Imagine that you bought 100 shares of Google ($GOOG) on Dec 31, 2015 (about 4.5 years ago).
At that time, $GOOG was trading at $758.88 per share. Today, it's at $1,515.55.
Assuming you're still holding on to your 100 shares, what's your rate of return on this investment?
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Well, your initial investment was $75,888 (100 shares times $758.88 per share).
That $75,888 has now grown to $151,555 (100 shares times $1,515.55 per share).
This growth has happened between 2015-Dec-31 and 2020-Jul-17 -- a period of 1,660 days.
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So you just plug these numbers into the standard compound interest formula:
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And if you rearrange the terms in the formula, you get your CAGR.
In this case, your CAGR works out to about 16.43%.
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Essentially, what this 16.43% CAGR means is:
Suppose you found a bank that paid 16.43% interest annually.
And suppose you deposited $75,888 into that bank on 2015-Dec-31.
Then, your money would have grown to $151,555 by 2020-Jul-17 -- same as your $GOOG investment.
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But you don't usually just buy a stock once and hold on forever.
You add to your position over time. Sometimes, you trim your position.
How do you calculate your CAGR in such cases?
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Let's take an example.
Say you bought 100 shares of $GOOG on 2015-Dec-31.
Then, on 2017-Apr-25, say you bought 50 more shares.
Then, on 2018-Jul-27, say you sold 70 shares.
This left you with 100 + 50 - 70 = 80 shares, which we'll assume you're still holding today:
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As the table above shows, your actions result in some cash inflows and outflows over time.
Every time you buy stock, there's a cash outflow. And every time you sell, there's a cash inflow.
Here they are, marked on $GOOG's stock chart:
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You can create a "cash flow timeline" that neatly captures these inflows and outflows. Like so:
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Going back to the bank analogy: every cash *outflow* is like a *deposit* you make into a hypothetical bank account that's compounding your money.
And similarly, every cash *inflow* is like a withdrawal you make from the same bank account.
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Imagine what would happen if you actually made these deposits and withdrawals.
Each deposit will give you a "benefit" that compounds over time, and each withdrawal will negate some of the benefit created by the deposits.
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For example, buying 100 shares of $GOOG on 2015-Dec-31 gives you the benefit of $75,888 compounding for 1,660 days.
And selling 70 shares on 2018-Jul-27 erases the benefit of $86,695 compounding for 721 days.
This all follows from the "cash flow timeline" above.
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These compounding benefits and negations all ultimately add up to your end balance ($121,244 in this case).
To capture this, we write a "CAGR equation".
The left side (LHS) of the equation takes care of all the compounding. The right side (RHS) is your end balance.
Pic:
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All that remains is to solve this equation to find R (our CAGR).
But this equation is *not* like the earlier equation we solved.
In the earlier equation, we could just rearrange the terms to solve for R.
But here, that's not possible. We need more powerful techniques.
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Graphing the LHS and the RHS is a good way to solve this equation.
You take R on the X-axis.
And on the Y-axis, you graph both the LHS and the RHS.
Wherever the LHS and RHS meet, that R is your CAGR.
Like so:
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There's also another way to solve the CAGR equation.
You start with a "guess" for R.
Say, R = 10%.
If R was really 10%, what would your end balance (on 2020-Jul-17) be after the compounding effect of all the deposits and withdrawals?
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Easy. This is just your LHS evaluated at R = 10%. In this case, it works out to $32,808.
This is *lower* than your real end balance: $121,244 (the RHS).
So your CAGR has got to be better than 10%.
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OK. Let's try 20% then.
Now, the LHS works out to $128,209.67.
This is *higher* than your end balance of $121,244.
So your CAGR can't possibly be as high as 20%.
That means your CAGR lies somewhere between 10% and 20%.
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OK. What about 15%?
Turns out the LHS for R = 15% is only about $97,534 -- also *lower* than your end balance.
So your CAGR has to be higher than 15% but lower than 20%.
Carrying on this way, you can narrow down your CAGR into a very tight range.
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This is called "the bisection method" of solving the CAGR equation.
You start with a CAGR guess. At the beginning, this guess may be wide off the mark. But you keep iterating and improving upon it. Pretty soon, you're really close to the right answer.
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Here's the bisection method in action for this particular example. As you can see, bisection finds the same answer as the graphical method (CAGR = 18.94%), but with much higher precision.
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Dividends can complicate the picture a little. But don't worry: the principle is the same.
Just treat each dividend as a cash inflow on its pay date. Add these inflows to the cash flow timeline. Then use the same procedure (graphical or bisection) to calculate CAGR.
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Here's a picture summarizing the 3 steps of a CAGR calculation:
1) Prepare a cash flow timeline, 2) Write down the CAGR equation, and 3) Solve this equation either graphically or via bisection.
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CAGR is a key metric we investors use to judge the performance of individual investments, portfolios, funds, and even fellow investors.
It's super important to know how to calculate CAGRs correctly. I hope this thread helps.
Thanks for reading. Enjoy your weekend!
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