Methodological Considerations in the Statistical Modeling of Catastrophe Bond Prices

• Published date: 01 March 2019
• Author: John A. Major
• DOI: http://doi.org/10.1111/rmir.12114
• Date: 01 March 2019

Risk Management and Insurance Review

C

Risk Management and Insurance Review, 2019, Vol.22, No. 1, 39-56

DOI: 10.1111/rmir.12114

FEATURE ARTICLE

METHODOLOGICAL CONSIDERATIONS IN THE STATISTICAL

MODELING OF CATASTROPHE BOND PRICES

John A. Major

ABSTRACT

The problem of specifying and fitting a statistical model of the pricing of prop-

erty catastrophe risk is addressed from a methodological perspective. Notable

21st century published efforts to do this are reviewed. The problem is framed

in a business context and various strategic and tactical issues are investigated.

Ana

¨

ıve application of ordinary least squares regression is seen to have unde-

sirable consequences. Alternative approaches are offered, including weighted

least squares with weights inversely proportional to capital requirements, and

alternative functional forms. Recommendations are offered.

INTRODUCTION

Insurance risk pricing is typically not modeled in the same way that corporate secu-

rities and derivative pricing is usually modeled. Only a small portion of insurance

risk––catastrophe bonds (often referred to as CAT bonds) and other insurance-linked

securities––is traded in public markets, and even there, the volume and breadth of offer-

ings is tiny compared to stock and bond markets. Thus, no-arbitrage or efficient portfolio

principles, justifiable from complete-market assumptions, do not apply. See Major and

Kreps (2002) for a more extensive discussion.

Statistical modeling, relating the empirically observed price to other variables including

some imputed or estimated underlying physical risk metric, becomes the preferred

approach. This can be seen in Lane (2000), Major and Kreps (2002), Wang (2004), Lane

and Mahul (2008), Bodoff and Gan (2009), Galeotti et al. (2013), Braun (2016), and G¨

urtler

et al. (2016).

In the academic finance literature, a “risk-neutral, structural” approach has tended to

dominate. This is exemplified by the Merton (1974) approach which in effect relates

price to risk through a Wang (2000) transform. See Berg (2010) for connections between

various approaches. However, statistical modeling of “real world” pricing and risk has

also sometimes been used in the corporate securities and credit risk market context, for

example, Heynderickx et al. (2016), Berndt et al. (2005), and Denzler et al. (2006).

The purpose of this article is to address some methodological issues that bear upon the

specification and fitting of an empirical insurance risk pricing model. Substantive advice,

John A. Major is Director of Actuarial Research, Guy Carpenter & Company, LLC, 1166 Avenue

of the Americas, New York, NY 10036; e-mail: john.a.major@guycarp.com.

39

40 RISK MANAGEMENT AND INSURANCE REVIEW

e.g., what predictors may be expected to perform better than others, is outside the scope

of this article.

To be more concrete, we restrict ourselves to situations where:

1. The instrument being priced is a contingent claim of finite duration (typically

1–5 years) where the buyer pays a fee that is agreed-upon prior to the contract

period, and the seller’s payment of the claim is a random variable between zero

and some fixed maximum amount.

2. The criterion (dependent) variable being modeled is the fee.

3. Predictor (independent) variables include an estimate of the expected value of the

claim payment.

4. Historical data are used to fit the model.

This situation is typical of catastrophe risk, notably catastrophe bonds, but also holds

for other risks including corporate bonds. In the insurance/reinsurance context, the risk

contract is the direct object of the transaction. In the context of catastrophe or corporate

bonds, the buyer of the bond is the seller of an implicit, embedded risk contract, and the

risk fee is part of the stream of coupons paid by the bond issuer. See Galeotti et al. (2013)

or Braun (2016) for a description of catastrophe bond mechanics.

For a given contract, the fixed maximum payment is referred to as the limit,andthe

ratio of the fee to the limit is referred to as the Rate on Line (ROL). The ratio of the

expected value of the claim payment to the limit is referred to as the Expected Loss (EL).

In the reinsurance industry, this last ratio is usually referred to as “loss on line,” with the

EL being dollar-denominated, but in this article we adhere to the conventional labeling

used in the academic literature on catastrophe bond pricing. For catastrophe reinsurance

contracts and some catastrophe bonds, payments begin when the firm’s covered losses

exceed a certain amount, known as the retention or attachment point. This is analogous to

the deductible in personal insurance. The probability of losses exceeding that point, or,

more generally, the probability that the contract or cat bond will pay for any losses at

all, is known as the attachment probability or probability of first loss (PFL).

The modeling task, then, is to draw conclusions about ROL given EL and ancillary

information about the contract. These conclusions may take several forms. The fullest

expression would be a conditional probability distribution of ROL, reflecting observed

variability in the historical data as well as uncertainty arising from the estimation process.

More restricted forms might be an expected value or median value of ROL.

Getting from the data to such an answer involves settling certain strategic and tactical

questions. The strategic questions revolve around the goals of the exercise:how is success

to be defined and measured, etc. The tactical questions revolve around techniques to be

applied to achieve the goals.

This article is divided into four sections. Section “Literature Summary” reviews the liter-

ature cited above. Section “Strategic Issues and Tactical Choices” examines the strategic

and tactical issues, in light of the literature, in an attempt to identify appropriate design

choices. Section “Conclusion” concludes.