Risk‐Free Rates and Variance Futures Prices

• Published date: 01 October 2016
• Author: Leonidas S. Rompolis
• Date: 01 October 2016
• DOI: http://doi.org/10.1002/fut.21767

Risk-Free Rates and Variance

Futures Prices

Leonidas S. Rompolis*

This paper investigates the relation between risk-free rates and ex-ante market volatility. It

derives a theoretical model implying a negative linear relation between risk-free rates and

variance futures prices. The latter are employed as a direct market-based ex-ante estimate

of risk-neutral volatility. Empirical analysis, conducted using LIBOR and variance futures

prices written on the S&P 500 index, indicates that the predictions of the model are sup-

ported by the data. The paper also provides evidence that, first, this negative relation varies

smoothly over time following business cycles, and, second, the variance risk premium is a sig-

nificant component of this documented relation. ©2015 Wiley Periodicals, Inc. Jrl Fut Mark

36:943–967, 2016

1. INTRODUCTION

Standard asset pricing theory claims that current risk-free rates are related to the ex-ante

conditional volatility of the stochastic discount factor, known as the pricing kernel. This

relation should be negative. It implies that the higher the variance of the stochastic discount

factor, the more investors are willing to invest in risk-free bonds, thus increasing their prices

and decreasing their yields. In a consumption-based asset pricing model, the volatility of

the stochastic discount factor is related to the volatility of consumption growth. When con-

sumption is more volatile, investors are more worried about the low-consumption states than

they are pleased by the high-consumption states. Therefore, they want to save more, driving

down interest rates. Thus, consumption volatility reflects precautionary savings (Cochrane,

2001).

The above theoretical predictions of a consumption-based asset pricing model are tested

in several studies. Weil (1989) examined the above relation in an unconditional setup using

ex-post data. His results lead to the well-known “risk-free rate puzzle.” However, his results

are only ex-post indications, as the model sets a conditional ex-ante equilibrium. Del Castillo

and Fillion (2002) and La Bruslerie and Fouilloux (2007) propose an econometric framework

of consumption volatility assuming that the latter follows a GARCH model. As this framework

estimates the ex-ante volatility of the consumption process, a conditional version of the

theoretical model can be examined. However, even in that case historical consumption data

are required to estimate the ex-ante forecast conditional volatility.

Leonidas S. Rompolis is the Assistant Professor at the Department of Accounting and Finance, Athens Uni-

versity of Economics and Business, 76 Patissionstreet, 10434 Athens, Greece. The author would like to thank

Aggeliki Noti for excellent research assistance. I also thank Elias Tzavalis, George Chalamandaris, and two

anonymous referees for comments and suggestions that greatly improved the paper. Financial support from

Research Center of Athens University Economics and Business (EP-1787-01) is gratefully acknowledged.

*Correspondence author,Assistant Professor, Department of Accounting and Finance, Athens University of Eco-

nomics and Business, 76 Patission street, 10434 Athens, Greece. Tel: +30-2108203465, Fax: +30-2108228816,

e-mail: rompolis@aueb.gr

Received September 2013; Accepted October 2015

The Journal of Futures Markets, Vol. 36, No.10, 943–967 (2016)

©2015 Wiley Periodicals, Inc.

Published online 23 December 2015 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21767

944 Rompolis

In this paper, to test the relation between risk-free rates and ex-ante conditional market

volatility, we develop a model based on the pricing kernel projected onto the return states

of a particular asset like, for example, the market portfolio (Rosenberg & Engle, 2002).

Assuming a specific form of this projected pricing kernel, motivated by the well-known

power/logarithmic utility function, we demonstrate that current risk-free rates are nega-

tively related to the ex-ante conditional variance of the asset log-return, where the pricing

kernel is projected onto its space under the risk-neutral measure. In contrast to the ex-ante

conditional variance of consumption, the latter is nowadays traded in the market. Thus, its

value can be directly observed. The financial instrument through which variance is traded is

the variance futures contract. Therefore, the model implies that risk-free rates and variance

futures prices are inversely related. As variance futures prices increase, indicating either

that ex-ante conditional variance under the physical measure or the variance risk premium

increases in magnitude, investors increase their precautionary savings, which drives down

interest rates. The intensity of this relation depends on the projected relative risk aversion co-

efficient. As this coefficient increases, the effect of variance futures prices on risk-free rates

also increases. The advantage of this approach, compared to that followed by the above-

mentioned consumption-based ones, is that we can empirically test the relation between

risk-free rates and ex-ante conditional market volatility without relying on model-based esti-

mates of volatility using historical data. This comes from the fact that variance futures prices

can be used as direct and model-free estimates of ex-ante conditional variance provided by

the market.

In the empirical part of the paper, we investigate if our theoretical predictions are sup-

ported by market data. Tothis end, we use the British Banker’s Association USD LIBOR, with

various maturity intervals, as a proxy of risk-free rates and variance futures prices written on

the S&P 500 index traded on the CBOE. The sample period covers the years 2004–2010,

which includes both business cycle expansions and contractions. We then examine both the

long- and short-run dynamic relation between the above two series using cointegration anal-

ysis as these are found to be I(1). The results of the empirical analysis support the predictions

of the model. We find a significant long-run negative relation between LIBOR and variance

futures prices across different maturity intervals examined. These results also indicate that

short-term rates are more affected by a change in variance futures prices than the longer-

term ones. This empirical finding provides evidence that investors are less averse toward risk

in long-run than in short-run (Bliss & Panigirtzoglou, 2004). Our results also reveal that LI-

BOR respond to the deviations from the long-run equilibrium, while variance futures prices

are weakly exogenous. These responses could be implemented through monetary policy. A

positive shock in variance futures prices, distorting the long-run equilibrium relationship,

may be a signal for monetary authorities to react by lowering real interest rates as suggested

in the literature (Rigobon & Sack, 2003). They also indicate that the variance futures market

offers higher error correcting power on the long-term LIBOR quotes than on the short-term

ones. This may be due to the fact that investors are more exposed to volatility risk in long-run

than in short-run (Bakshi & Kapadia, 2003).

In a second part of the empirical study,we extend the above framework employing a time-

varying cointegration analysis. This is motivated by the fact that a number of studies provide

evidence that the projected relative risk aversion coefficient, which is included in the cointe-

grating vector, is time-varying (Rosenberg & Engle, 2002; Bliss & Panigirtzoglou, 2004). The

results of this analysis indicate that the relation between LIBOR and variance futures prices

is indeed time-varying and follows business cycle conditions. From the beginning of the

sample period to December 2007, where the U.S. economy slipped into recession (following

the National Bureau of Economic Research definition), we have found that their relation

is stable. After December 2007, the strength of the relation between LIBOR and variance