This chapter examines the usual argument that adverse selection makes insurance work less well. It shows that from a public policy perspective, the absolutism of this argument is misconceived. Some degree of adverse selection in insurance is generally beneficial to society. The optimal level of adverse selection is the level which maximises loss coverage, the expected losses compensated by insurance for the population as a whole.
Although this proposition is unorthodox, there are some analogies which lend it intuitive appeal and plausibility. In many biological contexts, the analogy is hormesis: the phenomenon of stressors which are beneficial to an organism in small doses, but become harmful in higher doses. Examples of hormesis in the human body include the response to most drugs and nutrients (alcohol is a vivid example, but the same pattern applies for substances as ubiquitous and innocuous as water); the response to physical exercise; and the response to stress (cf. ‘eustress’ and ‘distress’). By analogy, we can think of the insurance system being ‘stressed’ by adverse selection. Small doses ‘stretch’ the system and make it work better, in the sense of increasing loss coverage, but higher doses make the system work less well.
The argument that some degree of adverse selection makes the insurance system work better will be presented in three ways: first, a verbal summary of the key argument; second, numerical examples; and third, mathematical details. The summary and examples are presented in this chapter, and the mathematics in Chapters 4–6.
The Key Argument
The following verbal summary repeats that given in Chapter 1, with only footnotes and a final paragraph added.
Consider an insurance market where individuals can be divided into two risk-groups, one higher risk and one lower risk, based on information which is fully observable by insurers. Assume that all losses and insurance are of unit amount (this simplifies the discussion, but it is not necessary). Also assume that an individual’s risk is unaffected by the purchase of insurance, i.e. there is no moral hazard.Footnote 1
If insurers can, they will charge risk-differentiated prices to reflect the different risks. If instead insurers are banned from differentiating between higher and lower risks, and have to charge a single ‘pooled’ price for all risks, a pooled price equal to the simple average of the risk-differentiated prices will seem cheap to higher risks and expensive to lower risks. Higher risks will buy more insurance, and lower risks will buy less.
To break even, insurers will then need to raise the pooled price above the simple average of the prices. Also, since the number of higher risks is typically smaller than the number of lower (or ‘standard’) risks, higher risks buying more and lower risks buying less implies that the total number of people insured usually falls.Footnote 2 This combination of a rise in price and a fall in demand is usually portrayed as a bad outcome, for both insurers and society.
However, from a social perspective, it is arguable that higher risks are those more in need of insurance. Also, the compensation of many types of loss by insurance appears to be widely regarded as a desirable objective, which public policymakers often seek to promote, by public education, by exhortation and sometimes by incentives such as tax relief on premiums. Insurance of one higher risk contributes more in expectation to this objective than insurance of one lower risk. This suggests that public policymakers might welcome increased purchasing by higher risks, except for the usual story about adverse selection.
The usual story about adverse selection overlooks one point: with a pooled premium and adverse selection, expected losses compensated by insurance can still be higher than with fully risk-differentiated premiums and no adverse selection. Although pooling leads to a fall in numbers insured, it also leads to a shift in coverage towards higher risks. From a public policymaker’s viewpoint, this means that more of the ‘right’ risks – those more likely to suffer loss – buy insurance. If the shift in coverage is large enough, it can more than outweigh the fall in numbers insured. This result of higher expected losses compensated by insurance – higher ‘loss coverage’ – can be seen as a better outcome for society than that obtained with no adverse selection.
Another perspective on this argument is that a public policymaker designing risk classification policies in the context of adverse selection normally faces a trade-off between insurance of the ‘right’ risks (those more likely to suffer loss) and insurance of a larger number of risks. The optimal trade-off depends on the response of higher and lower risk-groups to different prices (technically, the demand elasticity of different risk-groups), and will normally involve at least some adverse selection. The concept of loss coverage quantifies this trade-off, and provides a metric for comparing the effects of different risk classification schemes.
Numerical Examples
Numerical examples can illustrate the argument sketched above. These are similar in nature to the two scenarios illustrating ‘no adverse selection’ and ‘some adverse selection’ in the toy example in Chapter 1. But now I use more realistic numbers, and also introduce a third scenario of ‘too much’ adverse selection.
Suppose that in a population of 1,000 risks, 16 losses are expected every year. There are two risk-groups; 200 high risks have a probability of loss 4 times higher than the other 800 low risks. We assume that all losses and insurance are of unit amount, and that there is no moral hazard. An individual’s risk-group is fully observable to insurers.
Under the initial risk classification regime, insurers operate full risk classification, charging actuarially fair premiums to members of each risk-group. We assume that the proportion of each risk-group which buys insurance under these conditions – the ‘fair-premium demand’ – is 50%, which is realistic for life insurance in the UK and the USA.Footnote 3 Table 3.1 shows the outcome, which can be summarised as follows:
– There is no adverse selection. The average of the insurance premiums, weighted by numbers of insurance buyers at each price, is 0.016 (final column, fourth line). This is the same as the population-weighted average risk (final column, first line). Dividing the first by the second, we index the adverse selection as 0.016/0.016 = 1.0, indicating a neutral position (i.e. no adverse selection).
– Half the losses in the population are compensated by insurance. We heuristically characterise this as a ‘loss coverage’ of 0.5.
Table 3.1 Full risk classification: no adverse selection (base outcome)
| Low risk-group | High risk-group | Aggregate | |
|---|---|---|---|
| Risk | 0.01 | 0.04 | 0.016 |
| Total population | 800 | 200 | 1 000 |
| Expected population losses | 8 | 8 | 16 |
| Break-even premiums (risk-differentiated) | 0.01 | 0.04 | 0.016 |
| Numbers insured | 400 | 100 | 500 |
| Insured losses | 4 | 4 | 8 |
| Adverse selection | 1 | ||
| Loss coverage | 0.5 |
Now suppose that a new risk classification regime is introduced, where insurers are obliged to charge a common ‘pooled’ premium to members of both the low and high risk-groups. One possible outcome is shown in Table 3.2, which can be summarised as follows:
– The pooled premium of 0.02 at which insurers make zero profits is calculated as the demand-weighted average of the risk premiums: (300 × 0.01 + 150 × 0.04)/450 = 0.02).
– The pooled premium is expensive for low risks, so 25% fewer of them buy insurance (300, compared with 400 before). The pooled premium is cheap for high risks, so 50% more of them buy insurance (150, compared with 100 before). Because there are 4 times as many low risks as high risks in the population, the total number of policies sold falls (450, compared with 500 before).
– There is moderate adverse selection. The pooled premium of 0.02 exceeds the population-weighted average premium of 0.016, giving adverse selection = 0.02/0.016 = 1.25.
– The resulting loss coverage is 0.5625. The shift in coverage towards high risks more than outweighs the fall in number of policies sold: 9 of 16 losses (56%) in the population as a whole are now compensated by insurance (compared with 8 of 16 before).
Table 3.2 Risk classification banned: moderate adverse selection leading to higher loss coverage (better outcome)
| Low risk-group | High risk-group | Aggregate | |
|---|---|---|---|
| Risk | 0.01 | 0.04 | 0.016 |
| Total population | 800 | 200 | 1 000 |
| Expected population losses | 8 | 8 | 16 |
| Break-even premiums (pooled) | 0.02 | 0.02 | 0.02 |
| Numbers insured | 300 | 150 | 450 |
| Insured losses | 3 | 6 | 9 |
| Adverse selection | 1.25 | ||
| Loss coverage | 0.5625 |
Table 3.2 exhibited moderate adverse selection. Another possible outcome under the restricted risk classification scheme, this time with more severe adverse selection, is shown in Table 3.3, which can be summarised as follows:
– The pooled premium of 0.02154 at which insurers make zero profits is calculated as the demand-weighted average of the risk premiums: (200 × 0.01 + 125 × 0.04)/325 = 0.02154.
– There is severe adverse selection, with a further increase in the pooled premium and a significant fall in numbers insured.
– The loss coverage is 0.4375. The shift in coverage towards high risks is not sufficient to outweigh the fall in number of policies sold: 7 of 16 losses (43.75%) in the population as a whole are now compensated by insurance (compared with 8 of 16 in Table 3.1, and 9 out of 16 in Table 3.2).
Table 3.3 Risk classification banned: severe adverse selection leading to lower loss coverage (worse outcome)
| Low risk-group | High risk-group | Aggregate | |
|---|---|---|---|
| Risk | 0.01 | 0.04 | 0.016 |
| Total population | 800 | 200 | 1 000 |
| Expected population losses | 8 | 8 | 16 |
| Break-even premiums (pooled) | 0.02154 | 0.02154 | 0.02154 |
| Numbers insured | 200 | 125 | 325 |
| Insured losses | 2 | 5 | 7 |
| Adverse selection | 1.34625 | ||
| Loss coverage | 0.4375 |
Taking the three tables together, we can summarise by saying that compared with an initial position of no adverse selection in Table 3.1, moderate adverse selection leads to higher expected losses compensated by insurance (higher loss coverage) in Table 3.2; but too much adverse selection leads to lower expected losses compensated by insurance (lower loss coverage) in Table 3.3.
This argument that moderate adverse selection increases loss coverage is quite general: it does not depend on any unusual choice of numbers for the examples. It also does not assume any bias by the policymaker towards (or against) compensating the losses of the high risk-group in preference to those of the low risk-group. The same preference is given to compensation of losses anywhere in the population ex post, when all uncertainty about who will suffer a loss has been resolved. This implies giving higher preference to insurance cover for higher risks ex ante, before we know who will suffer a loss, but only in proportion to their higher risk.
Loss Coverage in Context
The remainder of this chapter makes a range of comparisons and analogies which helps to set the concept of loss coverage in context.
The Hirshleifer Effect
The idea that restricting risk classification and hence inducing a degree of adverse selection can produce a better outcome for society as a whole is analogous to the Hirshleifer effect.Footnote 4 This is named after the economist Jack Hirshleifer, who pointed out that too much information about future investment returns can reduce opportunities for risk sharing, because a risk which has been resolved cannot be shared. Although some people can make better decisions when more information becomes available, society as a whole can end up worse off. In insurance, the analogous phenomenon is that the use of too much information for risk classification can reduce risk sharing: although some people get cheaper insurance because of the increased use of information, the overall quantum of risk transferred to insurers falls, so society as a whole ends up worse off.
Loss Coverage and Compulsory Insurance
One way of maximising loss coverage is to make insurance compulsory – either through social insurance, or by laws which compel people to buy commercial insurance. Healthcare is often provided by social insurance, and liability insurances which protect the interests of unidentified third parties or the public at large are often made compulsory (e.g. third-party car insurance, employers’ liability insurance). The potential losses compensated by these insurances are largely independent of the personal circumstances of the insured, and so compulsion is a well-targeted approach to achieving policymakers’ desired (universal) loss coverage.
In other classes of insurance, such as life insurance, the policy motivations to increase loss coverage may be less conclusive, but are still present to some degree. For example, policymakers may consider that through incomplete information or behavioural bias, most people under estimate their needs for life insurance, and so too little life insurance is bought.Footnote 5 However, in life insurance the potential loss depends on the personal circumstances of the insured. A compulsory requirement for unmarried persons to purchase life insurance may not increase loss coverage (because there is often no financial loss to a survivor if the unmarried person dies); the cost of premiums to the unmarried person may itself represent an uncompensated loss. One might exclude unmarried persons from compulsion; but some unmarried persons do need life insurance, and equally some married persons do not. Overall, for insurances where the potential loss depends on personal circumstances, compulsion may be a poorly targeted and politically unpopular means of increasing loss coverage. It may be both more effective and more politically acceptable to leave purchasing decisions to individuals, and increase loss coverage by regulating risk classification.
There may be a few classes of insurance where there is little policy motivation to promote the compensation of losses by insurance. A clear example would be insurance against the cost of legal or regulatory penalties; a less clear example might be insurance of pet animals. In these cases, policymakers may view loss coverage as a less important objective than in markets such as liability, health and life insurance. Indeed, policymakers may even seek to reduce loss coverage, by discouraging the insurance. In extreme cases this might mean banning the insurance. In the UK, the Financial Conduct Authority bans the firms which it regulates from insuring against the cost of regulatory penalties.Footnote 6
Loss Coverage versus ‘Social Welfare’
There is an extant literature in economics which considers the effect of restrictions on risk classification. Economists typically take a utility-based approach: individuals make insurance choices to maximise their expected utility according to some utility function, and the outcomes of different risk classification schemes are evaluated by social welfare, typically defined as the sum of expected utilities over the entire population.Footnote 7
One way of describing the difference in this book’s approach compared with the economics literature is that rather than assigning equal weights to expected utility levels across low and high risks, I assign equal weights to loss coverage levels across low and high risks. Under my approach, if all losses are of unit amount, and high risks have a probability of loss twice that of low risks, then coverage of one high risk is considered to be worth the same ex ante as coverage of two low risks.
Another way of describing the difference between loss coverage and social welfare is to note that loss coverage focuses on maximising the benefit of insurance, framed solely as the compensation of losses. Minor details about preferences – who underpays and overpays for insurance, and how much they care (how it affects their utility) – are disregarded, provided that the insurance system as a whole equilibrates.
This disregard seems reasonable in the typical scenario where insurance premiums are small relative to wealth. It is also practical in the sense that loss coverage depends solely on observable claims, whereas social welfare depends on utilities which are never observable. Loss coverage also has the advantage of being a simpler concept, and so more conducive to wide understanding in policy discussions.
Although loss coverage and social welfare are distinct criteria, they can be shown to be tightly related under simple assumptions about utility (and hence insurance demand) functions. Specifically: for power utility functions (and hence iso-elastic insurance demand), either criterion – loss coverage or social welfare – gives the same rank ordering of different risk classification schemes. Those who prefer the criterion of social welfare in principle may then wish to think of loss coverage as an observable proxy, under certain assumptions, for social welfare.Footnote 8
Loss Coverage versus ‘Coverage’
The criterion of loss coverage and the criterion of social welfare can be further contrasted with the criterion of simple coverage, which is often implicit in informal discussions of risk classification. In informal discussions, adverse selection is often said to be a bad thing because it leads to a ‘reduction in coverage’ compared with that attained in the absence of adverse selection. Economists typically say that any reduction in coverage is ‘inefficient’.Footnote 9 But if loss coverage is increased, the quantum of risk transferred to insurers is increased. Why should an arrangement under which more risk is voluntarily traded and more losses are compensated be disparaged as ‘inefficient’?
Such discussions of the ‘reduction in coverage’ arising from adverse selection implicitly treat coverage of one high risk as equivalent to coverage of one low risk. But if the purpose of insurance is to compensate the population’s losses, this does not seem a valid equivalence: covering one high risk contributes more in expectation to this purpose than covering one low risk. In other words, these informal discussions are typically predicated on an inaccurate measure of the benefit of insurance.
Another succinct description of the contrast between loss coverage and the common informal reference to just ‘coverage’ is to note that loss coverage corresponds to risk-weighted insurance demand, and coverage to unweighted insurance demand.
Probabilistic Goods versus Reassurance Goods
The risk-weighted nature of loss coverage is predicated on the notion that the good provided by insurance is the contingent compensation of losses. This good materialises only in a particular future state of the world, which has different probabilities for higher and lower risks. That is, insurance is a probabilistic and individually heterogeneous good.
For such a good, one unit of sales to a higher-risk individual is a different good compared with one unit of sales to a lower-risk individual. In loss coverage, this difference is reflected in the risk-weighting of coverage. In utilitarian social welfare, it is reflected in higher probabilities attached to losses in expected utility calculations.
While I believe this framing of insurance as a probabilistic good is appropriate when considering public policy at a population and objective level, I acknowledge that at an individual and subjective level, insurance can also be framed as a non-probabilistic good. Insurance goods can be framed as non-probabilistic if the good is conceived as ‘peace of mind’ or ‘freedom from worry’. Think of car insurance, home insurance or life insurance: in every case, at an individual and subjective level, the good might just as well be framed as freedom from worry in the present, rather than as the contingent compensation of losses in the future. Insurance framed as ‘freedom from worry’ is a non-probabilistic experience good. In the specific insurance context, we can call it a reassurance good (reassurance being the experience which insurance provides).
If insurance is framed as a reassurance good, one unit of sales to a higher-risk individual is closer to the same good as one unit of sales to a lower-risk individual. I say ‘closer to’ rather than ‘equal to’ because although both higher and lower risks experience similar reassurance, it might be felt that the quantum of this reassurance increases for higher underlying probability of loss. This is certainly arguable, but it is not obvious that the increase should be in one-to-one proportion to the probability. And it seems at least equally arguable that for any class of insurance (say car, home or life), the quantum of reassurance is invariant to individual variations in probabilities of loss (or alternatively, increasing in less than one-to-one proportion, or at a sublinear rate, in those variations).
I acknowledge that the framing of insurance as a reassurance good may often be psychologically salient to individuals; and that in this framing, the risk-weighting in loss coverage might overstate the benefit of coverage of higher risks. Nevertheless, I believe that the framing of insurance as a probabilistic good is a better basis for public policy. This is for the same reason as I prefer loss coverage to social welfare: I believe that it is more practical to base public policy on observables (such as probabilities of loss), not on individuals’ unobservable states of mind (such as feelings of reassurance, or increases in utility).Footnote 10
Loss Coverage and Partial Risk Classification
The discussion above has considered only two possibilities for risk classification: fully risk-differentiated premiums or complete pooling. In practice, partial restrictions on risk classification are common, where only some risk variables (e.g. gender, family history, genetic test results) are banned. Loss coverage is maximised when there is an intermediate level of adverse selection, not too low and not too high. It seems plausible that in some markets, complete pooling generates too much adverse selection; but partial restrictions on risk classification generate an intermediate level of adverse selection, and hence higher loss coverage than under either pooling or fully risk-differentiated premiums. Loss coverage can then provide an explanation or rationalisation of the partial restrictions often seen in practice. Partial risk classification is considered further in Chapter 6.
Different Weights for Different Risk-Groups?
Loss coverage as defined in this book always places equal weight on equal expected losses, irrespective of the part of the population from which they are expected. No preference is given to achieving compensation of losses arising from higher or lower risks. This solution to the trade-off between the interests of higher and lower risks has obvious intuitive and egalitarian appeal. But in some circumstances, a policymaker might wish to give higher priority to compensating the losses of higher (or lower) risks – perhaps because those with higher insurance risks also tend to suffer other economic or social disadvantages. If so, it is straightforward to redefine loss coverage as a weighted average of expected losses over the population, with the weights reflecting the priority the policymaker places on compensation of losses arising from different risks.Footnote 11
Other Public Policy Criteria for Risk Classification
Loss coverage is not the only criterion which a public policymaker may consider when setting policy on risk classification. In some markets, risk classification might provide an incentive for loss prevention: for example, if house insurance premiums are lower for homes with security features such as burglar alarms or better locks, perhaps home owners will be more inclined to install these features.Footnote 12 A policymaker might also place value on ensuring the optional availability of insurance to members of higher risk-groups, distinct from the actual take-up of the option. The possible ‘spillover’ effects of insurance risk classification on privacy, public health and discrimination in other contexts such as employment might also be considered. Restrictions on risk classification sometimes appear to be based on a principled objection to statistical discrimination per se, rather than on an assessment of insurance market consequences. These and other objections to risk classification are discussed further in Chapter 7. Nevertheless, to the extent that consequences in the insurance market itself are given weight, loss coverage is a useful metric for those consequences.
Loss Coverage: The Insurer’s Perspective
While this book focuses mainly on a public policy perspective, the loss coverage concept can also be viewed from the insurer’s perspective. Maximising loss coverage is equivalent to maximising premium income. If profit loadings are proportional to premiums, maximising loss coverage could be a desirable objective for insurers. Even from the insurer’s perspective, adverse selection is not always a bad thing! This might help explain why insurers often appear rather slower to make use of every scrap of marginal information for risk classification than economic theory predicts, even when information is observable at zero or at negligible cost and apparently relevant to the risk.
Finkelstein and Poterba (Reference Finkelstein and Poterba2014) note that UK insurers were very slow to adopt ‘postcode pricing’ of annuities, despite postcode information being available at zero cost and regional variations in annuitant mortality having been reasonably well known for many years. They offer four possible explanations, but do not consider the possibility that the small amount of adverse selection induced by ignoring postcode information may give higher loss coverage (from the insurers’ perspective, higher aggregate annuity premiums). They also note that once one large insurer (Legal and General) adopted postcode pricing in 2007, most other insurers quickly followed. If profit loadings are proportional to premiums, Legal and General’s competitive innovation and the resulting ‘competitive adverse selection’ (as discussed in Chapter 8) may have made insurers collectively worse off.
Loss Coverage versus Preconceptions: Hidden in Plain View
The observation that a larger fraction of the population’s losses can be compensated with some adverse selection involves only elementary arithmetic, but as far as I am aware it has not been pointed out before. Given its technical simplicity, this may seem surprising. It becomes less surprising when one reflects on how the insight clashes with extant economic and industry dogma. Economists seem strongly committed to the idea that asymmetric information always makes all markets work less well; in contrast, loss coverage says that that a limited amount of asymmetric information makes insurance markets work better. Similarly, insurers seem strongly committed to the idea that unrestricted risk classification is commercially vital; in contrast, loss coverage says that some restrictions on risk classification are likely to be beneficial, including for insurers themselves. An unrecognised simplicity which clashes with so many people’s preconceptions can remain hidden in plain view.
One example where the loss coverage perspective contrasts with an argument made by economists on behalf of insurers is a paper Why the use of age and disability matters to consumers and insurers, commissioned by the trade association Insurance Europe from the British economic consultants Oxera.Footnote 13 The document presents an example of the effect of banning risk classification by disability in life insurance. The example is hypothetical, with assumed sums assured of £100,000, and all other figures chosen to make a case to policymakers against restrictions on risk classification – but in my view inadvertently making the opposite case. Table 3.4 summarises the figures presented by Oxera as the hypothetical position before a ban on risk classification.
Table 3.4 Life insurance with risk-based premiums (based on Oxera, 2012)
| Risk | Mortality | Population | Insured population (clientele) | Exapected cost per customer (€) | Total claims (€) | Premium rate (€) | Total premium income (€) |
|---|---|---|---|---|---|---|---|
| Standard | 1.4% | 800 | 400 | 1 400 | 560 000 | 1 400 | 560 000 |
| Double | 2.8% | 150 | 75 | 2 800 | 210 000 | 2 800 | 210 000 |
| High | 14.0% | 50 | 0 | 14 000 | |||
| Total | 1 000 | 475 | 770 000 | 770 000 |
Table 3.5 shows the figures given as the hypothetical position after a ban on risk classification. Oxera notes that the premium for the ‘Standard’ risks more than doubles after the ban, and suggests that this shows that a ban is undesirable.
Table 3.5 Life insurance with pooled premiums (based on Oxera, 2012)
| Risk | Mortality | Population | Insured population (clientele) | Expected cost per customer (€) | Total claims (€) | Premium rate (€) | Total premium income (€) |
|---|---|---|---|---|---|---|---|
| Standard | 1.4% | 800 | 200 | 1 400 | 280 000 | 3 333 | 666 667 |
| Double | 2.8% | 150 | 75 | 2 800 | 210 000 | 3 333 | 250 000 |
| High | 14.0% | 50 | 40 | 14 000 | 560 000 | 3 333 | 133 333 |
| Total | 1 000 | 315 | 1 050 000 | 1 050 000 |
But on Oxera’s figures, loss coverage is increased by restricting risk classification. Before the ban, the expected number of deaths compensated by insurance is 770,000/100,000 = 7.7. After the ban, the corresponding number is 1,050,000/100,000 = 10.5. Banning risk classification in this example gives an improvement: the shift in coverage towards higher-risk lives more than compensates for the fall in numbers insured.
Of course, since Oxera’s figures are hypothetical, it is easy to construct an example corresponding to Scenario 3 earlier in this chapter, where adverse selection ‘goes too far’ and lower loss coverage results from a ban. But the fact that this was not done suggests that the loss coverage perspective was not considered.
Summary
This chapter has suggested that the usual arguments that adverse selection always makes insurance work less well, and that more adverse selection is always worse than less, are misconceived. Adverse selection implies a fall in numbers insured, and also a shift in coverage towards higher risks. If the shift in coverage outweighs the fall in numbers, the expected losses compensated by insurance are increased – that is the loss coverage is increased. From a public policy perspective, a degree of so-called ‘adverse’ selection in insurance is a good thing.
