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”

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”)

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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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Loss coverage in different markets

Market

Perceived social value of restitution of losses

Relevance of insured’s personal circumstances

Observed public policy

Motor liability, employer liability

Very high Nil Compulsory insurance (100% loss coverage)

FSA penalties Negative Nil Insurance banned (0%)

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Market



Observed Public policy



Pet insurance Low? High Laissez-faire


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Market



Observed public policy



Life insurance High

High (# dependants?)

Voluntary insurance, but regulate risk classification to maximise loss coverage

Pet insurance Low? High Laissez-faire


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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 –

1.Tabular examples – 3 scenarios

2.Parametric model

3.Graphical presentation of (2)

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Scenario 1: risk-differentiated premiums

Lower risk group Higher risk group

Population 10,000 2,000

Risk 1/100 4/100

50% take-up (members of group purchasing insurance)

5,000 1,000

Premium required 1/100 4/100

(A) Expected loss events 100 80

(B) Loss events insured 50 40

Loss coverage:

Σ(B) / Σ(A)

(50+40)/(100+80) = 0.50

Note: 6,000 policies issued

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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

=> adverse selection bad?

NO!

Loss coverage is increased

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Scenario 2: pooled premium rate, higher loss coverage



Risk 1/100 4/100

Take-up: 40% for lower risks,75% for higher risks (NB adverse selection)

4,000 1,500

Premium required 1.82/100



Loss coverage:

Σ(B) / Σ(A)

(40+60)/(100+80) = 0.56higher

Note: 5,500 policies issued (cf. 6,000 before)

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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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Scenario 3: pooled premium rate, lower loss coverage



Risk 1/100 4/100

Take-up: 20% for lower risks,75% for higher risks (severe adverse selection)

2,000 1,500

Premium required 2.29/100



Loss coverage:

Σ(B) / Σ(A)

(20+60)/(100+80) = 0.44 lower

Note: 3,500 policies issued (cf. 5,500 and 6,000 before)

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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 (θi) where the weights are the expected population losses (Pi μi), 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

ii

iii

P

P

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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)

)($

)($

lossespopulation

lossesinsured

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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 π

π = 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

τi = “fair-premium take-up” (assume 0.5 throughout – not critical – just need scaling factor <1)

λi controls shape of demand curve

ePd

ii

ii

i

1

)(

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0

2000

4000

6000

0.01 0.02 0.03 0.04

Pooled premium π

Total demand

λ = 0.5

λ = 0.8

λ = 1.5

Total demand curves examples (various λ)

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Elasticity of demand di with respect to price π

or equivalently

which is

…elasticity increases as the “relative premium” (π/μi)

gets dearer

…and λi is the elasticity when π = μi, that is the

“fair-premium elasticity”

(=> corresponds to empirical estimates of price elasticity from risk-differentiated markets)

ln

)(ln

id

)(

. i

i

d

d

iii

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Seems plausible that normally λ1 < λ2

– 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 λ1 = λ2, or sometimes even if λ1 > λ2; but it has more force if λ1 < λ2)

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Specifying the equilibrium

Total income: π (d1(π) + d2(π)) …(1)

Total claims: d1(π) μ1 + d2(π) μ2 …(2)

Profit = (1) – (2)

Equilibrium:profit = 0

Existence Uniqueness (but generally not troublesome, for plausible

λi)

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0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims

Income from lower risks

Income from higher risks

Total INCOME (under pooled premium)

Claims from lower risks

Claims from higher risks

Total CLAIMS (under pooled premium)

(3) Graphical presentation of model

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(3) Graphical presentation of model

0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income

& total claims

Total INCOME = Total CLAIMS(under differentiated premiums)Income from lower risks

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0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims







28

0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims







29

0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims







30

0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims







31

0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims







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0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims

Loss coverage under the pooled premium may be higher than under risk-differentiated premiums (λ1= 0.5, λ2 = 1.1)

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0

50

100

0.00 0.01 0.02 0.03 0.04Pooled premium π

Total income&

total claims

But as we increase elasticity in the lower risk group, loss coverage eventually becomes lower than under risk-differentiated premiums

(λ1= 0.8(was 0.5), λ2 = 1.1)

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Higher loss coverageλ1= 0.5, λ2 = 1.1

Lower loss coverageλ1= 0.8, λ2 = 1.1

0

50

100

0.00 0.01 0.02 0.03 0.04

0

50

100

0.00 0.01 0.02 0.03 0.04

Summarising, when risk classification is restricted, two stylised cases

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For given relative populations (P1/P2), risks (μ1/μ2) and fair-premium take-ups (τ1/τ2) –

“When risk classification is restricted, loss coverage increases if λ2 is sufficiently high compared with λ1” (but no simple conditions like “λ2/ λ1 > k” or similar)

Eg –

– for λ1 = 0.6, any λ2 > 0.76

– for λ1 = 0.4, any λ2 > 0.33

(…note, for λ1 low enough, even λ2 < λ1 may be sufficiently high).

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Plots of pooled premium & loss coverage against λ2, for given λ1

λ1 = 0.4 →

λ1 = 0.6 →

Dashed lines = reference levels under risk premiums (no adverse selection)

0.42

0.47

0.52

0.57

0.62

0.3 0.6 0.9 1.2 1.5

λ 2

λ2 = 0.76

0.014

0.017

0.020

0.023

0.3 0.6 0.9 1.2 1.5

λ 2

Pooled premium

0.014

0.017

0.020

0.023

0.3 0.6 0.9 1.2 1.5

Loss coverage

0.42

0.47

0.52

0.57

0.62

0.3 0.6 0.9 1.2 1.5

λ2 = 0.33

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Sensitivity of results to λ1 and λ2

Writing L for loss coverage,

∂L/∂λ1 < 0 (readily confirmed by thought experiment)

∂L/∂λ2 could be +/-…

…but ∂L/∂λ2 > 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 λ1 than λ2

(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

=> 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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0

40

80

0.00 0.01 0.02 0.03 0.04

Multiple equilibria

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Conditions for multiple equilibria

)1(...1

dg

r

rf

g

ggg

(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 λi

)2(... dg

r

rf

g

ggg

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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)

1

22

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P1 = 80% of total population →

P1 = 90% →

(sigmoid steepening)

P1 = 95% →

(Multiple solutionsfor 1.33 < λ < 1.40)

Pooled premium

0.005

0.020

0.035

0.050

0.2 0.8 1.4 2.0

λ

Loss coverage

0.05

0.25

0.45

0.65

0.2 0.8 1.4 2.0

λ

0.005

0.020

0.035

0.050

0.2 0.8 1.4 2.0

0.00

0.20

0.40

0.60

0.2 0.8 1.4 2.0

0.005

0.020

0.035

0.050

0.2 0.8 1.4 2.0

0.05

0.25

0.45

0.65

0.2 0.8 1.4 2.0

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A collapse in coverage requires extreme λ

Pooled premium

0.005

0.020

0.035

0.050

0.2 0.8 1.4 2.0λ

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)

Loss coverage

0.00

0.20

0.40

0.60

0.2 0.8 1.4 2.0λ

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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!

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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

loss coverage as a public policy objective for risk classification schemes

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