Wednesday, December 11, 2013

High deductibles versus high out-of-pocket limits.

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.

Some Notation:

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.

Out-of-pocket Comparisons

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


EXP= $13,000.

(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.


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