Optimum Hurricane Futures Hedge in a Warming Environment: A Risk–Return Jump‐Diffusion Approach

• Author: Jack S. K. Chang,Carolyn W. Chang,Min‐Ming Wen
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01492.x
• Published date: 01 March 2014
• Date: 01 March 2014

© The Journal of Risk and Insurance, 2014, Vol. 81, No. 1, 199–217

DOI: 10.1111/j.1539-6975.2012.01492.x

199

OPTIMUM HURRICANE FUTURES HEDGE IN A WARMING

ENVIRONMENT:ARISK–RETURN JUMP-DIFFUSION

APPROACH

Carolyn W. Chang

Jack S. K. Chang

Min-Ming Wen

ABSTRACT

We develop an optimum risk–return hurricane hedge model in a doubly

stochastic jump-diffusion economy. The model’s concave risk–return trade-

off dictates that a higher correlation between hurricane power and insurer’s

loss, a smaller variable hedging cost, and a larger market risk premium

result in a less costly but more effective hedge. The resulting hedge ratio

comprises of a positive diffusion, a positive jump, and a negative hedging

cost component. Numerical results show that hedging hurricane jump risks

is most crucial with jump volatility being the dominant factor, and the faster

the warming the more pronounced the jump effects.

INTRODUCTION

Economic losses from climate change are substantial and on the rise.1The risk of

property damage and loss from hurricanes has been progressively magnified by

the ongoing growth of coastal population and property values.2Scientific evidence3

further finds that an increasing level of greenhouse gas concentrations would lead

Carolyn W. Chang works in the Department of Finance, California State University, Fullerton.

Jack S. K. Chang and Min-Ming Wen work with the Department of Finance & Law, California

State University, Los Angeles. The authors can be contacted via e-mail: cchang@fullerton.edu.

We gratefully acknowledge the financial assistance from the Center for Insurance Studies at

California State University,Fullerton, for a CIS Faculty Research Award.

1Catastrophic weather-related insurance losses in the United States have been rising 10 times

faster since 1971 than premiums, population, or economic growth.

2See Derrig et al. (2008) for discussions regarding coastal property growth and the challenge

faced by the insurance industry,and Kriesel and Landry (2004) and Landry and Jahan-Parvar

(2011) for empirical evidence regarding flood insurance participation in the coastal zone.

3Among the evidence provided is a comprehensive idealized hurricane intensity modeling

study by Knutson and Tuleya(2004), which uses future climate projections from nine different

global climate models and four different versions of the Geophysical Fluid Dynamics Lab

(GFDL) hurricane model. According to this study,an 80-year buildup of atmospheric CO2at

1 percent/year with compounding leads to roughly a one-half category increase in potential

hurricane intensity on the Saffir-Simpson scale and an 18 percentincrease in precipitation near

200 THE JOURNAL OF RISK AND INSURANCE

to increased severity and frequency of tropical cyclones. These bleak projections,

combined with the limited capacity of the insurance industry due to regulations,

transaction costs, and geographic concentration of insured risk, subject insurers to

facing a new era of climate risks management needing new hedging strategies.

Traditionally, insurance companies seek protection against underwriting risk through

the mechanism of reinsurance, but worldwide capacity shortage coupled with the ris-

ing number of catastrophes in recent years have made securitization of catastrophic

losses on both exchanges and over-the-counter markets a timely and desirable func-

tional alternative. A number of catastrope (CAT)-linked instruments, such as Hurri-

cane Risk Landfall Options (HuRLOs) launched by Weather Risk Solutions in 2008,

Hurricane CATBonds (e.g., one launched by USAA Inc. in 1997), Hurricane CatEPuts,

and the exchange-traded Hurricane Futures and Futures Options have been devel-

oped as a result. Among them, the exchange-traded hurricane futures market took off

after the 2004–2005 storms saddled the insurance industry. For insurance companies,

hurricane futures can provide additional coverage even if they have already bought

traditional backup insurance. Similarly,banks, utilities, energy companies, state gov-

ernments, and pension funds have strong motivations to seek additional coverage if

a massive hurricane or a series of storms make landfall in their region.

One of the important issues in futures hedging is the variables considered in the

objective function of the optimization problem. The traditional risk-minimization

futures hedging approach developed by Johnson (1960) and Edrington (1979) for-

mulates optimum futures hedge by assuming a diffusion underlying process, and

then minimizing the one-period risk of the hedge portfolio using the least square

regression or its variations.4Corporate hedging literature has, however, consistently

found that actual hedge ratios for commodity firms are significantly lower than those

prescribed under the risk-minimization approach, because while making a hedging

decision, firms consider not only risk minimization but also the hedging costs in-

curred. These hedging costs comprise of the fixed cost of setting up and maintaining

a hedge program and the variable costs from the forgone expected return and from

the hurricane core. A 1 percent/year CO2increase is an idealized scenario of future climate

forcing. An implication of the study is that if the frequency of tropical cyclones remains the

same over the coming century,a greenhouse-gas induced warming may lead to an increasing

risk in the occurrence of highly destructive Category 5 storms. Elsner, Kossin, and Jagger

(2008), Webster et al. (2005), and Emanuel (2005) provide empirical evidence that global

warming has been increasing the severity and destructive power of hurricanes over the past

few decades.

4More recent literature has paid attention to the incorporation of the impacts of random jumps,

stochastic volatility,and stochastic basis. Brennan and Schwartz (1990) incorporate stochastic

mean-reverting basis when studying the optimum strategy of an arbitrageur. Chang, Chang,

and Fang (1996) derive a two-factor optimum futures hedge model in a jump-diffusion frame-

work with stochastic basis. Schwartz (1997) applies a Kalman filter empirical methodology

to estimate mean reversion in the basis change to investigate commodity futures pricing and

hedging. Recognizing that hedgers are only concerned with downside risk, Lien and Tse

(1998) employ a bivariate APARCH-M model for the spot and futures return to derive time-

varying lower partial moment hedge ratios. Hilliard and Reis (1998) develop a three-factor

jump-diffusion model incorporating stochastic interest rates and stochastic convenience yield

to value futures and futures options.