The importance of understanding maths and biases when investing

The world’s attention continues to focus on Covid-19, though sometimes, for better or worse, the elections have provided a distraction. We’ve decided it is appropriate to make lemonade from our Covid-19 lemons and see what financial lessons can be learned.

As you will have seen elsewhere, there have been a variety of international responses to address Covid-19. Part of the reason for these differing responses could be attributed to a mathematical misunderstanding, in particular about how fast exponential growth is.

As we are now all expert amateur epidemiologists we know that Covid-19 is spread when an infected (and infectious) person passes the virus on to those nearby. In this manner, one infected person can pass the disease on to a number of others. The infectiousness of a disease is referred to as the R number or R0 (R-nought). In simple terms, it represents the average number of people that one infected person will pass the virus to.

The R number varies depending on local conditions and how the virus mutates, but at the outset of Covid-19 it was estimated to be between 2 and 2.5. In other words, one infected person would pass the disease (on average) to roughly two others. This is worse than the flu (with an R0 of 1.3), but much less than highly infectious measles (with an R0 of 12-18).

Simply based on those rudimentary statements, it is easy to dismiss Covid-19 as just being a more infectious version of the flu. If one person has the virus, they will pass it to approximately two others – the disease is just doubling. Each year we have a flu season and for most it is an inconvenience – though some sectors of the community like the elderly or those with compromised immune systems have to be careful. But at first blush, why is a disease that only doubles on average so bad?

A good illustration, and a handy history lesson, comes from a story about the invention of chess. One tale names the inventor of this classic game as Sissa ibn Dahir, an Indian brahmin (priest). Upon inventing a version of the game, the local king decided that a reward was in order. Rather than apply a bit of effort to come up with a suitable reward, the king asked ibn Dahir what reward he would like. The reply surprised the king – ibn Dahir asked for a grain of wheat to be placed on the first square of the chess board, two on the second square, four on the third square, doubling each time until all 64 squares of the chessboard were filled.

The king laughed at this request – he saw it as a humble and modest request. He set his officials to organising the reward as kings have better things to do than count grains of wheat. Perhaps he had a game of chess. In time his officials reported back to him. The humble and modest reward was anything but. While the first few squares could be filled by a single stalk of wheat, the doubling got notably problematic later on.

For the sake of brevity, to fill the entire chessboard as ibn Dahir requested the king required 18,446,744,073,709,551,615 grains of wheat. A large chess board was needed because those grains would weigh almost 1,200 billion tonnes. For perspective, the global production of wheat in 2019 was 780.8 million tonnes.

For the gluten intolerant, a modern version of this question is “would you prefer one million dollars or one cent, doubling every day for a month?”. As you may guess from the tale about ibn Dahir, the single cent, doubling every day would be worth over five million dollars after 30 days, and if you were in a month with 31 days you would have twice that amount.

If you were surprised about how large these numbers became, you are not alone. I suspect most readers had a “gut feel” as to what the answer was, but their estimates were well below the actual answers. This mistake is known as “exponential growth bias” and does have real life implications.

If you underestimate how fast a virus like Covid-19 that doubles can spread, you are less likely to take preventative measures like hand washing and social distancing. In doing so, the virus can rapidly get out of hand.

Get back to the money!

As promised, there is a financial link to this story. Exponential growth bias leads people to intuitively assume that growth is linear (e.g. picking two apples a day means you have two after the first day, four after the second, six after the third etc), and this causes them to massively underestimate exponential growth.

Let’s look at a simple retirement savings account. If we assume a 5% interest rate, by investing $1,000 today you will have $1,050 in a year, but more than $7,000 in 40 years’ time[1]. A 6% interest rate would give you over $10,000. By failing to recognise how much funds can grow over time, they can short-change themselves in retirement by not taking advantage of compounding returns over a long term.

The corollary of this is also true – people can underestimate how much interest-bearing debts can swell to. Over time it is possible to pay more in interest for mortgages and credit cards than the original debt.

A further surprise is that this is not a case of insufficient education or poor decision making. It has been shown that even highly educated people are susceptible to this bias. One 2012 study[2] reported “In a laboratory study with a sample of elite college students we again find systematic bias but show that subjects are largely unaware and unwilling to pay for aid. This overconfidence suggests that markets will not correct for the bias, as consumers will not seek enough financial advice”.

More than one problem

Exponential growth bias is not the only problem – there are a myriad of biases that people are susceptible to. David Robson writes about this in his book “The Intelligence Trap”, which shows intelligent and educated people often have a “bias blind spot”, believing themselves to be less susceptible to error than others.

There are a myriad of cognitive biases that we are exposed to. Rather insidiously, many still affect us even if we try to look out for them. Even the recently announced Nobel Prize for Economics was based on biases. The 2020 winners won the prize for their work on auction theory. Their work identified that even auctions for large items such as radio spectrum licences or mining licences were affected by biases – in that case the bias being the “winner’s curse” (which was the risk of overpaying to win the auction).

The Corporate Finance Institute lists the top ten biases as:

1. Overconfidence Bias

Overconfidence results from someone’s false sense of their skill, talent, or self-belief. It can be a dangerous bias and is very prolific in behavioural finance and capital markets. The most common manifestations of overconfidence include the illusion of control, timing optimism, and the desirability effect. (The desirability effect is the belief that something will happen because you want it to.)

2. Self Serving Bias

Self-serving cognitive bias is the propensity to attribute positive outcomes to skill and negative outcomes to luck. In other words, we attribute the cause of something to whatever is in our own best interest. Many of us can recall times that we’ve done something and decided that if everything is going to plan, it’s due to skill, and if things go the other way, then it’s just bad luck.

3. Herd Mentality

Herd mentality is when investors blindly copy and follow what other famous investors are doing. When they do this, they are being influenced by emotion, rather than by independent analysis. There are four main types: self-deception, heuristic simplification, emotion, and social bias.

4. Loss Aversion

Loss aversion is a tendency for investors to fear losses and avoid them more than they focus on trying to make profits. Many investors would rather not lose $2,000 than earn $3,000. The more losses one experiences, the more loss averse they likely become.

5. Framing Cognitive Bias

Framing is when someone makes a decision because of the way information is presented to them, rather than based just on the facts. In other words, if someone sees the same facts presented in a different way, they are likely to come to a different conclusion about the information. Investors may pick investments differently, depending on how the opportunity is presented to them.

6. Narrative Fallacy

The narrative fallacy occurs because we naturally like stories and find them easier to make sense of and relate to. It means we can be prone to choose less desirable outcomes due to the fact they have a better story behind them. This cognitive bias is similar to the framing bias.

7. Anchoring Bias

Anchoring is the idea that we use pre-existing data as a reference point for all subsequent data, which can skew our decision-making processes. If you see a car that costs $85,000 and then another car that costs $30,000, you could be influenced to think the second car is very cheap. Whereas, if you saw a $5,000 car first and the $30,000 one second, you might think it’s very expensive.

8. Confirmation Bias

Confirmation bias is the idea that people seek out information and data that confirms their pre-existing ideas. They tend to ignore contrary information. This can be a very dangerous cognitive bias in business and investing.

9. Hindsight Bias

Hindsight bias is the theory that when people predict a correct outcome, they wrongly believe that they “knew it all along”.

10. Representativeness Heuristic

Representativeness heuristic is a cognitive bias that happens when people falsely believe that if two objects are similar then they are also correlated with each other. That is not always the case.

How to fix these biases

Cognitive biases affect your life and those around you in a variety of ways. There is nothing you can do to magically make cognitive biases just go away, but there are ways you can overcome them. The best way is to get independent advice.  Independent advisers often have the data or experience you may lack, and because no two people are exactly alike, they may see conflicting values and priorities in a different light.

Other ways to overcome bias include:

1. Pay attention to your internal environment. Use the HALT method to avoid making crucial decisions if you are Hungry, Angry, Lonely or Tired.

2. Consider who else is impacted by your decision or lack thereof. This can help clarify your response and give you a chance for a considered response.

3. Remember that biases rely on intuition. Taking the time for a considered response allows you to reframe and identify biases or highlight where you need more information.

So, what are the tangible benefits to getting advice?

The Financial Services Council released a report titled “From Money and You” looking into how New Zealanders think about money. Its key findings are set out below.

Key Findings

1. New Zealanders are in poor financial shape, and think they are in a financially better position than they actually are, with many unable to survive financial distress for much more than a month.

2. New Zealanders that get professional financial advice tend to be higher earners and are financially stronger. They have:

  • 50% more in their KiwiSaver
  • Save 3.7% more of their earnings
  • Travel 6x more frequently

3. New Zealanders that get professional financial advice on average receive 4% better investment returns.

4. There are many additional intangible benefits to taking advice including confidence, an understanding of the ‘language of money’ and a positive impact on wellbeing. This provides further proof as to the value of getting good advice. 2020 has been a difficult year but getting the right advice can make it a bit easier.

Britannia Financial Advisers are experts in retirement planning, investing and pension transfers. Give our team a call today on 0800 28 28 33 for a no obligation chat about your unique financial situation. Or you can complete a Free Assessment form and we’ll be in touch with you.

The information contained in this article is of a general nature only and does not take into account individual circumstances. It is not intended to provide comprehensive or specific financial advice. Before making an investment decision you should talk to your Authorised Financial Adviser.


[1] Annual compounding over 40 years and does not take account of taxes or inflation.

[2] Exponential-Growth Bias and Lifecycle Consumption, Levy & Tasoff, 2012



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Disclosure Statements for Britannia's Financial Advisers are available on request and free of charge. Product Disclosure Statements for the Britannia Retirement Scheme and the Integral Master Trust are available from the schemes’ issuer, Britannia Financial Services Limited, phone 0800 500 811. Britannia Financial Services Limited is a Registered Financial Service Provider.

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