Rates Of Return

From time to time, the subject of how to calculate a rate of return comes up. There are two kinds of return we're usually interested in from a Loot Lizards point of view:

how much the investment has grown/shrunk overall (the net gain or rate of return)
how much the investment has grown/shrunk per year while we've owned it (the compounded rate of return).

How can we tell how much the stock price has grown?

Example 1:

Let's say we have a stock that we paid $50.00 a share for. It's now worth $60.00 a share.

What we're really looking for here is "what number do I need to multiply my original investment by to get its current value?" Let's call that number r for the time being.

ALGEBRA FLASHBACK
50.00 * r = $60.00
r = $60.00 / $50.00 = 1.2

So now we know that if we multiply $50.00 by 1.2, we'll get the current stock price.

But 1.2 doesn't really sound like anything we've ever heard before as a rate of return, and it's not the standard way of expressing these things. Normally, these are expressed as a percentage. At least they are when we go to the bank, at any rate.

All percent means is "by 100" and that's exactly what we do with r - multiply it by 100 to get 120%. So if we started with $50.00, we've now got 120% of what we started with, or $60.00.

That 120% number tells us how much the investment as a whole is worth relative to what we started with, but doesn't tell us what amount of the actual growth was. To get that, we need to subtract 100%, since 100% was the investment we started with. The distinction we're making here is how much the stock is worth overall ($60.00, or 120% of what we started with) versus how much we'd come out ahead if we sold it ($10.00 more, or 20% of the original investment).

So the actual growth in the stock's price is 20%.

Example 2:

Let's say that the stock's price had gone from $50.00 to $72.00.

50.00 * r = $72.00
r = $72.00 / $50.00 = 1.44
144 % of what we started with, or 44% rate of growth

Example 3

Let's try an example going the other way, where we started with a stock that was $60.00 and it is now worth $50.00. What would its rate of loss be?

60.00 * r = $50.00
r = $50.00 / $60.00 = 0.83

So we've got 83 % left of what we started with, and if we subtract the 100 % we started with, that leaves us with a -17% gain or a 17% loss.

I'm still confused about this percent thing.

All of the above deals with net gain, which means the amount that something has changed, no matter how long we've owned it. But as canny investors, we recognize that an investment that grows 5% in 3 months is a very different thing from an investment that grows 5% in 5 years. To measure this, we use something called a compounded rate of return.

So what about this compounded return thing?

A compounded rate of return takes into account another factor, time, as in how long we've held onto whatever investment we hold. By convention, a compounded rate of return is expressed for a year, but this is only a convention. You can express a rate of return in whatever timeframe you want to, and if you're looking to invest somewhere, it's a good plan to be sure that the historical rates of return have the word annual or annualized in them somewhere.

At some level, all investors realize that exponential growth is a good thing. The technical term used by the Loot Lizards to describe this is "a slope you can ski off of." The phrase itself even sounds cool. But at some level, it's a rude awakening to discover that measuring exponential growth is going to require, well, exponents.

To calculate a compounded rate of return, you need to have another little mathematical flashback. We're returning to seventh grade, and the voice of Elaine Peterson is saying something about rates of growth, logarithms and exponents. And the seventh grade mind is saying "Why do we need to learn this? It's not good for anything. I'm never going to need this." Unfortunately, the seventh grade mind was wrong.

When we talk about measuring growth over time, what we're looking for is solving for an annual rate of growth, which we'll call R.

Example 4

As an example, let's continue with our $50.00 stock. After 1 year, it's worth $60.00. Our net gain was 20%. Since we've only held the stock for a year, our net gain is the same as the compounded gain. After 2 years, it's worth $72.00 and our net gain was 44%.

Year	1996	1997	1998
Price	$50.00	$60.00	$72.00
Net Gain	n/a	20%	44%

But if you calculate the net gain for 1997 to 1998, it's 20%. (We did the calculation in Example 2.) So what we're really looking at here is a stock that had two years of 20% growth, for a net gain of 44%. So our compounded annaul rate of gain is going to be 20% a year, even though our net gain is 44% after two years. What that tells us is that if the stock continues to perform the way it has, we can expect it to continue increasing at 20% a year. The reason that we're splitting hairs this way has to do with wanting to predict where that stock price is going to be in three years (or in ten years, or whatever) based on what we've seen so far.

(($50.00 * 1.2) * 1.2 ) = $72.00
($50.00) (R) * (R) = ($50.00) * (R)² = $72.00

Fortunately, we already know that R is 1.2, because otherwise, what we'd be doing right now would be solving the equation

(R)² = $72.00 / $50.00

which would involve doing a square root, another concept that didn't seem especially useful back in seventh grade.

Since we already know that R looks to be 20%, we could make a projection for what the stock price would be in 1999, after three years by calculating

($50.00) * (R)³ = $86.40

If we wanted to generate a projection to the year 2000, then we'd take R to the fourth power (4 years since our initial investment).a

($50.00) * (R)⁴ = $103.68

So if our investment has gained at 20% a year for four years, why didn't it gain only 80% at the end of four years? How did it manage to have a net gain of 107%? Where'd the extra 27% come from? The answer is that each year's gain after the first one was on more than $50.00. For the second year, we were computing a 20% gain of $60.00. For the third year we were taking a 20% gain of $72.00, and so on and so on. This is what exponents live for.

Example 5

I paid $40.00 a share for a stock. I held it for 5 years. After 5 years, it was worth $60.00. That's a 50% net gain. That sounds pretty good. What was the annual compounded rate of gain?

We want to solve the following

($40.00) * (R)⁵ = $60.00
(R)⁵ = $60.00/$40.00
(R)⁵ = 1.5

So to find R, we're going to have to find the fifth root of $60.00 / $40.00 = 1.5, another math concept that seemed monumentally useless when it was presented in class back when dinosaurs roamed the earth. But this is not that hard a problem when you get right down to it, not if you use logarithms.

Mrs. Peterson pounded the following thing into my impressionable young mind (and I imagine someone very like her tried to pound it into yours too):

log (x ^y) = y log (x)

(Like much of what she tried to impart about logarithms, it looked monumentally useless at the time.)

And if we were to take the logarithm of both sides of that equation, what we end up with doesn't look nearly as scary as what we started with, aside from that log word.

(R)⁵ = 1.5

log ((R)⁵⁾ = log (1.5)
5 log (R) = log (1.5)
log (R) = log (1.5) / 5

And when you get right down to it, all log says is "go look it up in a table somewhere or punch the log button on your calculator." The best part is that it doesn't matter if you use a natural log or a common log, as long as you use it consistently. I'm using common logs here (base 10) because I actually managed to find a set of tables online. So if we go and look up log (1.5) in a table somewhere, we find that it's 0.1760913. And I'll take a division operation over calculating that fifth root any day of the week.

log (R) = log (1.5) / 5
= 0.1760913 / 5
= 0.0352182

So now we need to get rid of that log word, and that's what exponents do best. And since all exp (log) really means is "go look it up in the same table going the other direction," when we do that, we get

R = exp (log (R))
= exp (0.0352182) (we actually used 0.03502928 since it was closest)
= 1.084

Well all of this looks a little too good to be true and since nobody's dusted off that log stuff in about 20 years, it'd be good to try and multiply it out and see what happens:

40.00 * 1.084 * 1.084 * 1.084 * 1.084 * 1.084 = 59.76

Which ain't that bad when you're using a bookful of tables and a piece of paper. (The missing 24 cents is what we call "rounding error." If we'd used 1.085 as our multiplier for rate of growth, we'd've gotten $60.80 as the result. The truth, as always, lies somewhere in between.)

However, we're not quite done yet. We still have to do that multiply by 100, then subtract the initial investment thing, which gives us a compounded annual growth rate of 8.5%.