Insurance Ratemaking and a Gini Index

• Author: Edward W. (Jed) Frees,A. David Cummings,Glenn Meyers
• Published date: 01 June 2014
• DOI: http://doi.org/10.1111/j.1539-6975.2012.01507.x
• Date: 01 June 2014

©

DOI: 10.1111/j.1539-6975.2012.01507.x

335

INSURANCE RATEMAKING AND A GINI INDEX

Edward W. (Jed) Frees

Glenn Meyers

A. David Cummings

ABSTRACT

Welfare economics uses Lorenz curves to display skewed income distribu-

tions and Gini indices to summarize the skewness. This article extends the

Lorenz curve and Gini index by ordering insurance risks; the ordering vari-

able is a risk-based score relative to price, known as a relativity. The new

relativity-based measures can cope with adverse selection and quantify po-

tential profit. Specifically, we show that the Gini index is proportional to a

correlation between the relativity and an out-of-sample profit(price in excess

of loss). A detailed example using homeowners insurance demonstrates the

utility of these new measures.

INTRODUCTION

In today’s world of readily available information and interest in business analytics,

insurers seek to utilize nontraditional information in their ratemaking and under-

writing processes. Table 1 shows some nontraditional rating variables that are being

considered and used by some insurers.

Tosee how insurers can use new information in the rate making process, we first con-

sider the traditional premium. To set notation, we wish to compare the distribution

of an insurance loss yto a price Pin a portfolio of risks. For each risk in the portfolio,

we assume that the analyst has available a set of known exogenous risk character-

istics xupon which both the loss and price distributions depend. We emphasize the

dependence of the price on the risk characteristics xthrough the notation P=P(x).

Ignoring expenses, in an efficient marketplace, the price Pwill be close to the cost y.

This is a difficult proposition for the marketplace to ensure because:

•yis random with a distribution of outcomes.

•The distribution of yis complex. To illustrate this complexity, in typical homeown-

ers insurance data described in the “Homeowners Example” section, 94 percent of

Edward W. (Jed) Frees is at the University of Wisconsin and ISO Innovative Analytics. Glenn

Meyers is at ISO Innovative Analytics. A. David Cummings is at ISO Innovative Analytics.

The authors can be contacted via e-mail: jfrees@bus.wisc.edu, ggmeyers@metrocast.net, and

DCummings@iso.com, respectively.

The Journal of Risk and Insurance, 2013, Vol. 81, No. 2, 335–366

336 THE JOURNAL OF RISK AND INSURANCE

TABLE 1

Nontraditional Rating Variables by Line of Business

Type of Insurance Nontraditional Characteristics

Personal automobile Credit score, homeownership, prior bodily injury limits

Homeowners Insurance credit score, prior loss information, age of home

Workers compensation Safety programs, number of employees, prior loss information

Commercial gen liability Insurance credit score, years in business, number of employees

Medical malpractice Patient complaint history, years since residency

Commercial automobile Driver tenure, average driver age, earnings stability

Source: Werner and Modlin (2010).

the losses are zeros (corresponding to no claims) and when losses are positive, the

distribution tends to be right skewed and thick tailed.

•There are many different sets of insureds, corresponding to a variety of xis.

•There are many different contract variations, corresponding to different de-

ductibles, limits, coverages, riders, and so forth.

One point of view is that the premium should be the expected loss. This viewpoint

is supported in the context of (1) many independent contracts and (2) a competitive

marketplace.

Wesuppose that the insurer introduces a new risk-based score, denoted by S(xi). This

score may be based on a new rating algorithm, for example, from a modern statistical

approach such as the generalized linear model. Alternatively or additionally, it may

be based on new information, for example, a new variable such as in Table 1. To

assess this score, it is common (e.g., Wernerand Modlin, 2010) for insurers to examine

a relative premium

R(xi)=S(xi)

P(xi),

known as a relativity.

Suppose that the score Sis a desirable approximation of the expected loss. Then, if

the relativity is small, we can expect a small loss relative to the premium. This is a

profitable policy but is also one that is susceptible to potential raiding by a competitor

(adverse selection). In contrast, if the relativity is large, then we can expect a large loss

relative to the premium. This is one where better loss control measures, for example,

renewal underwriting restrictions, can be helpful.

Using these relativities, this article shows how to form portfolios of policies and

compare losses to premiums to assess profitability. This is the purpose of the ordered

Lorenz curve. If the insurer can form profitable portfolios, then the competition may

also be able to do so, inviting potential raiding by competing firms. To provide

protection, the insurer may use the scores:

•to form the basis of a new rating algorithm (pricing), or

•as part of the underwriting guidelines, either at policy initiation or renewal.

INSURANCE RATEMAKING AND A GINI INDEX 337

These portfolios are natural extensions of the classic Lorenz curve and associated Gini

index that we review in the following.

Lorenz Curve and Gini Index

In welfare economics, it is common to compare distributions via the Lorenz curve,

developed by Max Otto Lorenz (1905). A Lorenz curve is a graph of the proportion

of a population on the horizontal axis and a distribution function of interest on the

vertical axis. It is typically used to represent income distributions. When the income

distribution is perfectly aligned with the population distribution, the Lorenz curve

results in a 45 degree line that is known as the “line of equality.” The area between

the Lorenz curve and the line of equality is a measure of the discrepancy between

the income and population distributions. Two times this area is known as the Gini

index, introduced by Corrado Gini in 1912. See, for example, Sen and Foster (1998),

for additional background on income equality. For readers interested in examining

current international inequality measures, see the online resource UNU-WiderWorld

Income Inequality Database (2008).

The contributions of Joseph Gastwirth in the 1970s (e.g., Gastwirth, 1971, 1972) helped

to emphasize the importance of the Lorenz curve and the Gini index as tools for com-

paring distributions, particularly in economic applications. The subsequent literature

is extensive. In one strand of the literature, researchers have sought to understand

differences in economic equality among population subgroups (e.g., Gastwirth, 1975;

Lambert and Decoster, 2005). In another strand, analysts have introduced weight

functions into the Lorenz curve (e.g., to account for the number of publications when

studying impact factors; Egghe, 2005). Yitzhaki (1996) describes how weighted re-

gression sampling estimators can be of interest in welfare economic applications.

Here, the idea is to adjust regression weights for social attitudes toward inequality.

In another stream of research, analysts have used the Gini index for model selection

in genomics (Nicodemus and Malley, 2009) and in classification trees (Sandri and

Zuccolotoo, 2008).

Example: Distribution of Homeowners Premiums. For an insurance example, Figure 1

shows a distribution of premiums. This figure is based on a sample of 359,454 policy-

holders with premiums that will be described in the “Homeowners Example” section.

The left-hand panel shows a right-skewed histogram of premiums. When plotting this

figure, premiums that exceeded 1,200 were ignored. The right-hand panel provides

the corresponding Lorenz curve, showing again a skewed distribution. For example,

the arrow marks the point where 60 percent of the policyholders pay 40 percent of

premiums. The 45 degree line is the “line of equality;” if each policyholder paid the

same premium, then the premium distribution would be at this line. The Gini index,

twice the area between the Lorenz curve and the 45 degree line, is 29.5 percent for

this data set.

Relating Premium to Loss Distributions

From the “Lorenz Curve and Gini Index” section, the Lorenz curve is a device that

is well known in welfare economics for displaying distributions. It is particularly

useful for interpreting skewed distributions, a shape that insurance analysts are well

acquainted with.