Category Archives for "Numbers"

Section 2.2 – The Dynamics of Exponential Growth

 This Nautilus shell is a fine example of exponential growth. As you recall from our discussion in Section 1.7 – The System of the World exponential growth is caused where we have a reinforcing feedback loop from the output of a process to its input. As the Nautilus grows larger it eats more; and the more it eats, the faster it grows.

‘Moore’s Law’, coined in 1965 by Gordon Moore the chairman of Intel, is a conjecture that the number of transistors in a single integrated circuit would double every two years. It has proved astonishingly accurate, and persisted over a far longer time-scale than most commentators have expected (including Moore himself). At the time he proposed it, digital chips contained a dozen or so transistors. By 1974 the Intel 8080 – the first successful 8-bit microprocessor – included about 8,000 transistors. Today’s latest Pentium processors contain well over a billion transistors.

This has been the driving force of the digital and communication revolution. In 1972 the computer that the Ford Motor Company used to run its entire European operation took up a large room. Its central processor was housed in a box the size of a filing cabinet. The 64 kilobytes of main memory were in another case the size of a wardrobe, and its two disc drives were each the size of a washing machine, with a capacity of 30 megabytes each. Modern smartphones fit in your pocket and have memory of 128 gigabytes – that’s two thousand times the size of Ford’s pair of disc drives; and two million times the size of their computer’s core memory!

It is the exponential growth caused by the process of compounding interest which makes it possible to acquire financial wealth within a reasonable time-scale. Suppose that when you were born, a generous grandparent invested £1000 for you. If it was in a fund that was receiving interest at the rate of 7%,  and the interest received was reinvested into the fund, how much do you think you would have at the age of 20, 40 and 60?…

The answer is that by the age of 20  you would have £4,000. At the age of 40 you would have £15,000. And at the age of 60 you would have £59,000.

Small differences in the rate of interest have a dramatic impact on the size of the fund that accumulates over time. For instance, suppose the interest had been 10% instead of 7% – what size do you think the fund would have grown to? In that case it would have reached £45,000 by the age of 40, and by the age of 60 it would be worth £304,000!

In the next section – The Difference between an Enterprise and a Ponzi Scheme – we will be examining how these kind of rates of growth of wealth can be accomplished legitimately and honestly.

It is compound interest that makes it possible – in principle at least – to build up enough over the course of a working lifetime to provide an income in retirement. Consider the example of a person earning £20,000 a year who would like to have the same income in retirement, and is prepared to save 10% of their income towards that goal. They would need £400,000 by the time they retire if the fund can return them an income of 5%. Saving £2,000 a year, it would take 200 years to amass that by simple accumulation. However if the annual savings are put into investments which yield 10% – and this income re-invested – they would reach that £400,000 target in just 39 years! Of course, as already mentioned, many retirees are making the painful discovery that their investments have not been making anywhere near that 10% return. That wouldn’t even have been an unrealistic expectation. Any reasonably well-run business ought to be able to make a profit of 10% on capital employed; and in fact stock market gains plus dividends paid have exceeded this performance over most of the last few decades. One reason for the poor performance of pension funds, as noted previously, is that much of the profits that ought to have accrued to the fundholders had been highjacked along the way and diverted into the pockets of salesmen, fund-managers, stockbrokers and bankers – not to mention into the grotesquely inflated remuneration of senior executives in large publicly-quoted corporations.

And there’s another reason too: since 1997 the UK government has had its nose in the trough. One of Gordon Brown’s first moves as Chancellor was to eliminate the tax relief on dividends paid into retirement accounts, in a breathtakingly dishonest and cynical move. On reaching office, Brown realised that he couldn’t balance the books without an increase in taxation. On the other hand, if he increased general tax levels he would alienate the voting population who had just brought the Labour government into power; and if he increased tax on the super-rich he would antagonise the billionaire media moguls whose support was desperately needed. So he scooped up a windfall from a move that would not have been noted much at the time, but which he must have known would have disastrous longer term consequences. With most pensions already under-provided, this could only make the situation far worse by the time these investments reached maturity. Public perception of the heist was minimised with the co-operation of the media, whose owners knew which side their own bread was buttered on.

Over longer time-scales the results are even more extraordinary. What would you imagine an investment would return at 3% compound interest over 350 years? Keynes gave an amusing real-life illustration, writing in 1930. He observed that when Francis Drake returned to England in 1580 and presented the gold he had looted from Spanish galleons to Queen Elizabeth, she used it to pay off the national debt and was left with a balance of £42,000. She used this to set up the Levant Company, the East India Company and the Hudson Bay Company. These formed the basis of Britain’s global trading empire, and Keynes suggested that they might typically return a 6.5% profit per year on capital employed and re-invest half of that in expanding the enterprises. That 3% cumulative growth over 350 years would magnify the capital by a factor of about 100,000 – to a figure of £4.2 billion – which was the approximate value of Britain’s overseas investment holdings at the time he was writing.

Growth in any reinforcing-feedback system follows the classic exponential curve: 

Although a common perception of the nature of exponential growth is that nothing much happens for a while, and then it suddenly takes off, this is actually an illusion. In reality, the curve is self-similar along its entire length. If you took any segment – whether in the ‘flat’ part or the ‘taking-off’ section, and re-scaled it to fit in the same frame, it would look identical.

This is clearly seen if we re-plot the graph with a logarithmic vertical axis – one that has a constant ratio between each marker rather than the familiar linear scale, with its constant increments. A logarithmic scale is like the shutter speed dial on a camera: each change of one stop doubles the amount of light that reaches the sensor. So when the speed is 1 millisecond, one more millisecond will cause a certain brightening of the image, but when it is 125 milliseconds (1/8 of a second), it needs to be extended by another 125 to get the same brightening effect.

Although we often have trouble envisioning logarithmic scales, in fact they are very common, especially in relation to our sensory perception. At a low volume setting on your hi-fi there might be one tenth of a watt of power being fed to the loudspeakers; turn it up slightly to get a couple of just-perceptible increases of level and it will be sending one watt, and for another couple the output will be running at ten watts. The same subjective increase again would require 100 watts.

It’s a defining feature of exponential growth that a certain period will cause the value to double, and then the same period will double it again, and so on. There is a simple relationship between the percentage rate of growth and the doubling period.

The doubling time can be calculated by dividing 70 divided by the rate of growth. For example:

  • With a return of 7% per annum, an investment would double in 10 years;
  • with a return of 10% it would double in 7 years;
  • and for 2% it would take 35 years to double in value.

Declines in value – for example the reduction in purchasing power of currency due to inflation – can be thought of as a negative rate of interest, and the same logic applies. So losses accumulate dramatically as time passes. Assuming that the central banks achieve their “target rate of inflation” of 2%, that would mean that if you put away £1000 at the age of twenty-five, it would be worth only £500 by the time you are sixty. More detail on this in a later section on inflation within the Finance thread.

Exponential growth is a mathematical abstraction, but projection beyond a certain point always leads to an unsustainable position whenever we are talking about physical quantities in the real world.

To return to the example of breeding rabbits from the discussion in Section 1.7, each breeding pair typically produces six litters of six pups in a year, and rabbits become fertile after about six months (the exact numbers depend on the breed and the circumstances). So one pair would produce about 36 offspring in a year, plus the first litter will have added another 12 grandchildren and the second litter another 6, making 54 in all. Let’s ignore that last six to allow for the fact that older rabbits will be dying off, and call it 48.

After two years there would be 48 squared, or 2304 rabbits.

After 3 years it would be 48 cubed, or 110,592.

At this rate, in just eight years the rabbit population would grow to 1,352 trillion – about one rabbit for each square foot if habitable land area on the earth! Clearly other factors would have had to come into play before then.

A few more doublings of the density of transistors on a silicon chip and we will be down to a single atom between one element and the next. Some time before that, Moore’s Law will necessarily have run out of steam.

Investors who were lucky enough to buy Apple stock at the time of Steve Jobs’ return in 1997 and hang on to it will have seen the value of their holding grow by a factor of 400 in twenty years – an astonishing 34% average compound rate of growth (even including the several dramatic setbacks). Expectations of future growth are factored into the current share price. However another nine years of 34% growth rate would take Apple’s valuation to a level that exceeded the entire GDP of the United States. Another four years after that and it’s valuation would surpass the GDP of the whole world!

Section 1.3 – Big Numbers

Section 1.3 – How big is a million – a billion – a trillion?

“We used to call these astronomical numbers, but now the national debt is bigger than that! Maybe we should call them economical numbers!” – Richard Feinman

All the time we are given statistics on the news and elsewhere mentioning sums of money or other figures measured in millions, billions and trillions.  Do you just allow these numbers to wash over you without really comprehending what is being communicated? Do you just assume that somebody somewhere can make sense of this?

I’m going to give you a handle on visualising these sorts of magnitudes.

First of all let’s take a volume the size of a small sugar cube, 1 cm on each side: how big would 1 million of those be?
… It would be a cubic metre – the size of the jumbo bag of gravel.

What about a billion of them? …This would take up the volume of an Olympic swimming pool, 50 metres long. 20 metres wide, and one metre deep.

What about a trillion? They would fill a reservoir – half a kilometre long, 200 metres wide and 10 metres deep.

Let’s think about $1 million in money: suppose it was in $100 bills. You could fit $1 million in $100 bills into a briefcase.>

How about  $1 billion? To carry $1 billion worth of hundred dollar bills, it would take a 20 foot shipping container, stuffed completely full.

And for $1 trillion? You would need a freight train 6 km long, with all the boxcars stacked tight with hundred dollar bills.

What could you buy with $1 million? Well, you could buy a small flat in central London, about fifteen terraced houses in Rotherham, or three fairly high end Ferraris.

With $1 billion you could buy enough Ferraris to sit together nose to tail in a traffic jam on the motorway three lanes wide, and 4 km long.

With $1 trillion you could buy enough Ferraris to sit in a traffic jam three lanes wide all the way from Miami to San Francisco!

How long does it take to earn $1 million? For a typical worker earning $25,000 a year it would take an entire working lifetime of 40 years. For the chief executive of an average Fortune 500 corporation it would take five weeks. For football player Carlos Tevez it would take about nine days. For the world’s top financiers, about 6 hours.

How long would it take to earn $1 billion? Well, for that typical worker it would take it would take 40,000 years. The top 20 sports stars in the world together earn about $1 billion between them in a year. Hedge fund boss James Simon earns $1 billion every eight months.

Examples in the millions:

  • $1m –  cost of arming three Reaper drones, with two bombs and two Hellfire missiles on each one.
  • $1.5 – cost of one cruise missile.
  • $20m – cost to upgrade one nuclear warhead. The United States is currently planning to upgrade its entire stock of 3000 of them.
  • $85m – cost of one intercontinental ballistic missile.
  • 1m people – the number of lives that could be saved each year if clean water and sanitation were available to them.

Some things in the billions:

  • $8Bn – the cost of a nuclear submarine.
  • $10Bn – the amount Europeans spend annually on ice cream.
  • $20Bn – the United States spends currently each year on nuclear weapons.
  • $20Bn – the amount that American citizens spend annually on pet food.
  • $100Bn – the amount Europeans spend annually on alcoholic beverages.
  • $10Bn – the cost of providing clean water and sanitation everywhere on the planet

Examples in the trillions:

  • $8 trillion – the cost of American military expenditure during the Cold War. Presumably a similar amount was also spent by the Soviet Union.
  • $5.5 trillion – amount the United States spent developing and building nuclear weapons between 1940 and 1996.
  • $18.6 trillion – annual GDP of the United States.
  • $14.6 trillion – the net assets owned by the 50 richest individuals in the world.
  • $??? – the total assets owned by the poorest 4 billion inhabitants of the world are difficult to estimate, but are almost certainly less than $14 trillion.
  • $76 trillion – the total value of goods and services traded in the whole world in 2014.
  • $1200 trillion – the total value of financial derivatives held by banks and other financial institutions.

This is Section 1.3 of my forthcoming book The World in 2100: What might be Possible for Humanity? It is the opening segment of the ‘Patterns and Numbers’ theme. When we return to this thread, the next topic will be The Dynamics of Exponential Growth.

If you haven’t already done so, you can register to receive a free review copy just before it goes on general sale later this summer. Registering will also take you straight to Chapter 1 – The Foundations which will give you more idea of what the book will cover.