## Wednesday, April 9, 2014

### Incentives for three health plans.

Situation:   The table below lists characteristics (deductible, coinsurance rate, and out-of-pocket expense limits) for three different health plans

 Three Health Insurance Plans Plan Number One Plan Number Two Plan Number Three Deductible \$2,500 \$4,000 \$8,000 Coinsurance Rate 20% 30% 0% Out of-pocket Expense  Limit \$5,000 \$10,000 \$8,000

The table below describes total health expenditures (the amount paid by the insured and the amount paid by the insurance company) for a sample of 35,000 people.

 Health Expenditures Number of People \$0 8,000 \$500 7,000 \$3,000 6,000 \$6,000 5,500 \$12,000 5,500 \$25,000 2,800 \$150,000 200 Total 35,000

Questions:

What is the mean and the total health expenditure paid by the insurance firm for the three health insurance plans and this group of individuals?

A company offers all three insurance policies to its employees.  Discuss why average health expenditures will differ across these three plans?

Analysis:

We are given total health expenses (both insurance payment and payment by the insured).    We need insurance payments only for each of the three plans.

In order to get the insurance payment function we need the threshold where the insurance company pays 100% of the bill.   This calculation is explained in the post below.
http://www.dailymathproblem.com/2018/07/basic-algebra-of-two-insurance-plans.html

The calculation for the threshold triggering 100% payments is laid out in the table below.

 Plan 1 Plan 2 Plan 3 Formula OUTP_MAX \$5,000 \$10,000 \$8,000 a1 DEDUC \$2,500 \$4,000 \$8,000 a2 Coinsurance Rate% 0.2 0.3 NA a3 THRESH \$15,000 \$24,000 \$8,000 (a1-a2+a3*a32)/a3 CHECK \$5,000 \$10,000 \$8,000 a2+a3*(a4-a2)

The insurance plan parameters and the frequency distribution for the data on total health expenditures are used to get figures for insurance payments under the three plans.

The insurance company payout formula differs in three intervals.

First Interval:

Insurance expenditures are zero for health expenditures below the deductible.

Second Interval:

When total health expenses are between the deductible and the threshold triggering the maximum out-of pocket expense the insurance payment is

(1-CO_RATE) x (HE-DEDUC)
Third Interval:

Where CO_RATE is the coinsurance rate, HE is health expense and DEDUC is deductible.

When health expense is greater than the threshold triggering the maximum allowable out-of expenses the insurance payout is the sum of the payout at the threshold and the difference between total health expenditures and the threshold.

Payment at the threshold is

(1-C0_RATE) x (THRESH-DEDUC)

These numbers are fairly easy to program with a single if statement in SAS but much harder to do in EXCEL   I don’t have SAS so I hard wired the three intervals in EXCEL

 Health Expenditures Number of People Insurance payments plan 1 Insurance payments plan 2 Insurance Payments Plan 3 \$0 8,000 \$0 \$0 \$0 \$500 7,000 \$0 \$0 \$0 \$3,000 6,000 \$400 \$0 \$0 \$6,000 5,500 \$2,800 \$1,400.0 \$0 \$12,000 5,500 \$7,600 \$5,600 \$4,000 \$25,000 2,800 \$20,000 \$15,000 \$17,000 \$150,000 200 \$145,000 \$140,000 \$142,000 SUMPRODUCT \$144,600,000 \$108,500,000 \$98,000,000 Average Insurance Company Expenditure \$4,131.43 \$3,100.00 \$2,800.00

Discussion:

The high-deductible health plan is the least expensive of the three health plans.

More people have an incentive to at least partially economize on health expenditures under the second health plan than under the other two health plans.  (Under health plan two there is some incentive to economize until health expenditures reach \$24,000.   The incentive to economize under health plans one and three end at total health expenditures of \$15,000 and \$8,000 respectively.)

Insurance payouts are lower under health plan three than the other two health plans.  High deductibles are highly effective at reducing total costs because all insured individuals must pay the entire deductible prior to receiving payments.

There are two reasons to believe that the cost differential across health plans will be larger than the ones shown here if an individual is free to choose among all three health plans.

Both health plans two and three provide a greater incentive for individuals to economize on consumption of health care than health plan one.

Healthier people will tend to sign up for the least expensive health plan.

The cost figures presented here are a lower bound on the reduction in insurance payouts caused by moving away from plan one towards a plan with greater cost sharing.  Actual reduction in payouts is affected by the incentive to economize and adverse selection – the incentive for healthy people to choose the least expensive plan.

Other health insurance math problems found here.
http://www.dailymathproblem.com/p/piecewise-lines.html