Predicting Interest Rate Volatility Using Information on the Yield Curve

• DOI: http://doi.org/10.1111/irfi.12053
• Published date: 01 September 2015
• Date: 01 September 2015
• Author: Hideyuki Takamizawa

Predicting Interest Rate Volatility

Using Information on the

Yield Curve*

HIDEYUKI TAKAMIZAWA

Graduate School of Commerce and Management, Hitotsubashi University,

Kunitachi, Tokyo, Japan

ABSTRACT

This study examines whether information on the yield curve is useful for

predicting volatility of the yield curve. The information is used within

dynamic models by specifying the covariance matrix of changes in yield

factors as nonlinear functions of the factors. Using such models, it is found

that the information (i) is useful for predicting volatility of the slope factor,

achieving the accuracy comparable with the GARCH model; (ii) has incre-

mental value for predicting volatility of the curvature factor when combined

with a volatility-specific factor; and (iii) does not much improve prediction of

volatility of the level factor once the volatility-specific factor is introduced.

JEL codes: C58, E43, G12, G17.

I. INTRODUCTION

It does not seem unreasonable to think that the current yield curve contains

some information on the volatility of changes in interest rates. In making bond

portfolios or managing interest rate risks, investors will take account of condi-

tional second moments of bond returns or yield changes. The resulting shape of

the yield curve will then reflect investors’ views toward volatility. The purpose

of this study is to examine whether information on the yield curve is useful for

predicting the volatility of the yield curve.

The idea of relating interest rate volatility to the yield curve is not new.

Brown and Schaefer (1994), Christiansen and Lund (2005), Joslin (2010),

Litterman et al. (1991), and Phoa (1997) relate the volatility to the curvature, or

convexity, of the yield curve. Time series studies using long historical data on

US interest rates find a relation between the volatility and the level of a

particular yield, especially the short-term rate, such that high volatility is

accompanied by high level (see, e.g., Chan et al. 1992; Andersen and Lund

1997a; Gallant and Tauchen 1998; Ball and Torous 1999; Durham 2003).

* I am grateful to the anonymous referee for comments, which enabled me to significantly improve

this manuscript. This work was supported by JSPS KAKENHI Grant Number 23730212, SEIMEIKAI

research grant, and NOMURA Foundation research grant.

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International Review of Finance, 15:3, 2015: pp. 347–386

DOI: 10.1111/irfi.12053

© 2015 International Review of Finance Ltd. 2015

This simple level–volatility relationship, however, no longer seems to be a

decisive feature for relatively recent data. Figure 1 shows the time series of the

level and realized volatility of the first principal component (PC) constructed

from US dollar LIBOR and swap rates over 1991–2009: The details of these data

are provided in section II. Note that the first PC (PC1) is interpreted as a level

0.00

0.05

0.10

0.15

0.20

91 93 95 97 99 01 03 05 07 09

0.00

0.01

0.02

0.03

0.04

0.05

91 93 95 97 99 01 03 05 07 09

(a) Level

Annualized 4-week Standard Deviation

(b)

Figure 1 Time series of the level and realized volatility of first principal component

(PC1) over 1991–2009. Panel (a) presents the time series of the level of the PC1. Panel

(b) presents the time series of the realized volatility (annualized 4-week standard

deviation) of PC1, which is calculated by summing squared daily changes in PC1 over

the 4 weeks, dividing the sum by 4/52, and taking the square root. The vertical dotted

line separates the in-sample and out-of-sample periods.

International Review of Finance

348 © 2015 International Review of Finance Ltd. 2015

factor of the yield curve. It is observed that sharp rise in the volatility of PC1

around 2001–03 and 2008–09 is actually accompanied by the fall in the level of

PC1.

It is therefore not surprising that more recent studies using these data are

skeptical about the possibility of extracting volatility information from the

yield curve. Andersen and Benzoni (2010) test affine spanning conditions that

yield variances, both ex ante and ex post, can be expressed by some linear

combinations of yield levels if affine term structure models are true, and reject

these conditions. A direct implication of this result is that the relationship

between the volatility and the curvature of the yield curve is not supported by

the data because the curvature is normally measured by a linear combination of

yields. Collin-Dufresne et al. (2009) and Jacobs and Karoui (2009) reported that

yield variances extracted from the cross-section of yields through affine term

structure models do not behave similarly to typical variance measures com-

puted from time series data.

It is too early to conclude, however, that the yield curve is of little relevance

to the volatility. Although information on the cross-section of yields alone may

not be rich enough to identify volatility-specific factors, it may still be useful if

it is combined with information on the time series of yields. Furthermore,

nonlinear relationships between variances and levels of yields may exist even

though a linear relationship as implied by the affine models is not supported.

This study explores these possibilities that are not fully studied by the

earlier work. To combine information on the cross-section of yields with infor-

mation on the time series of yields, dynamic models of yield factors, rather

than regression models, are employed. Then, nonlinear relationships between

variances and levels of yields are incorporated into the dynamic models. Spe-

cifically, the covariance matrix of changes in yield factors is specified as non-

linear functions of the factors themselves. This is how information on the

yield curve is used: It is used for specification, but not for extraction, of the

volatility.

As such, the approach of examining information content of the yield curve

with respect to the volatility is different from that in the earlier work. Bikbov

and Chernov (2011) and Thompson (2008) use no arbitrage affine models with

particular attention to whether model-implied behavior of the volatility

changes by modifying estimation methods or adding option data. This study

uses both affine and nonaffine models and estimates them using only time

series dimension of interest rate data with particular attention to whether

information on the yield curve is useful for predicting the volatility. Also, the

dynamic models of the yield curve are different from those in the earlier work.

Christiansen (2005) embeds the GARCH volatility. Christiansen (2004) and

Pérignon and Smith (2007) incorporate regime switching into volatility mod-

eling. This study considers a volatility-specific factor that has a similar role to

the GARCH volatility, but not regime switching. Instead, the models explored

here are characterized by more flexible level-dependent specifications of vola-

tility. Using such models, this study uncovers both usefulness and limitations

Interest Rate Volatility

349© 2015 International Review of Finance Ltd. 2015