Piecewise Lines
This post provides an
interesting and practical example of piecewise lines. The example should be of interest to a
diverse group including algebra students, consumers, and insurance
executives.
Situation: Consider two health plans with deductibles,
coinsurance rate, and maximum allowable out-of-pocket limit as presented in the
table below.
Characteristics of two health plans
|
||
Health Plan One
|
Health Plan Two
|
|
Deductible
|
$5,000
|
$1,000
|
Coinsurance Rate
|
0.2
|
0.5
|
Maximum Allowable
Out-of-Pocket Limit
|
$10,000
|
$15,000
|
Definitions:
Deductible: A specified amount of money that a person must pay
before an insurance company will pay a claim.
Coinsurance Rate: The percent of cost on a claim
the insured person pays after the deductible is met and prior to the customer
meeting the maximum allowable out-of-pocket limit.
Maximum-allowable-out-of-pocket limit. The most an
insured person is required to pay for claims during a calendar year. Once this limit is reached the insurance
company pays 100% of all approved claims.
Questions:
At what level of total health
expenditures does the insurance company start paying 100 percent of approved
claims?
Write a piecewise linear function
where out-of-pocket health care expenses is the Y variable and total health
expenses is the X variable for the two health plans.
Write a piecewise linear
function where insurance company payment is the dependent variable and total
health expenses is the X variable?
Demonstrate that the sum of
the piecewise linear out-of-pocket expense function and the piecewise insurance
payout function equals total health expenses in each part of the domain of the
functions.
Piecewise Lines Answers
At what level of total health expenditures does the
insurance company start paying 100 percent of approved claims?
The level of total health
expenses triggering the maximum allowable out-of-pocket limit can be solved by solving
the following equation
OPM = D + r*(TEM-D)
Where
OPM is out-of-pocket maximum,
r is the coinsurance rate,
D is the deductible,
TEM is the level of TE
triggering the maximum allowable out-of-pocket expense limit. (Once TEM is hit the insurance company pays
100 percent of all claims)
For equation one plug D=$5,000,
r=0.2, and OPM into the equation. Solve
for TEM and get TEM = $30,000
Confirm that when TEM=$30,000
the out-of-pocket expense level is indeed $10,000 by plugging $30,000 into the
OPM equation and obtaining out-of-pocket expenses equal to $10,000.
For equation two plug
D=$1,000, r=0.5 and OPM=$15,000 into the equation. Rearrange for TE. I get TEM=$29,000.
I check my answer by plugging
TEM=$29,000 into the out-of-pocket equation with D and r and gat out-of-pocket
expenses equal to $15,000.
The piecewise lines for out-of-pocket
expenses:
Write a piecewise linear function where out-of-pocket
health care expenses is the Y variable and total health expenses is the X
variable for the two health plans.
There are three sections of
the piecewise out-of-pocket expense function.
The first section is for total expenses from 0 to the deductible. The second section is from the deductible to
the value of total expenses triggering the out-of-pocket expenses. (We just
calculated the trigger points.) The third
section is for total expenses over the level triggering the out-of-pocket
expense limit.
The piecewise out-of-pocket line is
presented below.
Piecewise Functions for Out-of-Pocket Expenses
|
|
Health Plan One
|
|
TE
|
OPE
|
$0 to $5,000
|
TE
|
$5,000 to $30,000
|
$5.000+0.20*(TE-$5,000)
|
>$30,000
|
$10,000
|
Health Plan Two
|
|
$0 to $1,000
|
TE
|
$1,000 to $29,000
|
$1,000+o.5*(TE-$1,000)
|
>$29,000
|
$15,000
|
Write a piecewise linear function where insurance
company payment is the dependent variable and total health expenses is the X
variable?
The insurance company pays $0
when total expenses are under the deductible, r*(TE-D) once the deductible is
met and until the out-of-pocket limit is met.
At out-of-pocket maximum, the
insurance company pays (1-r)*(TEM-D).
Above the out-of-pocket-maximum
the insurance company pays
(1-r)*(TEM-D)+(TE-TEM).
(The term (1-r)*(TEM-D) is a
constant. For the first health plan this
constant is .8*(30000-5000) or $20,000.) Confirm the constant is $14,000 for the
second health plan.
The piecewise Insurance Payment Lines
are below
Piecewise Insurance Payout Lines
|
|
Health Plan One
|
|
TE
|
IP
|
$0 to $5,000
|
$0
|
$5,000 to $30,000
|
0.2*(TE-$5,000)
|
>$30,000
|
20,000+(TE-30000)
|
Health Plan Two
|
|
TE
|
IP
|
$0 to $1,000
|
$0
|
$1,000 to $29,000
|
0.5*(TE-D)
|
>$29,000
|
14,000+(TE-29000)
|
Analyzing the sum of the two Piecewise Functions:
I will analyze the sum of the
out-of-pocket and insurance payout piecewise functions for the first health
plans and encourage the reader to do so for the second health plan.
For total health expenses
between $0 and $5,000 the out-of-pocket expenses are $TE and the insurance
payout is $0. The sum of the two is
$TE.
For total health expenses
between $5,000 and $30,000 out-of-pocket expenses are $5,000 +0.2*(TE-5000) and
insurance payouts are 0.8*(TE-5000). The
sum of two expressions is TE.
For total health expenses
greater than $30,000, out-of-pocket expenses are $10,000 (the maximum) and
insurance payouts are 20,000+TE-30000.
The sum of the two is again TE.
So the sum of total
out-of-pocket expenses and insurance paid expenses is equal to total
expenses.
Authors Note: I am planning a blog utilizing the piecewise
functions derived here and information on claims to calculate expected out-of-pocket
costs and expected insurance payouts for the two plan. Follow this blog In order to insure that you
obtain this information
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