Piecewise Lines

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