interest rate modeling getco

7/29/2019 Interest Rate Modeling Getco

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Work on Term structure

modeling

Sankha Banerjee

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Introduction Background Modeling Results Summary

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The?

Panic of 2008

Whatassetshaveheldtheirvaluebest US Treasury bills (stable value)

Nominal US Treasury bonds (increasing value) WhyhavenominalTreasuriesbeensuch good hedges?

Flight to quality helps safe assets, but why are nominal Treasuries regarded as safe?

They have no credit risk, but they do have inflation risk

Have nominal Treasuries always hedged investors against other risks?

Understanding Bond Risks Thislectureexplorestime-variationininflation risk and its effect on the

nominal Treasury yield curve

TheanalysismakessomeuseofTIPSdata but TIPS are not the main focus

Attheend,abriefanalysisofcurrencies along the same lines


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What do we know from the 2008?

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US Treasury bills and Nominal US Treasury bonds hold their value during a high

volatility event

They act as good hedge, and the phenomenon of "Flight to quality helps safe

assets.

Why are nominal Treasuries regarded as safe? They have no credit risk, but they

do have inflation risk


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Time series of the stock-bond covariance and the CAPM beta of the 10 year nominal bond


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Time series of the stock-deflation covariance and the CAPM beta of deflation


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Time series of the US 10-year inflation-indexed yields

Estimated time series of the real rate


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Time series of the real bond second moments


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Time series of the nominal bond second moments


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Estimated time series of permanent and transitory components of expected inflation


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Estimated time series of covariance of the inflation with interest rate.

Response of the nominal expected excess returns to covariance of the inflation with

Interest rate.


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Estimated time series of expected excess returns for 10 year nominal bonds.


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Response of the yield curves to covariance of the inflation with Interest rate.


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Changing Conventional Wisdom

Late1970sandearly1980s:

Bonds are exposed to the risk of stagflation Avoid them unless the term premium

is high

2000s:

Bonds are hedges against the risk of deflation Anchor to windward Hold them even at a low term premium

Changing CW reflects changing reality Bonds as hedges in 2007-2008

Changing Inflation Behavior

The changesinmeasuredbondrisksappear to be related to changing behavior of

the Phillips Curve WhenthePhillipsCurveisstable(early 1960s, 2000s), inflation falls when

unemployment rises

Then bonds do well in bad times and hedge macroeconomic risk

WhenthePhillipsCurveisunstable(1970s and early 1980s), inflation and

unemployment move together (stagflation)

Then bonds do badly in bad times and are risky


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Term structure modeling

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Real stochastic discount factor (SDF):

xt

is real rate:

zt

drives time-variation in volatility of SDF:

xt

and zt

follow AR(1) processes:

d i k d d li l


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Modelling the Yield Curve

Changingbondriskdoesseemtomatterover the long run

Intheshortrun,however,thereareother influences on the yield curve

Tocaptureitsmovements,weneedto consider more traditional factors as well:

The real interest rate

Investor attitudes towards risk Expected inflation Campbell,Sunderam,andViceira2008 undertakes this project

A Bond Pricing Model

Weconsiderfivefactorsthatmoveindifferent ways:

Real interest ratext(transient)

Risk aversion zt(persistent)

Long-run expected inflation t (permanent) Temporary expected inflation t(transient)

Covariance of inflation with recession t(persistent, can change sign)

Thefivefactorsarenotdirectlyobserved,so we back out their implied values from data we

do observe

Nonlinear Kalman filtering

I d i B k d M d li R l S


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Fisher equation Inflation risk premium

I t d ti B k d M d li R lt S


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Date

CAPMbeta

Introd ction Backgro nd Modeling Res lts S mmar


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Stagflation risk

Deflation risk



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Implications for the Yield Curve

Weplottheyieldcurveatthesamplemeanofall the state variables

Thenwevaryeachstatevariabletoitssample minimum and maximum, while holding the

other state variables at their sample mean



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This variable is the main innovation of our model, and plays important role:

(t)2 drives time variation in the conditional volatility of both realized

inflation and expected inflation.

zt tdrives time variation in the covariance of the real economy with

inflation, and thus determines nominal bond risk premia.

This covariance (and thus bond risk premia) can switch sign as ttakes

positive or negative values (zt is always positive in the data)



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Nested Models

Both ztand tconstant

Two-factor affine yield model of Campbell and

Viceira (2001, 2002).

Both real bond risk premia and nominal risk premia are constant. ztvarying and tconstant

Three-factor affine yield model ((Bekaert et al.,

2004, Buraschi and Jiltsov 2006, Wachter 2006).

Both real bond risk premia and nominal bond risk premia vary with aggregate risk

aversion

ztconstant and tvarying Single-factor affine yield model for the real term structure, and a linear-quadratic

model for the term structure for nominal interest rates.

Constant real bond risk premia, time-varying nominal bond risk premia.

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Estimation

Maximum likelihood via nonlinear Kalman filter because state variables are

unobserved.

UnscentedKalmanfilter(JulierandUhlmann 1997, Wan and van der Merwe 2000,Koijen and Binsbergen 2008)

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Observed Variables

Nominalyieldcurveatmaturities3months,1 year, 3 years, 10 years

TIPSyield

Realized inflation Equityreturnsanddividendyield(proxyforrisk aversion)

Realized bond variance and bond-equity covariance in daily data



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Implications for the Yield Curve

Realinterestrateandtemporaryexpected inflation move the short end

Riskaversionmovesthelongend

Permanentexpectedinflationmovestheyield

curve up and down in parallel

Inflation-recessioncovariancedrivesthe curvature of the yield curve

Implications for the Yield Curve Fixed-incomepractitionersanalyzetheyield curve using level, slope, and curvature factors

Theydonotrelatethesefactorstoexternal market conditions

Ourmodeldoes:

Real interest rate and permanent expected inflation

drive the level factor

Real interest rate, risk aversion, and temporary expected inflation drive the slope factor

Inflation-recession covariance drives the curvature factor

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Implications for Bond Returns Whathappenswheninvestorsbecomemorerisk averse?

Ifbondsarerisky,theninvestorssellbothstocks and bonds

Ifbondsarehedges,theninvestorssellstocks and buy bonds (flight to quality)

Thusmovementsinriskaversionamplifythe covariance of bonds and stocks

If the covariance is positive, it becomes more positive

If the covariance is negative, it becomes more negative

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Implications for Term Premia

Expectedexcessbondreturns(termpremia)are determined by

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