2 DAR Formulas

Objectives

In this section, we will discuss:

The three numbers that you will need to calculate Days in AR.
What those three numbers mean and where to find them.
The steps in the calculation of Days in AR.
How to create new formulas from the base Days in AR formula.
What you can learn from these new formulas and why you might want to do so.
The relationship between the variables involved in Days in AR.

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2.1 The Three Numbers

You need only three numbers to calculate the average Days in AR. They are the Number of Days in the Period, the total Gross Charges for that period, and the Ending AR Balance. Technically, you will also need the Average Daily Charge, but you can derive this number from two of the three essential numbers:

2.1.1 Number of Days in the Period

This is literally the number of days within the period for which you are measuring Days in AR. DAR is typically measured in monthly and quarterly increments, so this number will usually hover around 30, 60, 90, etc. The following table contains a list of common reporting periods and the number of days in each:

2.1.2 Total Gross Charges

Gross charges, which are the full fee schedule⁵ charges, are all charges generated by the practice, regardless of actual reimbursement. The Gross Charges figure is the total dollar amount charged during the Number of Days in the Period that you are measuring. Whether billed to insurance on a claim or to a self-pay patient, all charges for the period are included here.

2.1.3 Average Daily Charge

This number is the total Gross Charges amount divided by the Number of Days in the Period.

2.1.4 Ending AR Balance

This is the Accounts Receivables balance ($USD) at the close of business (COB) on the final day of the Period that you are measuring.

2.2 Step-by-Step Example

Let’s go through an example of the Days in AR calculation, step-by-step. The following table contains the three data points you need to calculate Days in AR:

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2.2.1 Calculate the Average Daily Charge

First, we need to calculate the Average Daily Charge. The formula is as follows:

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\[\begin{equation} \dfrac {\textrm{Gross Charges}}{\textrm{Number of Days in Period}} = {\textrm{Average Daily Charge}} \tag{2.1} \end{equation}\]

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Take the Gross Charges from the table above, divide it by the Number of Days in the Period and our Average Daily Charge will be $4,381.34:

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\[\begin{align} \dfrac {\textrm{Gross Charges}}{\textrm{Number of Days in Period}} &= {\textrm{Average Daily Charge}}\\ \\ \\ \dfrac{131,440.30}{30}&= 4,381.34 \tag{2.2} \end{align}\]

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total Gross Charges / Number of Days in the Period = Average Daily Charge.

131440.30 / 30

## [1] 4381.343

(Note: This is an R language coding chunk. I will be using these to show the outputs of calculations for the sake of accuracy.)

2.2.2 Calculate the Days in AR

Now that we have the three variables required by the DAR formula, we can calculate the Days in AR. The formula is as follows:

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\[\begin{equation} \dfrac {\textrm{Ending AR Balance}}{\textrm{Average Daily Charge}}= {\textrm{Days in AR}} \tag{2.3} \end{equation}\]

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Substitute the Ending AR Balance and Average Daily Charge into the formula and divide:
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\[\begin{align} \dfrac {\textrm{Ending AR Balance}}{\textrm{Average Daily Charge}}&= {\textrm{Days in AR}}\\ \\ \\ \dfrac{203,460.50}{4,381.34}&= 46.44 \tag{2.4} \end{align}\]

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Ending AR Balance / Average Daily Charge = Days in AR.

203460.50 / 4381.34

## [1] 46.43796

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This gives us an average Days in AR of 46.44 days. Now our report table looks like this:

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2.3 Visualizing Days in AR

In terms of Days in AR, the figures usually involve time-series data, i.e. month-to-month, quarter-to-quarter, etc. For time-series, best practices generally involve a line plot, tracing the Days in AR figures from left to right (in English-speaking cultures.)

If the stakeholders that you are presenting the data to have an established Days in AR target for their organization (for instance, it’s 39.445 in the graph below), it would help to map it across the time series, as I’ve mapped it as the black dotted line.

Another suggestion I would make is to differentiate between those months/quarters that were under (Passed) or over (Failed) the target threshold.

Days in AR Report: 2021 Month-to-Month. (DAR Target = 39.445)

Another way to visualize Days in AR is to chart the month-to-month variance from the target Days in AR:

Days in AR Variance: 2021 Month-to-Month. (DAR Target = 39.445)

2.4 Building on the DAR Formula

Now that we have a basic understanding of the formula for Days in AR, we can expand upon and rearrange it to do many other things. The formulas we will derive from the basic DAR calculation in this chapter will allow us to perform many, many different kinds of data analysis. To start off, we will include the calculation for the Average Daily Charge in the denominator of our basic DAR formula:

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\[\begin{equation} {\textrm{Days in AR}}=\dfrac {\textrm{Ending AR Balance}}{\textrm{total Gross Charges} \div \textrm{Number of Days in Period}} \tag{2.5} \end{equation}\]

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To simplify things, I’m going to begin using mathematical variables and acronyms for these numbers. They are as follows:

Variable	Acronym	Description
$n$	`NDiP`	Number of Days in the Period
$x$	`GCt`	total Gross Charges for `NDiP`
$y$	`EARB`	Ending AR Balance
$c$	`ADC`	Average Daily Charge for `NDiP`
$z$	`DAR`	Days in AR
$t$	`DARt`	Target Days in AR

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The Days in AR formula can now be represented in several ways:
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\[\begin{align} {\textrm{Days in AR}}&=\dfrac {\textrm{Ending AR Balance}}{\textrm{Gross Charges} \div \textrm{Number of Days in Period}}\\ \\ \\ {\textrm{DAR}}&=\dfrac {\textrm{EARB}}{\textrm{GCt} \div \textrm{NDiP}}\\ \\ \\ {z}&=\dfrac {y}{x \div n}\\ \tag{2.6} \end{align}\]

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2.4.1 EARB Required for $x$ DAR

What if we wanted to know the exact Ending AR Balance we would need to attain a certain number of Days in AR? First, we will need to find the formula for EARB. We have our DAR formula converted to variables to make it easier to visualize what is going on mathematically:

\[\begin{align} {\textrm{If:}} \ \ \ {\textrm{DAR}}&=\dfrac {\textrm{EARB}}{\textrm{GCt} \div \textrm{NDiP}}\\ \\ \\ {\textrm{Where:}}\ \ \ y&={\textrm{EARB}}\\ \ \ x&={\textrm{GCt}}\\ \ \ n&={\textrm{NDiP}}\\ \ \ z&={\textrm{DAR}}\\ \ \ \\ \\ {\textrm{Then:}} \ \ \ {z}&=\dfrac {y}{x \div n}\\ \tag{2.7} \end{align}\]

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We can find the formula for EARB by taking the formula for DAR and solving for $y$:

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\[\begin{align} {y}&=\dfrac{zx}{n}\\ \\ \\ \\ {\textrm{EARB}}&=\dfrac{\textrm{DAR} \times \textrm{GCt}}{\textrm{NDiP}}\\ \\ \\ \tag{2.8} \end{align}\]

\[\\[3pt]\] Let’s use our table of results from the example in Chapter 2 and say that we would like to have achieved a DAR of 39.44. What Ending AR Balance would we have needed?

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\[\\[3pt]\] Substituting the known variables from the table into our EARB formula and performing the calculation gives us a new EARB of $172,822.09:

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DARt ($t$) multiplied by GCt ($x$), then divided by NDiP ($n$), = EARB ($y$).

(39.445 * 131440.30) / 30

## [1] 172822.1

This is the Ending AR Balance we would need for 39.445 Days in AR.

2.4.2 GCt Required for $x$ DAR

We can perform the same steps if we wanted to know the total Gross Charges we would need for a certain number of Days in AR. We can find the formula for GCt by taking the formula for DAR and solving for $x$:

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\[\begin{align} {x}&=\dfrac{yn}{z}\\ \\ \\ \\ {\textrm{GCt}}&=\dfrac{\textrm{EARB} \times \textrm{NDiP}}{\textrm{DAR}}\\ \\ \\ \tag{2.9} \end{align}\]

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We would again like to have achieved a DAR of 39.445. What total Gross Charges would we have needed to achieve this result?

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If we plug our known variables into our formula for total Gross Charges, we get a GCt of $154,742.43.

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EARB ($y$) multiplied by NDiP ($n$), then divided by DARt ($t$) , = GCt ($x$).

(203460.50 * 30) / 39.445

## [1] 154742.4

The Average Daily Charge is determined in part by GCt, thus we will need to recalculate it since GCt has changed. Dividing the new GCt amount by the NDiP gives us a new ADC of $5,158.08.

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GCt ($x$) divided by NDiP ($n$) = ADC ($c$).

154742.43 / 30

## [1] 5158.081

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2.4.3 NDiP Required for $x$ DAR

We can perform the same steps if we wanted to know the Number of Days we’d need to attain a certain number of Days in AR. To find the formula for NDiP, we take the formula for DAR and solve for $n$, giving us:

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\[\begin{align} \frac {zx}{y}={n} \tag{2.10} \end{align}\]

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Looking back at our table, we would again like to have achieved a DAR of 39.44. What Number of Days in the Period would we have needed to achieve this result?

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If we plug our known variables into our formula for total Gross Charges, we get an NDiP of 25.48.

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DARt ($t$) multiplied by GCt ($x$), then divided by EARB ($y$), = NDiP ($n$).

(39.445 * 131440.30) / 203460.50

## [1] 25.4824

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As the Average Daily Charge is determined in part by NDiP, this calculation will also change it, to $5,158.08.

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GCt ($x$) divided by NDiP ($n$) = ADC ($c$).

131440.30 / 25.4824

## [1] 5158.082

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2.4.4 ADC Required for $x$ DAR

This formula is going to be a bit different from the previous three. Remember, our formula for the Average Daily Charge is:

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\[\begin{align} \dfrac {\textrm{Gross Charges}}{\textrm{Number of Days in Period}} = {\textrm{Average Daily Charge}} \tag{2.11} \end{align}\]

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Let’s take our DAR formula and make the appropriate adjustments:

\[\begin{align} {\textrm{If:}} \ \ \ \dfrac {\textrm{EARB}}{\textrm{ADC}}&={\textrm{DAR}} \\ \\ \\ {\textrm{Where:}}\ \ \ {\textrm{EARB}}&=y\\ \ \ {\textrm{ADC}}&=c\\ \ \ {\textrm{DAR}}&=z\\ \ \ \\ \\ {\textrm{Then:}} \ \ \ \dfrac {y}{c}&={z} \\ \tag{2.12} \end{align}\]

To find the formula for ADC, we take the formula for DAR and solve for $c$, giving us: \[\\[3pt]\]

\[\begin{align} \dfrac {y}{z}={c} \tag{2.13} \end{align}\]

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Looking back at our table, we would again like to have achieved a DAR of 39.445. What Average Daily Charge would we have needed to achieve this result?

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If we plug our known variables into our formula for Average Daily Charge, we get an ADC of $5,158.74.

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EARB ($x$) divided by DARt ($t$) = ADC ($c$).

203460.50 / 39.444

## [1] 5158.212

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As a change in ADC will effect GCt, we’ll use the ADC-centric formula for GCt to calculate it, giving us $154,746.36.

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NDiP ($n$) multiplied by ADC ($c$) = GCt ($x$).

30 * 5158.212

## [1] 154746.4

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2.4.5 ADC-Related Formulas

For completeness’ sake I’m going to include the formulas for EARB, NDiP, and GCt that include the ADC variable ($c$):

\[\begin{align} {\textrm{Where:}}\ \ \ {\textrm{EARB}}&=y\\ \ \ {\textrm{NDiP}}&=n\\ \ \ {\textrm{GCt}}&=x\\ \ \ {\textrm{ADC}}&=c\\ \ \ {\textrm{DAR}}&=z\\ \ \ \\ \\ {\textrm{Then:}} \ \ \ {z \times c}&={y} \\ {x \div c}&={n} \\ {n \times c}&={x} \\ \tag{2.14} \end{align}\]

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We’ll test these out to make sure everything’s in order:

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DAR ($z$) multiplied by ADC ($c$) = EARB ($y$).

46.43793010499292 * 4381.343

## [1] 203460.5

GCt ($x$) divided by ADC ($c$) = NDiP ($n$).

131440.30 / 4381.343

## [1] 30

NDiP ($n$) multiplied by ADC ($c$) = GCt ($x$).

30 * 4381.343

## [1] 131440.3

And, as you can see, everything adds up. One note is that you do want to be mindful to include the entire number and every one of its decimals in the calculations for accuracy, then round up afterwards. Hopefully by now, you’re beginning to see some patterns emerging that begin to illustrate the relationship between all of the variables that make up the DAR calculation.

2.4.6 Relationships Between the Variables

In this section, we will discuss the empirical relationship between DAR and all of it’s related variables. In the last section, you might have noticed that we established two different formulas for the Average Daily Charge:

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\[\begin{align} \dfrac {\textrm{GCt}}{\textrm{NDiP}} &= {\textrm{ADC}} \\ \\ \\ \dfrac {\textrm{EARB}}{\textrm{DAR}} &= {\textrm{ADC}} \tag{2.15} \end{align}\]

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If these are both true, then we can assume that the following equations are also true:

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\[\begin{align} \dfrac {\textrm{GCt}}{\textrm{NDiP}} &= \dfrac {\textrm{EARB}}{\textrm{DAR}} \\ \\ \\ \dfrac {\textrm{DAR}}{\textrm{NDiP}} &= \dfrac {\textrm{EARB}}{\textrm{GCt}} \\ \\ \\ \dfrac {\textrm{GCt}}{\textrm{EARB}} &= \dfrac {\textrm{NDiP}}{\textrm{DAR}} \tag{2.15} \end{align}\]

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Or, in mathematical terms:

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\[\begin{align} \dfrac {x}{n}&=\frac {y}{z} \\ \\ \\ \frac {z}{n}&=\frac {y}{x} \\ \\ \\ \frac {x}{y}&=\frac {n}{z} \tag{2.16} \end{align}\]

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Let’s test these out with our original report values:

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GCt ($x$) divided by NDiP ($n$) = ADC ($c$).

131440.30 / 30

## [1] 4381.343

EARB ($y$) divided by DAR ($z$) = ADC ($c$).

203460.50 / 46.43793010499292

## [1] 4381.343

EARB ($y$) divided by GCt ($x$) = the Actual EARB Ratio

203460.50 / 131440.30

## [1] 1.547931

DAR ($z$) divided by NDiP ($n$) = the Ideal EARB Ratio

46.43793010499292 / 30

## [1] 1.547931

GCt ($x$) divided by EARB ($y$) = the Actual GCt Ratio

131440.30 / 203460.50

## [1] 0.6460237

NDiP ($n$) divided by DAR ($z$) = the Ideal GCt Ratio

30 / 46.43793010499292

## [1] 0.6460236

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Again, you’ll want to include all of the decimals in the calculation for absolute accuracy, but:

the total Gross Charges divided by the Number of Days in the Period is equal to the Ending AR Balance divided by the average Days in AR. Not only that,
the average Days in AR. divided by the Number of Days in the Period is equal to the Ending AR Balance divided by the total Gross Charges.

This is an important fundamental relationship between these variables called the DAR Ratio that we’ll explore in depth in the next chapter.

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2.5 Formula Quick Reference

The following tables contain summaries of the variables and formulas that we have discussed in this chapter.

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Variable	Acronym	Description
$n$	`NDiP`	Number of Days in the Period
$x$	`GCt`	total Gross Charges for `NDiP`
$y$	`EARB`	Ending AR Balance
$c$	`ADC`	Average Daily Charge for `NDiP`
$z$	`DAR`	Days in AR
$t$	`DARt`	Target Days in AR

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Name	Formula	Variable Form
DAR¹	`EARB` / (`GCt` / `NDiP`)	$y \div (x \div n) = z$
DAR²	`EARB` / `ADC`	$y \div c = z$
GCt¹	(`EARB` x `NDiP`) / `DAR`	$(y \times n) \div z = x$
GCt²	`NDiP` x `ADC`	$n \times c = x$
EARB¹	(`DAR` x `GCt`) / `NDiP`	$(z \times x) \div n = y$
EARB²	`DAR` x `ADC`	$z \times c=y$
NDiP¹	(`DAR` x `GCt`) / `EARB`	$(z \times x) \div y = n$
NDiP²	`GCt` / `ADC`	$x \div c={n}$
ADC¹	`GCt` / `NDiP`	$x \div n = c$
ADC²	`EARB` / `DAR`	$y \div z = c$
DAR Ratio¹	`EARB` / `GCt`	$y \div x$
DAR Ratio²	`DAR` / `NDiP`	$z \div n$
DAR Ratio³	`GCt` / `EARB`	$x \div y$
DAR Ratio⁴	`NDiP` / `DAR`	$n \div z$

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1 Introduction

3 DAR Ratios

The Art of DAR

2 DAR Formulas

Objectives

2.1 The Three Numbers

2.1.1 Number of Days in the Period

2.1.2 Total Gross Charges

2.1.3 Average Daily Charge

2.1.4 Ending AR Balance

2.2 Step-by-Step Example

2.2.1 Calculate the Average Daily Charge

2.2.2 Calculate the Days in AR

2.3 Visualizing Days in AR

2.4 Building on the DAR Formula

2.4.1 EARB Required for \(x\) DAR

2.4.2 GCt Required for \(x\) DAR

2.4.3 NDiP Required for \(x\) DAR

2.4.4 ADC Required for \(x\) DAR

2.4.6 Relationships Between the Variables

2.5 Formula Quick Reference

Variable	Acronym	Description
\(n\)	`NDiP`	Number of Days in the Period
\(x\)	`GCt`	total Gross Charges for `NDiP`
\(y\)	`EARB`	Ending AR Balance
\(c\)	`ADC`	Average Daily Charge for `NDiP`
\(z\)	`DAR`	Days in AR
\(t\)	`DARt`	Target Days in AR

Name	Formula	Variable Form
DAR¹	`EARB` / (`GCt` / `NDiP`)	\(y \div (x \div n) = z\)
DAR²	`EARB` / `ADC`	\(y \div c = z\)
GCt¹	(`EARB` x `NDiP`) / `DAR`	\((y \times n) \div z = x\)
GCt²	`NDiP` x `ADC`	\(n \times c = x\)
EARB¹	(`DAR` x `GCt`) / `NDiP`	\((z \times x) \div n = y\)
EARB²	`DAR` x `ADC`	\(z \times c=y\)
NDiP¹	(`DAR` x `GCt`) / `EARB`	\((z \times x) \div y = n\)
NDiP²	`GCt` / `ADC`	\(x \div c={n}\)
ADC¹	`GCt` / `NDiP`	\(x \div n = c\)
ADC²	`EARB` / `DAR`	\(y \div z = c\)
DAR Ratio¹	`EARB` / `GCt`	\(y \div x\)
DAR Ratio²	`DAR` / `NDiP`	\(z \div n\)
DAR Ratio³	`GCt` / `EARB`	\(x \div y\)
DAR Ratio⁴	`NDiP` / `DAR`	\(n \div z\)