# What is the correct answer for percent change when the start amount is zero dollars \$0?

Percent change is a very common calculation in finance. It helps us track growth. The formula is:

`````` [(y2 - y1)/y1] x 100 = percent change
``````

If the start period is \$0 (i.e. no money was made in the first period, so `y1` is 0), the formula divides by 0, which is mathematically meaningless.

However, pragmatically, we understand that there is still meaning in the change. Some people think it should reflect a 100% change, an option that seems sensible to me. Others say infinity, which is not sensible at all. I've seen some opt to just change the start period to 1, than calculate, which could yield anything, but that can lead to skewed values in more complex calculations, not to mention changes that are orders of magnitude different than what you might expect. Then the purists insist that the only correct answer is `undefined`.

In finance, what is the convention for this issue. If I have `y1 = 0` and `y2 = 896`, what is the percent increase? What do my manager and investors expect to see?

Personally, as a manager and business owner, I want to see `100%` with an asterisk note: `y1 = 0`.

• @Michael I guess that supports the "infinity" answer. Pretty sure my investors don't want to hear "infinity percent increase over last month".
– user26460
Commented Aug 30, 2017 at 18:58
• 100% gain means the quantity doubled. It is certainly not the right number to show for this situation. Commented Aug 30, 2017 at 19:18
• I'd use "n/a" or "-" in this situation.
– quid
Commented Aug 30, 2017 at 19:55
• Why do you say infinity is not sensible? To me it seems the only sensible answer that is even quasi-numerical. Better still is to just put "N/A" or the like. Commented Aug 31, 2017 at 4:13
• Look at it this way. If Andy's investment has figures of y1=224, y2=896 the percent change would be 300% do you suggest that is three times better than Brian's investment of y1=0, y2=896? Commented Aug 31, 2017 at 7:25

There is no numerical convention in finance that I have ever seen. If you look at statements or reports that measure growth when the starting value is negative or zero, you typically see "n/a" or "-" or "*" as the result. Any numerical result would be meaningless. Suppose you used 100% and another company had a legitimate 150% gain - where would the 100% change rank?

What do my manager and investors expect to see?

As a financial analyst - I would not want to see 100%. I would instead rather see something that indicates that the % change is meaningless.

As an example, here's the WSJ documentation on change in Net Income:

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.

Thinking about it in another context:

Yesterday you and your friend had no apples. Today you have 1 and your friend has 20. What percentage increase did you both have? Did you both have a 100% increase? How can you indicate that your friend had a larger "increase"?

In that case (and in finance), the context needs to turn from a percentage increase to an absolute increase. A percentage increase is that scenario is meaningless.

• Yes, when comparing to other companies that did not have `y1 = 0`, there is a major issue. That shows that forcing it to say something else skews rankings. I think I just need to adjust my thinking.
– user26460
Commented Aug 30, 2017 at 19:33
• This. As a side note, the reason you can't divide by zero is because it really doesn't make sense; not because mathematicians want to work with awkward rules. Commented Sep 2, 2017 at 1:08

A value of zero or a negative value makes the percent change meaningless. Saying 100% when going from 0 to some other value is simply wrong.

I have seen a similar situation several times when looking at a public company with a loss last quarter. On Google Finance or some other service, the PE ratio will be blank, N/A, or something like that. If the company does not currently have earnings, then the PE ratio is meaningless. Likewise, if the company previously did not have earnings, then the percent change of the earnings is meaningless.

Also consider the example where the previous value was negative. If the previous value was negative 1 and the current value is positive 99, then this happens:

``````[(99 - -1)/-1] x 100 = -100%
``````

A negative change? But the value went up! Obviously that value does not make sense and should not be shown.

• Good point. It also doesn't work with negative values.
– user26460
Commented Aug 30, 2017 at 19:34
• Or indeed where last years loss was £1m but this years loss £2m. Could one say their profit is 200% of last year? (Even if technically true, or if one would usually say their loss is 200% of last years). Commented Aug 30, 2017 at 20:49
• @Stilez Mathematically, profit and loss is the same – except of the sign. Commented Aug 31, 2017 at 7:16
• @glglgl: It makes sense mathematically but it's useless for comparing company performance, since it fails to distinguish a company that went from making a loss into making a profit from a company that went from making a profit into making a loss. Commented Aug 31, 2017 at 7:49
• @IlmariKaronen Also the transition from "huge loss" to "barely profitable" is significantly different from "barely a loss" to "hugely profitable". Could be the same amount, though. Commented Aug 31, 2017 at 10:28

I'd personally display "n/a" The only other answer that makes sense to me other is "infinity" (phone keyboard doesn't allow me to input the symbol). This would at least allow you to show direction by using positive and negative infinity and mathematical as the the initial value approaches zero the percentage change approaches infinity which is the closet you can get to a meaningful value

• Agreed, though, while infinity is the closest value to being correct, it's still not quite right, since the initial value is actually zero and not just approaching zero in a limit. The limit of 1/x as x approaches 0 (from the right) is infinity, but 1/0 itself is undefined. Of course, the limit of 1/x as x approaches 0 from the left is actually negative infinity, rather than positive infinity. Commented Aug 31, 2017 at 5:12

In general, when dealing with quantities like net income that are not restricted to being positive, "percentage change" is a problematic measure. Even with small positive values it can be difficult to interpret.

For example, compare these two companies:

Company A:

• Y1 net income \$100m
• Y2 income growth -99.9%
• Y3 income growth +100,000%

Company B:

• Y1 net income \$100m
• Y2 income growth -99.7%
• Y3 income growth +40,000%

At a glance, I think most people would come away with the impression that both companies did badly in Y2, but A made a much stronger recovery. The difference between 99.7 and 99.9 looks unimportant compared to the difference between 100,000 and 40,000.

But if we translate those to dollars:

Company A: Y1 \$100m, Y2 \$0.1m, Y3 \$100.1m

Company B: Y1 \$100m, Y2 \$0.3m, Y3 \$120.3m

Company B has grown by a net of 20% over two years; Company A by only 1%.

If you're lucky enough to know that income will always be positive after Y1 and won't drop too close to zero, then this doesn't matter very much and you can just look at year-on-year growth, leaving Y1 as undefined.

If you don't have that guarantee, then you may do better to look for a different and more stable metric, the other answers are correct: Y1 growth should be left blank. If you don't have that guarantee, then it might be time to look for a more robust measure, e.g. change in net income as a percentage of turnover or of company value.

• @mch the comma in 100,000% is a thousands seperator. so 100% is off by 3 orders of magnitude. Commented Aug 31, 2017 at 20:35
• @mch Right you are, I wrote it out and then tweaked the numbers and didn't update everything. Will edit that.
– G_B
Commented Sep 1, 2017 at 0:34
• The example would also have worked with +100% and -30% in Y3 Commented Sep 1, 2017 at 8:11
• @mch argh! Fixed now... I hope.
– G_B
Commented Sep 1, 2017 at 9:01

## "New" will suffice.

Anyone who has any business looking at growth numbers will know thay are meaningless in the first year, So all they need to know is that it's the first year.

It's no different than the Billboard music charts' treatment of the "last week's chart ranking" and "movement up/down" columns.

It will help with visual layout if the figure used is about the same size as a percentage number. "New" fits nicely.

As has been said before, going from nothing to something is an infinite percent increase! It is not 100%. Maybe you had a dollar and now have \$101 that is a 10000% increase! Quite remarkable.

I often work with symmetrized percent changes like: spc = 100 * (y2-y1)/(0.5 * (y1+y2)) Where I compute the percent with respect to the average. First this is more stable as often measurements can have noise, the average is more reliable.

Second advantage is also that this is symmetric. So going from 95 to 105 is a 10 % increase while going from 105 to 95 a 10% decrease.

Of course you need to explain what you show.

• This is a very non-standard approach (I have never seen it); using a non-standard calculation to display financial information might be seen as an attempt to deceive, or at least, it could be quite confusing for a casual reader of the information. Commented Aug 31, 2017 at 14:54

In computing, you'd generally return naa%, for 'not a number'. Could you not put '-%' to show there is no value at this point? Surely the people seeing this aren't idiots and understand the charge on 0 is 0?

• Not correct. In computing you would return a positive infinity. Commented Sep 3, 2017 at 0:47
• @user207421 what system are you using that can handle dividing by zero and returns infinity? Commented Sep 7, 2022 at 22:16

What is the probability of a real occasion (meaning not just an example) being exactly zero? Even if you have 0.1 you can still do the math. Also, it is kind of depending on the occasion. For example, you want to calculate the ROI of an investment for which you had zero capital and you made that investment with leverage, meaning you got a loan. In order to get that loan you should have provided a collateral, so in this case as a starting sum you use the collateral. In another example, say EAT it's difficult to have exactly zero. So, in most cases you won't have to deal with zero values, only positives and negatives.

• It is very common for financial reference points to not be present in prior years. ie: this year, your company opens a branch in the US. You have a chart showing sales growth for all divisions. Your year-over-year sales growth in the US will be x/0. Also using assets/loans as reference points for income gain is not practical or reasonable in most accounting situations. Commented Aug 31, 2017 at 14:53
• @Grade'Eh'Bacon about the assets/loans, could you give an example of the most accounting situations? i gave this example after a conversation with a professor of mine about finding the leverage with zero equity. it went like: "how are you going to finance it with zero capital?" "with a loan" "then as your equity you will use the value of the collateral". I know it's not applicable in all situations and probably the OP is business oriented but i wrote it as an example (since he didn't specify the exact situation). Commented Aug 31, 2017 at 15:45
• Okay I think I misread what you were saying about calculating the % of income off of assets/loans; I thought you were saying that if you were calculating % income growth from prior year you could use your asset base instead, if you had no prior year income to compare to. Your professor's point is interesting and it seems like in some circumstances it might provide useful information but I'm not sure how any of this applies to the OP's question. Commented Aug 31, 2017 at 15:59
• In some kinds of accounts it's quite plausible to have exact zeroes in a mostly-non-zero series even after the first year. One branch is out of action for a year due to a disaster, with operations temporarily relocated to other branches, so that branch has zero sales; company experiences a financial crunch so they institute a hiring freeze and the figures for recruitment spend are zero; their revenue for widget sales is frozen pending a legal case over who owns the widget patents; etc. etc.
– G_B
Commented Sep 1, 2017 at 0:40

There are some assumptions which can be made in terms of the flexibility you have - I will start with the least flexible assumption and then move to more flexible assumptions.

If you must put down a number

1, your go-to for this("Change the start period to 1"), is pretty good, and it's used frequently for other divide-by-zero calculations like kda in a video game. The problem I have with '1' is that it doesn't allow you to handle various scales. Some problems are dealt with in thousands, some in fractions, and some in hundreds of millions. Therefore, you should change the start period to the smallest significantly measurable number you could reasonably have. Here, that would take your example 0 and 896 and give you an increase of 89,500%. It's not a great result, but it's the best you can hope for if you have to put down a number, and it allows you to keep some of the "meaning in the change."

If you absolutely must put something

This is the assumption that most answers have taken - you can put down a symbol, a number with a notation, empty space, etc, but there is going to be a label somewhere called 'Growth' that will exist. I generally agree with what I've seen, particularly the answers from Benjamin Cuninghma and Nath. For the sake of preservation - those answers can be summarized as putting 'N/A' or '-', possibly with a footnote and asterisk.

If you can avoid the measurement entirely

The root of your question is "What do my manager and investors expect to see?" I think it's valuable to dig even further to "What do my manager and investors really want to know?". They want to know the state of their investment. Growth is often a good measurement of that state, but in cases where you are starting from zero or negative, it just doesn't tell you the right information. In these situations, you should avoid % growth, and instead talk in absolute terms which mention the time frame or starting state. For example:

• First-x (year, month, quarter, etc.) revenue was 896
• Revenue started at 896
• Revenue has managed to grow to 896 in the first (year, month, quarter, etc.)
• Net revenuse has turned around and reached 896. This is a X% increase/decrease from back in (some time)
• Net revenue is now positive at 896
• Inserting "1" instead of the actual "0" just makes for an arbitrarily large %, which does not provide value for the reader. Commented Aug 31, 2017 at 19:44