A Model for an Insurance Market

Throughout Chapters 4–6, I assume a population of risks can be divided into a low risk-group and a high risk-group, based on information which is fully observable to insurers. Just two risk-groups is not, of course, a realistic model of most insurance markets, but it is enough to illustrate principles.Footnote ¹

Let μ₁ and μ₂ be the underlying risks (probabilities of loss) for the low and high risk-groups.

Let p₁ and p₂ be the population fractions for the low and high risk-groups, that is the proportions of the total population represented by each risk-group. This means a risk chosen at random from the entire population has a probability p₁ of belonging to the low risk-group.

Throughout Chapters 4–6, I assume that all losses and insurance cover are of unit size; this simplifies the presentation, but it is not necessary. I also assume no moral hazard, that is, given the risk-group, the probability of loss is not affected by the purchase of insurance.Footnote ²

All quantities defined below are for a single risk sampled at random from the population (unless the context requires otherwise).

The expected loss is denoted E[L] (‘L’ for ‘loss’) and given by:

(4.1) $\text{E}[L]={\displaystyle \sum _{i=1}^2{\mu }_i{p}_i}$

E[L] corresponds to a unit version of the third row of the tables in the examples in Chapter 3.

In the absence of limits on risk classification, insurers will charge risk-differentiated premiums equal to the probabilities of loss, π₁ = μ₁ and π₂ = μ₂ for risk-groups 1 and 2, respectively.

The expected insurance demand is denoted E[Q] (‘Q’ for ‘quantity’) and given by:

(4.2) $\text{E[}Q\text{]=}{\displaystyle \sum _{i=1}^{2}d({\mu }_i,{\pi }_i)p_i}$

where d(μ_i,π_i) is the proportional demand for insurance for risk-group i when premium π_i is charged, that is the probability that an individual selected at random from the risk-group buys insurance.Footnote ³

The expected premium is denoted by E[Π] (‘Π’ for ‘premium’) and given by:

(4.3) $\text{E}\left[\Pi \right]={\displaystyle \sum _{i=1}^{2}d({\mu }_i,{\pi }_i)}{p}_i{\pi }_i$

The expected insurance claim is given by:

(4.4) $\text{E[}QL\text{]=}{\displaystyle \sum _{i=1}^{2}d({\mu }_i,{\pi }_i)p_i{\mu }_i}$

Adverse Selection

The phrase ‘adverse selection’ is typically associated with positive correlation (or equivalently, covariance) of cover Q and loss L; most papers testing for adverse selection in the economics literature use this definition.Footnote ⁴ This section makes this definition precise, in a form which will later help to highlight the relationship between adverse selection and loss coverage.

By the standard definition, the covariance of Q and L is:

(4.5) $\begin{array}{c}Cov\left(Q,L\right)=\mathrm{E}\left[\left(Q-\mathrm{E}\left[Q\right]\right)\left(L-\mathrm{E}\left[L\right]\right)\right]\\ =\mathrm{{\rm E}}\left[QL\right]-\mathrm{{\rm E}}\left[Q\right]\mathrm{{\rm E}}\left[L\right].\end{array}$

While the economics literature typically uses covariance (Q, L) > 0 as a test for adverse selection, it is more convenient to note that when the covariance is zero, the two terms in Equation (4.5) must be the same. We can then define the adverse selection as:

(4.6) $\text{Adverse selection},A=\frac{\mathrm{{\rm E}}\left[QL\right]}{\mathrm{{\rm E}}\left[Q\right]\mathrm{{\rm E}}\left[L\right]}$

This enables us to index different types of selective behaviour by insurance customers as follows:

• A < 1: advantageous selection

• A = 1: no selection

• A > 1: adverse selection.Footnote ⁵

To compare the severity of adverse selection under different risk-classification regimes, we need to define a reference level of adverse selection. Adverse selection under alternative schemes can then be expressed as a fraction of adverse selection under the reference scheme. A convenient reference scheme is risk-differentiated premiums (actuarially fair premiums). Then using subscript 0 to denote quantities evaluated under risk-differentiated premiums, we define the adverse selection ratio as:

(4.7) $\text{Adverse selection ratio}=\frac{A}{A_0}$

In words, adverse selection ratio is the ratio of the expected claim per policy under the actual risk classification scheme to the expected claim per policy under risk-differentiated premiums.

Adverse selection ratio can also be thought of as the ratio of the demand-weighted average premiums required for insurers to break even under each risk classification scheme.

Loss Coverage

Loss coverage in this model is defined as the expected insurance claim (as previously evaluated in Equation (4.4)):

(4.8) $\text{Loss coverage}=\mathrm{{\rm E}}\left[QL\right]={\displaystyle \sum _{i=1}^{2}d\left({\mu }_i,{\pi }_i\right)}{p}_i{\mu }_i$

The product of random variables Q and L can alternatively be thought of as the following ‘indicator’ random variable:

(4.9) $QL=\left\{1\text{if the individual both incurs a loss and has cover},0\text{otherwise}\right\}$

Loss coverage can then be thought of as indexing the ‘overlap’ of cover Q and losses L in the population. It represents the extent to which insurance cover is concentrated over the ‘right’ risks (those most likely to suffer loss). It measures the efficacy of insurance in compensating the population’s losses.

The right-hand side of Equation (4.8) also shows that loss coverage can be thought of as the risk-weighted insurance demand for the randomly selected member of the population.

This concept of loss coverage as risk-weighted insurance demand can be contrasted with unweighted insurance demand, which corresponds to the ‘number of risks insured’ often referenced in informal discussions of adverse selection.

Loss Coverage Ratio

When comparing alternative risk classification schemes, it is often helpful to define loss coverage to be 1 under some suitable reference scheme. Loss coverage under alternative schemes can then be expressed as a fraction of loss coverage under the reference scheme. It is convenient to use the same approach as for adverse selection above, that is, I use risk-differentiated premiums as the reference scheme. Then using subscript 0 to denote quantities evaluated under risk-differentiated premiums, I define the loss coverage ratio (LCR) as:

(4.10) $\text{LCR}=\frac{\mathrm{{\rm E}}\left[QL\right]}{{\mathrm{{\rm E}}}_0\left[QL\right]}$

Note that in the numerical examples in Chapters 1 and 3, the fraction between 0 and 1 which I heuristically labelled ‘loss coverage’ was, more precisely, a loss coverage ratio with the reference scheme (i.e. under which loss coverage = 1) defined as compulsory insurance of the whole population. Clearly the choice of reference scheme – risk-differentiated premiums, compulsory insurance or something else – does not matter, provided we use a consistent reference when making comparisons of different proposed risk classification schemes.

Now note that loss coverage ratio in Equation (4.10) can also be expanded into:

(4.11) $\text{LCR}=\frac{\left(\frac{\mathrm{{\rm E}}\left[QL\right]}{\mathrm{{\rm E}}\left[Q\right]\mathrm{{\rm E}}\left[L\right]}\right)}{\left(\frac{{\mathrm{{\rm E}}}_0\left[QL\right]}{{\mathrm{{\rm E}}}_0\left[Q\right]\mathrm{{\rm E}}\left[L\right]}\right)}\times \frac{\mathrm{{\rm E}}\left[Q\right]}{{\mathrm{{\rm E}}}_0\left[Q\right]}$

Then by noting that expected population losses are the same irrespective of the risk classification scheme (i.e. E[L] = E₀[L]), we see that the first term on the right-hand side of Equation (4.11) is the adverse selection ratio in Equation (4.7). The second term on the right-hand side of Equation (4.11) is the ratio of demand under the actual risk classification scheme to demand under risk-differentiated premiums, which I call the demand ratio. So Equation (4.11) can then be interpreted as:

(4.12) $\text{LCR = Adverse selection ratio × Demand ratio}$

We can illustrate this decomposition of loss coverage ratio by applying it to the numerical examples in Chapter 3. The decomposition is shown in Table 4.1. In each of the three columns in the table, loss coverage ratio (the third line) is the product of adverse selection ratio and demand ratio (the first and second lines).

Table 4.1 Decomposition of loss coverage ratio into adverse selection ratio and demand ratio for Tables 3.1–3.3

	Table 3.1	Table 3.2	Table 3.3
$\begin{array}{l}Adverseselectionratio:\\ \left(\frac{\text{risk}-\text{weighted average premium}}{\text{population}-\text{weighted average premium}}\right)\end{array}$	1.0	1.25	1.3461
$\begin{array}{l}Demandratio:\\ \left(\frac{\text{numbers insured under pooled premium}}{\text{numbers insured under risk}-\text{differentiated premiums}}\right)\end{array}$	1.0	0.90	0.65
$\begin{array}{l}Losscoverageratio\left(\text{product of above}\right):\\ \left(\frac{\text{loss coverage under pooled premium}}{\text{loss coverage under risk}-\text{differentiated premiums}}\right)\end{array}$	1.0	1.125	0.875

The decomposition of loss coverage ratio into adverse selection ratio and demand ratio is sometimes helpful in correcting casual intuitions and commentary about restrictions on risk classification. Casual intuitions and commentary often reference rising average prices (adverse selection) and falling demand (numbers insured), but without considering how the two effects interact. But depending on the product of the two effects, loss coverage ratio may be higher or lower than 1 (that is, loss coverage may be increased or decreased). The decomposition highlights that predicting or observing a rise in average price and a fall in demand under a new risk classification scheme is not sufficient to demonstrate a worse outcome. The outcome in terms of loss coverage depends on the product of the two effects.

An illustration of the trade-off between loss coverage and adverse selection is provided by Figure 4.1. This graph plots loss coverage ratio against adverse selection ratio, based on the two risk-group model in this chapter with a plausible form for the demand function.Footnote ⁶ It can be seen that the maximum point for loss coverage corresponds to an intermediate degree of adverse selection, not too low and not too high.

The shape of this graph, with the interior maximum showing that loss coverage is maximised by an intermediate level of adverse selection, is the most important image in this book. A similar inverted U-shape is obtained for any reasonable demand function. Note that the highest loss coverage is obtained not despite the adverse selection, but because of the adverse selection. In moderation, adverse selection is a good thing.

Figure 4.1 Loss coverage ratio as a function of adverse selection ratio

Figure 4.1 is based on a relative risk β = μ₂/μ₁ = 4. If the relative risk is lower, the maximum value of loss coverage is lower, and this maximum is attained with a lower level of adverse selection. This is illustrated in Figure 4.2, which shows the plot of loss coverage ratio against adverse selection ratio for two values of relative risk, β = 3 and β = 4. The maximum of the dashed curve for β = 3 lies below and to the left of the maximum of the solid curve for β = 4.

Note that the right-hand terminal points of each curve in Figures 4.1 and 4.2 correspond to limiting values, not points at which I arbitrarily chose to stop drawing the curve. These limiting values represent the scenario where all lower risks have dropped out of insurance and only higher risks remain; clearly, adverse selection then cannot increase any more. In Figure 4.2, the terminal point for β = 3 lies to the left and below the terminal point for β = 4. To understand this, note that when all higher risks have dropped out of the market and only higher risks remain, lower relative risk β = 3 implies a lower break-even premium (lower adverse selection), and also that a lower fraction of the total risk in the population is covered (lower loss coverage).

Figure 4.2 Loss coverage ratio as a function of adverse selection ratio, for two relative risks

Summary

The main points of Chapter 4 can be summarised as follows:

1. 1. Loss coverage:

   $\begin{array}{c}\text{Loss coverage}=\mathrm{{\rm E}}\left[QL\right]\\ ={\displaystyle \sum _{i=1}^2{d}_i\left({\pi }_i,{\mu }_i\right)}{p}_i{\mu }_i\end{array}$

   Loss coverage is the expected losses compensated by insurance for the whole population. It represents the extent to which insurance cover is concentrated on the ‘right’ risks (those most likely to suffer loss). It measures the efficacy of insurance in compensating the population’s losses.

   Loss coverage under alternative risk classification schemes can be compared by loss coverage ratio:

   $\text{LCR}=\frac{\mathrm{{\rm E}}\left[QL\right]}{{\mathrm{{\rm E}}}_0\left[QL\right]}$
   where subscript 0 denotes a quantity calculated under some base or reference risk classification scheme, e.g. risk-differentiated premiums, or compulsory insurance.

2. 2. Adverse selection:

   $\text{Adverse selection},A=\frac{\mathrm{{\rm E}}\left[QL\right]}{\mathrm{{\rm E}}\left[Q\right]\mathrm{{\rm E}}\left[L\right]}$
   $\text{Adverse selection ratio}=\frac{A}{A_0}$
   where subscript 0 denotes a quantity calculated under some base or reference risk classification scheme, e.g. risk-differentiated premiums, or compulsory insurance.

3. 3. Decomposition of loss coverage ratio into adverse selection ratio and demand ratio:

   $\text{Loss coverage ratio = Adverse selection ratio × Demand ratio}$

This decomposition highlights that predicting or observing a rise in average price and fall in demand under a new risk classification scheme is not sufficient to demonstrate a worse outcome. The outcome in terms of loss coverage depends on the product of the two effects.