Loss coverage as a public policy objective for risk classification schemes

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Loss coverage as a public policy objective for
risk classification schemes

• (to appear in The Journal of Risk and Insurance)

www.guythomas.org.uk
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• 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 ü
• 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) ü

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• 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)

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

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• 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

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

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• 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
• gt adverse selection bad?
• NO!
• Loss coverage is increased

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• 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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• 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

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• Formal definition
• Loss coverage
• A weighted average of the take-ups (?i) where
  the weights are the expected population losses
  (Pi µi), both insured and non-insured, for each
  risk group
• Suggested policymakers 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

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• 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)
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• 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
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• (2) Parametric model for insurance demand
• Demand from population i at premium p
• p pooled premium charged (no risk
  classification)
• µi true risk for group i (i 1 lower risk, 2
  higher risk)
• Pi total population for group i
• ti fair-premium take-up
• (assume 0.5 throughout not critical just need
  scaling factor lt1)
• ?i controls shape of demand curve

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Total demand curves examples (various ?)
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• Elasticity of demand di with respect to price p
• or equivalently
• which is
• elasticity increases as the relative premium
  (p/µi)
• gets dearer ü
• and ?i is the elasticity when p µi, that is
  the
• fair-premium elasticity
• (gt corresponds to empirical estimates of price
  elasticity from risk-differentiated markets)

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• Seems plausible that normally ?1 lt ?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 ?1 ?2, or sometimes
  even if ?1 gt ?2 but it has more force if ?1 lt
  ?2)

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• Specifying the equilibrium
• Total income p (d1(p) d2(p)) (1)
• Total claims d1(p) µ1 d2(p) µ2 (2)
• Profit (1) (2)
• Equilibrium profit 0
• Existence ü
• Uniqueness û (but generally not troublesome, for
  plausible ?i)

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(3) Graphical presentation of model
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Loss coverage under the pooled premium may be
higher than under risk-differentiated premiums
(?1 0.5, ?2 1.1)
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But as we increase elasticity in the lower risk
group, loss coverage eventually becomes lower
than under risk-differentiated premiums (?1
0.8(was 0.5), ?2 1.1)
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Summarising, when risk classification is
restricted, two stylised cases
Higher loss coverage ?1 0.5, ?2 1.1
Lower loss coverage ?1 0.8, ?2 1.1
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• For given relative populations (P1/P2), risks
  (µ1/µ2) and fair-premium take-ups (t1/t2)
• When risk classification is restricted, loss
  coverage increases if ?2 is sufficiently high
  compared with ?1
• (but no simple conditions like ?2/ ?1 gt k or
  similar)
• Eg
• for ?1 0.6, any ?2 gt 0.76
• for ?1 0.4, any ?2 gt 0.33
• (note, for ?1 low enough, even ?2 lt ?1 may be
  sufficiently high).

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Plots of pooled premium loss coverage against
?2, for given ?1
?1 0.4 ?
Dashed lines reference levels under risk
premiums (no adverse selection)
?1 0.6 ?
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• Sensitivity of results to ?1 and ?2
• Writing L for loss coverage,
• ?L/??1 lt 0
• (readily confirmed by thought experiment)
• ?L/??2 could be /-
• but ?L/??2 gt 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 ?1 than ?2
• (for typical case of a larger population with
  lower risk)

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• 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
• gt 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))

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• 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

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Multiple equilibria
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• Conditions for multiple equilibria

(for demand r, risk µ and density of risk f, all
indexed by a risk parameter g) but
unfortunately doesnt lead to any simple
conditions on the ?i
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• Multiple equilibria
• Want to plot equilibrium premium and loss
  coverage against a single elasticity parameter
• So define a base ? and then set
• with a 1/3 say
• (a not criticalsimilar pattern of results for
  other plausible a)

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P1 80 of total population ?
P1 90 ? (sigmoid steepening)
P1 95 ? (Multiple solutions for 1.33 lt ? lt
1.40)
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A collapse in coverage requires extreme ?
Provided the real-world ? is in the green-arrow
range, no multiple solutions, no collapse in
coverage (But if there are particular markets
where the real-world ? may plausibly be in the
red-arrow range ? different policy for those
markets)
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• Summary next steps
• Some adverse selection may be good
• Stop telling policymakers (and students) its
  always bad!
• Done the poetry now do the plumbing!

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• 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