Why Do People Buy Health Insurance?

The Arithmetic of Risk-Pooling

• 10-percent chance of hospitalization

that costs $10,000

• 90-percent chance of no hospitalization

• 100 policyholders

10 x $10,000 + 90 x $0

100 = $1000

$100,000

100=

With N policyholders

.10 N x $10,000 + .90 N x $0

N

.10 x $10,000 = $1000

(.10 x $10,000) + (.90 x 0)

N (.10 x $10,000 + .90 x $0)

N

(factor out N)

(N’s cancel)

“Actuarially fair premium” or“actuarial value”

(.10 x $10,000) + (.90 x 0) = $1000

= total claims divided by number of policies

= expected (or average) claims per policy

= break-even premium

Risk-pooling => Averaging

(.10 x $10,000) + (.90 x 0) = $1000

• 10 out of 100 consumers pay $1000, get $10,000 of hospital care

• 90 out of 100 policyholders pay $1000, get nothing back

Insurance reduces risk

(.10 x $10,000) + (.90 x 0) = $1000

Suppose everyone has $7000 income

• 10 out of 100 policyholders pay $1000,have $6000 to live on

• 90 out of 100 policyholders pay $1000, have $6000 to live on

Without insurance

(.10 x $10,000) + (.90 x 0) = $1000

Suppose everyone has $7000 income

• 10 out of 100 policyholders have -$3,000 to live on

• 90 out of 100 policyholders have $7,000 to live on

The Insurance Trade-off

• Give up premium ($1000)

• To avoid risk of much bigger loss(10-percent chance of $10,000) ??

Group Exercise (Part 1)

• What premium will cover claims if everyone has a 20% chance of $8,000 hospitalization?– What is the insurance trade-off?

• What additional premium will cover claims if add 2 regular dental checkups @ $75 each (and everyone goes to the dentist)?– What is the insurance trade-off for the dental

coverage?

Group Exercise (Part 2)

• What premium will cover claims for 1000 policyholders if– 500 have 5-percent risk of $10,000 hospitalization

– 500 have 10-percent risk of $10,000 hospitalization

• What premium will cover claims for 1000 policyholders if– 800 have 5-percent risk of $10,000 hospitalization

– 200 have 10-percent risk of $10,000 hospitalization