Decomposing Yield to Maturity
Liuren Wu
Joint work with Peter Carr
Baruch College
The Role of Derivatives in Asset Pricing
June 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, future
bond 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 single
tent-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 premium
factor in a multi-factor setting (Calvet, Fisher, Wu (2015))
Before running away with regressions and term structure modeling, it is
useful to start from the very beginning and understand the basic,
fundamental composition of the yield curve.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
2 / 37
Decomposing yield to maturity
Yield to maturity represents a monotonic, but nonlinear, transformation of
the bond price.
Fundamentally, regardless of modeling assumptions, yield to maturity can
always be decomposed into three components,
1
2
3
Expectation: market expectation about future interest rate movements
Risk premium: compensation for bearing the risk of interest rate
fluctuation and its impact on bond returns,
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 premium
from the other two components in predicting bond excess returns, without
resorting to a forecasting regression
while hopefully shedding light on existing excess return forecasting
regression results.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
3 / 37
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 bond
maturing at a fixed date T ≥ t,
Let Mt,T denote the pricing kernel that links value at time t to value at time
T and let rt be the continuously-compounded short interest rate at time t.
The value of the zero-coupon bond can be written as
R
h RT
i
T
dQ
− t ru du
e − t ru du = EQ
Bt (T ) = EPt [Mt,T ] = EPt
e
.
t
dP
Et [·] denotes expectation under time-t filtration,
P denotes the real world probability measure,
Q denotes the so-called risk-neutral measure,
dQ
dP defines the measure change from P to Q. It is the martingale
component 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))
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
4 / 37
Yield decomposition of a zero coupon bond
Combining the yield transformation with the zero pricing equation,
yt (T ) ≡ −
h RT
i
1
ln Bt (T )
− t ru du
=−
ln EQ
,
t e
T −t
T −t
we can decompose the yield into three components
i
hR
T
yt (T ) = T 1−t EPt t ru du
i
h
RT
+ T 1−t EPt dQ
−
1
r
du
u
dP
h
tR
i
T
QRT
− T 1−t ln EQ
exp
−(
r
du
−
E
r
du)
u
u
t
t t
t
(Expectation)
(Risk premium)
(Convexity)
Derivation is based on simple addition/subtraction
Given the short-rate-based pricing framework, the three components are
all different expectations of future short rates over the bond horizon.
We henceforth use τ = T − t to denote time to maturity.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
5 / 37
Yield decomposition based on short-rate dynamics
yt (T )
=
+
−
1
2
3
i
hR
T
r
du
u
h t
i
RT
dQ
1 P
E
− 1 t ru du
τ t
dP
i
h
R
T
QRT
Q
1
r
du)
r
du
−
E
ln
E
exp
−(
u
u
t
t
τ
t
t
1 P
τ Et
(Expectation)
(Risk premium)
(Convexity)
The 1st term is the investor expectation of the average future short rate.
The 2nd term measures the covariance between dQ
dP and the average future
short rate, capturing the risk premium on the interest rate risk.
The 3rd term measures convexity induced by short rate variation.
RT
If t ru du is normally distributed with variance V , the convexity term
is simply half of the variance − 12 V .
More generally, it captures the sum of variance and higher-order
RT
P∞
cumulants, − n=2 κn!n , where κn denotes the nth-cumulant of t ru du.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
6 / 37
Short-rate based decomposition under a stylized example
To gain intuition, it is illustrative to go through an extremely stylized example
that 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 reversion
drt = κ(θ − rt )dt + σdWt and hence non-flat expectation.
The three-term decomposition for the yield curve is
1
1
yt (T ) = rt − γστ − σ 2 τ 2 .
2
6
The 1st term rt denotes the flat expectation under random walk.
Risk premium increases linearly with maturity. The market price of
interest rate risk (γ) tends to be negative, leading to positive bond
expected excess returns, and an upward sloping yield curve.
The convexity effect drives long yield down, and becomes increasingly
important at long maturities due to quadratic maturity dependence.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
7 / 37
EH regression is not a test of EH
yt (T )
=
+
−
i
hR
T
ru du
t
h
i
RT
dQ
1 P
E
−
1
r
du
u
t
τ
dP
i
h
tR
RT
T
Q
1
ln
E
exp
−( t ru du − EQ
t
t t ru du)
τ
1 P
τ Et
(Expectation)
(Risk premium)
(Convexity)
Expectation hypothesis (EH)
hR says ithat long rate is equal to expectation of
T
1 P
short rate: yt (T ) = τ Et t ru du
EH regressions are often formulated based on the null hypothesis of zero risk
premium. For example,
1
yt (T ) − rt
(yt+h (T ) − yt (T )) = a + b
+ et+h
h
τ
with null a = 0, b = 1, based on the hypothesis that expected bond excess
return is zero EPt [yt τ − yt+h (τ − h) − rt h] = 0.
EH does not imply zero risk premium, but implies that the risk premium is
equal to the magnitude of the convexity effect — this cannot hold across all
maturities!
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
8 / 37
Expectation hypothesis can lead to arbitrage
EH implies that risk premium=convexity, but this cannot hold across all
maturities 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 = σt dWt .
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 riskfree
excess return of σt2 dt.
If one maintains a constant dollar investment in the three bonds up to
expiry of the shorter-term bond, one would receive the following P&L,
R T −1 2
PL = t
σs ds.
The payoff of the butterfly looks just like that of a straddle, or a
delta-hedged long option position.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
9 / 37
Value variation of zero-cost, zero-duration butterflies
0.1
Bond portfolio value change, %
0.09
0.08
T=5, r=5%
T=10,r=5%
T=30,r=5%
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
−50
0
Percentage rate change, %
50
very much like the payoff of a delta-neutral straddle, delta-hedged long option, or
a variance swap..
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
10 / 37
Operationalize the decomposition
yt (T )
=
+
−
i
hR
T
ru du
t
i
h
RT
dQ
1 P
r
du
E
−
1
u
τ t
dP
h
tR
i
RT
T
Q
1
ln
E
exp
−( t ru du − EQ
t
t t ru du)
τ
1 P
τ Et
(Expectation)
(Risk premium)
(Convexity)
Take some observable short-term rate (e.g., 3-month Treasury bill) as the
short rate rt .
Use professional forecasts (e.g., bluechip) on the short rate to construct the
expectation.
Construct historical variance estimators on the short rate changes to
compute the convexity effect based on normality & random walk
assumption.
Under the normal/random walk assumption, convexity = − 16 σ 2 τ 2 , with
σ 2 being an annualized variance estimator on short rate changes.
Use the remainder of the yield as the risk premium, without the need to run
a forecasting excess return regression.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
11 / 37
Data and construction details
Data
Daily spot rate curve stripped from Treasuries from three month to 10
years, Jan, 1986-March, 2012
Monthly bluechip forecasts on the Treasury bill at horizons of 6 and 12
months, Jan, 1992-March, 2012.
Semi-annual bluechip long-term forecasts on Treasury bill, 6-11 year
horizon (Take as approximately 10 years).
Estimation
At each date, linearly interpolate the forecasts across horizons to
RT
generate forecasts (FR) on t ru du for τ = 9 months, 15 months, and
10 years.
Estimate annualized Treasury bill rate daily change variance σ 2 , using a
one year window. Construct the convexity component (CR) as − 16 σ 2 τ 2 .
The risk premium (RP) on each spot rate is computed as
RPt (τ ) = yt (τ ) − FRt (τ ) − CRt (τ ).
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
12 / 37
Short-term Treasury yield decomposition
9-month
15-month
8
8
9−Month Yield Decomposition, %
6
5
4
3
2
1
0
Yield
Expectation
Risk premium
Convexity
7
15−Month Yield Decomposition, %
Yield
Expectation
Risk premium
Convexity
7
6
5
4
3
2
1
0
−1
92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12
−1
92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12
Spot rates at short maturities are mainly driven by expectation.
Convexity effect is virtually zero.
Risk premium fluctuates around zero, probably reflecting more estimation
noise than actual risk premium.
Longstaff (2001): Expectation hypothesis holds well at super short term.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
13 / 37
Long-term Treasury yield decomposition
10−Year Yield Decomposition, %
8
Yield
Expectation
Risk premium
Convexity
6
4
2
0
−2
−4
92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12
Convexity effect becomes more significant, especially during volatile times.
Risk premium on long-term bond stays positive most of the time, can
account for half of the yield, but can vary significantly over time.
Caveat: The accuracy of long-run forecasts is hard to verify.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
14 / 37
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, the
accuracy 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-term
behaviors of both short and observed long rates.
We observe both short and long rates.
We know their near-term behavior better than long term.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
15 / 37
Near-term dynamics of all rates
Instead of being short-rate centric, we specify the near-term dynamics for rates
across 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 ) − γt σt (T )) dt + σt (T )dWtQ
Instantaneous yield changes are driven by a single shock (dWt ), matching
Cochrane&Piazessi single risk premium factor: A single market price of
risk 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 left
unspecified — Only their current levels and hence near-term dynamics are
specified.
Over longer term, yield curve variation can be driven by many factors that
govern the variation of expectation, risk, and risk premium.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
16 / 37
Decomposition based on near-term dynamics of all rates
Combining near-term dynamics of all rates,
dyt (T ) = (µt (T ) − γt σt (T )) dt + σt (T )dWtQ
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
∂B
1 ∂2B 2
+
(µt (T ) − γt σt (T )) +
σ (T ) = rt B,
∂t
∂y
2 ∂y 2 t
Re-arrangement leads to the following three-term decomposition:
1
yt (T ) = [rt + µt (T )τ ] − γt σt (T )τ − σt2 (T )τ 2 ,
2
a new, alternative decomposition based on near-term dynamics of all yields,
rather than the full-dynamics of the short rate.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
17 / 37
Different starting point, similar structure
No-arbitrage and continuous near-term dynamics of all yields lead to a new
decomposition:
1
yt (T ) = [rt + µt (T )τ ] − γt σt (T )τ − σt2 (T )τ 2 ,
2
Expectation is based on a linear interpolation between the short rate
and the near-term drift of the particular yield yt (T ).
Convexity is driven by the current volatility estimate of the particular
bond, 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 do
not matter for today’s decomposition.
Different starting points, similar term structure effects: The yield curve is
linear risk premium and quadratic in convexity if µ(T ), σ(T ) ∼ o(T )
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
18 / 37
Revist the EH regression under the new decomposition
The null hypothesis (a = 0 and b = 1) of the following EH regression
yt (T ) − rt
1
(yt+h (T ) − yt (T )) = a + b
+ et+h
h
τ
implies
1
yt (T ) − rt
(Et [(yt+h (T )] − yt (T )) = µt (T ) =
h
τ
Our no-arbitrage yield curve decomposition implies
yt (T ) − rt
1
+ γt σt (T ) + σt2 (T )τ,
τ
2
Even in the absence of risk premium (γt = 0)
µt (T ) =
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 are
negatively correlated — as they are, by no-arbitrage construction,
more so at longer maturities.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
19 / 37
New decomposition under the stylized example
drt = σdWt implies
dyt (T ) =
1
1 2
γσ + σ (T − t) dt + σdWt
2
3
Given flat expectation (zero drift) on short rate, the drift of fixed-expiry
yields is purely driven by the yield curve shape.
The decomposition becomes:
Expectation: r + µt (T )(T − t) = rt +
Risk premium: −γστ
Convexity: 12 σ 2 τ 2
1
2 γσ
+ 13 σ 2 τ ) τ
Combining them leads to yt (T ) = rt − 21 γστ − 16 σ 2 τ 2 as before.
The stylized example provides a link between
the partial, short-term dynamics of all long rates and
the full, long-term dynamics of the short rate.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
20 / 37
Decomposing yields of coupon-bonds (bond portfolios)
The yield to maturity on a coupon bond or bond portfolio can be defined
analogously,
X
m
Cj e yt τj
B(t, ytm ) ≡
j
where Cj denotes its cash flow at time t + τj .
We apply the same one-factor diffusion assumption to coupon-bond yields,
m
dytm = µm
t dt + σt dWt ,
where m is just an indicator of a particular coupon bond.
The same market price of risk γt would induce a drift adjustment γt σtm on
each bond under the risk-neutral measure.
While (σtm , µm
t ) can be bond-specific, the market price of the same risk γt
should be the same across all bonds for consistency.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
21 / 37
Coupon-bond yield decomposition
Assume dynamic no-arbitrage between bonds and the money market
account, we obtain an analogous decomposition for coupon-bond yields,
m
m
ytm = [rt + µm
t τ ] − γt σt τm −
1 m 2 2
(σ ) τm ,
2 t
(1)
where τm and τm2 denote value-weighted maturity (duration) and maturity
squared, respectively,
τm =
X Cj e −yt τj
j
Bm
τj ,
τm2 =
X Cj e −yt τj
j
Bm
τj2 .
With the weighted-average maturity and maturity squared definition, the
decomposition 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 only
depend on the current drift and volatility level of the yield-to-maturity on
each particular bond or bond portfolio.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
22 / 37
Predicting excess returns with no prediction on long rates
m
m
ytm = [rt + µm
t τ ] − γt σt τm −
1 m 2 2
(σ ) τm
2 t
The decomposition is equally applicable to spot rates or coupon-bond yields.
We can use historical variance estimators of daily yield changes to proxy σtm ,
without worrying about its long-run behavior.
It is well known that long-term, constant-maturity interest rates are virtually
impossible to predict (Duffee, 2002).
We impose this no-prediction principle on long-term rates and infer the
constant-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 examine
its behavior and implications.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
23 / 37
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 since
US: 2004/11/12 for 40 & 50 years
UK: 1999/1/19 for 20 & 30 years, 2003/08/08 for 40 & 50 years
US
UK
9
8
2y
10y
30y
50y
9
8
7
6
Swap Rates, %
Swap Rates, %
7
10
2y
10y
30y
50y
5
4
3
6
5
4
3
2
2
1
1
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
24 / 37
Swap rate variance estimators
Estimate variance σtm on each swap rate series with a 1y rolling window.
US
UK
2
2
2y
10y
30y
50y
1.8
1.4
1.2
1
0.8
0.6
1.6
1.4
1.2
1
0.8
0.6
0.4
0.4
0.2
0.2
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
2y
10y
30y
50y
1.8
Swap Rate Volatility, %
Swap Rate Volatility, %
1.6
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
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)
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
25 / 37
Convexity effects
Convexitym
t =
1 m 2 2
(σ ) τm
2 t
Treat swap rates as par bond yields, and compute value-weighted maturity
and maturity squared (τm and τm2 )
US
UK
2
2
2y
10y
30y
50y
1.8
1.4
1.2
1
0.8
0.6
1.6
1.4
1.2
1
0.8
0.6
0.4
0.4
0.2
0.2
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
2y
10y
30y
50y
1.8
Convexity Effect, %
Convexity Effect, %
1.6
0
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
Convexity effects are negligible on 2-year swap, but become quite significant
for longer-term swap rates, particularly so during crises.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
26 / 37
Extracting risk premium from long rates
Volatility is reasonably flat across maturities at the long end: σt0 (τ ) ≈ 0.
With no prediction on long-term constant-maturity rates, the drift of
long-dated constant-expiry rates is dictated by the yield curve shape:
µt (T ) = 0 − yt0 (τ )
Given yt (τ ) = rt + µt (T )τ − γt σt τ − 12 σt2 τ 2 , we have
µt (T ) = −µt (T ) − µ0t (T )τ + γt σt + σt2 τ
1
1
γt σt + σt2 τ
2
3
We can thus infer the market price of interest risk from long bonds based on
its slope and its variance estimator:
⇒
µt (T ) =
λt ≡ −γt =
ytm − rt + 16 (σtm )2 τm2
1 m
2 σt τm
A negative market price of rate risk γt implies positive expected excess
return 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 market
price of bond risk from swap rates with maturities 10 years and longer.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
27 / 37
Earning excess returns from long bonds
λt =
ytm − rt + 16 (σtm )2 τm2
1 m
2 σt τm
With no views on directions of long rate movements, one expects to make money
from long bonds in two forms:
1
The interest-rate differential between the long rate (which one receives) and
the 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 proportional
to the realized variance over the horizon of the bond, analogous to the
delta-neutral P&L of an option.
If rates across maturities are at the same level and move in syn in the
future, one makes more money from longer-term bonds due to
convexity, similar to delta-hedged gains from a long option posiiton.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
28 / 37
Market price of bond risk
UK
1.5
1
1
Market price of bond risk
Market price of bond risk
US
1.5
0.5
0
−0.5
0.5
0
−0.5
−1
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
−1
95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16
The market price of bond risk extracted from different rates are similar in
magnitude and move together, consistent with the one-factor assumption.
Over the common sample, the cross-correlation estimates among the
different 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, and
2007, but tended to be high during recessions.
In the UK, the market price became quite negative during 1998 and
2007-2008.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
29 / 37
Predicting bond returns with no-prediction on long rates
Correlation between ex ante risk premium (γt σtm ) and ex post excess returns on
each par bond, with the average denoting the correlation between the average risk
premium and the average bond excess return over the common sample period
Maturity
10
15
20
30
40
50
Average
Horizon: 6-month
US
0.31
0.28
0.26
0.24
0.26
0.25
0.28
UK
0.18
0.22
0.22
0.18
0.19
0.22
0.23
Horizon: One year
US
0.36
0.36
0.34
0.31
0.31
0.29
0.33
UK
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 lead
to significant prediction on bond excess returns.
The predictors (risk premium) are generated based purely on a variance
estimator and the current slope of the yield curve, without fitting of
predictive regressions.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
30 / 37
Predicting stock returns with market price of bond risk
Correlation between ex ante market price of bond risk and
on S&P 500 index and FTSE 100 index, respectively
Maturity
10
15
20
30
40
Horizon: 6-month
SPX
0.04
0.06
0.06
0.06
0.15
FTSE
0.23
0.28
0.28
0.35
0.35
Horizon: One year
SPX
0.12
0.13
0.13
0.13
0.24
FTSE
0.28
0.31
0.33
0.39
0.38
ex post excess returns
50
Average
0.13
0.37
0.16
0.35
0.22
0.40
0.25
0.39
The market price of bond risk reflects general market sentiment on risk
attitude, 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 available
sample-length difference.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
31 / 37
Link interest rate volatility to the yield curve curvature
1 m 2 2
(σ ) τm
2 t
Since interest rate volatility estimates are reasonably flat across maturities
σ 0 (τ ) ≈ 0, the curvature of the interest rate curve is at least partially driven
by the interest rate volatility.
m
m
ytm = [rt + µm
t τ ] − γt σt τm −
Absent from interest rate predictability, bond risk premium is chiefly driven
by 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 the
interest rate curve, we can represent the bond risk premium purely as a
function of the interest rate curve shape.
This may allow us to represent Cochrane and Piazzesi (2005)’s (CP) single
bond risk premium factor within our dynamic valuation framework.
CP represent risk premiums on 2-5 year bonds in terms of a single risk
factor made of 1-5 year forward rates.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
32 / 37
Map interest rate volatility to the forward rate curve
US
UK
−3
0.02
0.015
0.01
0.005
0
−0.005
−0.01
x 10
2
3
4
5
6
Variance loading
Variance loading
8
2
3
4
5
4
2
0
−2
1
2
3
4
Forward rate maturity, Years
5
−4
1
2
3
4
Forward rate maturity, Years
5
Strip the forward rate curve (with half-year tenor) from the LIBOR and swap
rates, and regress the historical 2-5 year swap rate variance estimators
against 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-implied
market price of risk.
Even with constant market price of risk, risk premium varies with variance,
and accordingly with the tent-shaped forward-rate portfolio.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
33 / 37
Map interest rate volatility to the forward rate curve
US
UK
−3
0.025
0.02
Variance loading
0.015
0.01
0.005
0
−0.005
6
x 10
10
15
10
30
40
50
4
Variance loading
10
15
10
30
40
50
2
0
−2
−0.01
−4
−0.015
−0.02
1
2
3
4
Forward rate maturity, Years
5
−6
1
2
3
4
Forward rate maturity, Years
5
Variance on longer-dated rates have similar “tent-shaped” loading on
forward rates, which capture the forward curve curvature.
Regression R 2 :
Mat 2
3
4
5
7
10
15
20
30 40
50
US 0.54 0.52 0.51 0.50 0.49 0.43 0.42 0.36 0.33 0.43 0.39
UK 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 between
between cap/swaption implied volatility levels and the level of the interest
rate curvature factor.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
34 / 37
Unspanned volatility and the convexity effect
USV says that short-term interest rate volatility changes cannot be explained
by 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 is
proportional to the level of interest rate variance.
Our theoretical decomposition
Our empirical evidence: regress variance levels on forward rates
Heidari & Wu (2003): Relation between option implied vol levels and
the level of the interest rate curvature factor
Cochrane&Piazessi (2005): the tent shaped forward rate portfolio as
the single risk premium factor
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
35 / 37
Concluding remarks
We decompose bond yields into three components:
1
2
3
expectation,
risk premium, and
convexity
We perform the decomposition from two perspectives:
1
2
The long dynamics of the short rate
The short dynamics of long rates
The decompositions reveal several insights:
Convexity effect is negligible at short maturities, but can become very
important for super long rates.
From the decomposition, we can extract bond risk premium from
historical variance estimators and either professional survey forecasts on
future short rates or the assumption of no prediction on long rates.
The extracted bond risk premium predicts future excess returns on
both bonds and stock indexes.
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
36 / 37
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 purely
be driven by the convexity effect, even in the absence of risk premium.
The Cochrane and Piazessi (2005) tent-shaped forward rate portfolio risk
premium factor can be a proxy, at least partially, for the convexity effect, or
equivalently 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, risk
premium on long-term bonds is a combination of the yield curve slope
and the convexity effect.
While the EH literature misses the convexity term, the CP bond excess
return regression may have just picked it up.
Trading a butterfly is just like trading a straddle...
Carr and Wu (NYU & Baruch)
Decomposing Yield to Maturity
June 4th, 2016
37 / 37