COVID-19, financial decisions & those tricky exponential numbers

This is a guest post by David Steele, Policy and Research Manager at The Money Charity (currently furloughed).
He was previously Policy Manager Financial Services for Age UK and Head of the Research and Statistics Unit at the UK Film Council.
This blog is written in David’s personal capacity and does not necessarily represent the views of his current employer.

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With the Sunday Times feature Coronavirus: 38 days when Britain sleepwalked into disaster and other similar evidence, it is now widely accepted that the UK missed critical weeks, from late January to mid-March 2020, in our preparations for the COVID-19 pandemic. Readers will have their own selection, but here are some of the things people said to me during this period:

• ‘We’re waiting for Public Health England to raise the risk level before we do any resilience planning’ (19 February)
• ‘Maybe it will turn out to be like Y2K’ (4 March)
• ‘I’m relaxed. The numbers will go up, then they will come down again.’ (13 March)
• ‘I’m not scared. It’s just like the flu.’ (14 March).

By 14 March, the UK had 1,140 detected cases of COVID-19 and case numbers were growing at a compound rate of 31.4% per day. Deaths were growing at over 40% per day. On top of that, we had seen on the web and on our TV screens:

• The extraordinary lockdown of Wuhan and Hubei province in China (from 23 January onwards).
• The time-lapse pictures of cranes building a new hospital in Wuhan at breakneck speed (30 January)
• The disastrous outbreak of COVID-19 in Northern Italy (21 February onwards)
• The NASA images of reduced air pollution over China, which showed that the whole country was in some degree of lockdown (29 February).

The rates of change we were seeing in the UK (30%+ per day) had already shown up in China and Italy and other places where COVID-19 was establishing itself.

Yet, as a country, we gazed at this and it somehow didn’t register. A tsunami was approaching, but we saw it as a small wave about to break on Brighton beach. How could this be? Were our government and its scientific advisers paralysed by fear? Were we subconsciously wishing doom upon ourselves? Or was it a cognitive problem of some sort – we were looking at something that lay outside our direct experience and somehow couldn’t compute it?

What happens to a number that expands at a rate of 30% per day? 30% would be high for most things if it were an annual rate of change. As a daily rate it is astronomical. In the course of a year a number growing at 30% per day expands by 3 x 10⁴¹ times, billions of times more than the number of stars in the visible universe.

As soon as the UK saw a 30% per day expansion rate, every possible alarm bell should have been ringing. Every emergency procedure we have should have been leaping into action. Yet somehow it didn’t register.

Q1: What are the next three numbers in the following series?

37 – 49 – 63 – 82 … … …

You can find the answer at the bottom of this blog. You will probably find that your prediction, particularly for the final number, is not very good. This is normal.

There are many potential reasons why the UK did not react quickly to the COVID-19 crisis but one possible reason is straightforwardly cognitive: that we suffer from a bias known as the “linear prediction heuristic.”  Putting this in lay terms, we tend to make linear forecasts even when presented with exponential data.

A linear forecast assumes the same absolute increase in each period.

An exponential forecast assumes the same percentage change in each period.

It has been found experimentally and in field research that the way our brains typically work is to take the last two numbers in a series (in Q1 these are 63 and 82), work out the difference (in this case 19), then keep adding that difference to get the next numbers in the series, so in the Q1 example, we would tend to go: 37 – 49 – 63 – 82 – 101 -120 – 139.

Here’s how Katie Parker, a behavioural psychologist, explains it:

People treat all numbers as absolute values and apply arithmetic operations to yield the absolute difference between the last two data points. They then add or subtract the absolute difference from the last observation to predict the next observation. This method acts to yield judgments which linearly project the trends, even when some data are percentages, displayed with ‘%’ signs and trending exponentially rather than linearly.

Experimentally, this bias has been found in experts as well as lay people and applies even when strong hints are dropped, such as including the % sign in the question. We tend to ignore the earlier numbers in the series, fail to calculate percentage changes (we can’t do this in our heads and many would struggle even with a calculator) and substitute a linear forecast instead.

When we make a linear forecast for a number growing at 30% per day, we drastically misunderstand what is happening to us. What felt to many people in early March 2020 to be a minor threat from COVID-19, became by mid-April our biggest national crisis since World War II.

Yet, anyone with training in maths, statistics, epidemiology or finance will tell you that calculating a series of exponentially growing numbers is not one of the hard questions of science. Once you know what to do, it can be done pretty quickly on a calculator app or spreadsheet.

The strange paradoxes of the human brain

Rocket science is the science of calculating the trajectories of bodies accelerating in gravitational fields. It is generally considered to be very hard. So hard, in fact, that it required a maths genius, Dr Kathleen Johnson, to calculate the orbits of the early American spaceflights.

Here is a rocket science question:

Q2: You are about to cross the road when eighty yards away a car comes round the corner. How long does it take you to work out whether you can safely cross the road before the car arrives?

(a) Five minutes.
(b) Five seconds.
(c) One second.
(d) It seems to happen instantaneously.

Here is the paradox: we struggle to recognise an exponential series, which is an easy maths problem, but we can answer some rocket science questions instantaneously in our heads, seemingly without even having to think about it. We simply “know”.

Q3: You have question 2 again, but this time you are given the car’s speed and acceleration and your potential speed and acceleration and you must work out the answer using a calculator or spreadsheet. How long would it take you?

(a) Five hours.
(b) Five minutes.
(c) Five seconds.
(d) I give up.

What has this got to do with money?

Those of us who have worked in the fields of economics and personal finance know that exponential numbers are everywhere: the rate of inflation, GDP growth, the growth in atmospheric CO₂, interest rates, credit card repayments etc. These are all numbers that change by a given percentage (roughly or exactly) from one period to the next, rather than by the same absolute amount. They all “compound”, that is, the growth in the next period applies to the larger number at the end of the previous period, rather than to the smaller number at the beginning of the previous period.

In economics and finance, percentage changes are usually calculated on an annual basis, as in “GDP grew by 1.2% this year,” “the rate of inflation last year was 1.6%” or “the interest rate on my credit card went up to 25% p.a. last month”.

For small rates of change (eg of GDP) the compounding effect can be ignored in the short term. Over a short period of time, a linear forecast is nearly as good as an exponential forecast. Evolutionally, this is one of the reasons we are poor at spotting exponential numbers. But the higher the percentage rate of change, the more important it is to recognise that the number is exponential.

It is most important with high-cost credit: credit cards, payday loans, buy-now-pay-later etc. Once penalties and fees are added to short-term credit, amounts owed can rocket out of control.

Good financial management also has exponential characteristics. Increasing mortgage or credit card payments has a compounding effect – the term of the loan is substantially reduced. Adding contributions to your pension gives you a compound addition to your pension savings. Compound effects are a double-edged sword: they can cut in your favour, or they can cut against you.

But how to encourage the positive use of exponential numbers when experiments and field research show that our brains can’t recognise them when we see them, that we suffer from the linear prediction bias?

We should expect that strong regulation will be required. If we are fighting a natural predisposition of the human mind, mild measures are unlikely to have a significant effect. Strong regulation will be needed to (a) directly control aspects of finance where the risk of harm is high, and (b) prescribe the way in which information is presented, so as to neutralise as far as possible the linear prediction bias, or to use it to positive effect.

On this point, behavioural psychologists come to our aid – to some extent. Katie Parker explains that under controlled experimental conditions, if people are offered two numbers rather than one they will tend to ‘split the difference’ between the two numbers. For example, if someone taking out a mortgage is offered one repayment amount for a 20-year term and a larger monthly payment to reduce the term to 10 years, they will tend to plump for a repayment mid-way between the two.

Our bias is to do simple, linear maths using two numbers. This leads us astray when faced with an exponential series, but by reframing financial choices as a choice between two numbers that can be averaged, people will make more thrifty financial decisions than under the standard commercial defaults such as the credit card minimum payment.

This method can be applied to all sorts of personal financial decisions: payday lending, mortgage repayments, credit card repayments etc. It could also be applied to savings, by reframing savings and pension decisions into binary choices between numbers that can be averaged.

For financial aficionados, this might seem crude, but given the mathematical limitations of the human brain, it will result in better financial decisions overall than allowing commercial defaults to prevail.

Graphical and colour aids to comprehension

In most fields of human activity, people do not rely on words and numbers on paper to get their message across. Think lockdown art, environmental colour codes for kitchen appliances, infographics, YouTube how-to videos, TikTok memes… the examples are endless.

But in finance, nearly all communications remain in boring black text and numbers on paper – often (still) in small print with long sentences and paragraphs. The mere sight of a communication from our banks has a tendency to put us to sleep.

These types of communications hinder understanding and conspire with poor financial decision-making. Where defaults are used, these tend to maximise lender earnings rather than consumer welfare, like the credit card minimum repayment or the payday loan slider bar.

Financial communications need to be redesigned to work for consumers, using colour, graphics and numbers attuned to the human brain, as in the mortgage example above. For example, interest rates could be colour-coded to indicate the difference between low, medium, high and dangerous.

Q4: Which of the following two presentations is likely to have the most positive effect on consumer financial decision-making?

(a) Future interest rates may rise or fall. Be sure you have chosen the repayment terms most suitable to your budget and repayment preferences.

(b) Pay only the minimum payment per month and the term of your loan will be: 25 years.  Pay £80 per month and the term of your loan will be: 5 years.

 In the post-Covid-19 pandemic world, let’s make this a priority.

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Answers to Questions

Q1: 107 – 139 – 180   (an expansion rate of 30% per number)

Q2: (d) It seems to happen instantaneously.

Q3: This depends on you. Probably (a) or (d).

Q4: (b)

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The graph is from Neil Ferguson et al, Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College London, March 2020.

 

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