1/

Get a cup of coffee.

(Or, as the wise @JohnLegere says, get many cups of coffee!)

In this thread, I& #39;ll show you how to do CAGR calculations.

For those unfamiliar, CAGR = Compounded Annual Growth Rate. It& #39;s also called IRR (Internal Rate of Return). https://twitter.com/JohnLegere/status/1283210193694842886?s=20">https://twitter.com/JohnLeger...
2/

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& #39;s at $1,515.55.

Assuming you& #39;re still holding on to your 100 shares, what& #39;s your rate of return on this investment?
3/

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.
4/

So you just plug these numbers into the standard compound interest formula:
5/

And if you rearrange the terms in the formula, you get your CAGR.

In this case, your CAGR works out to about 16.43%.
6/

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.
7/

But you don& #39;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?
8/

Let& #39;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& #39;ll assume you& #39;re still holding today:
9/

As the table above shows, your actions result in some cash inflows and outflows over time.

Every time you buy stock, there& #39;s a cash outflow. And every time you sell, there& #39;s a cash inflow.

Here they are, marked on $GOOG& #39;s stock chart:
10/

You can create a "cash flow timeline" that neatly captures these inflows and outflows. Like so:
11/

Going back to the bank analogy: every cash *outflow* is like a *deposit* you make into a hypothetical bank account that& #39;s compounding your money.

And similarly, every cash *inflow* is like a withdrawal you make from the same bank account.
12/

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.
13/

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.
14/

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:
15/

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& #39;s not possible. We need more powerful techniques.
16/

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:
17/

There& #39;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?
18/

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%.
19/

OK. Let& #39;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& #39;t possibly be as high as 20%.

That means your CAGR lies somewhere between 10% and 20%.
20/

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.
21/

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& #39;re really close to the right answer.
22/

Here& #39;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.
23/

Dividends can complicate the picture a little. But don& #39;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.
24/

Here& #39;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.
25/

CAGR is a key metric we investors use to judge the performance of individual investments, portfolios, funds, and even fellow investors.

It& #39;s super important to know how to calculate CAGRs correctly. I hope this thread helps.

Thanks for reading. Enjoy your weekend!

/End
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