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Calculating Percent Change when the Base is a Negative Number

Date: 01/18/2002 at 14:29:55
From: Cindy Carrigan
Subject: Calculating percent change when the base is a negative number

In preparing financial statements, I must calculate the percent 
change of the net income between years. If in year 1 we have a net 
loss (negative number) of -$100 and at the end of year two we make a 
profit of $50, what is the net change? 

I would take the difference beteen year 1 and year 2 of $150 and 
divide by the base of -$100 to get -150%. But this doesn't make sense 
when we've actually grown the profit; it should be an increase. 

So that leads me to believe that I should divide by the absolute value 
of the base to get the percent increase. Does this make sense? Are 
there specific rules to deal with this situation? 


Date: 01/18/2002 at 15:02:17
From: Doctor Peterson
Subject: Re: Calculating percent change when the base is a negative 

Hi, Cindy.

In my opinion, percent change is a meaningless statistic when the 
underlying quantity can be positive or negative (or zero). The actual 
change means something, but dividing it by a number that may be zero 
or of the opposite sign does not convey any meaningful information, 
because the amount by which a profit changes is not proportional to 
its previous value. Yet, such a percentage is often requested, and in 
reasonable cases seems useful. So what do we do?

I've never found any "official" statement that it should not be used, 
or how it should be handled. But I just did a search to see if anyone 
reports a percent change in profit, and ran across this interesting 
explanation from the Wall Street Journal:

    Help: Digest of Earnings   

    Net Income: Income after a company's taxes and all other expenses
    have been paid. Net Income is listed in thousands of U.S. dollars
    in the digest, unless otherwise indicated. The detailed earnings
    report presents whole U.S. dollar amounts, unless otherwise

    Net Income percent change is the change from the same period from
    a year ago. Percent change is not provided if either the latest
    period or the year-ago period contains a net loss. On the digest
    page, if a company posts a profit in the latest period against a
    loss in the year-ago period, the percent change is represented as
    a "P". Similarly, if a company posts a loss in the latest period
    against a profit in the year-ago period, the percent change is
    represented as a "L".

So although they do report percent change in net income, they don't 
present it as a number when the sign changed, but just indicate that 
it did change from a profit to a loss or vice versa. That seems 

- Doctor Peterson, The Math Forum   

Date: 01/18/2002 at 15:36:29
From: Cindy Carrigan
Subject: Calculating percent change when the base is a negative number

Thank you for the rapid reply. It has stimulated much conversaton 
within our Finance Department. Of particular value is the WSJ 
reference. You've nudged us to think about this financial presentation 
in a new way. Analyzing the data beyond the pure mathmatical accuracy 
is extremely important to us. 

Thanks again, 
Cindy Carrigan

Date: 01/18/2002 at 16:07:43
From: Doctor Peterson
Subject: Re: Calculating percent change when the base is a negative 

Hi, Cindy.

This question has come up many times in the past, and I've often been 
curious as to what is the best solution. I don't find any other 
discussions in our archives, but I did find several unarchived 
answers. You might be interested in these:

>I am trying to show a comparison between business plan profit targets
>(units=dollars) and actual profit results. I want to show the 
>comparison in terms of percentage points. For example: target=$1 
>mil. --> actual=$1.5 mil. --> percentage=150%. 
>Here's the problem. What if the profit target is negative and the 
>actual result is positive? Or, what if both the target and the result
>are negative? How do I compute a percentage-based comparison? In 
>other words, what effect does the negative sign have on the method of
>computing percentages?

The method of computing percentages will not change, but the 
significance of the statistic will be less than obvious. You divide 
the actual profit by the business plan profit and multiply by 100%. 
Negative divided by positive, or positive divided by negative, is 
negative; negative divided by negative is positive.

You'll need to explain that, if the target is negative, then a 
positive percentage greater than 100 means you did even worse than you 
expected; a positive percentage less than 100 means you didn't do as 
badly as you expected; and  a negative percentage means you had a 
profit after all.

If your business plan has profits that vary so widely that some are 
negative, then I would question the utility of percentages. The 
smaller the planned profit, the bigger the actual figure will appear 
as a percentage of plan. In the worst case, a unit planned to break 
even will have an infinite percentage if it makes any profit at all, 
and a percentage of negative infinity if it has any loss at all. 

A simple difference between actual and planned profits would be more 
informative in such a case. If any scaling of the profit figures is 
needed, it might be more meaningful to show the difference as a 
percentage of some figure that does not hang around zero, like total 
costs or net worth or something ... but I'm not an MBA or accountant, 
I really don't know what I'm talking about here.

- Doctor Rick, The Math Forum

In the next answer, negative values don't arise in the question, but 
are referred to at the end:

>I am the Laboratory information system supervisor in a hospital. Our
>computer system calculates the change of specific results to monitor 
>a patient's status. But there is a problem with the calculation in 
>the base code of the program. 
>If one day the patient has a Blood count of 100 (P for Previous) and
>the next day the Blood count is 75 (C for current), the computer 
>should calculate the change as -25% using the calculation |P-C|/P= D
>(D for Delta) and then check for positive or negative by the equation
>If P>C then D*-1 or If P<C then D*1.
>The program is calculating using the equation: ((P/C)-1)*100 in the
>above situation this would calculate the Delta as 33.33333% 
>Referenced in a Laboratory manual published before computers were 
>used to calculate the delta changes.
>The computer's calculation would mark a change from 100 to 50 as a 
>100% change! 
>If I had a 100% loss I should have nothing remaining.
>So I am trying to find any information or Documentation to send to 
>the company that created the program code. Is there a one-step 
>calculation? Or is the 2-step needed? Is there any general Delta 
>check Formula documentation? Thank you for any help in this matter.

From the information that you've provided, the computer is not 
computing the percent change (which I'll call %Delta), but rather 
calculating a quantity that is closely related to it. In fact, it is 
the percent change in the other direction:

   A 25% loss (from blood count 100 to 75) is being reported 
   as 33.3% gain. This is what you would compute if you were
   going from 75 to 100.

The calculation for percent change that you wrote is correct, but we 
can simplify it a bit, to make the two-step procedure more 
understandable, and make it seem like a one-step calculation.

   %Delta = (C-P)/P

For many situations, the variables are never negative:  P>0, C>0.
Then %Delta as specified above can be a positive number (C>P, 
"increase"), a negative number (C<P, "decrease"), or zero (C=P). You 
can see then that %Delta is computed according to the equation above 
in a single step. Whether or not the variable increased or decreased 
depends on the sign (+ or -) of %Delta, and this agrees with our 
intuition: a %Delta of +100% means that something doubled in size, 
while a %Delta of -50% means that something was halved.

However, things can get a bit tricky if you allow variables to take on
negative values. In such cases, I favor leaving the definition of 
%Delta as above, namely %Delta = (C-P)/P, with no absolute value 
signs. If P>0, then this formula works just as before. Now suppose 
P = -80 < 0, and C = -60. The percent change by our definition is 
(-60 - (-80))/(-80) = 20/(-80) = -0.25 = -25%. What does this mean?  
The percent change is negative (which should represent a decrease), 
but C is greater than P. The explanation is that if P<0 (e.g. a loss, 
or a deficit), then the magnitude of the loss _decreased_ (by 25%, in 
our example), just as claimed. 

There is an alternate way to define the percent change so that
increases (e.g. -80 to -60) are always given by a plus sign in the
percent change; the formula for this is %Delta = (C-P)/|P|. This might
lead to confusion similar to that seen in the computer program. The 
definition for %Delta that you should use will depend on your 
situation, but in general I favor the method in which one simply 
dispenses with the absolute value signs and carefully interprets what 
is happening when P<0.

Thanks for writing, and please write back if you need additional help.

- Doctor Douglas, The Math Forum

That may provoke still more discussion. I'd be interested in any 
conclusions you reach, so we can share them with others who ask.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Negative Numbers

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