Decomposing Yield to Maturity - Baruch College

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 Expectation: market expectation about future interest rate movements 2 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 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

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

dQ dP

e-

T t

ru du

= EQt

e-

T t

ru du

.

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 )

- ln Bt (T ) T -t

=

- T

1 -

t

ln EQt

e-

T t

ru du

,

we can decompose the yield into three components

yt (T ) =

T

1 -t

EPt

T t

ru du

+

T

1 -t

EPt

dQ dP

-

1

T t

ru du

-

1 T -t

ln EQt

exp

-(

T t

rudu - EQt

T t

ru du )

(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 = T - t to denote time to maturity.

Carr and Wu (NYU & Baruch)

Decomposing Yield to Maturity

June 4th, 2016 5 / 37

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