On this page is a compound annual growth rate calculator, also known as CAGR. It takes a final dollar amount as input, along with a time frame and starting amount. The tool automatically calculates the average return per year (or period) as a geometric mean.
The compound annual growth rate is a special label applied in the business world to the so-called Geometric Mean.
For us investors, it is the percentage which applied equally to every period would leave us with the final amount. Since investing almost always means volatility, with portfolios moving up and down based on value in the market, CAGR strips out that volatility to only concentrate on the starting and ending point. You ignore the path and only see what constant percentage would have left your investment in the current state.
Literally, a geometric mean is the central tendency of the product of a set of numbers, while a simple average is based on the sum of a set of numbers.
A simple average doesn't work in the investing case because it doesn't have the same concept of history as the CAGR; simple averages can't deal with volatility. It's best illustrated in a simple example:
You start with $1,000. In the first year you lose 50% of your money. In the second year, you gain 30%. For the third year, you gain 20%.
A simple average of the three gains would give you: -50% + 30% + 20% = 0% gain a year... implying you still have $1,000. This is wrong.
The real answer? You finished with $780, or a compound annual growth rate of -7.948% a year:
$1,000 * (1 - (-7.948)) = $920.52
$920.52 * (1 - (-7.948)) = $847.36
$847.36 * (1 - (-7.948)) = $780.01
As you can see, "-50%, +30%, +20%" and "-7.948%, -7.948%, -7.948%" are equivalent from the perspective of your investment.
(Of course, maybe not from the perspective of your stomach!)
As we said in the last section, the geometric mean is based on the product of a set of numbers, so the Geometric Mean formula looks like this:
Let's work through the example given in the last section:
( (.50) * (1.30) * (1.20) ) ^ (1/3) <- In this case, the '3' is the number of terms. We are doing three years here.
( .78 ) ^ (1/3) = .92052
Now, as we are describing a percentage, we can subtract '1' to convert it:
.92052 - 1 = -.07948 or -7.948%
The compound annual growth rate formula is essentially the same thing, just simplified to use for business and investing. We can use it to get the same result with only the starting and ending values along with the number of periods; we'll use years for consistency:
Let's walk through the same example again using this formula with a 3 year timeframe, a $1,000 starting point, and a $780 ending point:
( (780/1000) ^ (1 / (3 - 0) ) ) - 1 =
( .78 ^ (1/3) ) - 1 =
0.92051 - 1 = -.07948 or -7.948%
As you can see, either formula gets us to the same point. Depending on whether you know the final percentage change or the final total, you can pick the easier formula to get the answer.
We're partial to OpenOffice ourselves, but the idea is exactly the same: you will use the GEOMEAN function.
In Excel: =GEOMEAN(0.5,1.3,1.2)-1
In OpenOffice and similar: =GEOMEAN(0.5;1.3;1.2)-1
Alternatively, you can build up the CAGR formula like this:
= ( ( (Cell with Final Value) / (Cell with Starting Value) ) ^ ( 1 / Number Periods ) ) - 1
You can also use XIRR and IRR to get the CAGR as well. Use the inverse of the starting amount as the initial investment, and have a withdrawal of the final amount on the last day. These would be equivalent to our by-now infamous example:
(Bonus, ignore if you aren't advanced: Yes, the dates I picked were deliberate. IRR and XIRR will return slightly different results if the period chosen has a leap year!)
If you want to go in the other direction, the investment calculator models the future growth of an investment based on a return. Try all our finance calculators, or see other tools in our financial basics series:
Dollar cost averaging: an excellent strategy for the investor with money to deploy, but worried about the risk of going into the market all at once. With that in mind, this article will try to set your mind at ease and explain how dollar cost averaging might just be the psychological trick you need to get your money working for you in stocks!
Let's say you first entered the workforce in 1988. Let's also say that on every 15th of the month since you started working, you invested $500 a month into an S&P 500 Index Fund. You never sold any of your fund.
That's 308 purchases, equaling about 26 years in the workforce, making you right around 50 years old.
308 purchases, $500 at a time - that's $154,000 invested in the stock market.
Well, how much do you think your purchases would have been worth on February 7, 2014? $200,000? $300,000?
Try $545,504.97.
That behavior, of course, lent the name to this article. 'Dollar Cost Averaging' is the process of automating your investments by just sticking to your investment plans over a long period of time, and buying on a schedule. Often this can be once a month, or perhaps, once a paycheck.
"No, I can't make it. I need to time this $500 buy perfectly!"
Inspired by an interesting question on the Bogleheads forum, (which sent a fair amount of traffic to our S&P 500 reinvestment calculator) we set out to quickly answer poster sls239's question about market timing while investing monthly... and teaching an important lesson about Dollar Cost Averaging and Overanalysis. And... ulcers.
We, of course, tacked on a few assumptions - namely, this mystical S&P 500 mutual fund had no fees, no taxes owed, reinvested dividends for free, and our investors never sold. So, we ran three scenarios corresponding to the questions in the forum, using the S&P 500 Total Return Index:
Investor C must have destroyed the other two, right? Wrong again!
Here's how our heroes performed:
| Investor A | Invest B | Investor C | |
| (Steady) | (Bad Luck) | (Brilliant Timer) | |
| Ending Balance | $545,504.97 | $532,220.84 | $561,289.10 |
| Average Yearly Return | 8.745% | 8.595% | 8.923% |
Shockingly, the difference between perfectly timing the S&P 500 every single month through 26 years of a career and the bad luck version of that strategy was a mere $29,068.26 cents. Investor C's edge was a miniscule .33% over Investor B. Here's a graph showing Investor A's (our steady hero's) account value over time:
Why Bother Overoptimizing?
Seriously now, what's the point? While you expected to see results similar to the famous traffic jam scene in the beginning of Office Space, it turns out that when investing in a large diversified index fund over a long period of time, your timing barely matters at all.
As a matter of fact, Investor A comes out way on top in this scenario - without spending any time attempting to calculate fair values or using technical analysis to nail his entry points in his month to month investing, he merely took his paycheck and bought his $500 of S&P 500 in the middle of the month without fail. While his over-optimizing friends B and C wasted precious heartbeats on a pointless problem, Investor A showed us what comparative advantage means... and became B and C's boss.
Seriously... Just Invest... Early and Often!
Your takeaway? That's a simple one. If you're like the majority of Americans (or Canadians, or Russians, or <insert your country's people>), you get your paycheck from a steady job. When you get your paycheck, you should invest said paycheck in a broad mutual fund (perhaps even in the S&P 500? Your call!).
Worrying about perfectly timing every paycheck is going to eat into your time which can be better spent elsewhere - like, say, in climbing the career ladder or starting your own business. Suffering and stuttering over every purchasing decision is just going to give you an ulcer.
Seriously - your call. Do you want ulcers from straining your brain every trading day for 26 years in a row, or can you live with an iron stomach and without $15,784.13?
Yes, you're welcome - you can thank me for your good health in 26 years. And, yes, I'm glad to settle this question.
On this page you'll find a Kelly Criterion Bet Calculator. Enter your assumptions on
We automatically calculate your ideal bet size with the Kelly Criterion and your assumptions.
The Kelly Criterion is a formula to determine the proper size of a bet with known odds and a definite payout. With hand waving and basic math you can also use it to help guide your investment decisions.
It's most useful to determine the size of a position you should take.
The Kelly Criterion bet calculator above comes pre-filled with the simplest example: a game of coin flipping stacked in your favor.
Hit calculate, and see that you should definitely take the bet. Your optimal bet size is 25% of your bankroll.
(Now, find a casino stupid enough to offer those odds!)
Of course, you can see practical the practical value of Kelly betting when it comes to things with discrete results and obvious probabilities - say pot odds in a poker hand. Your mileage may vary.
What do you think about simple Kelly betting? Even though it is designed to never let you go bankrupt, Kelly still allows wild volatility swings.
Do you prefer another strategy? Perhaps half or quarter Kelly methods?
The Kelly Criterion is the brilliant summation of a betting strategy first discovered by Information Theorist John Kelly. Kelly came up with a betting system which optimizes bankroll growth based upon known odds and a definite payout. If you can find an exploitable, repeatable edge, Kelly's system tells the maximum you should bet based upon that criteria.
Here's a calculator which applies the concepts in this post to come up with an allocation:
Extending Kelly a bit further (like Ed Thorp, author of two math bibles for the investor/bettor Beat the Dealer and Beat the Market, has done) we can do a bit of hand-waving and make it work for the stock market.
Some derivations of "Stock Market Kelly" involve using back-looking numbers such beta to approximate the continuous returns of securities. We're going to do it in a discrete way, and use discrete numbers for wins and losses.
Let's say that you're investing with a 10 year time-frame – you want to buy a house or retire, for example. You have an extra $100,000 and are trying to determine the best allocation between stocks and treasury bonds.
Let's try to calculate is your 'edge' and your 'odds'.
It's true: garbage in, garbage out. All we can do is take an educated guess and hope that it is close enough to reality to guide our choices. (See: past performance is no guarantee of future results.)
As they say, history doesn't repeat itself but it often rhymes.
Odds: The S&P 500 beats 10 Year Treasuries roughly 85% of the time over rolling 10 year periods. We'll then enter .85 for our odds of stock out-performance.
Edge: Edge is tough, but for arguments sake, let's use 5%.
Historically 5% is a decent choice; sometimes authors will take average earnings yield and subtract Treasury yield. Change it as you desire.
'Normal' or 'Full' Kelly is
\frac{probability*(1+odds\ offered)-1}{odds\ offered}We need to modify the Kelly Criterion a bit to take into effect the fact that generally a security won't 'go to zero'. (Even a losing 'bet' almost always has some value in the stock market).
We simplify the equation to
\frac{expected\ value}{odds\ offered}Here's the math using the assumptions in the previous section:
\frac{expected\ value}{odds\ offered} = \\~\\
\frac{.85*.05*(5\%) - .15*.02*(-2\%)}{79\%} = \\~\\
\frac{0.0395}{5\%} = 79\%
So, in this theoretical portfolio with your historic estimate of odds and edge, aim for 79% stocks and 21% bonds. The standard disclaimer applies: these numbers are guesses, so adjust your expectations accordingly.
For traditional Kelly applications, also try the Kelly Calculator for bet sizing.