What is the correct answer for percent change when the start amount is zero dollars $0?
"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." 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: A negative change? But the value went up! Obviously that value does not make sense and should not be shown. "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" "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:  Company B:  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." 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. "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." 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? 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. "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:"
The optimal time period is unambiguously zero seconds.  Put it all in immediately.  Dollar cost averaging reduces the risk that you will be buying at a bad time (no one knows whether now is a bad or great time), but brings with it reduction in expected return because you will be keeping a lot of money in cash for a long time.  You are reducing your risk and your expected return by dollar cost averaging. It's not crazy to trade expected returns for lower risk.  People do it all the time.  However, if you have a pot of money you intend to invest and you do so over a period of time, then you are changing your risk profile over time in a way that doesn't correspond to changes in your risk preferences.  This is contrary to finance theory and is not optimal.  The optimal percentage of your wealth invested in risky assets is proportional to your tolerance for risk and should not change over time unless that tolerance changes. Dollar cost averaging makes sense if you are setting aside some of your income each month to invest.  In that case it is simply a way of being invested for as long as possible.  Having a pile of money sitting around while you invest it little by little over time is a misuse of dollar-cost averaging. Bottom line: forcing dollar cost averaging on a pile of money you intend to invest is not based in sound finance theory.  If you want to invest all that money, do so now.  If you are too risk averse to put it all in, then decide how much you will invest, invest that much now, and keep the rest in a savings account indefinitely.  Don't change your investment allocation proportion unless your risk aversion changes. There are many people on the internet and elsewhere who preach the gospel of dollar cost averaging, but their belief in it is not based on sound principles.  It's just a dogma. The language of your question implies that you may be interested in sound principles, so I have given you the real answer. It appears very possible that Google will not have to pay any class C holders the settlement amount, given the structure of the settlement.  This is precisely because of the arbitrage opportunity you've highlighted.  This idea was mentioned last summer in Dealbreaker. As explained in a Dealbook article: The settlement requires Google to pay the following amounts if, one year from the issuance of the Class C shares, the value diverges according to the following formula: If the C share price is equal to or more than 1 percent, but less than 2 percent, below the A share price, 20 percent of the difference; If the C share price is equal to or more than 2 percent, but less than 3 percent, below the A share price, 40 percent of the difference; If the C share price is equal to or more than 3 percent, but less than 4 percent, below the A share price, 60 percent of the difference; If the C share price is equal to or more than 4 percent, but less than 5 percent, below the A share price, 80 percent of the difference.” If the C share price is equal to or more than 5 percent below the A share price, 100 percent of the difference, up to 5 percent. ... If the Class A shares trade around $450 (after the split/C issuance) and the C shares trade at a 4.5 percent discount during the year (or $429.75 per share), then investors expect a payment of: 80 percent times $450 times 4.5 percent = $16.20. The value of C shares would then be $445.95 ($429.75 plus $16.20). But if this is the new trading value during the year, that’s only a discount of less than 1 percent to the A shares. So no payment would be made. But if no payment is made, we are back to the full discount and this continues ad infinitum. In other words, the value of a stock can be displayed as: {equity value} + {dividend value} + {voting value} + {settlement value} = {total share value} If we ignore dividend and voting values, and ignore premiums and discounts for risk and so forth, then the value of a share is basic equity value plus anticipated settlement payoff.  The Google Class C settlement is structured to reduce the payoff as the value converges.  And the practice of arbitrage guarantees (if you buy into at least semi-strong EMH) that the price of C shares will be shored up by arbitrageurs that want the payoff. The voting value of GOOGL is effectively zero, since the non-traded Class B shares control all company decisions.  So the value of the Class A GOOGL voting is virtually zero for the time being.  The only divergence between GOOGL and GOOG price is dividends (which I believe is supposed to be the same) and the settlement payoff.  Somebody who places zero value on the vote and who expects dividend difference to be zero should always prefer to buy GOOG to GOOGL until the price is equal, disregarding the settlement. So technically someone is better off owning GOOG, if dividends are the same and market prices are equal, just because the vote is worthless and the nonzero chance of a future settlement payoff is gravy. The arbitrage itself is present because a share that costs (as in the article) $429.75 is worth $445.95 if the settlement pays out at that rate.  The stable equilibrium is probably either just before or just after the threshold where the settlement pays off, depending on how reliably arbitrageurs can predict the movement of GOOG and GOOGL. If I can buy a given stock for X but know that it's worth X+1, then I'm willing to pay up to X+1.  In the google case, the GOOG stock is worth X+S, where S is an uncertain settlement payment that could be zero or could be substantial.  We have six tiers of S (counting zero payoff), so that the price is likely to follow a pattern from X to X+S5 to X-S5+S4 to X-S4+S3, and climbing the tier ladder until it lands in the frontier between X+S1 and X+S0.  Every time it jumps into X+S1, people should be willing to pay that new amount for GOOG, so the price moves out of payoff range and into X+S0, where people will only pay X. I'm actually simplifying here, since technically this is all based on future expectations.  So the actual price you'd pay is expressed thus: {resale value of GOOG before settlement payoff = X} + ( {expectation that settlement payoff will pay 100% of difference = S5} * {expected nominal difference between GOOG and GOOGL = D} ) + ({S4} * {80% D}) + ({S3} * {60% D}) + ({S2} * {40% D}) + ({S1} * {20% D}) + ({S0} * {0% D})  = {price willing to pay for Class C GOOG = P} Plus you'd technically have to present value the whole thing for the time horizon, since the payoff is in a year.  Note that I've shunted any voting/dividend analysis into X.  It's reasonable to thing that S5, S4, S3, and maybe S2 are nearly zero, given the open arbitrage opportunity.  And we know that S0 times 0% of D is zero.  So the real analysis, again ignoring PV, is thus: P = X + (S1*D) Which is a long way of saying: what are the odds that GOOG will happen to be worth no more than 99% of GOOGL on the payoff determination date? See Started new job. Rollover previous employer 401k to new 401k, IRA or Roth IRA? for a start.  Kevin, the discussion is far more complex than you might think. Say your account grows by X, (pretend it's 10 if you wish) and your tax rate is Y (25%?). If you take the initial sum, tax it at Y, but then grow it X, the result is identical to doing it in the reverse order. So $1000 to start can grow to $10,000, then after tax, $7500. Or $1000 taxed to $750, then grow to $7500.  For pretax deposits, the key is that you deposit those contributions at your marginal rate, i.e. the rate you'd pay on the last $X taxed. But withdrawals start at zero.  In the perfect scenario, you will save 25-28% tax on deposits, but at retirement, enjoy taxation at 0%,10%,15% for a large portion or all of the withdrawals.  (Note, others can suggest rates will rise, and they may be right. My answer is based on the current tax structure.) A new earner, at 10 or 15% may be better off starting with Roth, and as they earn their way to 25% or higher slide over to pre-tax deposits. My 14 year old baby sits, and makes enough to fund a Roth, but pays no tax as she earns less than her own standard deduction for what that's worth. I too was very confused when I tried to be tricky and paid down my balance BEFORE the bill date. I thought this would be a great thing because it would show my utilization near zero percent. The opposite happen, it dropped my credit score from 762 to 708. Here is the best example I can come up with when it comes to utilization. Lets pretend you are an insurance company and you trying to figure out who are the best risk drivers. The people that drive 10% of the day are a better risk than the people that drive 50% of the day. The people that drive 50% of the day are a better risk than the people that drive 90% of the day. Here is the rub when people drive 0%. When you look at the people at 0% they appear to be walking, busing or flying. What they are NOT doing is driving. Since they are not driving (using Credit) they are viewed as POOR drivers since they are not keeping up on their driving skills. (Paying bills, watching how they spend, and managing their debt). So, now before the billing date I pay down my balance to something between 5 to 10% of my utilization. After the bill is issued, I pay it off in FULL. ( I am not going to PAY these crazy interest rates). What shows up on my credit report is a person that is driving his credit between 5 and 10% utilization. It shows I know I how to manage my revolving accounts. I know it's dumb, you would think they reward people that have zero debt, I don't hate banks I hate the game. ( I do love me some reward points =)) [] As you can see from the graphs above, as absolute distance from ATM (At The Money) increases, the ratio represented by Delta begins to approach 0:1 or 1:1. Meaning, as Delta approaches 1 the option price moves up 1 dollar for every 1 dollar the stock price moves up. As Delta approaches 0 the option price does not move as the stock price moves. As the absolute distance from ATM increases Gamma approaches zero (0). Meaning, as the price of the option increases or decrease the change in delta is at its highest as the option price is nearest the money. As the price of the option moves away from the money, in either direction, it changes less drastically. Interestingly, Delta is the first derivative of the value of the option price with respect to the underlying asset price. And Gamma is the first derivative of Delta. Definition of Delta Definition of Gamma "If your income stream goes up, it would usually increase both your ""income"" term and your ""assets"" term since that money sits in your bank account as an asset. (Even more likely a combination of assets and expenses go up if you have cost associated with the increase in income.)  In this case, they balance in the equation and your equity doesn't change. The question as you posed it is true mathematically, but the ""paradox"" happens because you're not taking into account where the money form the increased income falls in other terms of the equation." Line one shows your 1M, a return with a given rate, and year end withdrawal starting at 25,000.  So Line 2 starts with that balance, applies the rate again, and shows the higher withdrawal, by 3%/yr.  In Column one, I show the cumulative effect of the 3% inflation, and the last number in this column is the final balance (903K) but divided by the cumulative inflation.  To summarize - if you simply get the return of inflation, and start by spending just that amount, you'll find that after 20 years, you have half your real value.  The 1.029 is a trial and error method, as I don't know how a finance calculator would handle such a payment flow. I can load the sheet somewhere if you'd like.  Note: This is not exactly what the OP was looking for. If the concept is useful, I'll let it stand. If not, downvotes are welcome and I'll delete. "To your first comment: yup. To your second comment, A = L + E. If E goes down, and L goes up, the net effect is 0. Then, if L goes down, and A goes up, the net effect is 0 and we are balanced once again. There is no ""rebalancing"" equity. You just have to make sure that, at the end of your journal entries, the accounting equation holds.  It's a very unintuitive concept to wrap your head around, but spend some time mapping out the flow of various journal entries. Once it clicks, you'll really understand the logic."