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Nanyang Business School Forum on Risk Management and Insurance

Ordinary and Markov-Switching Autoregressive Models for Firm-Level Underwriting Data

by | Oct 16, 2018 | Actuarial Pricing, Non-life insurance, Statistics, Underwriting | 0 comments

Tags: Underwriting-profitability, Regimes, Cycles, Autoregressive Process, Markov-Switching
More from: Frank Y. Feng, Michael R. Powers

Editor’s Note: Posted by Michael R. Powers, Zurich Group Chair Professor of Risk Mathematics, SEM Professor of Economics and Business, Schwarzman College, Tsinghua University; Frank Y. Feng, School of Finance and Shanghai Institute of International Finance and Economics, Shanghai University of Finance and Economics. This paper received the 2018 APRIA Harold D. Skipper Best Paper Award.

The analysis of underwriting-profitability regimes has formed an important topic in property-liability (P-L) insurance research since the mid-1980s (see Stewart, 1984; and Venezian, 1985). The conventional wisdom among both practitioners and researchers is that P-L markets are characterized by patterns of alternation between two distinct regimes: “hard” markets, in which insurance prices rise and coverage becomes more restricted; and “soft” markets, in which prices decrease and capacity increases. Most of the scholarly literature infers, either explicitly or implicitly, that these alternating regimes follow a regular, sinusoidal pattern that can be described as an “underwriting cycle” (see Weiss, 2007, for a survey of relevant articles). However, recent critiques have questioned the cyclical nature of such time series: Powers (2011) cited the empirical work of Venezian and Leng (2006) to assert that true cycles were relatively rare in aggregate market , and that the assumption of cyclicality is an example of “scholarly pareidolia”; whereas Boyer, Jacquier, and Van Norden (2012) argued that parameter estimates from conventional autoregressive models are insufficient to support conclusions of cyclicality.

Whether or not underwriting regimes are truly cyclical, the occurrence of alternating hard and soft markets has important implications for insurance practice. In particular, the price fluctuations associated with changing regimes make it difficult both for insurance companies to establish long-term strategies to manage their exposure portfolios, and for government regulators to make timely decisions regarding the companies’ financial health. For these reasons, insurance researchers have long sought economic explanations for the alternating underwriting regimes. Moreover, the assumption of cyclicality has made the problem even more puzzling, because regularly occurring profitability patterns would suggest a violation of the efficient-market hypothesis.

Two widely discussed theoretical explanations for the existence of alternating underwriting regimes are: the arbitrage theory proposed by Cummins and Outreville (1987), which attributes the ups and downs of insurance-company profitability to regulatory requirements and other institutional lags; and the capacity-constraint theory of Winter (1994), which suggests that occasional exogenous shocks affect insurance-company leverage, thus triggering a series of alternating hard- and soft-market phases as companies adjust to the new environment. Recently, Henriet, Klimenko, and Rochet (2016) integrated these two approaches into a comprehensive, dynamic mathematical model, from which they predicted an asymmetry in the duration of hard and soft markets (namely, that soft markets should be somewhat longer than hard markets).

Empirical research generally has focused on efforts to confirm the existence of underwriting cycles – and estimate their regular, sinusoidal periods – by fitting AR(2) models to aggregate P-L profitability time series (usually either combined ratios or underwriting rates of return). Articles in this extensive literature have explored: differences in underwriting patterns by country; the impact of exogenous macroeconomic variables (especially interest rates) on underwriting results; and the statistical characteristics of profitability , including nonstationarity, cointegration with other variables, and structural changes. (See, e.g., Venezian, 1985; Cummins and Outreville, 1987; Haley, 1993; Chen, Wong, and Lee, 1999; Leng, Venezian, and Powers, 2002; Harrington and Yu, 2003; Leng and Meier, 2006; and Meier and Outreville, 2006.)

Motivated by the model of Henriet, Klimenko, and Rochet (2016), which offered a theoretical explanation for alternating hard and soft markets without any explicitly cyclical mechanism, the present authors investigated the cyclicality and symmetry of underwriting regimes in Feng et al. (2017). The latter paper, which was among the first to apply a Markov-switching autoregressive (MS-AR) model to historical underwriting-profitability, showed that a univariate MS-AR(2) model with two alternating regimes fits U.S. P-L market better than an ordinary AR(2) model. Based upon this result, Feng et al. (2017) argued that a Markovian regime-switching mechanism is more appropriate than autoregressive/cycle formulations, thus supporting the conclusions of Powers (2011) and Boyer, Jacquier, and Van Norden (2012). The authors also found insufficient evidence to support a prediction of Henriet, Klimenko, and Rochet (2016) that, on the average, the durations of hard markets tend to be shorter than those of soft markets.

The present work extends that of Feng et al. (2017) by applying both ordinary and Markov-switching AR models to firm-level P-L from the United States. The investigation employs both univariate and multivariate methods to address the following important questions:

(1) Are individual-firm underwriting patterns heterogeneous, deviating significantly from industrywide patterns? and
(2) Do relationships among individual-firm underwriting patterns show evidence of firm competition and/or collaboration?

In addition to reinforcing the conclusions of Feng et al. (2107), our analysis argues against the existence of distinct, firm-level regimes in the U.S. property-liability market, but offers evidence of cross-company interactions over time. More specifically:

• Question (1) is answered in the negative by our univariate analyses, which provides evidence against the existence of distinct, firm-level underwriting regimes in the U.S. property-liability market; and

• Question (2) is answered in the affirmative by our multivariate analyses, which identifies the presence of intertemporal cross-company interactions.

These observations offer new insights into the nature of underwriting-profitability patterns, thus affording potential tools for both insurance companies and regulators to enhance the management of risk.

In addition to the above conclusions, we also find further support for the arguments of Feng et al. (2107) that: (a) MS-AR(2) models fit U.S. P-L underwriting-profitability better than ordinary AR(2) models; and (b) there is insufficient evidence to reject a null hypothesis of durational symmetry between hard and soft markets.

The complete paper is available at:


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