3 DAR Ratios
Objectives
In this section, we will discuss:
The meaning of ratios and their different types.
The DAR ratios and how to use them.
Describing a line on a graph with a ratio.
Graphing the DAR ratios and the benefits of doing so.
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3.1 Overview of Ratio Types
A ratio is a mathematical parameter used to express the relationship of one quantity to another. To calculate a ratio, one quantity is divided by another. The quotient can be greater than 1 or less than 1. For example, if seven men and five women were in a group, the ratio of men to women would be \(\dfrac {7}{5}\). This may also be written as 7:5 and verbalized as “7 to 5.”
The numbers 7 and 5 have no common factors, so this ratio cannot be simplified any further. However, if the group consisted of 6 men and 10 women, the ratio would be 6:10. Because the numbers in this ratio have a common factor of 2, the ratio can be simplified by dividing each number by 2, which simplifies the ratio to 3:5.
One more thing of note: since a ratio can be expressed as a fraction, the quotient of the fraction is another viable way of displaying a ratio:
A proportion is a type of ratio in which \(x\) is a portion of the whole \((x+y)\). In a proportion, the numerator is always included in the denominator. For example, if two women out of a group of 10 over the age of 50 have had breast cancer, where \(x = 2\) (women who have had breast cancer) and \(y=8\) (women who have not had breast cancer), the calculation would be 2 divided by 10. The proportion of women who have had breast cancer is \(0.2\) or \(20\%\).
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A rate6 is another type of ratio in which there is a distinct relationship between the numerator and denominator and the denominator often implies a large base population. A measure of time is often an intrinsic part of the denominator. The basic rule of thumb for calculating a rate is to indicate the number of times something actually happened in relation to the number of times it possibly could have happened (actual/potential).
For example, let’s say you have been eating out often in the past few weeks. To calculate the rate of meals you have eaten out in one week, divide the number of meals that you ate out (for example, 13) by the number of meals you could have eaten out (21). The calculation is:
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3.1.1 A Note on Percentages
As you can see above, the solutions to the proportions and rates can be expressed as a percentage, while the ratio cannot. There is a very good reason for this, though not one that immediately presents itself. Let’s look at the definitions again:
| Ratio | A mathematical parameter used to express the relationship of one quantity to another, where one quantity is divided by another. |
| Proportion | A type of ratio in which the numerator (\(x\)) is a portion of the whole (\(x+y\)). In a proportion, the numerator is always included in the denominator. |
| Rate | A type of ratio in which there is a distinct relationship between the numerator and denominator and the denominator often implies a large base population. |
| Percentage | A unit of proportion expressed as a fraction of 100. A percentage is a dimensionless number that has no unit of measurement. |
A proportion and a rate can be represented as a percentage because their numerators are inherently part of their denominators. This is not so for a ratio as it’s only comparing one number to another, even though the two numbers might be of the same group.
Something about each quantity separates it from the other (in this case, gender) and, as such, we want to know how many with attribute A there are compared to how many with attribute B there are. With anything so simple, it can be easy to overthink and misunderstand the concept.
Look at the ratio example again. There are seven men and five women in a group. We want to know how many men there are and how many women there are. Nothing more, nothing less. Thus, the ratio of men to women in this group is 7 to 5.
To compare, if we wanted to know the proportions of men and women in this group, we would have to involve a numerical representation of the group as a whole. There are seven men and five women in the group. This group, then, contains a total of twelve people. We can now calculate these proportions, representing the results as percentages:
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Financial ratios play an important role in analyzing the financial health and performance of a business. In terms of those most useful to a medical practice or healthcare organization, several are excellent measures of the performance of Accounts Receivable.
3.2 The “1.5x” Rule
Let’s test out one of the many AR management “best practices” that I have come across time and again:
your monthly Ending Accounts Receivable Balance should never be more than 1.5 times your total Gross Monthly Charges.
In other words, if your GCt is $1.00 at the end of the month, then your EARB can be no more than $1.50 or you are in trouble. Are you though? Assuming our organization’s target DAR is 35 days, I am going to calculate the DAR for each of the three monthly NDiPs to find out:
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Confirmed, you are in BIG trouble. Because you need to find a better measure of monthly AR performance.
But why is this 1.5x rule so bad? Two very big reasons: it doesn’t consider 1) a DAR benchmark of any kind or 2) the number of days in the period that it is measuring.
3.2.1 Bad Advice, Great Example
Even though the 1.5x rule is comically bad advice, it is a great example of a ratio and a prime opportunity to showcase many of the things calculating a ratio can do. Let me explain.
If, according to this rule, your EARB should never be more than 1.5 times your monthly GCt, then the ideal ratio of EARB to GCt is 1.5 to 1. Since you can represent this as a fraction and divide, the ideal ratio is also 1.5 (\(1.5 \div 1 = 1.5\)).
Let’s test out this ideal to show why is it not what it says it is. Suppose that your EARB is $203,460.50 and your GCt is $131,440.30. Therefore, your actual ratio is 203,460.50 to 131,440.30. Represent this as a fraction and divide:
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Your actual ratio of EARB to GCt is 1.548. In other words, your EARB is 1.548 times your GCt, breaking the 1.5x rule. We can check this by multiplying the GCt by 1.5 and seeing whether or not the result is less than our EARB (spoiler: it is):
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\[\\[3pt]\] Calculating and comparing this ‘actual’ ratio and it’s ‘ideal’ counterpart is the beginning of a set of tools that I’ve come to call the DAR Ratio.
3.3 The DAR Ratios
In Chapter 3, we established three formulas showing the interrelationship between the DAR variables:
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The second equation is the foundation of this so-called DAR Ratio. The left side of the equation represents the Ideal Ratio for \(x\) DAR, consisting of the ratio of your organization’s desired DAR target (or DARt) to the Number of Days in the Period (NDiP). The right side is your Actual Ratio, which is the ratio of your EARB to your GCt. Let’s set this up in equation form:
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I will use the table of figures from our earlier 1.5x example to demonstrate a good use case for the DAR ratios:
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Our desired DAR target (DARt) was 35 days, our EARB was $1.50, and our GCt was $1.00. Let’s divide:
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\[\begin{align} {\textrm{Actual Ratio}} &= \dfrac {\textrm{EARB}}{\textrm{GCt}} \\ \\ \\ {1.50} &= \dfrac {1.50}{1.00} \\ \\ \\ \\ {\textrm{Ideal Ratio}} &= \dfrac {\textrm{DARt}}{\textrm{NDiP}} \\ \\ \\ {1.25} &= \dfrac {35}{28} \\ \\ {1.166667} &= \dfrac {35}{30} \\ \\ {1.12903226} &= \dfrac {35}{31} \\ \\ \tag{3.8} \end{align}\]\(\\\)
Now our table looks like this:
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We now know that if our DAR target is 35 days and we are in a month with 28, 30 or 31 days, then our EARB needs to be, respectively, 1.25, 1.17, or 1.13 times our GCt. This is why it is called the Ideal Ratio.
In the same way that we used the formula for EARB Required for \(x\) DAR in Chapter 3, we can use the Ideal Ratio to get the same result.
We simply take the GCt amounts in our table above and multiply them by the Ideal Ratio. This will give us the EARB amounts required for a DAR of 35 days:
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\[\begin{align} {\textrm{GCt}} \times {\textrm{Ideal Ratio}} &= {\textrm{EARB Required}} \\ \\ \\ {1.00} \times {1.25} &= {1.25} \\ \\ \\ {1.00} \times {1.17} &= {1.17} \\ \\ \\ {1.00} \times {1.13} &= {1.13} \\ \\ \tag{3.9} \end{align}\]\(\\\)
To make sure these EARB amounts are correct, we will plug them into our DAR formula:
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\[\begin{align} \dfrac {\textrm{EARB}}{\textrm{GCt} \div \textrm{NDiP}}&={\textrm{DAR}} \\ \\ \\ \dfrac {1.25}{1.00 \div 28} &= {35} \\ \\ \\ \dfrac {1.17}{1.00 \div 30} &= {35} \\ \\ \\ \dfrac {1.13}{1.00 \div 31} &= {35} \\ \\ \tag{3.10} \end{align}\]\(\\\)
And update our table:
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3.3.1 Ratio Variables Do Not Move Together as One
Remember that the “Actual” Ratio we’re calculating here is EARB divided by GCt, and not GCt divided by EARB. This is important because it makes EARB dependent on GCt.
In other words, GCt will never change because of a change in EARB, which is true in the real world.
EARB changes on a daily basis because of GCt. This is because every Gross Charge is added to the AR Balance whether that charge is paid immediately (and is subtracted from the AR Balance) or 90 days later.
Another important reason to mention this is that, just as you can use the formulas for EARB Required for \(x\) DARt and GCt Required for \(x\) DARt, you can also calculate both of these using the DAR Ratios. However, calculating one will not tell you the other. They are completely independent of each other.
Calculating the
EARBrequired will tell you ONLY theEARBrequired for thatGCt.Calculating the
GCtrequired will tell you ONLY theGCtrequired for thatEARB.
Making these amounts work in harmony, given \(x\) DARt, requires a proportion and is the subject of the next chapter, the DAR Percentages.
The following tables and graphs illustrate the relationship between EARB and GCt, given that one or the other is constant.
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EARB Required for a DARt of 35 Decreases Rapidly over 365 Days.
For a larger version of the interactive graph, click HERE.
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As you can see from the above graph and table, the EARB amount required for a DARt of 35 days decays rapidly, only beginning to stabilize around day 35 (of course.) This is in stark contrast to GCt:
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GCt Required for a DARt of 35 Increases Steadily over 365 Days.
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The GCt amount required for a DARt of 35 days rises steadily in a linear fashion over the same period of time. These results are strikingly different to the graph you will see in Chapter 5, wherein GCt and EARB are graphed as parts of a whole, i.e. interacting with each other.
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3.4 Using the DAR Ratios
Let’s examine some sample data and apply the DAR ratios to this data. The following table contains 12 months of financial data from a physician practice.
The accompanying graph charts the EARB and GCt across those months. As you will see, DAR failed four months in a row, from March to June.
What jumps out at you when looking through the data?
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total Gross Charges & Ending AR Balance 12-Month Comparison. DARt is 39.445.
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Some Quick Takeaways from the Data
Examining the four months that failed on the graph, you can see that:
When
GCtfalls too far belowEARB,DARexceeds the 39.445 target.When the two get close enough,
DARis under the target.
Looking at the data in the table, you can see that:
If the Difference between the Actual and Ideal Ratio is positive,
DARis over the target set by the Ideal Ratio (39.445, in this case) and is failing.If the Difference is negative,
DARis under the target or passing.
Note:
DARwould meet the target (39.445) if the Difference was zero, but this will never be the case, as theGCtandEARBwill never be the exact same number.
Another interesting find is that if you plot each month’s Ratio Difference and Days in AR, they line up in an exact one-to-one relationship:
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Ratio Difference & Days in AR 12-Month Comparison. DARt is 39.445.
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3.5 A Ratio Describes The Slope of a Line
The Ideal ratio represents the slope of a line. Actually, any ratio represents the slope of a line.
The slope of a line (also called its gradient) is equal to the ratio of the change in y-coordinates to the change in x-coordinates. This slope shows the rise of a line in the plane along with the distance covered in the x-axis.
The Ideal ratio is equal to the change needed in EARB relative to the change in GCt, given a certain DARt and NDiP.
Let’s try to unpack all of that. The most common form of a linear equation (an equation describing a line) is the slope-intercept form, represented as:
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How does this apply to Days in AR? First, this formula is yet another way of calculating the EARB or GCt needed for \(x\) DAR. Let’s re-label the formula:
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What Ending AR Balance would you need for a target DAR of 35 days on the 25th day of the Period, with a Gross Charges total of $80? First, calculate your Ideal ratio (\(35 \div 25 = 1.4\)). Now substitute these figures into the formula:
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So, if your Gross Charges total was $80, your Ending AR would need to be $112 (or less) to meet a target DAR of 35 days on the 25th day of the period. Note: The Ideal ratio represents the slope of a line that intercepts the y-axis at (0,0), so the y-intercept will always be zero.
Reminder: If your Ideal Ratio is 1.4, that means that your Ending AR Balance must be equal to or less than 1.4 times your total Gross Charges (for an
NDiPof 25 and aDARtof 35.) In the above example, this means that your Ending AR would need to be $112 or less.
As the
NDiPclimbs, the Ideal Ratio will dip below 1. A targetDARof 35 with anNDiPof 90 results in an Ideal Ratio of approximately 0.38888. In other words, your Ending AR must be equal to or less than 0.38888 times yourGCt. For anNDiPof 90, aDARtof 35, and aGCtof $80, yourEARBwould need to be equal to or less than $31.11.
3.5.1 Graphing the Ideal Slope
Visualing data can be immensely helpful in understanding what’s going on. Below is a table of data and a graph of that data.
The diagonal dashed line represents the Ideal ratio for a DARt of 35 days and an NDiP of 25.
GCt is the x-coordinate and EARB is the y-coordinate. What jumps out at you from the data on the graph?
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NDiP = 25, DARt = 35. y-intercept (\(b\)) = 0, slope (\(m\)) = 1.4.
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First, you can see that if the GCt/EARB combination is below the DARt (passing Days in AR), it’s point on the graph is to the RIGHT and BELOW the dashed diagonal line.
If the GCt/EARB combination is above the DARt (failing Days in AR), it’s point is to the LEFT and ABOVE the dashed diagonal line.
Moreover, you can tell just by looking at the points how close a GCt/EARB combo is to failing or passing Days in AR.
For the row with a GCt of $4 and an EARB of $9, you can see that the GCt needs to increase by just $1.00 to pass Days in AR.
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3.5.2 A More Thorough Example
Below is a dataset of 88 rows, each one representing one 31-day month. Each month’s GCt and EARB is plotted on the scatter plot (GCt on the x-axis, EARB on the y-axis). The slope for a target Days in AR of 39.445 has been mapped to the dashed black line. Explore the dataset and the interactive plot.
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Sample Financial Data for for 31-Day Months.
(intercept \(b\) = 0, slope \(m\) = 1.2724193548)