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Underestimating compounding

We all underestimate the power of compound growth – which leads investors astray in many ways.

One common criticism of the government is that it delayed the lockdown for too long because it failed to appreciate just how quickly the coronavirus would spread – a daily R0 of two means that one case becomes more than 16,000 within a fortnight. One scientific adviser says the Sage group “were not sure the politicians understood its exponential spread”.

This accusation is plausible because it is a common error to underestimate the power of compound growth. The late Albert Allen Bartlett, a physics professor at Columbia University, said: “The greatest shortcoming of the human race is our inability to understand the exponential function.” He was exaggerating: there’s plenty of competition for the title of the greatest shortcoming of the human race. But he had a point. And it is a shortcoming that can cost investors dearly.

For one thing, underestimating the power of compounding causes us to underestimate the importance of reinvesting dividends. Since December 1999 the All-Share index has risen a mere 1 per cent. But if you had reinvested your dividends from it, you would have doubled your money. Even low exponents multiply a lot over time.

The same mistake leads us to underrate the importance of saving from an early age. If you save £1,000 a year at a real return of 3 per cent a year you’ll have just under £37,000 after 25 years. But after 30 you’ll have over £47,000. Saving for an extra five years thus gives you a 100 per cent return on your additional £5,000 of saving.

Of course, younger readers will object that they can’t save much due to low incomes and high rents. True, but this just shows how much long-term damage these do.

Even when we start saving and investing, however, failing to appreciate compound growth can cost us a fortune because we don’t see just how much fund managers’ fees grow over time. An extra percentage point in charges per year doesn’t sound much. But over 20 years it could easily cost you £4,000 on a £10,000 fund holding.

When we take charge of our own stockpicking, though, we run into other perils by underestimating compounding.

To see one, ask why it is that so many big investments such as HS2 so often go over time and budget.

Imagine a project with 50 separate tasks, each with a 95 per cent chance of being completed on time. The odds seem good, don’t they?

No. There’s only a one-in-12 chance of them all being done. And if delay on one means delaying the whole project, this means it’s overwhelmingly likely it will over-run. Not understanding this is an error with its own name – the planning fallacy.

This fallacy afflicts our attitudes to growth stocks. For a small company to get big, a lot must go right. It must avoid cannibalising its existing sales; it must control costs as it grows; it must exclude rivals from its market; it must get marketing and pricing right; and so on. Even if the company has a decent chance of achieving each of these, it has a low chance of achieving all of them.

It’s for this reason, says the University of Liverpool’s Charlie Cai, that investors have in the past paid too much for growth stocks – they’ve underestimated how much growth will depress asset turnover.

In the same spirit, New York University’s Aswath Damodaran and UCLA’s Bradford Cornell have described what they call the “big market delusion”. Investors, they show, pay too much for companies with potentially big markets because they overestimate the chances of parlaying potential sales into actual profits.

Smaller growth stocks – the sort listed on Aim – have underperformed for decades because of this variation on the planning fallacy, the failure to see that lots of high probabilities compound to give us a low one. It’s for this same reason that book makers love taking accumulator bets.

There’s another way in which stockpickers can neglect compounding. To see it, think about a company you know well. You should have a good idea of its management’s strengths and weaknesses, how much competition it faces, the risks to its business model and so on. But what will these look like in 10 or 20 years’ time? If you’re honest, you don’t have a clue. Uncertainty compounds over time.

Worse still, so too does what Joseph Schumpeter called creative destruction. Only 23 of the original members of the FTSE 100 when it was launched in 1984 are still in the index. Granted, some left because they were taken over at a decent price, but others just declined such as Thorn EMI, MFI and Hanson. In the long run, then, even big blue-chips are brought low.

Which means you cannot be a genuine long-term, buy-and-hold stockpicker. If you try, you’ll end up with a portfolio like that of Montgomery Burns in the The Simpsons, holding stocks such as Transatlantic Zeppelin, Confederated Slaveholdings and the Baltimore Opera Hat Company.

A failure rate of only 1 or 2 per cent per year compounds hugely over an investment career, and so creative destruction grinds finely over the long run. For me, this is a case for long-run investors to hold not individual stocks but rather tracker funds and private equity – the latter because we cannot assume that the beneficiaries of creative destruction will be companies that are listed today.

All I’m saying here is a variant on a point made by the US futurist Ray Amara in the 1960s: “We overestimate the impact of technology in the short term and underestimate the effect in the long run.”

Our short term overestimation happens because of the planning fallacy: we underestimate how multiple low probabilities of failure compound to give us a high probability.

Over the long term, however, we underestimate upside compounding. Although any individual company has a big chance in the short term of failing to exploit its opportunities, in the long run somebody will succeed – in the same way that if you buy enough lottery tickets you’ll win a prize. Amazon made internet shopping pay in a way that hundreds of firms in the late 90s did not.

“Eventually”, however, can be a long time. In their book, The Second Machine Age Erik Brynjolfsson and Andrew McAfee describe how it took 30 years for electricity to raise manufacturing productivity because it took companies that long to reorganise factories to improve workflow: electricity facilitated the assembly line whereas steam power did not. Once a few companies had learned that trick, however, most others followed.

Which highlights another exponential process we underestimate – the fact that knowledge sometimes compounds over time. If you’re learning a musical instrument or foreign language and improve by just 1 per cent a week the improvement will seem imperceptible from day to day. But over a year you’ll improve by two-thirds and over five you’ll become 13 times better. What the cycling coach Sir Dave Brailsford called the aggregation of marginal gains is a powerful force.

And it’s one we don’t often harness. One reason we don’t do so as investors is that we focus on expiring information – facts that go out of date and don’t lead anywhere. For years after the EU referendum in June 2016 many of us obsessed about what type of Brexit we’d get and when. Most of that effort was wasted – it made us no money at the time, and I doubt it has taught us much of value for the future.

Equally, investors can waste time gathering company-specific short-lived information that doesn’t predict returns. What matters instead is general purpose knowledge – and correct mental habits – that can help us make better decisions in future, and which we can build upon.

It’s not just individual investors who miss out on compounding knowledge. So too do companies. If knowhow compounded over time we’d expect to see productivity grow steadily. But in fact it has stagnated in the past 10 years. A big reason for this was pointed out by Jonathan Haskel (now a member of the monetary policy committee) and colleagues. They showed that a huge chunk of productivity growth comes not from companies upping their game but from inefficient firms closing and better ones replacing them. That tells us that many individual firms don’t compound their knowledge – which might be one reason why so many fail over time.

Professor Bartlett, then, was right. We often fail to appreciate exponential processes, both their dangers and benefits.

Why? To see one reason, give yourself a few seconds to estimate the answer to this equation: 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8. The answer is at the bottom of the page*.

Most people give a big underestimate. This is because they calculate 1 x 2 x 3 x 4 in their heads and get a small answer and adjust it upwards. But they don’t adjust it enough. They are misled by the anchoring heuristic: their answers are anchored by 1 x 2 x 3 x 4. They thus fail to see how the equation explodes upwards.

I fear, though, that there might be another reason. Conventional economics underemphasises the possibilities of explosive growth. Of course, gross domestic product (GDP) growth and equity returns (if you reinvest dividends!) compound over time. But the dominant paradigm in economics is one of stability, in which negative feedback dampens down “shocks” and return us to equilibrium – when, for example, small falls in share prices attract value investors who bid prices back up.

In this world, we don’t need to understand explosive compounding, because it doesn’t happen.

In the real world, however, it does sometimes. We do occasionally see explosive growth in asset prices – for example in Bitcoin or in tech stocks in the late 90s – as confidence spreads. Yale University’s Robert Shiller has shown that some economic narratives spread in the same way that viruses do. And – as we saw in March – we also see implosions in prices as initial falls lead to more selling by risk-parity traders or leveraged investors. In underestimating such positive feedback processes which amplify initial disturbances, conventional economics distracts us from the need to grasp exponential processes.

Of course, it only takes a few seconds with a calculator to teach us how even low growth compounds quickly. But this is not enough. Bill Shankly once told a young player: “the trouble with you son is that your brains are all in your head.” He had a point. Brain knowledge of compounding is not enough. We need a visceral intuitive grasp of how numbers work – which is something many of us lack

*It’s 40320.