1. 1 Loss coverage as a public policy objective for risk classification schemes (to appear in The Journal of Risk and Insurance) www.guythomas.org.uk

2. 2 Main point From a public policy perspective, some adverse selection may be good Roughly: “The right people, those more likely to suffer loss, tend to buy (more) insurance” More technically: even if fewer policies are sold as a result of adverse selection, it may increase the proportion of loss events in a population which is covered by insurance (the “loss coverage”)

3. 3 Plan of talk Background Idea of loss coverage Perceived relevance in different insurance markets Three presentations of main point – tabular, parametric, graphical Multiple equilibria (if time)

4. 4 Background Poets and plumbers Poetry! Insurance economics… …zero-profit equilibrium …assume adverse selection is a material issue, worthy of theoretical attention

5. 5 More background Dissatisfaction & distress with public policy statements about risk classification (eg genetics & insurance) In my view, often malign But today, not talking specifically about genetics – wider perspective Benevolent, utilitarian public policymaker Main motivation : reduce aggregate suffering

6. 6 Loss coverage “The proportion of loss events in a population which is covered by insurance” (assume all losses size 1, insurance either 1 or 0) Given objective of reducing suffering, higher loss coverage often a reasonable objective for a public policymaker (ie insurance = “good thing”) (and cf. eg. tax relief on premiums, & public policymakers’ statements) But by observation, importance as perceived by policymakers seems to vary for different markets

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10. 10 Will now show that “right amount” of adverse selection can increase loss coverage, even if fewer policies are sold Three alternative presentations – 1.Tabular examples – 3 scenarios 2.Parametric model 3.Graphical presentation of (2)

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12. 12 Now suppose we charge a single pooled premium rate Take-up (previously 50%) –rises to 75% for higher risks –and falls to 40% for lower risks (NB adverse selection) Fewer policies issued => adverse selection bad? NO! Loss coverage is increased

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14. 14 Now suppose the adverse selection is more severe – Assume take-up – rises to 75% for higher risks (as above), – but falls to only 20% for lower risks (cf. 40% above) Fewer policies issued AND Loss coverage is reduced

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16. 16 Summary of scenarios Loss coverage is increased by the “right amount” of adverse selection (but reduced by “too much” adverse selection) In examples above, the outcome when risk classification is restricted depends on response of each risk group to change in price – demand elasticity Outcome also depends on relative population sizes and relative risks

17. 17 Formal definition Loss coverage = “A weighted average of the take-ups (θ i ) where the weights are the expected population losses ( P i μ i ), both insured and non-insured, for each risk group” Suggested policymaker’s objective: higher loss coverage Equal weights on coverage of higher & lower risks ex-post, so 4x weight on coverage of 4x higher risks ex-ante

18. 18 Alternative definitions of loss coverage In our model, loss always 1, and insurance 1 or 0 More generally, could have loss coverage = Or prioritise losses up to a limit (eg moratorium) Or could place greater weight on restitution of higher risks’ losses, even ex-post (like a spectral risk measure, but weighted by risk not severity)

19. 19 Other observed public policy objectives Public health (eg take-up of genetic tests & therapies) Privacy (eg perception that genetic data private & sensitive) Optional availability of insurance to higher risk groups, irrespective of actual take-up Moral principle of solidarity / equality, rejection of principle of statistical discrimination Incentives for loss prevention (eg flood risk) …..still, loss coverage a useful idea for an insurance-focused public policymaker

20. 20 (2) Parametric model for insurance demand Demand from population i at premium π π = pooled premium charged (no risk classification) μ i = true risk for group i ( i = 1 lower risk, 2 higher risk) P i = total population for group i τ i = “fair-premium take-up” (assume 0.5 throughout – not critical – just need scaling factor <1) λ i controls shape of demand curve

21. 21 Total demand curves examples (various λ )

22. 22 Elasticity of demand d i with respect to price π or equivalently which is …elasticity increases as the “relative premium” ( π / μ i ) gets dearer …and λ i is the elasticity when π = μ i, that is the “fair-premium elasticity” (=> corresponds to empirical estimates of price elasticity from risk- differentiated markets)

23. 23 Seems plausible that normally λ 1 < λ 2 – because for higher risks, insurance is dearer relative to the prices of other goods and services – and so given a common budget constraint, small proportional ↓ in price leads to a larger ↑ in demand for higher risks than for lower risks (the story still works if λ 1 = λ 2, or sometimes even if λ 1 > λ 2 ; but it has more force if λ 1 < λ 2 )

24. 24 Specifying the equilibrium Total income: π (d 1 (π) + d 2 (π))…(1) Total claims: d 1 (π) μ 1 + d 2 (π) μ 2 …(2) Profit = (1) – (2) Equilibrium:profit = 0 Existence Uniqueness   (but generally not troublesome, for plausible λ i )

25. 25 (3) Graphical presentation of model

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32. 32 Loss coverage under the pooled premium may be higher than under risk-differentiated premiums (λ 1 = 0.5, λ 2 = 1.1)

33. 33 But as we increase elasticity in the lower risk group, loss coverage eventually becomes lower than under risk-differentiated premiums (λ 1 = 0.8(was 0.5), λ 2 = 1.1)

34. 34 Higher loss coverage λ 1 = 0.5, λ 2 = 1.1 Lower loss coverage λ 1 = 0.8, λ 2 = 1.1 Summarising, when risk classification is restricted, two stylised cases

35. 35 For given relative populations ( P 1 /P 2 ), risks (μ 1 /μ 2 ) and fair-premium take-ups (τ 1 /τ 2 ) – “When risk classification is restricted, loss coverage increases if λ 2 is sufficiently high compared with λ 1 ” (but no simple conditions like “ λ 2 / λ 1 > k” or similar) Eg – – for λ 1 = 0.6, any λ 2 > 0.76 – for λ 1 = 0.4, any λ 2 > 0.33 (…note, for λ 1 low enough, even λ 2 < λ 1 may be sufficiently high).

36. 36 Plots of pooled premium & loss coverage against λ 2, for given λ 1 λ 1 = 0.4 → λ 1 = 0.6 → Dashed lines = reference levels under risk premiums (no adverse selection)

37. 37 Sensitivity of results to λ 1 and λ 2 Writing L for loss coverage, ∂L/∂λ 1 < 0 (readily confirmed by thought experiment) ∂L/∂λ 2 could be +/-… …but ∂L/∂λ 2 > 0 is typical for plausible parameters (note, contrary to possible casual intuition that higher elasticity is always going to make adverse selection “worse”) Results are much more sensitive to λ 1 than λ 2 (for “typical” case of a larger population with lower risk)

38. 38 Insurers’ perspective Maximise loss coverage = maximise premium income So in this setting, depends whether prefer to maximise market size by number of policies, or by premium income => possible explanation of why insurers’ lobbying on risk classification regulation is internationally incoherent (But once we drop assumption of zero profits, many actions of insurers are concerned with minimising loss coverage (eg claims control, policy design))

39. 39 Empirical estimates of fair-premium elasticity Term insurance: 0.4-0.5 (Pauly et al, 2003) 0.66 (Viswanathan et al, 2007) Private health insurance: USA: 0-0.2 (Chernew et al., 1997; Blumberg et al., 2001; Buchmueller et al, 2006) Australia: 0.36-0.50 (Butler, 1999) Not seen estimates for other classes Conclusion: (limited) evidence could be consistent with story

40. 40 Multiple equilibria

41. 41 Conditions for multiple equilibria (for demand r, risk μ and density of risk f, all indexed by a risk parameter g ) …but unfortunately doesn’t lead to any simple conditions on the λ i

42. 42 Multiple equilibria Want to plot equilibrium premium and loss coverage against a single elasticity parameter So define a “base” λ and then set with α = 1/3 say ( α not critical…similar pattern of results for other plausible α)

43. 43 P 1 = 80% of total population → P 1 = 90% → (sigmoid steepening) P 1 = 95% → (Multiple solutions for 1.33 < λ < 1.40)

44. 44 A collapse in coverage requires extreme λ Provided the real-world λ is in the green-arrow range, no multiple solutions, no collapse in coverage (But if there are particular markets where the real-world λ may plausibly be in the red-arrow range → different policy for those markets)

45. 45 Summary & next steps Some adverse selection may be good Stop telling policymakers (and students) it’s always bad! Done the poetry – now do the plumbing!

46. 46 References (2007) Some novel perspectives on risk classification. Geneva Papers on Risk and Insurance, 32: 105-132. (2008) Loss coverage as a public policy objective for risk classification schemes. Forthcoming in The Journal of Risk & Insurance. (2008) Demand elasticity, risk classification and loss coverage: when can community rating work? Working paper. www.guythomas.org.uk