Financial Mathematics Text

Thursday, October 3, 2013

Growth at 2% Per Year

So one thing that frequently occurs is that we assume growth will continue at a constant rate. I mention 2% because this is often seen as the long-term real growth rate of an economy. What I want to do is get some perspective on that.

Let's suppose we're living 6000 years ago, circa the 4th century BCE. If I started with 1 of something (it doesn't matter at this point what it is) 6000 years ago, and it grew at 2% per year, how much would I have today? The answer is simple:
1 \times 1.02^{6000} &\approx 4\times 10^{51}\\
&= 4,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
Mathematicians have a technical term for numbers this size: a lot!

So I think we need some perspective on how much that actually is. Let's consider some other really large numbers.

1) The number of grains of sand in the world.

So I found two sources estimating the number of grains of sand in the world. The first source is here and the estimate is  $7.5 \times 10^{18}$.

The second estimate is here and the estimate is $5.6 \times 10^{21}$.

Now even if you took the larger estimate, 1 grain of sand growing at 2% per year would still be $7.1\times10^{29}$ greater or . . .
. . . assuming I counted all of those 0's correctly.

2) The number of bacteria in the world.

Bacteria are a bit smaller than grains of sand and they're just about everywhere. So perhaps there are more bacteria in the world. And in fact, there are, at least according to this estimate. Apparently there are $5\times10^{30}$ bacteria in the world.

Unfortunately that's still quit significantly smaller than compounding 2% interest for 6000 years. To get this number closer we would have to put all of the bacteria in the world on one grain of sand and do that for each grain of sand. Then we'd be somewhere in the ballpark.

So the moral so far is that we need to think bigger.

3) The size of the universe.

While the size of the universe is not known, the size of the observable universe (per wikipedia) is 46 billion light-years.

Well, that's not a big number. So let's use a much smaller scale. If we use meters, the size of the universe becomes about $4.35 \times 10^{26}$. So that's a little bigger than the number of bacteria in the world. But we're still not there yet.

But can we use a smaller scale?

The smallest scale I'm aware of is the Planck length. So let's see how big the observable universe is in Planck lengths:
$$ 4.35 \times 10^{26}m \frac{1 \ell_{P}}{1.616\times 10^{-35}m} =2.69 \times 10^{61}\ell_{P}$$
So there you have it. We finally found a number big enough to capture 2% growth. We had to take the (known) universe and measure it using the smallest length scale that we currently have.

But what if we just increase that growth rate to 2.5%? Then we'd be back to the drawing board. Since 2.5% growth for 6000 years would, yet again, outpace the size of the universe:
$$1.025^{6000} \approx 2.2\times 10^{64}$$
Oh, the powers of compounding growth. But that should give some perspective here. Long term growth projections are not reasonable.


  1. I'm imagining we're talking about a DCF valuation here, and assuming a 2% growth rate for the terminal value. And of course you're right, the assumption itself is unrealistic. No business will grow cash flows at 2% forever. Very few businesses will even exist 100 years from now. But the actual effect on the valuation is not that concerning. If cash flows are $1 now, we assume a constant growth rate of 2%, and our discount rate is 8% then our PV of the year 100 cash flow is 32/100 of a cent. And what happens 200, 300 years from now is basically irrelevant to the PV we get. So while philosophically this is an important point, mathematically it doesn't present a real issue (assuming your discount rate is much higher than growth rate, at least)

  2. I think that's an excellent point on why we can get away with that assumption for DFC models. That's certainly one place that assumption comes in and the discount factor grows at a much faster rate so the, like you said, 300 years out and it's effect on value will be negligible.

    But I also had in mind, for example, real GDP growth which in recent years has been around 2% (and 4% during much stronger growth periods). Those assumptions I believe are unrealistic as well. It might work for a few more years but with resource constraints, I have hard time imagining the kind of growth we've seen in the past (mainly industrial era) will continue for much longer. I guess time will tell.


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