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Decomposing Yield to Maturity

Liuren WuJoint work with Peter Carr

Baruch College

The Role of Derivatives in Asset PricingJune 4th, 2016

Carr and Wu (NYU & Baruch) Decomposing Yield to Maturity June 4th, 2016 1 / 37

Overview

How to predict future interest rate movements, and more importantly, futurebond excess returns, has been a perennial topic in the academic literature.

Expectation hypothesis (EH): Long-rates are expectations of short rates.

Violations are regarded as evidence of (time-varying) risk premium.

Predict excess bond returns using the current yield curve slope

Most EH regressions reject EH (e.g., surveys by Campbell (95)...)except at very short maturities (e.g., Longstaff, 2000).

Cochrane&Piazzesi (2005): Excess bond returns can be predicted by a singletent-shaped forward rate factor. Many follow-up works on

Empirical replication and enhancement (Cieslak&Povala (2015))

Term structure modeling to accommodate such a single risk premiumfactor in a multi-factor setting (Calvet, Fisher, Wu (2015))

Before running away with regressions and term structure modeling, it isuseful to start from the very beginning and understand the basic,fundamental composition of the yield curve.


Decomposing yield to maturity

Yield to maturity represents a monotonic, but nonlinear, transformation ofthe bond price.

Fundamentally, regardless of modeling assumptions, yield to maturity canalways be decomposed into three components,

1 Expectation: market expectation about future interest rate movements2 Risk premium: compensation for bearing the risk of interest rate

fluctuation and its impact on bond returns,3 Convexity: an effect induced by the nonlinearity of the transformation

between bond price and yield-to-maturity.

We operationalize the decomposition and strive to separate the risk premiumfrom the other two components in predicting bond excess returns, withoutresorting to a forecasting regression

while hopefully shedding light on existing excess return forecastingregression results.


The pricing and yield transformation of a zero-coupon bond

Let Bt(T ) be the price at time t ≥ 0 of a default-free zero-coupon bondmaturing at a fixed date T ≥ t,

Let Mt,T denote the pricing kernel that links value at time t to value at timeT and let rt be the continuously-compounded short interest rate at time t.

The value of the zero-coupon bond can be written as

Bt(T ) = EPt [Mt,T ] = EP

t

[(dQdP

)e−

∫ Tt

rudu

]= EQ

t

[e−

∫ Tt

rudu].

Et [·] denotes expectation under time-t filtration,P denotes the real world probability measure,Q denotes the so-called risk-neutral measure,dQdP defines the measure change from P to Q. It is the martingalecomponent of the pricing kernel that defines the pricing of various risks.

The yield-to-maturity of the bond is defined via the following transformation:

Bt(T ) ≡ exp(−yt(T )(T − t))


Yield decomposition of a zero coupon bond

Combining the yield transformation with the zero pricing equation,

yt(T ) ≡ − lnBt(T )

T − t= − 1

T − tlnEQ

t

[e−

∫ Tt

rudu],

we can decompose the yield into three components

yt(T ) = 1T−tE

Pt

[∫ T

trudu

](Expectation)

+ 1T−tE

Pt

[(dQdP − 1

) ∫ T

trudu

](Risk premium)

− 1T−t lnEQ

t

[exp

(−(∫ T

trudu − EQ

t

∫ T

trudu)

)](Convexity)

Derivation is based on simple addition/subtraction

Given the short-rate-based pricing framework, the three components areall different expectations of future short rates over the bond horizon.

We henceforth use τ = T − t to denote time to maturity.


Yield decomposition based on short-rate dynamics

yt(T ) = 1τ E

Pt

[∫ T

trudu

](Expectation)

+ 1τ E

Pt

[(dQdP − 1

) ∫ T

trudu

](Risk premium)

− 1τ lnEQ

t

[exp

(−(∫ T

trudu − EQ

t

∫ T

trudu)

)](Convexity)

1 The 1st term is the investor expectation of the average future short rate.

2 The 2nd term measures the covariance between dQdP and the average future

short rate, capturing the risk premium on the interest rate risk.

3 The 3rd term measures convexity induced by short rate variation.

If∫ T

trudu is normally distributed with variance V , the convexity term

is simply half of the variance − 12V .

More generally, it captures the sum of variance and higher-order

cumulants, −∑∞

n=2κn

n! , where κn denotes the nth-cumulant of∫ T

trudu.


Short-rate based decomposition under a stylized example

To gain intuition, it is illustrative to go through an extremely stylized examplethat assumes simple random walk dynamics on the short rate:

drt = σdWt ,

with constant market price of dWt being γ.

It is a simplified version of Vasicek (1977), who allows mean reversiondrt = κ(θ − rt)dt + σdWt and hence non-flat expectation.

The three-term decomposition for the yield curve is

yt(T ) = rt −1

2γστ − 1

6σ2τ 2.

The 1st term rt denotes the flat expectation under random walk.Risk premium increases linearly with maturity. The market price ofinterest rate risk (γ) tends to be negative, leading to positive bondexpected excess returns, and an upward sloping yield curve.The convexity effect drives long yield down, and becomes increasinglyimportant at long maturities due to quadratic maturity dependence.


EH regression is not a test of EH

yt(T ) = 1τ E

Pt

[∫ T

trudu

](Expectation)

+ 1τ E

Pt

[(dQdP − 1

) ∫ T

trudu

](Risk premium)

− 1τ lnEQ

t

[exp

(−(∫ T

trudu − EQ

t

∫ T

trudu)

)](Convexity)

Expectation hypothesis (EH) says that long rate is equal to expectation of

short rate: yt(T ) = 1τ E

Pt

[∫ T

trudu

]EH regressions are often formulated based on the null hypothesis of zero riskpremium. For example,

1

h(yt+h(T )− yt(T )) = a + b

yt(T )− rtτ

+ et+h

with null a = 0, b = 1, based on the hypothesis that expected bond excessreturn is zero EP

t [ytτ − yt+h(τ − h)− rth] = 0.

EH does not imply zero risk premium, but implies that the risk premium isequal to the magnitude of the convexity effect — this cannot hold across allmaturities!


Expectation hypothesis can lead to arbitrage

EH implies that risk premium=convexity, but this cannot hold across allmaturities because of their different orders of dependence on maturity.

Risk premium increases linearly with maturity, convexity quadratically.

Suppose the short rate follows random walk (no prediction): drt = σtdWt .

EH implies flat yield curve yt(T ) = rt for all T .

A flat curve with parallel shifts allows arbitrage:

Construct a zero-cost, zero duration butterfly portfolio by investing($1,-$2,$1) to zeros expiring at (T − 1,T ,T + 1).

The zero-cost portfolio generates a positive instantaneous riskfreeexcess return of σ2

t dt.

If one maintains a constant dollar investment in the three bonds up toexpiry of the shorter-term bond, one would receive the following P&L,

PL =∫ T−1t

σ2s ds.

The payoff of the butterfly looks just like that of a straddle, or adelta-hedged long option position.


Value variation of zero-cost, zero-duration butterflies

−50 0 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Percentage rate change, %

Bond p

ort

folio

valu

e c

hange, %

T=5, r=5%

T=10,r=5%

T=30,r=5%

very much like the payoff of a delta-neutral straddle, delta-hedged long option, ora variance swap..


Operationalize the decomposition

yt(T ) = 1τ E

Pt

[∫ T

trudu

](Expectation)

+ 1τ E

Pt

[(dQdP − 1

) ∫ T

trudu

](Risk premium)

− 1τ lnEQ

t

[exp

(−(∫ T

trudu − EQ

t

∫ T

trudu)

)](Convexity)

Take some observable short-term rate (e.g., 3-month Treasury bill) as theshort rate rt .

Use professional forecasts (e.g., bluechip) on the short rate to construct theexpectation.

Construct historical variance estimators on the short rate changes tocompute the convexity effect based on normality & random walkassumption.

Under the normal/random walk assumption, convexity = − 16σ

2τ 2, withσ2 being an annualized variance estimator on short rate changes.

Use the remainder of the yield as the risk premium, without the need to runa forecasting excess return regression.


Data and construction details

Data

Daily spot rate curve stripped from Treasuries from three month to 10years, Jan, 1986-March, 2012

Monthly bluechip forecasts on the Treasury bill at horizons of 6 and 12months, Jan, 1992-March, 2012.

Semi-annual bluechip long-term forecasts on Treasury bill, 6-11 yearhorizon (Take as approximately 10 years).

Estimation

At each date, linearly interpolate the forecasts across horizons to

generate forecasts (FR) on∫ T

trudu for τ = 9 months, 15 months, and

10 years.

Estimate annualized Treasury bill rate daily change variance σ2, using aone year window. Construct the convexity component (CR) as − 1

6σ2τ 2.

The risk premium (RP) on each spot rate is computed asRPt(τ) = yt(τ)− FRt(τ)− CRt(τ).


Short-term Treasury yield decomposition

9-month 15-month

92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12−1

0

1

2

3

4

5

6

7

8

9−

Mo

nth

Yie

ld D

eco

mp

osi

tion

, %

Yield

Expectation

Risk premium

Convexity

92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12−1

0

1

2

3

4

5

6

7

8

15

−M

on

th Y

ield

De

com

po

sitio

n,

%

Yield

Expectation

Risk premium

Convexity

Spot rates at short maturities are mainly driven by expectation.

Convexity effect is virtually zero.

Risk premium fluctuates around zero, probably reflecting more estimationnoise than actual risk premium.

Longstaff (2001): Expectation hypothesis holds well at super short term.


Long-term Treasury yield decomposition

92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12−4

−2

0

2

4

6

8

10

−Y

ea

r Y

ield

De

com

po

sitio

n,

%

Yield

Expectation

Risk premium

Convexity

Convexity effect becomes more significant, especially during volatile times.

Risk premium on long-term bond stays positive most of the time, canaccount for half of the yield, but can vary significantly over time.

Caveat: The accuracy of long-run forecasts is hard to verify.


Practical limitations of the short-rate based decomposition

Short-rate forecasts are available only at a few horizons.

Decomposing long-term rates need long-term forecasts of short rates, theaccuracy of which is hard to verify.

The convexity effect of long-term rates can be sensitive to random walk (v.mean-reversion) assumptions on short rate.

Ideally, we can rely less on the short rate, but more on the near-termbehaviors of both short and observed long rates.

We observe both short and long rates.We know their near-term behavior better than long term.


Near-term dynamics of all rates

Instead of being short-rate centric, we specify the near-term dynamics for ratesacross all maturities:

Continuous dynamics on yields of all fixed-expiry zero bonds:

dyt(T ) = µt(T )dt + σt(T )dWt

With market price of risk γt on dWt , the Q dynamics become,

dyt(T ) = (µt(T )− γtσt(T )) dt + σt(T )dWQt

Instantaneous yield changes are driven by a single shock (dWt), matching

Cochrane&Piazessi single risk premium factor: A single market price ofrisk factor γt drives the excess returns on all bonds.USV: Volatility shocks are unspanned by short-term yield changes.

γt , µt(T ), σt(T ) can all change randomly, but their dynamics are leftunspecified — Only their current levels and hence near-term dynamics arespecified.

Over longer term, yield curve variation can be driven by many factors thatgovern the variation of expectation, risk, and risk premium.


Decomposition based on near-term dynamics of all rates

Combining near-term dynamics of all rates,

dyt(T ) = (µt(T )− γtσt(T )) dt + σt(T )dWQt

with the yield transform of bond value: B(t, yt(T )) ≡ e−yt(T )(T−t).

No dynamic arbitrage between bonds and money market implies:

∂B

∂t+∂B

∂y(µt(T )− γtσt(T )) +

1

2

∂2B

∂y2σ2t (T ) = rtB,

Re-arrangement leads to the following three-term decomposition:

yt(T ) = [rt + µt(T )τ ]− γtσt(T )τ − 1

2σ2t (T )τ 2,

a new, alternative decomposition based on near-term dynamics of all yields,rather than the full-dynamics of the short rate.


Different starting point, similar structure

No-arbitrage and continuous near-term dynamics of all yields lead to a newdecomposition:

yt(T ) = [rt + µt(T )τ ]− γtσt(T )τ − 1

2σ2t (T )τ 2,

Expectation is based on a linear interpolation between the short rateand the near-term drift of the particular yield yt(T ).

Convexity is driven by the current volatility estimate of the particularbond, instead of aggregate risk of the future short rate.

Risk premium is driven by the current level of market price of risk.

γt , µt(T ), σt(T ) can all change tomorrow randomly, but their dynamics donot matter for today’s decomposition.

Different starting points, similar term structure effects: The yield curve islinear risk premium and quadratic in convexity if µ(T ), σ(T ) ∼ o(T )


Revist the EH regression under the new decomposition

The null hypothesis (a = 0 and b = 1) of the following EH regression

1

h(yt+h(T )− yt(T )) = a + b

yt(T )− rtτ

+ et+h

implies1

h(Et [(yt+h(T )]− yt(T )) = µt(T ) =

yt(T )− rtτ

Our no-arbitrage yield curve decomposition implies

µt(T ) =yt(T )− rt

τ+ γtσt(T ) +

1

2σ2t (T )τ,

Even in the absence of risk premium (γt = 0)

The null of EH regression is violated as long as rates are random.

The violation becomes larger for longer maturities.

Slope is biased downward (b̂ < 1) when variance and rates arenegatively correlated — as they are, by no-arbitrage construction,more so at longer maturities.


New decomposition under the stylized example

drt = σdWt implies

dyt(T ) =

(1

2γσ +

1

3σ2(T − t)

)dt + σdWt

Given flat expectation (zero drift) on short rate, the drift of fixed-expiryyields is purely driven by the yield curve shape.

The decomposition becomes:

Expectation: r + µt(T )(T − t) = rt +(12γσ + 1

3σ2τ))τ

Risk premium: −γστConvexity: 1

2σ2τ 2

Combining them leads to yt(T ) = rt − 12γστ −

16σ

2τ 2 as before.

The stylized example provides a link betweenthe partial, short-term dynamics of all long rates andthe full, long-term dynamics of the short rate.


Decomposing yields of coupon-bonds (bond portfolios)

The yield to maturity on a coupon bond or bond portfolio can be definedanalogously,

B(t, ymt ) ≡

∑j

Cjeymt τj

where Cj denotes its cash flow at time t + τj .

We apply the same one-factor diffusion assumption to coupon-bond yields,

dymt = µm

t dt + σmt dWt ,

where m is just an indicator of a particular coupon bond.

The same market price of risk γt would induce a drift adjustment γtσmt on

each bond under the risk-neutral measure.

While (σmt , µ

mt ) can be bond-specific, the market price of the same risk γt

should be the same across all bonds for consistency.


Coupon-bond yield decomposition

Assume dynamic no-arbitrage between bonds and the money marketaccount, we obtain an analogous decomposition for coupon-bond yields,

ymt = [rt + µm

t τm]− γtσm

t τm −1

2(σm

t )2τ 2m, (1)

where τm and τ 2m denote value-weighted maturity (duration) and maturitysquared, respectively,

τm =∑j

Cje−ytτj

Bmτj , τ 2m =

∑j

Cje−ytτj

Bmτ 2j .

With the weighted-average maturity and maturity squared definition, thedecomposition in (1) is equally applicable to the yield to maturity of zeros,coupon bonds, or bond portfolios.

The decomposition does not rely on any long-term projections, but onlydepend on the current drift and volatility level of the yield-to-maturity oneach particular bond or bond portfolio.


Predicting excess returns with no prediction on long rates

ymt = [rt + µm

t τm]− γtσm

t τm −1

2(σm

t )2τ 2m

The decomposition is equally applicable to spot rates or coupon-bond yields.

We can use historical variance estimators of daily yield changes to proxy σmt ,

without worrying about its long-run behavior.

It is well known that long-term, constant-maturity interest rates are virtuallyimpossible to predict (Duffee, 2002).

We impose this no-prediction principle on long-term rates and infer theconstant-expiry rate drift µm

t based on the current yield curve shape.

Infer the single market price of risk factor (γt) from long rates and examineits behavior and implications.


Data

Data: US and UK swap rates 1995.1.3-2016.5.11, 5378 business days

Based on 6-month LIBOR Maturity, 2,3,4,5,7,10,15,20,30

Extended maturity sinceUS: 2004/11/12 for 40 & 50 yearsUK: 1999/1/19 for 20 & 30 years, 2003/08/08 for 40 & 50 years

US UK

9596 97 9899 00 01 0203 04 05 0607 08 0910 11 12 1314 15 16 170

1

2

3

4

5

6

7

8

9

Sw

ap

Ra

tes,

%

2y

10y

30y

50y

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

9

10

Sw

ap

Ra

tes,

%

2y

10y

30y

50y


Swap rate variance estimators

Estimate variance σmt on each swap rate series with a 1y rolling window.

US UK

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sw

ap

Ra

te V

ola

tility

, %

2y

10y

30y

50y

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Sw

ap

Ra

te V

ola

tility

, %

2y

10y

30y

50y

Long rates vary as much as, if not more, than short rates.

Convexity effects are large on long rates.⇒ Super long term rates must fall?!v.s. ”Long Forward and Zero-Coupon Rates Can Never Fall” (Dybvig,Ingersoll, Ross, 1996)


Convexity effects

Convexitymt =1

2(σm

t )2τ 2m

Treat swap rates as par bond yields, and compute value-weighted maturityand maturity squared (τm and τ 2m)

US UK

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Co

nve

xity

Eff

ect

, %

2y

10y

30y

50y

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 160

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Co

nve

xity

Eff

ect

, %

2y

10y

30y

50y

Convexity effects are negligible on 2-year swap, but become quite significantfor longer-term swap rates, particularly so during crises.


Extracting risk premium from long rates

Volatility is reasonably flat across maturities at the long end: σ′t(τ) ≈ 0.

With no prediction on long-term constant-maturity rates, the drift oflong-dated constant-expiry rates is dictated by the yield curve shape:µt(T ) = 0− y ′t (τ)

Given yt(τ) = rt + µt(T )τ − γtσtτ − 12σ

2t τ

2, we have

µt(T ) = −µt(T )− µ′t(T )τ + γtσt + σ2t τ

⇒ µt(T ) =1

2γtσt +

1

3σ2t τ

We can thus infer the market price of interest risk from long bonds based onits slope and its variance estimator:

λt ≡ −γt =ymt − rt + 1

6 (σmt )2τ 2m

12 σ

mt τm

A negative market price of rate risk γt implies positive expected excessreturn on bonds. We use λt ≡ γt to denote the market price of bond risk.

We use six-month LIBOR as rt for the financing cost and extract the marketprice of bond risk from swap rates with maturities 10 years and longer.


Earning excess returns from long bonds

λt =ymt − rt + 1

6 (σmt )2τ 2m

12 σ

mt τm

With no views on directions of long rate movements, one expects to make moneyfrom long bonds in two forms:

1 The interest-rate differential between the long rate (which one receives) andthe short rate (which is the financing cost).

2 Convexity: Duration-neutral trading profits due to long-term rate variation

The duration-neutral P&L from trading long-term bonds is proportionalto the realized variance over the horizon of the bond, analogous to thedelta-neutral P&L of an option.

If rates across maturities are at the same level and move in syn in thefuture, one makes more money from longer-term bonds due toconvexity, similar to delta-hedged gains from a long option posiiton.


Market price of bond risk

US UK

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16−1

−0.5

0

0.5

1

1.5

Ma

rke

t p

rice

of

bo

nd

ris

k

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16−1

−0.5

0

0.5

1

1.5

Ma

rke

t p

rice

of

bo

nd

ris

k

The market price of bond risk extracted from different rates are similar inmagnitude and move together, consistent with the one-factor assumption.

Over the common sample, the cross-correlation estimates among thedifferent λt series average 99.67% for US, and 98.76% for UK.

In the US, market price of risk approached zero in late 1998, 2000, and2007, but tended to be high during recessions.

In the UK, the market price became quite negative during 1998 and2007-2008.


Predicting bond returns with no-prediction on long rates

Correlation between ex ante risk premium (γtσmt ) and ex post excess returns on

each par bond, with the average denoting the correlation between the average riskpremium and the average bond excess return over the common sample period

Maturity 10 15 20 30 40 50 AverageHorizon: 6-month

US 0.31 0.28 0.26 0.24 0.26 0.25 0.28UK 0.18 0.22 0.22 0.18 0.19 0.22 0.23

Horizon: One yearUS 0.36 0.36 0.34 0.31 0.31 0.29 0.33UK 0.36 0.42 0.40 0.32 0.33 0.37 0.39

Paradoxically, the assumption of no prediction on long-dated swap rates leadto significant prediction on bond excess returns.

The predictors (risk premium) are generated based purely on a varianceestimator and the current slope of the yield curve, without fitting ofpredictive regressions.


Predicting stock returns with market price of bond risk

Correlation between ex ante market price of bond risk and ex post excess returnson S&P 500 index and FTSE 100 index, respectively

Maturity 10 15 20 30 40 50 AverageHorizon: 6-month

SPX 0.04 0.06 0.06 0.06 0.15 0.13 0.16FTSE 0.23 0.28 0.28 0.35 0.35 0.37 0.35

Horizon: One yearSPX 0.12 0.13 0.13 0.13 0.24 0.22 0.25

FTSE 0.28 0.31 0.33 0.39 0.38 0.40 0.39

The market price of bond risk reflects general market sentiment on riskattitude, which also seems to show up in the stock market.

The market price of risk inferred from different swap rate are very similar.The different predictive correlation estimates reflect partially the availablesample-length difference.


Link interest rate volatility to the yield curve curvature

ymt = [rt + µm

t τm]− γtσm

t τm −1

2(σm

t )2τ 2m

Since interest rate volatility estimates are reasonably flat across maturitiesσ′(τ) ≈ 0, the curvature of the interest rate curve is at least partially drivenby the interest rate volatility.

Absent from interest rate predictability, bond risk premium is chiefly drivenby a combination of the interest rate slope and the interest rate volatility,the latter driven by the convexity effect.

If we can link the interest rate volatility to the shape (e.g., curvature) of theinterest rate curve, we can represent the bond risk premium purely as afunction of the interest rate curve shape.

This may allow us to represent Cochrane and Piazzesi (2005)’s (CP) singlebond risk premium factor within our dynamic valuation framework.

CP represent risk premiums on 2-5 year bonds in terms of a single riskfactor made of 1-5 year forward rates.


Map interest rate volatility to the forward rate curve

US UK

1 2 3 4 5−0.01

−0.005

0

0.005

0.01

0.015

0.02

Forward rate maturity, Years

Variance

loadin

g

2

3

4

5

1 2 3 4 5−4

−2

0

2

4

6

8x 10

−3


Variance

loadin

g

2

3

4

5

Strip the forward rate curve (with half-year tenor) from the LIBOR and swaprates, and regress the historical 2-5 year swap rate variance estimatorsagainst the five forward rates at 1-5 year maturities.

The “tent-shaped” loading is similar to the CP’s risk premium loading.

Superimposing this tent-shape on the slope would be our model-impliedmarket price of risk.

Even with constant market price of risk, risk premium varies with variance,and accordingly with the tent-shaped forward-rate portfolio.


Map interest rate volatility to the forward rate curve

US UK

1 2 3 4 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025


Var

ianc

e lo

adin

g

10

15

10

30

40

50

1 2 3 4 5−6

−4

−2

0

2

4

6x 10

−3


Var

ianc

e lo

adin

g

10

15

10

30

40

50

Variance on longer-dated rates have similar “tent-shaped” loading onforward rates, which capture the forward curve curvature.

Regression R2:Mat 2 3 4 5 7 10 15 20 30 40 50US 0.54 0.52 0.51 0.50 0.49 0.43 0.42 0.36 0.33 0.43 0.39UK 0.53 0.50 0.51 0.44 0.33 0.23 0.23 0.21 0.25 0.21 0.35

Literature: Heidari&Wu (2003) find strong positive correlation betweenbetween cap/swaption implied volatility levels and the level of the interestrate curvature factor.


Unspanned volatility and the convexity effect

USV says that short-term interest rate volatility changes cannot be explainedby short-term interest rate changes.

Our one-factor diffusion setting is in line with the USV evidence.

Convexity effect says that the level of curvature for the forward rate curve isproportional to the level of interest rate variance.

Our theoretical decompositionOur empirical evidence: regress variance levels on forward ratesHeidari & Wu (2003): Relation between option implied vol levels andthe level of the interest rate curvature factorCochrane&Piazessi (2005): the tent shaped forward rate portfolio asthe single risk premium factor


Concluding remarks

We decompose bond yields into three components:

1 expectation,2 risk premium, and3 convexity

We perform the decomposition from two perspectives:

1 The long dynamics of the short rate2 The short dynamics of long rates

The decompositions reveal several insights:

Convexity effect is negligible at short maturities, but can become veryimportant for super long rates.From the decomposition, we can extract bond risk premium fromhistorical variance estimators and either professional survey forecasts onfuture short rates or the assumption of no prediction on long rates.The extracted bond risk premium predicts future excess returns onboth bonds and stock indexes.


Concluding remarks

The decompositions shed light on the literature findings on bond risk premium:

Expectation hypothesis can lead to arbitrage.

The EH regressions findings may not be driven by risk premium, can purelybe driven by the convexity effect, even in the absence of risk premium.

The Cochrane and Piazessi (2005) tent-shaped forward rate portfolio riskpremium factor can be a proxy, at least partially, for the convexity effect, orequivalently interest rate risk.

Even with constant market price, risk premium increases with risk level,which is proportional to the curvature of the forward rate curve.Under the assumption of no-prediction on long-term rates, riskpremium on long-term bonds is a combination of the yield curve slopeand the convexity effect.

While the EH literature misses the convexity term, the CP bond excessreturn regression may have just picked it up.

Trading a butterfly is just like trading a straddle...


decomposing yield to maturity - baruch collegefaculty.baruch.cuny.edu/lwu/papers/yld_ov.pdf ·...

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