR,Pt

.

To keep the model arbitrage-free, Feller conditionsensure strict positivity.

Since this model allows for stochastic yield volatility, werefer to it as the SV model.

32 / 77

Treasury Yield Data and Model Estimation

The nominal Treasury yields:

Gürkaynak, Sack, and Wright (2007) from FRBOG.Off-the-run, zero-coupon Treasury bond yields.Weekly data from Jan. 1995 through Dec. 2010.8 maturities: 0.25, 0.5, 1, 2, 3, 5, 7, 10 years.

The real Treasury yields:

Gürkaynak, Sack, and Wright (2010) from FRBOG.Zero-coupon yields derived from TIPS prices.Weekly data from Jan. 2003 through Dec. 2010.6 maturities: 5, 6, 7, 8, 9, 10 years.To limit effects from the lagged inflation indexation, weavoid the short end of the TIPS curve.

Estimation:

Both models are estimated with the standard Kalmanfilter.

33 / 77

Preferred Model Specifications

Using the Bayesian Information Criterion, we select apreferred specification of K P for each model.

CV model:

K Pcv =

κP11 0 0 0κP

21 κP22 κP

23 00 0 κP

33 0κP

41 κP42 0 κP

44

.

Note: κP14 is insignificant relative to CLR (2010, 2012).

SV model:

K Psv =

κP11 0 0 κP

14

0 κP22 κP

23 00 0 κP

33 00 0 0 κP

44

.

Note: κP21 and κP

41 are insignificant, while κP42 is not admissible.

34 / 77

Model FitMaturity

in monthsCV model SV model

Nom. yields Mean RMSE Mean RMSE3 -0.54 9.53 0.75 19.236 0.00 0.00 -0.17 8.2312 1.79 5.80 0.00 0.0024 2.21 3.98 0.46 1.5636 0.00 0.13 0.00 0.0060 -2.66 3.72 -0.28 1.2784 0.09 3.37 0.24 0.59

120 9.54 12.03 -1.15 4.41

TIPS yields Mean RMSE Mean RMSE60 -3.99 20.27 -2.04 13.5872 -2.61 12.23 -0.51 5.8784 -1.31 5.64 0.00 0.0096 0.00 0.00 -0.38 4.72

108 1.35 4.94 -1.52 8.74120 2.75 9.32 -3.32 12.34

Max log L 52,558.88 54,470.81

Overall, the fit to the nominal yields is of similar quality.However, better fit of the TIPS yields in the SV model . 35 / 77

Update

2000 2004 2008 2012 2016

−2

02

46

8

Rat

e in

per

cent

Correlation: 96.2%

Ten−year nominal yield Ten−year real yield

2000 2004 2008 2012 2016

−1

01

2

Rat

e in

per

cent

Correlation: 77.5%

Ten− over five−year nominal yield difference Ten− over five−year real yield difference

I extend the TIPS data sample back to January 8, 1999,and forward to August 14, 2015.

Separate level effects in nominal and real yields despitehigh correlation.

The slopes of the two yield curves continue to exhibit therequired pattern. 36 / 77

Model Validation

I perform weekly rolling real-time estimations of both modelsover the period from January 6, 2005 to August 14, 2015.

This allows me to study model performance:

Inflation expectations and forecast evaluation;

Inflation risk premiums;

Nominal and real yield forecasts;

Nominal and real term premiums;

Nominal and real yield volatility projections.

37 / 77

Calculation of Model-Implied Inflation Expectations

The model-implied expected inflation is given by

πet (τ) = −1

τln EP

t

[e−

∫ t+τt (rN

s −rRs )ds].

For the affine Gaussian CV model, it holds that

EPt

[e−

∫ t+τt (rN

s −rRs )ds]= exp(Bπ(τ)′Xt + Aπ(τ)),

where Bπ(τ) and Aπ(τ) are solutions to a system of ODEs:dBπ(τ)

dτ= −ρ1 − (K P)′Bπ(τ), Bπ(0) = 0,

dAπ(τ)

dτ= −ρ0 + Bπ(τ)′K PθP +

12

3∑

j=1

(Σ′Bπ(τ)Bπ(τ)′Σ

)j,j ,A

π(0) = 0.

K P , θP , and Σ reflect the P-dynamics, while ρ0 and ρ1 are:

ρ0 = 0 and ρ1 =

11 − αR

0−1

.

38 / 77

Calculation of Model-Implied Inflation Risk Premiums

Now, the ODEs for the model-implied expected inflation aresolved with a standard fourth order Runge-Kutta method.

Note that a similar approach is used to calculate themodel-implied inflation expectations for the SV model, butthe system of ODEs is slightly different.

For both models it holds that we use the BEI decomposition


t (τ) = πet (τ) + φt(τ),

to calculate the inflation risk premium, once we have thecorresponding expected inflation, πe

t (τ), and fitted orobserved nominal and real yields.

39 / 77

One-Year Inflation Expectations (1)

Rat

e in

per

cent

2005 2007 2009 2011 2013 2015

−2

−1

01

23

45

6

Rat

e in

per

cent

CV model SV model Blue Chip forecast CPI one−year inflation realizations

Rather similar first moment characteristics for the two models.Models’ πe

t (τ)s are more responsive than the Blue Chipforecasts. 40 / 77

One-Year Inflation Expectations (2)

2005 2007 2009 2011 2013 2015

−3

−2

−1

01

23

45

6

Rat

e in

per

cent

Rat

e in

per

cent

CV model SV model One−year inflation swap rate CPI one−year inflation realizations

Models’ πet (τ)s are notably less variable than the one-year

inflation swap rate since they adjust for the risk premium.41 / 77

Evaluation of One-Year Inflation Forecasts

Model Mean MAE RMSERandom Walk -28.76 176.21 227.75Blue Chip -7.28 118.19 155.51Inflation swap 30.89 144.78 197.96CV model 30.49 136.29 164.57SV model 43.13 136.29 168.42

The table reports summary statistics of monthly real-timeone-year inflation forecasts from January 2005 to July2014, a total of 115 forecasts.The forecast dates are mapped to the 10th of eachmonth (the Blue Chip release date).The CV and SV models have similar performance.Both models easily beat the random walk and theone-year inflation swap rate.The Blue Chip survey is tough to beat (so far)!

42 / 77

Two-Year Inflation Expectations

2005 2007 2009 2011 2013 2015

−3

−2

−1

01

23

45

6

Rat

e in

per

cent

Rat

e in

per

cent

CV model SV model Two−year inflation swap rate CPI two−year inflation realizations

At the two-year forecast, the model expectations remain close.

Note that the two-year inflation swap rate is more stable.43 / 77

Evaluation of Two-Year Inflation Forecasts

Model Mean MAE RMSERandom Walk -41.41 113.00 146.89Inflation swap 16.21 101.41 130.15CV model 16.01 114.51 132.66SV model 36.67 112.00 135.22

The table reports summary statistics of monthly real-timetwo-year inflation forecasts from January 2005 to July2013, a total of 103 forecasts.

Now, all forecasts are mapped to the end of each month.

Again, the CV and SV models have similar performance.

Now, the two-year inflation swap rate is less variable andperforms slightly better than the two models.

All three methods beat the random walk as measured byRMSE.

44 / 77

Five-Year Inflation Expectations

2005 2007 2009 2011 2013 2015

−2

−1

01

23

45

6

Rat

e in

per

cent

CV model SV model Five−year inflatin swap rate SPF five−year inflation forecast CPI five−year inflation realizations

The models’ five-year inflation forecasts are also very similar!

They are notably more variable than the SPF forecasts.45 / 77

Inflation Risk Premiums

2005 2007 2009 2011 2013 2015

−3

−2

−1

01

23

Rat

e in

per

cent

CV model SV model

2005 2007 2009 2011 2013 2015

−3

−2

−1

01

23

Rat

e in

per

cent

CV model SV model

Estimated inflation risk premiums are similar acrossmodels.

Note that inflation risk premiums can be negative forextended periods, in particular more recently.

46 / 77

Yield Forecasts

I generate yield forecasts 26 weeks and 52 weeks aheadbased on the rolling real-time estimation of bothmodels—528 and 502 forecast errors, respectively.

To do so, I first calculate the conditional mean of the statevariables at the relevant horizons:

EPt [XT ] = (I − e−K P (T−t))θP + e−K P (T−t)Xt .

Now, the forecasted yields are:

EPt [y

NT (τ)] = BN(τ)′EP

t [XT ] + AN(τ),

EPt [y

RT (τ)] = BR(τ)′EP

t [XT ] + AR(τ).

The benchmark in the comparison is the random walkassumption.

Also, for the nominal yields, I will compare the models’forecast performance to the B-CR model.

47 / 77

Nominal Yield Forecast Evaluation

26 weeks 52 weeksThree-month yieldMean MAE RMSE Mean MAE RMSE

Random walk -13.29 41.56 72.54 -33.05 76.02 124.41B-CR model -30.50 41.89 72.85 -59.31 80.50 121.50CV model -44.32 64.09 86.60 -96.74 109.74 147.99SV model -38.74 52.75 72.78 -81.55 100.29 126.53

26 weeks 52 weeksTwo-year yieldMean MAE RMSE Mean MAE RMSE


26 weeks 52 weeksTen-year yieldMean MAE RMSE Mean MAE RMSE


48 / 77

Real Yield Forecast Evaluation

26 weeks 52 weeksFive-year yieldMean MAE RMSE Mean MAE RMSE

Random walk -6.67 54.77 75.57 -16.08 81.98 97.42CV model -19.49 54.06 72.64 -52.21 79.8 100.34SV model -44.50 69.52 84.32 -73.89 99.48 117.83

26 weeks 52 weeksTen-year yieldMean MAE RMSE Mean MAE RMSE

Random walk -6.42 42.37 56.83 -14.45 65.17 77.90CV model -3.13 49.43 65.49 -28.37 73.07 87.08SV model -37.04 58.54 70.56 -61.85 87.89 103.16

The CV and SV models are inferior to both the random walkand the B-CR model when it comes to forecasting yields.

A likely explanation is that the models have not been correctedfor any finite-sample bias in their estimated factor dynamicsunlike the B-CR model. 49 / 77

Ten-year Nominal and Real Term Premiums

2005 2007 2009 2011 2013 2015

−4

−2

02

46

Rat

e in

per

cent

CV model SV model B−CR model KW model

2005 2007 2009 2011 2013 2015

−3

−2

−1

01

23

4

Rat

e in

per

cent

CV model SV model

Nominal and real term premiums are calculated as:

TPNt (τ) = yN

t (τ)−∫ t+τ

tEP

t [rNs ]ds,

TPRt (τ) = yR

t (τ)−∫ t+τ

tEP

t [rRs ]ds.

Nominal term premiums are compared to KW and B-CR. 50 / 77

Predicted Yield Volatility

I use the estimated factors and parameters from thereal-time estimations to generate predicted yieldvolatilities.

Predicted one-month conditional volatilities are given bythe square root of

V Pt [yN

T (τ)] =1τ2

BN(τ)′V Pt [XT ]BN(τ),

V Pt [y

RT (τ)] =

1τ2

BR(τ)′V Pt [XT ]BR(τ),

where V Pt [XT ] is the conditional covariance matrix of the

state variables, T − t = 112 , and τ is the yield maturity.

For nominal yields, I compare the models’ performanceto the B-CR model.

51 / 77

Realized Yield Volatility

Realized yield volatility measures are constructed fromhigh frequency daily data.

I use realized standard deviations. For each businessday t , the number of trading days, N, over the following31 day period are identified and used to calculate

RV STDt,τ =

√√√√N∑

n=1

∆y2t+n/N(τ),

where ∆yt+n/N(τ) is the change in yield yt(τ) from tradingday (n − 1) to trading day n.

This measure is referred to this as the 31-day realizedyield volatility at time t .

52 / 77

Nominal Yield Volatilities

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

CV model SV model B−CR model One−month realized volatility of two−year yield

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

CV model SV model B−CR model One−month realized volatility of ten−year yield

The figures show conditional one-month nominal yieldvolatilities at the two- (left) and ten-year (right) maturities.

The CV and SV models are not able to match thevariation in realized nominal yield volatility measures.

Note that the B-CR model also has difficulties inmatching realized long-term yield volatilities.

53 / 77

Real Yield Volatilities

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

CV model SV model One−month realized volatility of five−year yield

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

2005 2007 2009 2011 2013 2015

020

4060

8010

012

0

Rat

e in

bas

is p

oint

s

CV model SV model One−month realized volatility of ten−year yield

The figures show conditional one-month real yieldvolatilities at the five- (left) and ten-year (right) maturities.

For real yields the match to realized yield volatilitymeasures is slightly better, at least on average, but stillclearly not satisfactory.

54 / 77

Summary of Model Validation

The CV and SV models are competitive at forecastinginflation at short- and medium-term horizons.

By implication, they generate reasonable inflation riskpremium estimates.

However, both models are relatively weak at forecastingnominal and real yields, likely due to finite-sample bias.

As a consequence, their nominal term premiumestimates are less credible than competing models.

Finally, both models are relatively weak at matching thevariation in realized yield volatility measures.

The remainder of the presentation will focus on the risk ofdeflation and how the models can be used to assess it.

55 / 77

Deflation Probabilities in the CV Model

To calculate deflation probabilities, we need the distributionalproperties of the price process

Πt+τ

Πt= e

∫ t+τt (rN

s −rRs )ds.

Taking logs, CLR (2012) show that in the CV model

Yt,t+τ = ln

(Πt+τ

Πt

)∼ N

(mP

Y (t , τ), σPY (τ)

2),

Probt

(Yt,t+τ ≤ ln(1 + q)

)= Φ

(ln(1 + q)− mP

Y (t , τ)σP

Y (τ)

).

To assess deflationary outcomes, q = 0, and

Probt(Yt,t+τ ≤ 0

)= Φ

(−mP

Y (t , τ)σP

Y (τ)

).

56 / 77

Deflation Probabilities in the SV Model (1)

Within the SV model we are still interested in the followingtype of probabilities

Probt

(Yt,t+τ ≤ ln(1 + q)

).

However, the introduction of stochastic volatility makes thesystem non-Gaussian.

To generate the needed probabilities, we use Fouriertransform analysis as per Duffie, Pan and Singleton (2000).

57 / 77


The change in the market-implied price index for the periodfrom t until t + τ is given by

Πt+τ

Πt= e

∫ t+τt (rN

s −rRs )ds.

We want to calculate the probability of the event that the netchange in the price index between t and t + τ is below acertain critical level q.

By implication, we are interested in the states of the worldwhere

Πt+τ

Πt= e

∫ t+τt (rN

s −rRs )ds ≤ 1 + q,

or, equivalently,∫ t+τ

t(rN

s − rRs )ds ≤ ln(1 + q).

58 / 77


Since the nominal and real instantaneous short rates are given by

rNt = LN

t + St ,

rRt = LR

t + αRSt ,

the distributional properties of the following process are of interest:

Y0,t =

∫ t

0(rN

s − rRs )ds =

∫ t

0(LN

s + Ss − LRs − αRSs)ds ⇒

dY0,t = (LNt + (1 − αR)St − LR

t )dt .

In general, the P-dynamics of the state variables Xt are given by

dXt = K P(θP − Xt)dt +ΣD(Xt)dW Pt .

Adding the Yt -process to this system, leaves us with a five-factorSDE ...

59 / 77


dLNt

dSt

dCt

dLRt

dY0,t

=

κP11 0 0 κP

14 0κP

21 κP22 κP

23 κP24 0

κP31 κP

32 κP33 κP

34 0κP

41 0 0 κP44 0

0 0 0 0 0

θP1

θP2

θP3

θP4

0

dt

−

κP11 0 0 κP

14 0κP

21 κP22 κP

23 κP24 0

κP31 κP

32 κP33 κP

34 0κP

41 0 0 κP44 0

−1 −(1 − αR) 0 1 0

LNt

St

Ct

LRt

Y0,t

dt

+

σ11 0 0 0 00 σ22 0 0 00 0 σ33 0 00 0 0 σ44 00 0 0 0 0

√

LNt 0 0 0 0

0√

1 0 0 00 0

√

1 0 00 0 0

√

LRt 0

0 0 0 0 0

dW LN ,Pt

dW S,Pt

dW C,Pt

dW LR,Pt

dW Y ,Pt

,

where Z0,t = (LNt ,St ,Ct ,LR

t ,Y0,t) represents the augmented statevector.

60 / 77


This is a system of non-Gaussian state variables. Thus, we use theFourier transform analysis described in full generality for affinemodels in Duffie, Pan, and Singleton (2000).

They provide a formula for calculating contingent expectations:

GB,b(y ;Zt,t , t ,T ) = EP[e−

∫ Tt ρ′

ψZs,T dseB

′

Zt,T 1{bZt,T≤y}

∣∣∣Ft

].

First, we define

ψ(B;Zt,t , t ,T ) = EP[e−

∫ Tt ρ′

ψZs,T dseB

′

Zt,T

]= eBψ(t,T )′Zt,t+Aψ(t,T ),

where Bψ(t ,T ) and Aψ(t ,T ) are solutions to a system of ODEs

dBψ(t ,T )

dt= ρψ + (K P

Z )′Bψ(t ,T )−12

5∑

j=1

(Σ′

Z Bψ(t ,T )Bψ(t ,T )′ΣZ )j,j(δjZ )

′

,

dAψ(t ,T )

dt= −Bψ(t ,T )′(K θ)P

Z −12

5∑

j=1

(Σ′

Z Bψ(t ,T )Bψ(t ,T )′ΣZ )j,jγjZ ,

with Bψ(T ,T ) = B and Aψ(T ,T ) = 0.61 / 77


Duffie, Pan, and Singleton (2000) show that

GB,b(y ;Zt,t , t ,T ) =ψ(B;Zt,t , t ,T )

2−1π

∫ ∞

0

Im[e−ivyψ(B + ivb;Zt,t , t ,T )]

vdv .

We are interested in the cumulative probability function of Yt,T

conditional on Zt,t , that is, EP[1{Yt,T≤y}|Ft ].

Thus, to get the desired probability function, we must fix

B = 0, b =

00001

, ρψ =

00000

, y = ln(1 + q),

γZ =

01101

, δZ =

1 0 0 0 00 0 0 0 00 0 0 0 00 0 0 1 00 0 0 0 0

, and ...

62 / 77


K PZ =

κP11 0 0 κP

14 0κP

21 κP22 κP

23 κP24 0

κP31 κP

32 κP33 κP

34 0κP

41 0 0 κP44 0

−1 −(1 − αR) 0 1 0

,

(K θ)PZ =

κP11 0 0 κP

14 0κP

21 κP22 κP

23 κP24 0

κP31 κP

32 κP33 κP

34 0κP

41 0 0 κP44 0

0 0 0 0 0

θP1θP

2θP

3θP

40

,

and

ΣZ =

σ11 0 0 0 00 σ22 0 0 00 0 σ33 0 00 0 0 σ44 00 0 0 0 0

.

63 / 77


Now, ψ(B + ivb;Zt,t , t ,T ) is obtained by solving the system ofODEs with boundary conditions:

Bψ(T ,T ) = B + ivb;Aψ(T ,T ) = 0.

This allows us to calculate the complex number

ψ(B + ivb;Zt,t , t ,T ) = eBψ(t,T )′Zt,t+Aψ(t,T ).

Next, we can calculate the real number

Im[e−ivyψ(B + ivb;Zt,t , t ,T )].

Finally, we repeat this for all v ∈ (0,∞) to numericallyapproximate (cutoff: ∞ ≈ 1,000)

∫ ∞

0

Im[e−ivyψ(B + ivb;Zt,t , t ,T )]

vdv

and GB,b(y ;Zt,t , t ,T ) can be evaluated.64 / 77

One-Year Deflation Probabilities

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

Lehman BrothersBankruptcy

Sept. 15, 2008

CV model SV model

The risk of deflation prior to September 2008 was basically zero.

In late 2008, the models assigned a high probability to net deflationconsistent with the subsequent negative y-o-y CPI inflation.

The probabilities from the SV model are markedly higher2009-2011 due to its higher conditional volatility estimates. 65 / 77

Robustness for Alternative SV Specifications (1)

2007 2008 2009 2010 2011

−4

−3

−2

−1

01

23

4

Rat

e in

per

cent

CLR(0) (CV) model CLR(1)−LN model CLR(1)−C model CLR(1)−LR model CLR(2)−L^NC model CLR(2)−L^NL^R (SV) model CLR(2)−SC model CLR(2)−CL^R model

We analyze all 8 admissible combinations with zero, one, andtwo stochastic volatility factors.

Shown are five-year inflation expectations from all 8 models.

Note the very similar pattern. Thus, first moment dynamics arerelatively insensitive to the volatility specification. 66 / 77

Robustness for Alternative SV Specifications (2)

2007 2008 2009 2010 2011

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

CLR(0) (CV) model CLR(1)−LN model CLR(1)−C model CLR(1)−LR model CLR(2)−L^NC model CLR(2)−L^NL^R (SV) model CLR(2)−SC model CLR(2)−CL^R model

We analyze all 8 admissible combinations with zero, one, andtwo stochastic volatility factors.

Shown are one-year deflation probabilities from all 8 models.

Note that the risk of tail events is clearly sensitive to thevolatility specification! 67 / 77

Pricing the Deflation Protection Option (1)

We follow the intuition presented in Wright (2009) andexamine the yield spread between two TIPS:

A seasoned one with accumulated inflation protection— such that its deflation protection option is worth zero.A recently issued TIPS without any accrued inflationprotection — its deflation option is at-the-money.

Since they have similar maturities and other payment terms,the spread should reflect the value of the deflation option.

Procedure:Determine CS par-coupon yield for seasoned bond.Determine C0 par-coupon yield for recently issued.Difference is model-implied value of the deflationprotection measured as a par yield spread.

68 / 77


First, consider a hypothetical seasoned TIPS with T yearsremaining to maturity that pays an annual coupon C semi-annually.

Assume this bond has accrued sufficient inflation compensation soit is impossible to reach the deflation floor before maturity.

The par-coupon bond satisfying these criteria has a coupon ratedetermined by the equation:

2T∑

i=1

C2

EQt

[e−

∫ tit rR

s ds]+ EQ

t

[e−

∫ t+Tt rR

s ds]= 1.

The first term is the sum of the present value of the 2T couponpayments using the model’s fitted real yield curve at day t .

The second term is the discounted value of the principal. Thecoupon rate that solves this equation is denoted as CS.

69 / 77


Next, consider a new TIPS with no accrued inflation compensation.

Since the coupon payments are not protected against deflation, thedifference is in accounting for the principal deflation protection:

2T∑

i=1

C2

EQt

[e−

∫ tit rR

s ds]

+EQt

{ΠT

Πt· e−

∫ Tt rN

s ds 1{

ΠTΠt

>1}

}+ EQ

t

{1 · e−

∫ Tt rN

s ds 1{

ΠTΠt

≤1}

}= 1.

The first sum term is the same as for the seasoned TIPS.The second term is the NPV of the principal paymentconditional on ΠT

Πt> 1. Full inflation indexation applies in this

case as indicated by the scaling with the price change term.The third term is the NPV of the principal conditional on netdeflation; i.e., when ΠT

Πtis less than 1, it is replaced by a value

of one to provide the promised deflation protection.The par-coupon yield of a new TIPS that solves this equationis denoted as C0. 70 / 77

Value of Deflation Protection Option

2003 2004 2005 2006 2007 2008 2009 2010 2011

050

100

150

200

Spr

ead

in b

asis

poi

nts

Lehman BrothersBankruptcy

Sept. 15, 2008

CV model SV model

Shown is the value of the deflation protection optionmeasured as par yield spread between syntheticseasoned and newly issued five-year TIPS. 71 / 77

Question: Is Deflation Risk Real? Or Just Illiquidity?

TIPS are known to be illiquid, see Christensen and Gillan(2012) and Fleckenstein et al. (2014) for discussions.

At times, when such effects are severe, this will tend to pushup TIPS yields and depress BEI rates - independent ofinvestors’ inflation expectations.

Since the financial crisis was characterized by extremeilliquidity in many markets, it raises the question: Is thedeflation risk we measure real?

In response, we run regressions to control for sources thatcould affect TIPS liquidity.

72 / 77

Regression Analysis

VIX: represents near-term uncertainty about the generalstock market as reflected in options on the S&P 500.

HPW measure: Hu, Pan, and Wang (2013) introduce anilliquidity measure that represents an economy-wideilliquidity risk factor.

Off-the-run spread: The off-the-run over on-the-runTreasury yield spread at the ten-year maturity.

AAA corporate bond credit: The yield spread ofAAA-rated U.S. industrial corporate bonds overcomparable Treasury yields. We use the two-yearmaturity.

TIPS trading volume: The TIPS trading volume seriescollected by the New York Fed on a weekly basis. Weuse the eight-week moving average.

73 / 77

Regression Results

SV modelExplanatory Var.(1) (2) (3) (4) (5) (6)

Constant -0.17∗∗ -0.01 -0.09∗∗ 0.03∗ 0.39∗∗ 0.00(-13.37) (-1.91) (-9.01) (2.42) (10.95) (-0.04)

VIX 1.74∗∗ 0.67∗∗

(32.41) (7.41)HPW measure 0.06∗∗ 0.04∗∗

(36.11) (10.17)Off-The-Run Spread 1.72∗∗ 0.10

(34.60) (0.82)AAA Credit Spread 0.36∗∗ -0.10∗∗

(17.01) (-3.88)TIPS Trading Vol. -0.03∗∗ -0.01∗∗

(-5.79) (-4.03)Adjusted R2 0.72 0.76 0.74 0.41 0.07 0.83

Note that the VIX is systematically significant.We use its coefficient as a measure of the trueunobserved option value.This produces a “liquidity-adjusted” option value series.

74 / 77

Liquidity-Adjusted Deflation Protection Option Value

2003 2004 2005 2006 2007 2008 2009 2010 2011

050

100

150

200

Spr

ead

in b

asis

poi

nts Lehman Brothers

BankruptcySept. 15, 2008

Deflation option value, no adjustment Deflation option value, with liquidity adjustment

The figure shows the actual and the liquidity-adjusted five-yeardeflation option values from the SV model .

Assuming that TIPS illiquidity only affects the estimated valueof the real level factor, LR

t , we use the liquidity-adjusted optionvalues to back out an alternative liquidity-adjusted path, LR

t . 75 / 77

Liquidity-Adjusted Deflation Probabilities

2003 2004 2005 2006 2007 2008 2009 2010 2011

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

Lehman Brothers

Bankruptcy

Sept. 15, 2008

Deflation probability, no adjustment Deflation probability, with liquidity adjustment

We use the liquidity-adjusted LRt to calculate liquidity-adjusted

deflation probabilities.

Liquidity-adjustment produces notably lower estimates.

After adjusting for liquidity effects, spells of elevated deflationrisk remain clearly identified—only their severity is reduced. 76 / 77

Conclusion

The deflation protection option in TIPS bonds is a uniqueprice-level derivative that has only drawn limited interest,mainly because it is usually out of the money.

In this paper, we propose a term structure model withspanned stochastic volatility that generates reasonabledeflation protection option prices and delivers smallerpricing errors than a similar model with constant volatility.

We hope to improve the model specification andpotentially use it to price other inflation derivatives.

Also, we use the model on an on-going basis to monitorthe outlook for price inflation, the risk of deflation inparticular, as implicitly assessed by bond investors.Two important caveats should be highlighted:

The models do not respect the zero lower bound for yields;

There is no explicit correction for TIPS liquidity premiums.On-going research is trying to address both issues. 77 / 77

inflation expectations and risk premiums in an arbitrage-free

Documents