Question: A health plan has a $3,000 deductible, a coinsurance rate of 20%, and a maximum allowable out-of-pocket expense limit of $9,000. What level of total health expenditures will result in the insurance company paying all further covered health care expenses during the year?
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.
Will denote total health expenditures EXPT and out of pocket expenditures OUTP.
Out of pocket expenditures under the plan are the minimum of $9,000 (the limit under the plan) and the amount stipulated by the benefit formula.
When EXPT is less than $3,000 the OUTP is equal to EXPT.
When EXPT is greater than $3,000 the OUTP is give by equation below.
OUTP=Min($9,000,$3000+0.20 x (EXPT- $3000))
When OUTP=$9,000 the insurance firm must pay all additional expenditures.
This occurs when
$9000 = $3000 + 0.20 X (EXPT - $3000)
($6000 + 0.20 X $3,000)/0.2 = OUTP = $33,000.
Note that when EXPT= $33,000 the OUTP = $3,000 + 0.20 x ($33,000- $3,000)
or OUTP = $9,000.
Happiness — Answer appears to check today.