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 outofpocket 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
OutofPocket 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 outofpocket limit.
Maximumallowableoutofpocket 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 outofpocket 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 outofpocket 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 outofpocket limit can be solved by solving
the following equation
OPM = D + r*(TEMD)
Where
OPM is outofpocket maximum,
r is the coinsurance rate,
D is the deductible,
TEM is the level of TE
triggering the maximum allowable outofpocket 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 outofpocket expense level is indeed $10,000 by plugging $30,000 into the
OPM equation and obtaining outofpocket 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 outofpocket equation with D and r and gat outofpocket
expenses equal to $15,000.
The piecewise lines for outofpocket
expenses:
Write a piecewise linear function where outofpocket
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 outofpocket 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 outofpocket expenses. (We just
calculated the trigger points.) The third
section is for total expenses over the level triggering the outofpocket
expense limit.
The piecewise outofpocket line is
presented below.
Piecewise Functions for OutofPocket 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*(TED) once the deductible is
met and until the outofpocket limit is met.
At outofpocket maximum, the
insurance company pays (1r)*(TEMD).
Above the outofpocketmaximum
the insurance company pays
(1r)*(TEMD)+(TETEM).
(The term (1r)*(TEMD) is a
constant. For the first health plan this
constant is .8*(300005000) 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+(TE30000)

Health Plan Two


TE

IP

$0 to $1,000

$0

$1,000 to $29,000

0.5*(TED)

>$29,000

14,000+(TE29000)

Analyzing the sum of the two Piecewise Functions:
I will analyze the sum of the
outofpocket 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 outofpocket 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 outofpocket expenses are $5,000 +0.2*(TE5000) and
insurance payouts are 0.8*(TE5000). The
sum of two expressions is TE.
For total health expenses
greater than $30,000, outofpocket expenses are $10,000 (the maximum) and
insurance payouts are 20,000+TE30000.
The sum of the two is again TE.
So the sum of total
outofpocket 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 outofpocket
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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