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Computing methods of compounded gains, Total Return
The AVERAGE annual gain is NOT Useful Nor Accurate for calculating the total gain
Computing Annualized Return that Includes the Incomplete Current Year Is NOT Acceptable Practice
Seven calculators below
The greater number of years of long-term investments makes the compounding of gains become a large exponential function (with years as the exponent power), which is an astounding big deal. A fixed 10% gain in each of 30 years becomes a final value of 1.1030 = 17.45x the initial value and 1645% overall gain. But the market gains do vary up or down each year (often large variations, even sometimes negative), so see below for Annualizing the actual overall gain so it can be compared with others.
In addition to the price gains, another major factor is that reinvested dividends add more free shares every year (usually four times a year). Many stocks pay dividends in some degree. The S&P 500 index fund dividends vary (very roughly maybe about 2% a year), but if reinvested, dividends add a few free shares to total returns each quarter. Then those gains see more gains, compounding every following year. Reinvested dividends are a Real Big Deal, whereas withdrawn dividends simply reduce your investment. So give a good think to long term investment always with reinvested dividends. Start young, so it will be waiting at retirement. The S&P 500 calculator on previous page shows this with 50 years of past S&P 500 history.
When we say something gained 1.5x times, it has 1.5X more value than your original investment. The 1x is the initial amount and the 0.5x is the gain. This 1x is extremely important in gain calculations, much more below. That 1x becomes 100% when multiplied by 100 for percentage. And if it gained to 2.5x, the investment principle is still 1x and the gain is 1.5x. The 1x is key in performance calculations. You need to realize that the formulas have this ± 1 to refer to the original investment. The -1 subtracts your original investment (to leave only the gain amount) without needing its actual dollar amount.
| Compounding | ||
|---|---|---|
| Rate | Years | Growth |
| 10% | 5 | 1.61x |
| 10 | 2.59x | |
| 15 | 4.18x | |
| 20 | 6.73x | |
| 25 | 10.83x | |
| 30 | 17.45x | |
| 35 | 28.10x | |
| 40 | 45.26x | |
| 45 | 72.89x | |
| 50 | 117.39x | |
| 20% | 5 | 2.49x |
| 10 | 6.19x | |
| 15 | 15.41x | |
| 20 | 38.34x | |
| 25 | 95.40x | |
| 30 | 237.38x | |
| 35 | 590.67x | |
| 40 | 1469.77x | |
| 45 | 3657.26x | |
| 50 | 9100.44x | |
Compounding is leaving the gain invested to grow exponentially even more, each year, year after year, so then a Fixed 10% is that more gain next year than it was this year (and it grows greater every future year). Every year's starting point is the previous years gain result. Dividends are just a withdrawal anyway, so putting it back as Reinvested dividends really adds up long term. For example, 40 years reinvested compounding at 10% results in (1.10)40 = 45.26x value. That is (1 + 0.10), which the 1 is the years beginning value (as 1x regardless of its actual value) from the preceding year, and the 0.10 is the 10% gain it will add each year (more detail below). Whereas if the dividend (which is actually a withdrawal) is not put back (reinvested), then the dividend value is not available next year. There is a chart on the previous S&P 500 page that clearly shows this difference if the dividend is not reinvested. It may not matter greatly the first few years, but it becomes frightfully alarming long term. The loss is well less than the value of all the dividends that were withdrawn.
Long term compounding makes a tremendous difference, not so much in early years, but extreme in long term, like 30 or 40 years, after it grows to millions. 10% of millions is a very large growth each year. You really should realize that always withdrawing the dividends has the corresponding tremendous loss of long term profit). So if any divided, always reinvest them if you have the choice.
Years of compounding is the largest gain effect of long term market investments. One year has earnings, which (if positive) then that increased working total also increases earnings the next year, continually repeating every year. The compounding drastically increases the gain, and the more years, the more increase. The early years are less impressive, but when it has grown long term for 30 or 40 years, the gains are awesome. Doubling the gain rate (which doubling the investment can effectively do) increases extremely more than even the years.
Do not waste your years. They only come once. Especially don't waste your retirement investment that will be required for your survival for maybe 30 years of retirement without any salary. That's way too late to fix it then. For your own best interests, unless you have actual better plans, you should be planning to provide that today, when it is the affordable plan. Be smart. The wise advice is to start that very early while you still have the years. The many years of compounding are your best and most affordable retirement tool. Take a minute to think about that.
But market gain is variably different every day, always going up and down, making it hard to judge actual performance numbers, but there is a mathematical way called Annualized Return (below). But first, the next is six years of example Total Return statistics. Price gains do Not include dividends, but reinvestment adds more shares, and so published Total Return does include reinvested dividends. Morningstar also shows Total Return %.
Total Return is the actual rate of return of an investment, with the word Total generally meaning also compounding of reinvested dividends.
You do pay tax on each years dividends (reinvested or not), so for tax purposes, reinvested dividends are also added to your cost basis (after 2012, your U.S. tax 1099 shows cost). That prevents paying tax on the same dividend again when sold later.
The same percentage numbers are computed regardless of any initial value, $10 or $10,000,000. Both methods A and B include the compounding. These are just made-up example gain numbers, accurate math but NOT representing any actual real stock.
| Example of Manual Method "A" of Overall Return | ||||||
|---|---|---|---|---|---|---|
| Yr | Initial Value | Total Return | Annual Gain | Final Result | Compound Gain | Annualized Return |
| 1 | $10000.00 | 15% | $1500.00 | $11500.00 | 15% | 15% |
| 2 | $11500.00 | 23.5% | $2702.50 | $14202.50 | 42.025% | 19.17424% |
| 3 | $14202.50 | 10.4% | $1477.06 | $15679.56 | 56.7956% | 16.17463% |
| 4 | $15679.56 | -5.2% | $-815.34 | $14864.22 | 48.64223% | 10.4169% |
| 5 | $14864.22 | 12.1% | $1798.57 | $16662.79 | 66.62793% | 10.75149% |
| $16662.79 | 20% | $3332.56 | $19995.35 | 99.95351% | ||
Manual Method A: (find gain % by knowing the result)
This uses the actual final accumulated dollars of value, regardless if it includes reinvested dividends or not. If there are any reinvested withdrawals, that money received should be added back to this new total. But if it includes money you added to the account at some time (that was not present all the time), the gain numbers will be accurate, but it won't be accurate gain percentage rates. But technically, all you need to know is the first initial value and the final value. You cannot use just the two share prices if you have added or withdrawn shares.
From this computed table, this overall example six year gain (initial to final amount) is
This is just the gain portion. The gain can be called the Total Return (of the period of time), here near 1x or 100% a year each year. It is before multiplying by 100 for percentage.
Multiply Total Return by 100 for the Gain Percentage.
0.9995351 × 100 = 99.95351% This case is 100% gain = 2x value. It is gain only, and does not include the initial investment.
These are the initial and final, or before and after values. The Result is only the gain (it has subtracted the initial value).
Add 1 to the Total Return as the gain multiplier of the investment. As a multiplier, the 1 represents the initial investment. First a simple example: If you had $100 and then it gained 50% (0.5), the new total value is
$100 x (1 + 0.5) = 1.5x x 100 = $150.
Or the above New value is (1x + Total Return) x Initial value = 1.9995351 x $10,000 = $19999.53
The 1x represents the Initial value to be multiplied. It is a very important simplification. Whatever its actual value, the result is 1 + (gain/100) multiplied by the initial value is the new final result.
If you already know the final and Initial values, then you can use this gain formula, or see the 1st Gain calculator below. It does not use Initial value. Stock price does Not include dividends, but the dollar result does include effect of your reinvested dividends. Any withdrawals are received value which should be added back into the final total, but withdrawals will drastically reduce long term gains.
In A, Total Return will include any dividends if the investment was including dividends into your total value.
In B, Morningstar Total Return will include dividends if any, even if your investment did not. That could mean your gain might be less than Morningstar shows. And Morningstar cannot know if you withdrew some of it before, so it could include more than you see.
Multiplied Method B: (find actual gain % knowing each years rate)
(1.15 x 1.235 x 1.104 x 0.948 x 1.121 x 1.20 - 1) x 100 = 99.95351% gain.
This 1 is the same 1x just mentioned for the initial value. We don't need to know its value to compute gain, like from stock prices. The first 1.15 is that 1 initial value plus its 15% gain (adding 0.15). The -1 at the end subtracts the initial value, leaving the gain. This comparison of the two 99.95351% numbers seems pretty close.
See the 3rd calculator below (Compounding of Yearly Total Return Percent) which does this. Morningstar.com shows ten past years of these annual Total Return % numbers (includes reinvested dividends, which for stocks are at the Price vs Fair Value tab, and for funds are at the Performance tab), and a direct copy and paste works.
The order of the years makes no final difference. Compounding is simply repeated multiplication of gains. This method does use the annual gain numbers with only 3 or 4 significant digits, so it is close, but not totally precise. From this source, knowing the initial dollar value does not affect the gain percentage. But then, the year's final Total Result value is (1 + gain/100) × initial invested value. For an initial $10,000 investment, then 1 + 99.95351% is 1.999535 × $10000 = $19,995.35 result value.
Terms again: Note that in this example, $10000 grew by 99.9535% to be $9995.35 gain to be $19995.35 new value to total 1.999535x more (double, 2x). Those are all different values. So gain can be referred to as gain percentage (99.95351%) or Total Return (0.9995351x) or new total of 1.99535x more. The point is it's important to realize if the initial value is included in the term or not. The initial value (whatever its actual value) can be thought of as 1x, and final value is some x% more at the end. We need to realize if the original value is contained in our values or not.
A couple of these equations (like B above) will have either a + 1 or a - 1 in them to add or remove the initial value of 1x, which is 100% of our initial value.
The 1 is the Tricky Part of the Math Details
Don't make this be hard, because it's not hard at all, and you'll really need this. It does seem clever, and it really makes it easy.
The +1 and -1 in these equations represents the initial 1x investment value, separate from the gain part. When you say something gained to 1.5x times value, the 1x is the initial amount and the 0.5x is the added gain. That 1x becomes 100% when multiplied by 100 for percentage (and is independent of the actual dollar amount). Realizing this makes the math easy.
For example, take the case of the third year in the table above, with 56.7956% gain then. Adding 1 represents the original value to be a Gain Multiplier, and if $10000 was the original investment, then it works this way: 10000 × (1 + 56.7956/100) = 10000 × 1.567956 = $15679.56 final value (original plus gain).
To convert a Total Return % gain to be the x times final value accumulation, then divide by 100 (simply move the decimal point 2 digits left) and then add 1. This 1 adds the 100% of initial value to the gain. Divide the gain percentage by 100 to get the multiplying factor.
You can do this in your head. Examples:
10% gain is (0.10 + 1) = 1.10x times the initial value (1x initial + 0.10x gain).
100% gain is (1.00 + 1) = 2x times the initial value (1x initial + 1x gain).
200% gain is (2.00 + 1) = 3x times the initial value (1x initial + 2x gain).
18915.2% gain is (189.152 + 1) = 190.152x times the initial value.
1.10x gain means the years final value is 1.10 × whatever its initial value was.
Vice versa, to convert the x times final accumulation to the Total Return %, then subtract 1 and multiply by 100. Subtracting the 1 subtracts the 1x initial investment value (100%), leaving only the gain. The 1x is always the initial value, whatever it was.
Whatever the initial amount was, it is 1x (which times the 100 is 100%). Subtracting the initial 1 (initial value) from final multiplied result leaves just the gain portion (in Annualized Return next below). Adding 1 to the gain gives the multiplier of initial dollars to the final dollars. The gain computes the same numbers regardless of whatever the value of initial dollars. The 99.95351% gain in this example is 1.9995351x value, which is essentially 2x total value. So the final value would be 200%, but subtracting 1x initial value, the gain was 2×100% - (2-1)×100 = 100% (1x less than value, for any gain). The 1st Gain calculator below tries to differentiate gain and value.
This tricky part is repeated again below. It is essential to work with the gain and the initial investment.
Each year's gain is an individual multiplier of the initial value. Each factor is (1 + gain/100), with 15% gain becoming a 1.15x multiplier of value. Negative gains use the same method, for example -5.2% is 1 + (-5.2/100) resulting in 0.948x value that year. That 1 + gives the new value, including the initial value. Then a - 1 subtracts the 1x initial value, to see just the gain portion. Or the 1 is added to gain (the 1 is actually 1×100, which is 100% of the original initial value), to see the final total value result. The amount of "gain" does not include the initial value, but the total value result does.
You should always reinvest dividends, because they are Not new income, they are simply withdrawals of previous unrealized gains you already had. You should reinvest them to preserve your investments total value, and keep receiving gain from them. Continual withdrawal of dividends will cause a drastic reduction in your investment. Note that a share price result does not include dividends or compounding, but a formal Total Return does include them, including reinvested dividends. If there were no withdrawals, the final value resulting from the investment is a clear total answer including everything that happened. So $50,000 value result from $25,000 invested is a 50000/25000 = 2x gain, pure and simple. And it is a (2x - 1x)×100 = 100% gain. However, the rate of gain depends on the time of duration. If a market result took 10 years to double, it is (21/10 - 1)×100 = 7.1773% Annualized Return. Meaning, (1 + 0.071773)10 = 2x value would be the same result if it had come from a fixed rate interest source. Annualized Return seems a useful way to compare varying market results. Annualized Return is the same result, but of a fixed gain rate.
Not even if all years are equal Fixed gain, because the Average does NOT include compounding.
For an example, ten years of market gain of
-20% 20% -30% 30% -40% 40% -50% 50% -60% 60%.
These numbers obviously Average to be 0.00%, but the actual gain result is
-0.80 x 1.20 x -0.70 x 1.30 x -0.50 x 1.50 x -0.60 x 1.60 = 0.3522x gain (1/3 of initial value), or ‑64.8% overall gain (2/3 loss), and is ‑9.91% annualized gain (which is the actual overall gain rate (loss), and the same result). Compounding makes a big difference.
Even if the years are 10% 10% 10% 10% 10%, the average year is 10% but the compounded gain of five years at 10% is 1.105 = 61.1%.
If you started over every year with the exact same investment value, then average could work as the sum of the gains. But the marketing uses compounding, each year starts by including the gain of the previous year.
So forget about using the Average for the gain, it's numerically meaningless in the compounded result. It may be nice to know, you might expect next year to be in that range (or it might not), but it does not compute total gain. Average is just a sum divided by years, but compounded gain is a multiplied product which includes compounding. The gain in each year's value is in the next years beginning principle.
The stock market varies every day, up and down, a little or a lot, positive or negative. Large or Negative gains in years containing more money have more effect than years when the balance is smaller (and vice versa). A down year of -50% then is then half the money working. So then it requires a gain of 100% to get back to even (Calculator #4 below).
The average gain is only the SUM of all years gain divided by the number of years, which does Not include the compounding which is a multiplication of all years gain. The Average of the fund's annual gains is NOT a valid measure of stock performance. An average might suggest a usual future years expectation, but the total gain of a multi-year investment is the Multiplied Product of all the years gain. The chronological order of the years does not matter in multiplication.
Average is just a sum, but real gain is a multiplied product of several years of gain, and that includes compounding.
Using Method B above, (1.15 × 1.235 × 1.104 × 0.948 × 1.121 × 1.20 - 1) × 100 = 99.95351% gain in the six years.
In this example, Averaged and Annualized values are in fact quite close (both about 12%), but forget that, it is not usable info. Because it really just varies with the range extent of the numbers. In comparing 180 stocks over 10 years, the ratio of Annualized/Averaged is NOT 1, but ranges from a minimum of -13x to a maximum of +88x. That is random, average has no corelation to the actual annualized gain that is achieved. You could say annualized provides the correct average, because Annualized Gain computes the Fixed Rate that computes the exact same gain result in the same time.
Annualized Return is the better way to compare variable stock performance, including reinvested dividends and compounding (however make no mistake, annualizing must only include complete whole years. For comparisons, it computes a Fixed rate that gives the Same gain in the Same time. The actual long term compounding result is the multiplication product of all the years gains (in the format of 1 + gains, meaning each with the 1.15x type of number instead of the 15% type number). This product times the initial investment is how many dollars you end up with. Examples of the method next below. Annualized Return is the computed Fixed Rate that produces the same gain result every year in the same time. Now that is an average, not of the numbers, but of an equivalent of the same gain every year creating the same result in the same time. Which might interest you.
The reasonable plan for comparing long term performance is to use Annualized Return, which is described below. But it is not an accurate or believable number if including the incomplete current year.
If you try to use annualized gain using a partial current year, like computing by calling it maybe 0.3 years, which will assume that rate continues the whole year. but it publishes as a complete year. Or if you call it to be 1 year anyway, then that is shown as the years final result. Neither idea is remotely believable in the market. We do not know the future.
It's a kludge, but to assume the example includes full years, we can show it nevertheless in calculator 3 below, To do that, you can add a final 0 (zero %) gain year (1.0x gain) to be the final incomplete, because in most of my calculators, the final year is assumed the partial current year and so is not to be included as Annualized. The purpose of adding the last dummy zero is to include it as the actual final year in this hypothetical calculation (the zero gain year is 1x gain and so does not change the result). It is only a WHAT IF the last actual year were a whole year. Just adding a final zero is Not a valid way of computing Annualized gain of an incomplete year. Because the Annualization will not be complete or accurate until the final year completes.
Total Return (for stock or bonds) is the value gain plus any dividends received and reinvested within the reported period. Morningstar.com tracks this Total Return % number for stocks and funds. (See more)
This point is simple, but crucial for understanding. Fist computing the gain, and then Annualized Return is next.
Gain % = Total Return X 100
Gain multiplier = 1 + Total Return (to include the initial dollars as 1x)
Example: If $2 becomes $3 (or if $2 Million becomes $3 Million)
Gain % = 0.5 X 100 = 50% gain
Value multiplier = 1 + 0.5 = 1.5x value
Original value is 1x (for any dollar value) and 1x value plus 0.5x gain is 1.5x value.
Dividends (reinvested or not) cause no change in value on the first day (dividends are withdrawals of past income), but withdrawn dividends keep reducing your investment, and reinvested dividends keep restoring it with free shares (at no additional cost) compounded over long term.
On the day of dividend distribution, the share price is reduced by the same dividend dollars per share. So yes, that price reduction does pay for the withdrawal, however you have the withdrawn dividend in your pocket, or reinvesting puts it back into the investment, so either way, there is no change (due to the dividend) in your overall value on that day (no gain, no loss, simply a distributed withdrawal). However the reinvested dividend adds more free shares (no additional cost, which I call free), and the added shares remain as long as you leave them there. So long term then has more free reinvested shares contributing earnings (more compounding shares every quarter is the most likely situation). Long term, this is a Big Deal.
But the market does not do nice even rates every year. The market goes up and down every day. Every year is probably widely different, even perhaps some are now and then negative for a loss. We know how much money we made, but the actual gain rate is not obvious.
Annualized Return math is just below, but first, we need the overall gain.
Annualized Rate: The overall amount of gain is of course important, but the number of years is also a large factor of that gain number. Annualizing it to show the equivalent fixed annual rate of gain each year allows understandable and meaningful comparisons. It includes reinvested dividends to compute the equivalent fixed rate of return of the investment giving same gain result. The annualized rate uses the then current values due to any cash either added or withdrawn along the way (so if any withdrawals, it may not match your actual results).
The terms sometimes do require a bit of attention. Morningstar publishes the Total Return % for the YTD, and that correctly is the Total Return YTD (year-to-date), but it is a single year and Not "Annualized" yet. That number will of course change by the end of the year. And at Trailing Returns, they show Total Return % for periods of 1 to 15 years, which if more than one year is Annualized for complete years from the current date, but are not year end numbers.
But for any gains, including variable market gains, over any number of years:
Example: $100 to $110 is (110 - 100)/100 × 100 = 10% gain (whether 1 year or 40 years).
Adding 1 + gain gives the final investment: (1 + 0.10) × 100 = (1.1 × $100) = $110 final value.
The Annualized Return procedure is based on the standard compounded Gain formula, which is a Fixed Rate to an exponent of years.
Percentage Gain = (1 + (fixed interest rate / 100))years
Fixed Rate example: Gain of 10% is (1.10)years = 33.1% final Percentage Gain
Fixed 10% for 3 years is the standard formula: 1.103 = 1.331x Total Return value, which then -1 (this -1 is subtracting the original principle) leaves percentage Gain 33.1% in 3 years, which is the overall compounded final result. 1x original principle plus 33.1% gain, in 3 years.
The Annualized Return uses the gain multiplier, 1 + gain multiplier. Then it reverses the fixed rate formula in that annualized rate is 1 + Percentage Gain to exponent of 1/years.
Annualized Return % = (1 + (Percentage Gain / 100)1/years - 1) × 100
((1.331)1/3 - 1) × 100 = 10.0% Annualized Return.
((1 + 0.331)1/3 - 1) is 0.10 which × 100 is 10.0% Annualized Return for each of 3 years, which becomes very meaningful since each years market gain was in fact very variable (confusing). Some market years are up big, some are up just a bit, and a few might even be a negative loss, which does not present any clear view of what the gain rate was. The computed Annualized rate (10% here) is a Fixed rate every year to the exact same result, which is NOT what actually happened, but it is the exact same overall gain in the same time, which becomes a very useful and accurate and useful representation of the gain rate over time. The Annualized rate is the computed Fixed rate that would have created the exact same result.
Here's a made-up example of three volatile years of gain (rounded a bit) that reached 33.1% total gain, which also has an Annualized gain of 10%. It may not have happened that way, but the final result is the same as if it did. Which is meaningful and usefully comparable.
| Year | Total Return | Year Gain Multipliers | Cumulative Gain | ||
|---|---|---|---|---|---|
| Individual | Cumulative | ||||
| 1 | 2022 | 20% | 1.2x | 1.2x | 20% |
| 2 | 2023 | -12% | 0.88x | 1.056x | 5.6% |
| 3 | 2024 | 26.05% | 1.2605x | 1.331x | 33.1% |
Cannot compute Annualized Return for an incomplete Year To Date, but 2022 to 2024 Annualized Return is 10.00% for each of 3 years. The 3 years Gain is 1.331x, or 33.1%
Annualized Return % = (1.3311/3 - 1) × 100 = 10%
Verification: (1 + 0.10)3 years - 1) × 100 = 33.1
Annualized Return % might be something handy you'd like to know, (which is AS IF the gain result was from a Fixed rate that gives the same result in the same time), computed from the Total Return % from the same final result. That's an understandable number which allows accurate comparison with other Annualized results.
Total Return % = (1 + (Percentage Gain / 100)1/years - 1) × 100
Most common is Annualized Percentage:
Annualized Return % = (1 + Percentage Gain / 100) 1/years) - 1) x 100)
The +1 and -1 are the initial investment
Caution: There is a mathematics problem of "fractional powers of negative numbers" involving complex numbers
(√-1, etc.), which is an annualization problem. The annualization exponent is fractional and negative gains can be a problem. That might not be current information, as I have not had any problem with it here. Handheld calculators are said to maybe have less problem with it than JavaScript. If you even get a result, you can verify the annualization result is correct by the verification reversal method back to the beginning gain. Positive gains should not be a problem.
Reversal meaning: (1 + (Annualized Number /100)years - 1) × 100 = the 0.331 should give the same first gain number.
Do NOT confuse Annualized rate with Average rate. Average rate is just a sum, does not compound, and is not useful for a market variable rate result. Years of 10%, -10%, 20%, -20%, 30%, -30% computes Zero average, but is instead -13.5% total gain and -2.39% Annualized Return. (See Average above)
Annualized is the same compounded value achieved from the years, starting from the SAME actual final gain, but computes the Annualized gain that would produce the same result AS IF from its Fixed percentage every year. That equal result makes gains easy to understand and to compare market performance. Market years do vary widely, some years might even be negative, making it hard to realize the rate of actual final gain long term. Annualized uses the Same Actual Overall Gain, and then is a more logical way to compare actual long term performance. It hides any volatility during the years, computing a smooth path of a Fixed Rate that would give the SAME result in the same time.
Caution: Annualized values are not complete nor accurate unless they include whole years of data.
What is the resulting performance of the first example six years shown above as:
15%, 23.5%, 10.4%, -5.2%, 12.1% 20%
The answer is shown above to be 99.95351% total gain, and 12.24186% Annualized Return, which annualized rate is more understandable and comparable as performance. Annualized Return (compounded, the same total gain viewed as the equivalent return each year if each year were equal) computes the equivalent result fixed interest rate AS IF it were that same fixed interest rate every year. It didn't actually happen that way, but this computed fixed rate is the real Fixed Rate that still gives the same final total result, still the same accurate result number if it had happened, which is a very good way to understand the gain. Annualization can be helpful because it is comprehensible, and otherwise very difficult to visualize the result of a string of variable years, maybe both large and small gains, some even negative. The Average of the years is NOT the correct answer.
However, if that last 20% value was the incomplete current year, then the Annualized answer is incomplete and wrong and invalid. Annualized has a different definition of complete years. Incomplete years simply cannot be annualized (because the future is unknown). All you can do is to leave out the incomplete year for now, and compute Annualization of only the first complete years. The Gain of all years of the data is valid, just not the Annualization if current incomplete year is included.
To compute the Annualized Return rate for the example in the above table with the above gain formula is:
Total gain is ($19995.35 - $10000) / $10000 = 0.999535×100 = 99.95351% gain in 6 Full years (Method A). The final total value in dollars would be (1 + 0.999535) × initial investment. The 1 is the initial 1x investment included in the final total value. You can think of it as 1x initial investment, first added, and then subtracted to leave just the gain. The percentage is the same for $1 or $1,000,000.
Note that Percentage Gain formula above (and calculator below) can enter units of either this method (like a 1.15 multiplier for 15% gain) or can use actual values like resulting dollars (for example. If the initial value was $1, the 1.15x gain is $1.15 value). Percentage comparisons compute regardless of the actual value, just meaning, $8 vs $1 or $800 vs $100 is exactly the same percentage as $8,000,000 vs $1,000,000. These are 8x or (8-1)×100 = 700% gain. The -1 subtracts out the 1x initial investment to show only the gain. The gain percentage is not affected by the initial investment value. Percent only reflects the degree of difference between Start and Final value.
Annualized Return % of this overall gain of 99.95351% is ((1 + 0.999535)(1/6 years) - 1) × 100 = 12.24186%
Annualized Return meaning AS IF this result were from the same fixed gain every year that would produce the same final result.
First we have to know the final gain (6 years), 0.999535 or 99.95351%, explained above.
The first +1 adds the initial investment value to the gain to compute final value (× 100 is 100% of initial value). If this may not be obvious at first, when the value of something grows to 1.5x more, the 1 is the original value, and the .5 is the gain. The 1 + .5 is the final value, and 1.5 - 1 is the gain.
The (1 + 0.999535)(1/6 years) reverses to compute the 1.1224186x final value.
The -1 subtracts the 100% of initial principle to leave just the gain, instead of the final value.
The × 100 is the percentage number (12.24186% annualized).
Again, if the final gain period is negative, like -8.2% over 6 years, still use 1 + (-8.2/100) which is (0.9181/6 - 1) × 100 = -1.4158% annualized.
You really need this part to use the math, and it is really easy if you just think a second.
In all these gain formulas, there are the initial and the final amounts. The -1 and the +1 represent the 1x initial amount , separate from the gain, and is independent of the dollar amount (it distinguishes between gain and resulting total value). When you say something gained to 1.5x times value, 1x is the initial value amount (dollars, but represented by the 1x amount) and the 0.5x is the gain. If it gained to 2.5x value, still the initial amount is 1x and the gain then is 1.5x (times the initial investment, and 2.5x is the total result). It doesn't matter if that initial amount was $100 or $100,000 or whatever, it is 1x of the initial amount, and 1x represents it, and we use percentages (1x × 100 is 100%) in the final result. We don't need to know an actual dollar amount, but it can be computed from either the times the initial or from its added % result.
So in the above Annualized Return formula, the first +1 adds the initial value of 1x to the gain (to be the 1 + 0.999535 gain result above). And then it applies the reversed exponent of 1/years (1/5 = 0.2), and then subtracts the initial amount (the -1 again) and multiplies by 100 for percent of only the gain part (less the initial amount).
One more example: If you had $1000, and it gained 20% fixed rate in each of 5 years, then:
(1 + 0.20)5 = 2.488x final total result (then -1 × 100 is 148.8% gain to $2488 total value). The exponent works for a fixed rate gain.
1.20 x 1.20 x 1.20 x 1.20 x 1.20 = the same 2.488x value (148.8% total gain). This way is used for all varied market years, Method B above.
Or Gain: ($2488 - $1000) / $1000 = 1.488x and × 100 = 148.8% gain to 2.488x total value. Could be varied market years.
It's not hard to get confused, so remember, if computing final Value (not dollars, but the x multiplier of initial), then you have to subtract the 1x (or 100%) initial to see percentage Gain. Or if you know percentage Gain, then add the 1x for the x multiplier. Realizing this relationship makes it easier. A 15% gain is a 1.15x multiplier. And it's the same 1x for multiple years (and the larger numbers).
Then (1 + 1.488)1/5 = 1.20x. Then -1 and × 100 = 20.0% Annualized Result, as if the exact Same Gain with this fixed rate. Could have been varied market years with the same gain result.
We already knew this fixed rate was 20%, but the same math works for very varied market years. It lets you compare your market gain as if the same result occurred from that fixed rate, to be better able to comprehend your actual market performance. A +15% year is 1.15x gain (1 + 0.15), and a Negative year gain of –15% is handled the same, as (1 + –0.15) = 0.85x gain.
The Annualized Return is of course Not what actually happened in each specific year, however it starts with the actual real final gain result and works backwards. The point is if an investment did actually achieve 1.9995351x value (99.95351% gain) in 6 years (again, this is speaking of the Total Return result including reinvested dividends), the resulting annualized 12.24186% interest in six years would be that same 1.9995351x value result. It is a very good way to visualize performance of investments that vary so much (+ and -) each year. It could still matter if comparing with a published fixed rate compounded differently, but compounding each year is what Annualized Return computes. Stocks are basically compounded on the date of every dividend and also with every market day's price, but annualization computes accurately using whatever final dollars are actually present at the end of the year (instead of how they got there). That is also how annual Total Return stock numbers are computed. So whenever Morningstar says the 10 year Total Return of a stock was say 12.5%, that means annualized, as if every year. That didn't literally happen every year, but the final result was exactly the same (annualized return is computed from the resulting gain). You may want to think of the fixed rate gain as the resulting dollars annualized too (should be the same, unless fees or other costs). The purpose of Annualization is for comparing the return of a varying rate as if it was a fixed interest rate of results, useful to better visualize a number for the varying market gain.
Global Investment Performance Standards (GIPS) accounting standards (Section 2.A.12, page 9) says "Returns for periods of less than one year must not be annualized".
Not for any incomplete year. The Annualized Return cannot be computed if the year span includes the current year, because the current year is incomplete, and the future is unknown. FWIW, including the current year will assume the year ends at that rate, which requires the assumption that the remaining months will all be zero gain (if entered as a whole year), or if entered as a fraction of the year will assume that the future months continue at the same gain rate. Either result is unbelievable nonsense in the variable market. Repeating, including an incomplete year in an annualized calculation is pointless, it will NOT be correct in any form.
Annualized data must include Whole Years Only (else it is incorrect math).
So the calculators on this site will omit Annualized Return if the current year is recognized to be included (as being the final year). You can of course still see the current year gain so far, and that will be correct, but the incomplete year cannot be in the Annualized Return, and it is not shown as such.
The order of the years does not matter to the math multiplying. Reversing the computed order of the years achieves the same gain accurately (but must not include any partial years).
The examples 1.999535x gain is very nearly 2x. And the Annualized Return 12.24% is near 12%. And the doubling in 6 years × 12% is the Rule of 72. Which is an approximation, but it does indicate the math is working OK. (Rule of 72 is below).
The large number of digits are shown in the calculators in case someone needs the precision to reverse calculate for verification (like initial defaults in Calculator 1 & 2). Values like $1,000,000.01 are 9 significant digits. If reverse calculations don't reach the same exact number, your factors need as many significant digits for equal precision (which is why more digits are shown in the calculators here). (Only 7 significant digits here in previous paragraph, but the final value is only 7 digits.) It is possible for a calculator or computer to use FULL precision of all the numbers, especially for exponent calculations. Meaning, don't round off the data until time to show it. For example, the initial values for calculator 1 & 2. Computing 9 digit values (like $1,000,000.01) needs at least that many significant digits all along for full precision. You would round off final results to show them, but while in the computer, compute with the full available precision, without any rounding. Rounding during calculation (of dollars and cents or of percent) limits result precision to that limit, however even a rough approximation might still be adequate as a ballpark comparison.
The Average of a fund's annual gains is Not the meaningful measure of stock performance. Because the average is just the sum of each years gain divided by the number of years. But the long term compounding result is the multiplication product of each years (1 + gains), which is the compounding. It's difficult to judge a final result just looking at several mixed year results (of maybe both + and -), but the Annualized Total Return is a very good way to realize a realistic and meaningful long term performance rate of a stock. Maybe it didn't actually literally happen that way, but annualized gives precisely the same gain in the same time, and computes a more useful comparison with a "continual" Fixed gain rate that works too.
For any of the calculators, see its extra notes below.
Legend in 1. below: See the "Tricky part" notes repeated two places above.
The "X" factor is the resulting multiple of the initial value (the Tricky part).
The "%" factor is the gain, less the initial value.
The "1" shown in the formulas represents the initial value, without having to know its dollar value.
Caution: Annualized Return cannot and will not be calculated if including the current incomplete year data. Gain calculations are OK as YTD, but Annualization is Whole Years Only.
Notes for these calculators:
Significant digits: For your own calculations, don't round off in your calculator until the final result you want to show. The apparently excessive number of digits shown is because large values like $1,000,000.01 have 9 significant digits, which needs that many significant digits in the other factors (like interest rate) to accurately match the same precision (years is an exponent of interest, which needs precision).
For example in calculator 2, the initial values shown (reverse computing the calculator 1 default) if using instead 12.2% (3 significant digits, instead of 12.20184546) computes $999,342.31 which is not 9 significant digits of the fully accurate number of $1,000,000.01.
Approximations might be a very adequately useful estimate, but not the exact final result. This one is less than 0.1% error, but a more significant number of dollars would be nice to know. Full precision does require the Necessary number of significant digits. The precision of future market result estimates is unknown anyway, but exactly matching a reverse verification of Annualized rate needs about all the digits you can manage (compute it before any rounding). I've usually shown the Annualized Return with 10 significant digits here, which may be excessive to view, but when the math has an exponent of 40 years for a result in the millions, it is more accurate to provide adequate accurate precision.
Any computed result will not have any more precision than the least significant digits in any number used to compute it. However, some numbers are Exact Numbers, like 3 apples or 10 people or a $20 bill, which are fully precise numbers. If you withdrew $8000, then 8000 is an Exact number, the best you can do. Or perhaps sometimes 2x investment or 5 years are fully precise numbers. The Morningstar annual Total Return percentage is shown with typically as few as 2 or 3 significant digits, which is Not exact, but that approximation might be close enough for the purpose. Typically, try Not rounding calculator numbers until the final number you want to show.
1st calculator, Compounded Gain and Annualized Return: Hopefully it is both self-explanatory and maybe the most useful. Technically, any units work (like price or distance or weight or time). Or Dollars or Euros or Yen, but gain calculations need Not be about money.
2nd calculator, Future Value: This one is perhaps less used, but if no withdrawals or additions, it might estimate expectations of final value, however a future variable market gain rate is not predictable. This calculator can show the importance of compounding time. Years are your best investment tool. To just see the gain rate compounding of the rate percent, you can use $1 value.
Or it can be used to reverse compute to verify an Annualized calculation is correct.
3rd calculator, Total Return from yearly gains, Two methods of data entry:
Annualizing only one year is pointless, as only one year will be the same number as the years total gain. Annualizing is about the effective result of compounding multiple years of gain.
The incomplete current year cannot be included in annualization. The concept of annualization is whole years of data. The calculator could ask about the status of the final year, but the final year entered is assumed to be Morningstar data, so the final year is then incomplete, and it is omitted from annualization. The overall gain is computed OK, but the final year will be omitted from annualization since an incomplete year computes invalid annualization.
If the final year value is in fact a complete year, then to let it compute as a complete year (which if Not complete, it will be Not be a valid future value), you can add one more fake year as 0 (0% gain) following the real final year, to then be assumed to be 2026 and ignored and omitted. Its zero gain does not affect the final gain (0 is converted to a 1.00x gain multiplier of that year, which has no effect). But then faking a new last one as 0% gain allows computing annualized gain as if it had the meaning that all the other years were complete years.
But warning, some of you may get the idea to add an additional final year as 0 just to see the current 2026 Annualized. If so, you may not understand yet, but that would assume that year is already the complete year, with the effect that your actual final year result is assumed to close the year at the same reported gain (with zero additional change, which is not a believable actual future market result). It might be approximate in December, but is plain nonsense in January. The future is not known, and a meaningless number is better omitted than to compute garbage.
The gain is a multiplied product, so the order of years does not change the final number, but here, the final year is assumed to be the current partial year and ignored. And you might better understand seeing the years in a correct order.
Morningstar Annual Total Return % charts for all funds (including ETF) now run the opposite ordered direction for all stocks, so there is an extra step above to reverse funds. ETF has the option of using the return on share Price, or the return on NAV, which includes reinvested dividends (which should be preferred).
Morningstar also shows another number called Trailing Returns for various year periods which is the same annualized method, but from the current date day instead of year end values as here (so all those years are complete then). It may be just as useful, but that means the both the starting point and the ending point data of each year period also varies every day, to use that same day for 5 or 10 years. It does bypass the incomplete year issue, but it includes calendar data not available here.
Each year's data percentage must be the years final Total Return %, which assumes reinvested dividends. Morningstar.com shows ten years plus YTD of these annual Total Return % numbers. If the current partial YTD is included (an incomplete year as of yet), then of course that last YTD value will change as the year progresses and we do not know a correct number. But if omitting that incomplete final year, this method makes getting histories compounded gain numbers be easy. The data is shown in a table as interpreted in case you tried editing the data and messed up somehow, it may help see the trouble. If any data trouble editing the Morningstar data, just start over with the simple Copy and Paste. That works.
Computing Annualized Return for an incomplete current year is just NOT properly done. It cannot be the correct current year result, because we do not know the future, and the gain will change every day until the year end. But Annualizing earlier complete years will be accurate.
Extra 3rd Calculator Details: The Copy and Paste from Morningstar uses tab keys for year separators (between each year value). And that tab works fine here, you can leave it just as copied, and Pasted it. Actually, this tab thing should also work from any web screen HTML table showing years of Total Return percentage. But a blank also can be used as the year terminator.
The concern was that if you edit the data, the calculator cannot type a tab into the browser's field. Tab means "next field" to the browser. So the calculator instead uses a space character between numbers to separate them, and shows it that way. The calculator will replace any tabs (from Morningstar Copy/Paste) with a space. So the calculator doesn't care which is present, either space or tab will work (but the separator used must be one of them). You will see the space in the displayed data, but the tab still works too (but you cannot type a tab, and any tab imported will become a space). Again, a Copy of the Morningstar data will have tabs in it, which works as is, but which will then become a space, which is fine, it works. Multiple embedded spaces in the data are OK, they will be combined into a single space. Simply Copy/Paste the Morningstar Total Return % data line, and it will work. One exception, Morningstar puts a - in an
4th Calculator, Gain to recover from a down market: Just to make the point that if a stock is down 50%, that takes 100% gain to recover to the previous value. That change is 50% of a larger value going down vs.100% of a smaller value going up.
5th Calculator, Tax Details: This calculator is for funds, which typically use the Averaged Cost of selling Shares that you can specify to manage the Cost Basic calculation. Stocks don't use Averaged cost, but multiple purchases have different costs. For stocks that requires knowledge of share lot numbers. Stocks may use FIFO (First In, First Out, oldest shares with lowest cost and the most gain and tax) is common and the oldest and is often the default unless you change yours. LIFO (Last In, First Out, newest stock, highest price, the least gain and tax) sells the newest shares. For funds, Average is usually the default method, which are bought as dollars instead of shares. Vanguard has a MinTax choice which evaluates the shares for stocks, looking for long term Capital Gains if possible. Capital gains are often taxed at a 15% rate, but can be zero tax in low income returns, and can be a maximum of 20% for high incomes. The large middle range is 15% tax.
Capital Gains are gains (or losses) on investments held over one year. Capital Gains tax is an entirely separate tax calculation, which is then added to your income tax. But Capital Gains tax rate does depend on your overall total taxable income to establish tax brackets. The regular IRS income tax tables do not include Capital Gains, but which will be added to the total tax. Capital Gain losses are subtracted from any Capital Gains gain. If the loss is greater than the gains, the remaining loss is subtracted from ordinary income tax at the rate of $3000 each year.
IRA withdrawals (or on anything not yet taxed) has Cost Basis of zero, meaning tax on every penny of it is owed at withdrawal, so there is no Capital Gains for IRA. A dividend is a withdrawal, reducing invested value and future earnings, however reinvested dividend dollars restore the Market value, and are then added to the subsequent stock Cost Basis next time. Dividends are withdrawals of your earned gains, which become realized gain and so are taxed then. Reinvested dividends become FREE additional shares, with same money you had before, so are no additional cost, so therefore are FREE added shares each time. Which is very worthwhile long term. Withdrawals reduce cost basis. Again, your brokerage should show current Cost Basis, and it affects the tax.
IRA withdrawals do not provide Capital Gains because IRA Cost Basis is zero (so the total IRA withdrawal amount is taxed at ordinary income tax rate). Every IRA penny withdrawn is taxed, for inheritances too. That will be a very big sum when your IRA has grown large, like to a few million. IRA will taxed at regular income tax rates instead of capital gains. Investors may see advantage in doing the wheeling and dealing in an IRA, because there is no tax until withdrawn, but be aware that tax will always be paid when withdrawn. Tax is due, and with a 10% penalty if IRA is withdrawn before age 59 1/2, with some exceptions.
Roth is not taxed, but there is 10% penalty if withdrawn before the Roth account is not 5 years old, or before age 59 1/2.
Unless you invested it before 2012, your brokerage shows both Market value and Cost Basis (of all shares, but Average Cost might still work for fewer shares) and your 1099 form will show the gain to be taxed. Cost Basis method depends on which you selected at your broker (FIFO, LIFO, Average, etc.). Average cost is assumed here (which is also used by most funds, because funds count buy and sell values as dollars instead of as shares, That causes fractional shares.)
Cautions: Do retirement in a Roth or at least in a non-IRA account for Capital Gains tax. IRA taxes on this amount will be expensive (a high rate on a 100% taxable large amount, taxed as regular income). If you have decades until retirement withdrawals, convert IRA (or 401k if possible) to Roth early, which is a taxable event, but is vastly cheaper before it grows so much. One difference though, IRA can be withdrawn anytime, and Roth is only withdrawn without penalty after age 59 1/2. But if wheeling and dealing in stock, IRA has no tax until withdrawn from the IRA, which seems a big advantage at the time. Buy and sell stocks without any tax (until withdrawn from IRA). However that IRA might grow to an enormous amount, which will be withdrawn and taxed then as ordinary income tax rates. But a Roth is taxed if the Roth account is Not five years old, likely not much problem.
If you have a 401k at company that you have left, you can move it to an outside IRA source (probably at any brokerage like Vanguard) where you will have complete control of it (what to invest in, conversions, withdrawals, etc).
At least currently, a retirement IRS required IRA RMD withdrawal (after age 72) may not be allowed except from a standard mutual fund (the fractional share problem), and moving it then may be a huge taxable event best avoided. Some EFT may allow withdrawals of either dollars or shares, but some EFT even refuse dividend reinvestment (fractional shares). The EFT rules are new yet.
A little different approach that was the first try, but due to FIFO and LIFO lots having different cost basis, it was not feasible here. The U.S. IRS 1099 forms show only the withdrawal amount, but the IRS does receive additional cost basis info from brokers. Since 2012, brokers do record Cost Basis on taxable accounts of regular stocks and funds. Cost Basis Method is declared at your broker, and is like FIFO (First In, First Out, meaning first shares bought are sold first, or an Averaged price for funds. This method can affect the Cost Basis number, but your broker does use your choice. If you made only one purchase, that buy will be the share cost, unless dividends have added to it. But for stocks, actual tax will depend on the Capital Gain method (FIFO, LIFO, etc) which will decide which shares will be sold. A stock will not use Average cost.
If your broker shows Cost Basis, then (Cost Basis / Total Value) will be the percentage of the total that is not taxed. If you withdraw 10% of it then you will owe tax on the (Cost Basis / Total Value) × (1 - 0.10) × 100 fraction of it, but again your Costs Basis Method will choose which shares are sold. Or easier if shown, (Unrealized Gain / Total value) × 100 is the taxable percentage, and you are selling some percentage of the total value. You need to record that total on the day of sale.
But for stocks, actual tax will depend on the Cost Basis method (FIFO, LIFO, etc) which will decide which shares will be sold. Funds use Average share cost, but a stock will not use Average cost, but for example, maybe FIFO or LIFO methods. If your broker shows Cost Basis, then (Cost Basis / Total Value) will be the percentage of the total that is not taxed (and 1 - that will what is taxed). If you withdraw 10% of it then you will owe tax on the (Cost Basis / Total Value) × (1 - 0.10) × 100 fraction of it, but again your Costs Basis Method will choose which shares are sold and their individual cost then. Or easier, if shown (Unrealized Gain / Total value) × 100 is the taxable percentage, and you are selling some percentage of the total value. But you are still at the mercy of your Cost Basis method, which decides which shares are sold. The 1099 from the broker will have all of that settled.
6th Calculator, How much money for a specific result?
I hope its use is self evident. Its purpose and intention is to warn that retirement is obviously coming some day, and then retirement could last 30 more years to age 95. That will be very difficult with no salary or savings, and far too late to start then. There are very many trying to scrimp by on Social Security, which is no fun. Few ever even think about retirement, other than maybe about traveling the world, but the least expensive way to provide for it is to not waste all of the compounding years you might have now. They won't come back. The calculator can't tell you how much retirement savings will be needed, because that depends on individual life style and medical condition and age now and where you live, etc, and no telling what else. We often see it said as $1.5 million (like they know). I've not seen any qualification of it, but it's getting bit out of date now. We don't know that inflation will do in several more decades, but my clueless guess is, depending on age now, is at least 2 to 4 times more then. And keep an eye on inflation. The affordable solution starts early and uses the years. It would seem very wise to be working on providing that now (even smarter, in previous years too).
You will need a big investment still earning for you all during retirement too. Start early, meaning NOW. Don't waste the compounding years, the years are your best tool. We hear that from age 25 on, everyone should be saving 15% of their salary. Few do though. And that is not sufficient any more anyway. 40 years (from age 25) of $15,000 at 11% is "only" 1.1140 = 65x, then 65 × 15000 = $975K. $25K computes 1.62 million, and $35K is $2.28 Million (from 11% S&P 500 at 40 years). Don't forget retirement can last 30 years to age 95, so that's 70 years for age 25. If you want to waste however many years you have left, maybe you might rethink that. You might think about how much inflation there might be, and about what might happen to you in retirement if your funds run out. Retirement is coming. Will you be prepared? If you should become over-prepared, your children will probably need the inheritance for their retirement.
The 6th calculator order just seemed better placed up with the top two.
Off-topic a little, and just for fun, but if checking that these numbers are reasonable, a simple rule of thumb approximation is the Rule of 72 that says an investment value about doubles if the years × fixed percentage gain = 72. Markets are Not fixed rates though, but annualized rates are. So 6 years × 12% = 72 would approximately double to be 2x value (1.126 = 1.9738, almost 2x). But the precise fixed rate that exactly doubles in 6 years is (21/6 - 1) × 100 = 12.24620483%, which is the Annualized formula, and that beginning 2 is (1 + 1) to be the initial value 1x + 100% gain doubled to then be 2x which is produced by 12.2446% gain for 6 years. But a few more significant digits helps exponent precision (12.246204836 = 1.999999997 and 12.24620483 x 6 = 73.47722899),
The Rule of 72 is an approximation said to date back to at least the first known mention in Summa de arithmetica by Pacioli in year 1494 (in the time of Columbus, long before calculators (or even logarithms), when calculating was quite difficult.
Actually, 6 years should be Rule of 73.47723 for 12.246205% doubling in 6 years (1.122462056 = 2.0000). The 12.246205% Annualized Return is an impressive rate of gain when compounded over years, doubling every six years. That's a 2% error, still pretty close, but 3 years is worse, and 9 years is better. The Rule of 72 math is compounded annually.
The precise result for the interest rate which doubles in N years is
(2(1/N) - 1) * 100, so if 10 years for example is shown next. (the 0.1 is 1/10)
Those doubling Steps: The gain each year is 20.1 = 1.07177346x, which is 1 + 0.07177346.
Then (gain - 1) x 100 = 7.177346% each year (the -1 subtracts the original principle) to leave just the gain.
The 2 is the 1x principle plus the 1x gain result, to double compounded each year.
Therefore 1.07177346x growth each year gives 1.0717734610 = 2.0x result.
This actual 10 year doubling "rule" would be the (2(1/10) - 1) * 100 = 7.177346% x 10 years = Rule of 71.77346, but is only precise for 10 years, and then it is a 71.773 Rule. But as an approximation, 72 is close.
This calculator purpose was to look at the accuracy of doubling with the Rule of 72, however you can also enter different multipliers here (other than 2x for double, such as 1.5x or 10x multipliers, giving corresponding approximate rules of 42 or 259, more or less).
In exploring the Rule of 72 (in this table), it became clear it is only a simple rough approximation. Speaking only of doubling, the worst accuracy with the Rule of 72 is if 6 or less years. The most accurate case for Rule of 72 doubling is for 9 years. But a Rule of 70 works better for doubling long term, 15 years or more. Still, just as an approximation, 72 might be generally close enough, except maybe for 5 or less years.
I suggest that in specific situations, the first or second Gain calculator above will be more useful and versatile and certainly more precise than the Rule of 72. For example, in that 1st calculator above, New = 2, Old = 1 is 2.0x Value and for 6 years is 12.246205% Annualized Return, and that Rule is 73.48. The calculated percentage rate numbers are about the gain ratio, independent of the actual amount of money which is just 1X (speaking of gain, it is 1X value).
Compounding is certainly a big deal in investments, making many long term years be the most profitable part. Only a year or two is not so dramatic, but compounding is exponential with time, becoming huge over many years. Long term can be exceptionably good. The S&P 500 (gain and reinvested dividends) has averaged an annual return around 12%. The future is not known, but the wide market (like the S&P 500) sure seems a good bet if considered "long term". Two facts though, worse times have happened several times over the years before, and 2, it always recovers and continues. The S&P 500 was down 25% at $3585.62 on Oct 3 2022, but 40 years ago it started with only about $122, which is an increase of about 30x so far, even if Not counting the dividends (45x with reinvested dividends).
Plug in your own numbers, but if your age is 40 years or less, then you still have at least 25 years before retirement at 65 (when you will then be needing a source of income). Years are your best investment tool, and TODAY is the best time to be considering that. And the investment can continue earning during 30 years of retirement withdrawals too. Fewer years can be very profitable too, but the years will be your best tool and the largest growth multiplier, so wake up, and get with it and take advantage of it now (the term Buy Low means, if the market is currently low, to make buying right now be the best and most profitable time, a very wise plan). The market always recovers.
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