Last week’s question led to a number of previous questions, which would have made it too long. Here we’ll look at the last couple references we gave, dealing with percentages of a negative base. This time, the problems will be mostly about money.
Percent change from a net loss: Skip the numbers!
First, we have this question from 2002:
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 between 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? Thanks.
As we saw last time, the formula is $$\text{percent change}=\frac{\text{new }-\text{ old}}{\text{old}}\times100\%,$$ which in this case (changing from \(-100\) to \(50\)) would give $$\text{percent change}=\frac{50-(-100)}{-100}\times100\%=-150\%.$$ A negative percent change seems wrong for an increase.
If we changed the formula to $$\text{percent change}=\frac{\text{new }-\text{ old}}{|\text{old}|}\times100\%,$$ we would get $$\text{percent change}=\frac{50-(-100)}{100}\times100\%=150\%.$$
I’ve seen people claim that using the absolute value fixes things; but if this were valid, then increasing \(-100\) by \(150\%\) should result in 50. We find that \(150\%\) of \(-100\) is \(-150\), and “increasing” \(-100\) by \(-150\) gives us \(-250\)! So this “fixes” the issue that an increase should always be positive, but it does not produce a number that is actually a percent increase.
I answered, having seen similar questions before:
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?
In the example, the actual change is $150 additional profit, which is very meaningful; but that this is -150% of the old value is not. Just looking at this number, we can’t tell whether we had a profit last year and lost 150% of it (and now have a loss), or had a loss last year, and gained 150% of that amount, changing it to a profit (as it was). In fact, a positive percent change could represent an even greater loss, which is again misleading.
We’ll see more of what I mean by “not proportional” in our last question below.
I’d seen such questions before, but not being an accountant, did not want to rely only on my mathematical impression:
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:
The part I excerpted here has not changed since then:
Help: Digest of Earnings http://www.wsj.com/public/resources/documents/doe-help.htm 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 indicated. 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 they just say “P” if a loss changed to a profit, or “L” if a profit changed to a loss (or a loss remained a loss?), without trying to give a numerical value to that change. This makes the most important fact visible!
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 sensible.
Cindy quickly wrote back:
Thank you for the rapid reply. It has stimulated much conversation 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 mathematical accuracy is extremely important to us.
Today, I find that Wikipedia agrees, more or less:
The relative change is not defined if the reference value (vref) is zero, and gives negative values for positive increases if vref is negative, hence it is not usually defined for negative reference values either. For example, we might want to calculate the relative change of −10 to −6. The above formula gives \(\frac{(−6)−(−10)}{−10}=\frac{4}{−10}=−0.4\), indicating a decrease, yet in fact the reading increased.
…
The domain restriction of relative change to positive numbers often poses a constraint. To avoid this problem it is common to take the absolute value, so that the relative change formula works correctly for all nonzero values of vref … It is common to instead use an indicator of relative change, and take the absolute values of both v and vref .
So when you can’t refuse to state the percent change, you have to say more to clarify the meaning. I would not say it is actually “working correctly”, in a mathematical sense.
Two more answers, for confirmation
I wanted to pass on some other opinions I’d found from the Math Doctors that had not been published:
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:
Percent of a negative target: Questionable utility
First, we had this question:
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?
This is like the question above, but is about a comparison rather than a percent change. Here, the percentage, comparing a value to a target, is just $$\text{percentage}=\frac{\text{actual}}{\text{target}}\times100\%,$$ As above, if the target was negative $1,000,000, and the actual value was (positive) $500,000, the percentage would be $$\text{percentage}=\frac{500,000}{-1,000,000}\times100\%=-50\%;$$ does that make sense?
Doctor Rick answered that:
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.
To illustrate these cases, if the target was negative $1,000,000, and
- the actual value was negative $1,500,000, the percentage would be $$\text{percentage}=\frac{-1,500,000}{-1,000,000}\times100\%=150\%,$$ a positive percentage representing a greater loss (bad);
- the actual value was negative $500,000, the percentage would be $$\text{percentage}=\frac{-500,000}{-1,000,000}\times100\%=50\%,$$ a positive percentage representing a smaller loss (good);
- the actual value was positive $1,000,000, the percentage would be $$\text{percentage}=\frac{1,000,000}{-1,000,000}\times100\%=-100\%,$$ a negative percentage representing a change to a gain (great!).
Your first impression would not be that a negative percentage is a good thing! Similarly, a positive 50% would sound bad, until you noticed that it meant you had half as much loss, which is good.
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.
… which is why I wanted to check what a financial source said!
Percent change in medical data: keep the same formula
In the next answer, negative values don't arise in the question, but are referred to at the end:
Here, I’ll skip the actual question (about a computer miscalculating percent change in a blood count), and jump to the part of Doctor Douglas’s answer that is relevant. (It’s actually irrelevant to the question in its context, but was suggested by the use of an absolute value in the question.)
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.
This is $$\%\Delta=\frac{\text{Current}-\text{Previous}}{\text{Previous}}\times100\%$$
As we’ve said, the formula doesn’t change, but the resulting number has to be carefully interpreted; a positive percentage doesn’t always mean things got better, but just that it became more of what it was. A positive percent change in a loss is more of a loss.
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.
This would be $$\%\Delta=\frac{\text{Current}-\text{Previous}}{|\text{Previous}|}\times100\%$$
Either approach makes some sort of sense; but either way, something has to be explained!
I concluded my reply to Cindy:
That may provoke still more discussion. I'd be interested in any conclusions you reach, so we can share them with others who ask.
Percent of a negative goal: an inappropriate calculation?
We got a similar question in 2003:
When Percentage Calculations are Inappropriate How do I best calculate the % of goal when at least one of the numbers is a negative number? With positive numbers, this is easy… Goal = 100 Actual = 98 % to Goal = 98 / 100 = .98 = 98% of goal but it appears more difficult when your goal is negative. Here is my example: A company that provides a monthly service brings on new customers every day, but unfortunately also has customers deactivate from their service each month as well. In these tough economic times, they have a quota to reach: –1000 customers ("negative" 1000 customers) this year (meaning that they will have 1000 more customers deactivate than those that activate). If they lose fewer than 1000 customers, they will have surpassed their goal. For this example, let's say they blow away their goal and bring on 100 MORE customers than they deactivate. What % did they beat their goal? If we use the calculation above, it would be … Goal = -1000 Actual = 100 % to Goal = 100 / -1000 = -.10 = -10% of goal?? Example 2: Goal = -1000 Actual = -500 % to Goal = -500 / -1000 = .50 = 50% of goal?? It seems simple. What am I doing wrong? Thank you.
I answered, starting with a link to the answer above, and then emphasizing the main conclusion:
Hi, David. This is not an uncommon problem; I've discussed something like it here in our archives: Calculating Percent Change when the Base is a Negative Number http://mathforum.org/library/drmath/view/55720.html The problem is that percentage is not the right way to measure achievement of this sort of goal. It's really inappropriate in any case, but it becomes obvious when the goal is zero or negative. What's wrong is that the goal is not proportional to the effort expended, so the percentage of the goal attained does not measure anything meaningful. In particular, attaining a negative goal does not require negative effort!
The mere possibility of a negative goal suggests that you really shouldn’t be using percentages in the first place. The problem of non-proportionality arises from the nature of the goal itself, which also leads to the possibility of a negative goal: What we attain does not depend solely on what we do, but on the economy in general. I’ll have a nice image of this below.
Your goal is to keep the loss from getting too great. If you lose half as many customers as you hoped, you have done very well, not half as well as you intended. (That's what your 50% example shows.) How can we adequately measure such performance? The best approach, I think, is to ignore percentages entirely, and just say that you retained 200 more customers than you had hoped, rather than that you lost only 800 of the 1000 you had expected.
This is, essentially, the WSJ approach. I wish I could see a good substitute for percentages in the entire concept (how to measure success in reaching a goal), but it seems to be expected.
If you need a percentage, you have to decide what to compare this with. You might just use the absolute numbers, rather than the loss: you had, say, 15,000 customers, and your goal was to have no less than 14,000 at the end of the year. If you have 14,200, then you accomplished 14,200/14,000 * 100% = 101.4% of your goal of customer retention. But although the numbers taken this way can't go negative, they still are not necessarily proportional to effort; if you had 15,000,000 customers to start with, and lost 800 rather than 1000, the percentage would look a lot less, namely 100.001%. That doesn't seem appropriate (but perhaps it is).
Perhaps what you need to do is to change perspective, as I did here, from “percent of planned numbers added” to “percent of planned customers retained”.
Now the analogy:
The problem is that you are running up a down escalator: as hard as you run, you are being carried backward by a force you have no control over. You want to measure how well you are running by how slowly you move backward, rather than by how fast you move forward. If you knew how fast the escalator was moving (how many customers would have left if you had made no effort), you could subtract that before measuring your progress. But you don't know that number. When the escalator was moving up, making it look as if you were making a big effort to gain customers even if you did nothing, you wouldn't have thought to subtract the amount of gain that could be attributed to "good times"; but your numbers then were just as meaningless as they are now! It's just harder to hide it now. Again, I recommend just reporting the number of extra customers retained, and not trying to compare it with any arbitrary base.
When the percentage seems to make sense, it may still not be appropriate. When the problem becomes obvious, it should trigger questions about the whole concept, rather than just a quick fix.
David replied:
What a stumper! I have floundered with this issue for years and couldn't find anyone who could answer my question. I read through the Wall Street Journal article that you linked and I can see that I'm not the only one who was stumped. Thank you for answering this difficult question for me.
In last week’s discussion, we looked into percent increases of temperature, finding issues of negative temperatures, and of arbitrary zero points.
There, we had to decide what it means to be “twice as hot” (a 100% increase), which makes no sense when the current temperature is negative, but really makes no sense just because we need to decide what 0 heat means. When is it not hot at all? Here, we have to think about what it means to accomplish “half the goal” of losing only 1000 customers. The escalator analogy is similar to measuring our accomplishment relative to an “absolute zero”, a baseline below which we can’t go. When have we accomplished nothing? Probably when we lose everyone we would have lost anyway.