Question: A person is considering two insurance plans. The first plan has a $5,000 deductible and a $5,000 maximum allowable out-of-pocket expense limit. The second plan has a $3,000 deductible, a 20% coinsurance rate, and a $9,000 out-of-pocket expense limit.
Write a function describing insurance payments received by the insured as a function of total health expenses incurred under the two health plans.
When are actual out-of-pocket expenses under the first insurance plan greater than out-of-pocket expenses under the second insurance plan?
First some Definitions:
Deductible — A deductible is the specified amount of money that an insured must pay prior to an insurance company paying a claim.
Coinsurance rate — the share of health expenses paid by the insured for expenses over the deductible but lower than the maximum allowable out-of-pocket expense limit.
The out-of-pocket expense limit — The maximum amount of covered claims that the insured is responsible for under a policy.
After the out-of-pocket expense limit is reached the insurance company pays all covered expenses.
I will denote EXPT as total Health Care Expenses, REIM_EXP as reimbursed expenses, and OUTP as out-of-pocket expenses.
Payments to insured under the first health plan:
The first plan is simple
REIM_EXP = 0 for EXPT<$5,000.
REIM_EXP=(EXPT-$5,000) for EXPT>$5,000.
Payments to insured under the second health plan:
REIM_EXP=$0 for EXPT< $3,000.
REIM_EXP = 0.80 x (EXP-$3,000) for $3,000<=EXPT<=$33,000.
At EXPT= $33,000, out of pocket expenses are $9,000. For this calculation go to my previous post.
When total covered health expenditures exceed $33,000 all health expenditures are reimbursed.
REIM_EXP= $24,000 + (EXPT-$33,000) for EXPT > $33,000.
All expenses are out-of-pocket when total expenditures are below the deductible. When EXPT is less than $3,000 the out-of-pocket expenses of both plans are identical.
For EXPT over $3,000 out-of-pocket expenses rise quicker for the first plan than the second plan because the second plan reimburses 80% of expenses.
Out-of-pocket expenses for the first plan max out at $5,000 when EXPT=$5,000.
Out-of-pocket expenses for the second plan for EXPT>$3,000 are given by the following equation.
OUTP=Min((3,000 + 0.2 x (EXP-3000),9,000)
Setting OUTP= $5,000 and solving we get:
5000= 3000 + 0.2 x (EXP-3000)
Rearrange the above and get
$2,600=0.20 x EXP
(Note that 5000 = 3000 + 0.20 x (13,000-3000))
When Health Expenses are between $3,000 and $13,000 out-of-pocket expenses are lower under health plan two than under health plan one.
When total health expenses rise over $13,000 unreimbursed health costs are lower for the first health plan with the larger deductible.
Concluding Remarks: What is better a health plan with a high deductible or a health plan with a lower deductible but a higher out-of-pocket expense limit? There is no correct answer to this question. High deductibles result in a large number of people receiving no or very little insurance payments However the reduction in payouts due to a high deductible allows insurance firms to offer a lower allowable out-of-pocket limit.
Interestingly, prior to ACA many Republicans supported higher deductible health plans linked to health savings accounts. Ironically, the ACA is forcing many younger people into higher deductible plans. See the WSJ article below.