and Annualized Return, Methods and 6 calculators

A menu into this page is:

Computing methods of compounded gains, Total Return and Annualized Return

Computing Annualized Return that Includes the Incomplete Current Year Is NOT Acceptable Practice

**Six calculators below**

The greater number of years of long-term investments makes the compounding of gains be 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 1.10^{30} = 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 Reinvested dividends are a Real Big Deal, whereas withdrawn dividends simply reduce your investment.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.** That 1x becomes 100% when multiplied by 100 for percentage. And if it gained to 2.5x, the original 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 point here is the -1 subtracts your original investment (to leave only the gain amount) without needing its actual dollar amount.

**Compounding** is leaving the gain invested, which is a growing investment, so then a Fixed 10% is that much more gain next year than it was this year (and it grows even more every future year). It really adds up long term. For example, 30 years reinvested compounding at 10% is (1.10)^{30} = 17.45x value, whereas if withdrawn, 30 × 0.10 is only 3x value. Long term compounding makes a tremendous difference. I hope you also realize that always withdrawing those dividends has the corresponding tremendous loss of profit, leaving only the 1x (for an overall total of 4x value, which is only 23% in this example).

Total Return is the actual rate of return of an investment, with the word Total generally meaning also compounding of reinvested dividends.

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. But market gain is variably different every day, making it hard to judge actual performance numbers, but there is a mathematical way called Annualized Return. Here's six years of example **Total Return** statistics. Price gains do Not include dividends, but reinvestment adds more shares, and so published Total Return does assume reinvested dividends. Morningstar also shows Total Return %. 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:**

Gain =

New value - Old value

Old value

× 100 %Old value

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.

From this computed table, this overall example six year gain (initial to final amount) is

Gain =

$19995.35 - $10000

$10000

× 100 = 99.95351% overall gain %$10000

If you already know the final and initial values, then you can use this gain formula, or see the 1st Gain calculator below. 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.

**Multiplied Method B:**

This uses each years annual **Total Return %**. These Morningstar Total Return % results assume reinvested dividends and no withdrawals. The format is that the first year gain of 15% in the table above means the first year result was 1.15 × (the initial 1x amount). That then becomes the initial amount for the second year. These same six individual yearly gains (of 15%, 23.5%, 10.4%, -5.2%, 12.1% and 20%, each as 1 + percent/100) will also compute total compounding result as

**(1.15 x 1.235 x 1.104 x 0.948 x 1.121 x 1.20 - 1) x 100 = 99.95351% overall gain**.

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, but it is close, but not exact. The initial dollar value does not affect the gain percentage. And 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.**

**The Tricky Part of the Math Details is the 1**

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 1 becomes 100% when multiplied by 100 for percentage (and is independent of the dollar amount). Knowing that 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.

You can do this in your head. Examples:

100% gain is (1.00 + 1) = 2.00x times the initial value (1x initial + 1x gain).

200% gain is (2.00 + 1) = 3.00x times the initial value (1x initial + 2x gain).

18915.2% gain is (189.152 + 1) = 190.152x times the initial value.

Vice versa, to convert the times final accumulation to the Total Return %, then subtract 1 and multiply by 100. Subtracting the 1 is the 1x initial investment value (100%), leaving the gain.

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 add the gain to 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. The -1 is subtracting 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.

Note that a **price** result does not include dividends or compounding, but a formal **Total Return** does, 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 (2^{1/10} - 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% gain overall, but the actual 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 ‑64.776% overall gain (2/3 loss), and is ‑9.91% annualized gain (which is the actual average gain, and the same result as 10% loss each year). Compounding makes a big difference.

So forget about using the the Average for the gain, it's numerically meaningless in the compounded result. Average is just a sum divided by years, but compounded gain is a multiplied product which includes compounding. The gain of each year is 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 ‑40% then is less money working. Then 40% gain is only the 0.6x money x 1.4x gain which is only 84% of the original money. So it instead requires a gain of 66.7% to get back to even (0.6 x 1.667 = 1x. Calculator #4 below).

The average gain is only the **sum** of all years gain divided by the number of years, and which does not include dividends or compounding. The Average of the fund's annual gains is NOT a valid measure of stock performance. The total gain is the **Multiplied Product** of all the years gain. And the chronological order of the years does not matter in multiplication.

If you try to compute the incomplete year as a complete year, the result assumes the year finished at that rate, which is of course unknown and unlikely. If you try to call it a partial year, like maybe as 0.3 years, it will assume the rest of the 0.7 year continues earning at that same rate. Neither idea is remotely believable in the market. We do not know the future.

It's a kludge, but we're assuming this example is ten complete years. So to show the example in calculator 3 below, I added a final 0 (zero %) gain year (1x gain) after the last year, **because in most of my calculators, the final year is assumed the partial current year and so can't be included as Annualized**. The purpose of adding the last zero was to include the actual final year in this hypothetical calculation (the zero gain year is 1x gain and so does not change the result). However this assumes that the example's actual last year is a complete 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.

**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. And Annualized Return is the equivalent Fixed Rate producing the same result.

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.

Annualized Return math is just below, but first, we need the overall gain.

**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)

Total Return X$ =

(Ending value - Beginning value) + Divs

Beginning value

Beginning value

Total Return % =

Total Return X$

Amount Invested

× 100Amount Invested

**Annualized Rate**: The overall amount of gain is of course important, but the number of years is also a large factor of that number. Annualizing it to show the equivalent fixed annual rate of gain each year allows understandable 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:**

Percentage Gain =

New value - Old value

Old value

× 100 = % gainOld value

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} final Percentage Gain

Fixed 10% for 3 years is the standard formula: 1.10^{3} = **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** reverses the fixed rate formula in that annualized rate is 1 + Percentage Gain to exponent of 1/years.

**Total Return = (1 + (Percentage Gain / 100) ^{1/years}) Annualized** (× 100 for percentage)

((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 if each years market gain was in fact very variable.

((1.331)^{1/3} - 1) × 100 = 10.0% Annualized Return.

**But do NOT confuse Annualized rate with Average rate.** Average rate is not useful for a compounded variable rate result.
Years of 10%, -10%, 20%, -20%, 30%, -30% computes Not Zero gain, 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 a fixed percentage every year**. That 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 as $1 initial investment, first added, and then subtracted to leave just the gain. The percentage is the same for $1 or $1,000,000.

Or using Method B, (1.15 × 1.235 × 1.104 × 0.948 × 1.121 × 1.20 - 1) × 100 = **99.95351% gain in the six years**.

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 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)} computes 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 (0.918^{1/6} - 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 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 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.

Includes the Incomplete Current Year

Is NOT Acceptable Practice

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 last 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.

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 are shown in the calculators here). (Only 7 significant digits here, 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 the years of (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 long term performance rate of a stock. Maybe it didn't actually literally happen that way, but annualized gives precisely the same gain, and computes a more useful comparison with an "average" gain rate that works too.

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.

**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.

**Significant digits:** In that reverse computing use, the apparently exorbitant number of digits shown is **because large values like $1,000,000.01 have 9 significant digits, which needs at least that many significant digits in the interest rate to accurately match the same precision** (years is an exponent of interest). For example in calculator 2, the initial values shown (trying to reverse compute the calculator 1 default) if using instead 12.2% (3 significant digits) computes $999,342.31 which is not 9 significant digits for 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. 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 first, before any rounding). I've shown the Annualized Return with 10 significant digits, 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. Or perhaps sometimes 2x investment or 5 years are fully precise. The Morningstar annual Total Return percentage is shown with typically as few as 2 to 4 significant digits, but it is pretty close though.

**3rd calculator, Total Return from yearly gains, Two methods of data entry:**

- These data values
**are NOT the regular share prices, but are**(which includes dividends and compounding). It accepts Copy and Paste of that data**Total Return %**of each year's result**direct from Morningstar screens**(or from other web tables too, if Total Return %). At Morningstar, this Total Return % data for stocks is at their*Price vs Fair Value*tab, and for funds are at their*Performance*tab. Any empty early years marked with a — can be copied or not, ignored either way.The calculator's initial data shown is the Morningstar Nvidia NVDA stock, 2014 to 3/7/2024, which was a Record High price. Note that its 2022 year end close at -50.3% was Not the bottom low, which was Oct 14 at -61.8% from year end 2021. So the hot growth stocks may have the highest gains, but are also among the greatest risks.

- Or you can simply enter each year's Total Return % separated by a space. Enter each yearly gain percent age (including dividends), in format like as 15% instead of 1.15x. The calculator will convert it to the 1.15 format.
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 last year, but

**the final year entered is assumed to be Morningstar data, so the last 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 2024 and omitted. Its zero gain does not affect the final gain (0 is converted to a 1.00x gain multiplier of that year). But then faking a new last one as 0% gain allows computing annualized gain as if all the other years were complete years. But

**warning**, some of you will get the idea to add a final 0 just to see the real 2024 Annualized. 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). The future is not known, but a meaningless number is better omitted than to report garbage.The gain is a multiplied product, so the order of years does not change the final number, but here, the last year is assumed to be the current partial year and ignored. And you might better understand seeing the years in a correct order.

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*. 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 5 or 10 years ago. It does bypass the incomplete year issue, but it includes 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 Calculator 3 Details:** The Copy and Paste from Morningstar uses **tabs** for year separators (between each year value). And that tab works fine here, you can leave it just as copied, and directly Paste it. Actually, this tab thing should also work from any web screen HTML table showing years of **Total Return** percentage.

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. 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.

**Calculator 4, Gain to recover a down market:** Just to make the point that if a stock is down 20%, it takes 25% gain to recover the previous value. That is 20% of a larger value going down vs. 25% of a smaller value going up. And if down 50%, it takes a 100% gain to recover.

**Calculator 5, Tax Details:** The U.S. IRS 1099 forms I see show only the withdrawal amount, but the IRS does receive additional info from brokers. Since 2012, brokers do show Cost Basis on taxable accounts. 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, all share prices are averaged. This method can affect the Cost Basis number, but your broker does report it.

Dividends or IRA (anything not yet taxed) has Cost Basis of zero, meaning tax on all of it is owed at withdrawal. 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. Prior withdrawals reduce cost basis. Again, your brokerage should show current Cost Basis, and it affects the tax.

6. Calculate the Fixed Interest Rate

and actual Rule Number that will

compound into x value

and actual Rule Number that will

compound into x value

**An Enter key** in either of these

two fields recomputes this table

Extend range to years

Off-topic a little, 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 (1.12^{6} = 1.9738, almost 2x). Or the precise rate that exactly doubles in 6 years is (2^{1/6} - 1) × 100 = 12.2446%, which is the Annualized formula, and that first 2 is (1 + 1) to be the initial value 1x doubled to 2x.

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 the calculating was pretty difficult. Actually, 6 years should be Rule 73.4772 for 12.246205% doubling in 6 years (1.12246205^{6} = 2). The 12.246205% Annualized Return is an impressive rate of gain when compounded over many long term years, doubling every six years. The Rule of 72 is compounded annually.

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, like 1.5x or 3x or 10x multipliers, with corresponding approximate rules of 41, 116, or about 259).

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 72 is if 5 or less years. The most accurate case for Rule 72 doubling is for 9 years. But a Rule of 70 works better for doubling long term, 20 years or more. Still, as an approximation, 72 seems generally close enough. That error is 3.26% at 5 years, 0.31% at 10 years, and -2.04% at 20 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 over 6 years is 2.0x Value of course, but at 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.

Compounding is certainly a real 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 it sure seems a good bet if you consider "long term"). Two facts though, worse times have happened several times over the years before, and 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, not even counting the 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 be needing a source of income). Today is the latest 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, now (the term *Buy Low* means, if the market is currently low, to make buying right now be the best and most profitable time, very wise). The market always recovers, but lost years cannot be recovered.