Lesson in Valuation from a 2200-Year-Old Greek Mathematician

By the fifth century BC, the Greeks had firmly established that the earth was a sphere. Although they knew it was a sphere, they did not know how big the sphere was.

The philosopher Plato (400 BC) declared the earth’s circumference to be 64,412 km. Some 150 years later, the mathematician Archimedes estimated it to be 48,309 km.

It is not known exactly how Plato or Archimedes arrived at their calculations, but Plato’s measurement was off by sixty percent and Archimedes’ by twenty percent. At least they were making progress.

Then, in around the third century BC, which is more than 2,200 years before modern scientists calculated it, the Greek mathematician Eratosthenes used just a tower and simple math to arrive at Earth’s circumference of about 40,000 km (less than one percent off the mark from the exact circumference of 40,075 km).

Eratosthenes lived in the city of Alexandria in northern Egypt. He knew that on a certain day each year, the Summer Solstice (21st June), in the town of Syene in southern Egypt, there was no shadow at the bottom of a well.

He realized that this meant the Sun was directly overhead in Syene at noon on that day each year.

Eratosthenes knew that the Sun was never directly overhead, even on the Summer Solstice, in his home city of Alexandria, which is further north than Syene.

The only possible reason he saw was that the surface of the Earth is curved. Not only that: the greater the curvature, the greater the difference in the shadow length. He also assumed that the Sun was so far away that its rays were essentially parallel when they reached the Earth.

Eratosthenes realized that he could determine how far away from directly overhead the Sun was in Alexandria by measuring the angle formed by a shadow from a vertical object.

He measured the length of the shadow of a tall tower in Alexandria and used simple geometry to calculate the angle between the shadow and the vertical tower. This angle turned out to be 7.2 degrees.

Next, Eratosthenes used a bit more geometry to reason that the shadow’s angle would be the same as the angle between Alexandria and Syene as measured from the Earth’s centre.

Now, 7.2 degrees is 1/50th of a full circle (50 x 7.2° = 360°). Eratosthenes understood that if he could determine the distance between Alexandria and Syene, he would merely have to multiply that distance by 50 to find the circumference of Earth.

He had the distance between the two cities measured (he probably hired walkers who were trained to measure distances by taking regular strides,), which came to around 5,000 stadia, or about 800 km.

Next, he multiplied this distance by 50 to get the earth’s circumference as about 40,000 km, which gives his estimate less than a one percent error. What an achievement that was!

Watch the great American astronomer and cosmologist Carl Sagan explain the genius of Eratosthenes –

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What is my purpose of telling you about a man in ancient Greece and how he arrived at Earth’s circumference 2,200 years ago using just a tower and two numbers – 7.2 degrees and 800 km?

Consider how analysts and investors arrive at target prices and intrinsic values of stocks, and you would know where I am coming from.

People use alpha, beta, CAPM, two, three, and five-stage models running into hundreds of rows of an excel sheet to arrive at business valuations that mostly turn out to be far from reality.

You tell them that the best valuations are done back-of-the-envelope – like Eratosthenes may have done back-of-the-stone – they look at you with suspicion.

I feel shivers run down my spine whenever I use even my simplistic 2-3 variable valuation models. And now, the way I know Eratosthenes arrived at Earth’s circumference 2,200 years ago, puts even my simple models to shame.

Ben Graham, who gave us the simple valuation formula –

Intrinsic value = Earnings per share × [(8.5 + (2 × Expected annual growth rate, g)]

– and who later decried it, wrote this in The Intelligent Investor –

There is a special paradox in the relationship between mathematics and investment attitudes on common stocks.

Mathematics is ordinarily considered as producing precise and dependable results; but in the stock market the more elaborate and abstruse the mathematics the more uncertain and speculative are the conclusions we draw therefrom.

I would like to offer this advice from Graham, and Eratosthenes’s method of calculating Earth’s circumference, to anyone wanting to learn about stock valuations.

One is that you must give importance to a stock’s valuations only after you have answered in ‘Yes’ to these two questions – (1) Is this business simple to be understood? and (2) Can I understand this business?

And two, you must not try to quantify everything. In stock research, the less non-mathematical you are, the more simple, sensible, and useful will be your analysis and results.

Good analysis and valuation, after all, is generally back-of-the-envelope.

All you need is an observing eye, some brain, a pencil, and a used envelope.

Thank you, Eratosthenes! You have been a great teacher in mathematical simplicity.

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A Few Stories You Shouldn’t Miss

• The Hidden World of Failure (Barry Ritholtz)
• FOMO is every investor’s worst enemy. Here’s how to fight it (Market Watch)
• Are We Trading Our Happiness for Modern Comforts? (The Atlantic)
• Lots of Overnight Tragedies, No Overnight Miracles (Morgan Housel)
• The Sense of an Ending (Arun Shourie)
• Reflections on 100 Baggers (Chris Mayer)
• The Best Investing Books for Every Kind of Investor (Nick Maggiulli)

Meditations

Two things are infinite: the universe and human stupidity; and I’m not sure about the universe. ~ Albert Einstein

Be the change that you wish to see in the world. ~ Mahatma Gandhi

A Question for You
Life isn’t just about figuring out what to do. The real challenge is (not so) simply doing the things we know we should be doing.

Ask yourself – What things I am not doing that I know I should be doing? Why?

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That’s about it from me for today.

Have a great weekend.

Stay safe.

With respect,
— Vishal