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.
the other insurance math problems can be found here.