Million Dollar Maths E-Book

Hugh Barker

• Herausgeber: Atlantic Books
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Veröffentlichungsjahr: 2018

Beschreibung

Million Dollar Maths is an invaluable guide to the straightforward and outlandish mathematical strategies that can make you rich. ____________ How can you turn $1000 into $1 million? What is the best way to beat the lottery odds? When is the best time to take out a loan? How did one group of gamblers bet on hole-in-ones to win £500,000? How can maths help you set up a successful tech start-up? What about proving the Goldbach Conjecture for $1 million? Learn the techniques for growing your everyday finances, as well as the common mistakes to avoid. Discover the skills, both fair and foul, that offer an additional edge when investing and gambling. And discover why we often misunderstand probability and statistics - with troubling financial costs. From making the most of special offers to utilising the power of exponential growth in your investments; from the art of card counting, to inventing the next Google, Million Dollar Maths is the quintessential primer to the myriad ways maths and finance intersect.

Leseprobe

First published in hardback in Great Britain in 2018 by Atlantic Books, an imprint of Atlantic Books Ltd.

Copyright © Hugh Barker, 2018The moral right of Hugh Barker to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

1 2 3 4 5 6 7 8 9

A CIP catalogue record for this book is available from the British Library.

Internal illustrations © Diane Barker

Hardback ISBN: 978-1-78649-322-4

EBook ISBN: 978-1-78649-323-1

Printed in Great BritainAtlantic BooksAn Imprint of Atlantic Books LtdOrmond House26–27 Boswell StreetLondonWC1N 3JZ

www.atlantic-books.co.uk

Contents

INTRODUCTION: Maths and Money, A Curious Relationship

CHAPTER 1: The Power of Exponential Growth

CHAPTER 2: Beating the Casino

CHAPTER 3: Gambling Systems and Strategies

CHAPTER 4: The Successful Investor

CHAPTER 5: Hacking, Cracking and Gaming the System

CHAPTER 6: Designing the Next Google

CHAPTER 7: Use Maths to Improve Your Performance

CHAPTER 8: Proving the Impossible

CONCLUSION: Being Maths Aware

INTRODUCTION

Maths and Money, A Curious Relationship

Annual income twenty pounds, annual expenditure nineteen [pounds] nineteen [shillings] and six [pence], result happiness. Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

Charles Dickens, David Copperfield

Like it or not, we live in a material world, in which money can help to create opportunities in life. We all know that money can’t buy you love or happiness. But the lack of money can certainly lead to deprivation and frustration. So it is only natural that people with reasonable maths skills may occasionally ponder how that know-how can be used to maximize their wealth. Can they, for instance, manage their financial affairs or business better? Can they come up with a brilliant new mathematical idea or a related piece of technology? Or can they use their maths skills for more nefarious purposes, such as gambling systems or hacking and cracking?

The quote above, from David Copperfield by Charles Dickens, points to the fact that solvency is always preferable to bankruptcy. This is not the most astounding insight, although it is sound advice in its own way. But most of us would prefer to put away a bit more than sixpence a year for a rainy day. If we’re really honest, most of us would like to become as rich as possible. The self-help industry is hugely profitable largely because it sells the dream of rapid wealth for minimal effort. I won’t be making that claim here, just exploring the many large and small ways that you can put maths to work for you.

I’ll explore the many connections between maths and finance and the opportunities this creates for significant moneymaking. I’ll include stories of famous investors, business people and gamblers who have used mathematical formulae or techniques in their work (I’ll mostly avoid getting bogged down in value judgements about the morality of gambling and speculation as opposed to investment, though I will acknowledge where there are potential legal issues or other risks in a financial strategy). Modern technology also relies increasingly on maths, whether it be in the algorithms used by social media companies, the complex maths that underpins Bitcoin, or the ongoing battle between hackers, crackers and internet security experts. I’ll also give quick summaries of things to do and things not to do as we go along.

The largest part of the book will focus on personal finance, gambling and investment, all of which can be easily understood by anyone with high school level maths. Some of this maths may already be obvious to you, but it is surprising how many people enjoy an occasional flutter without fully understanding the mathematics of the roulette table, or consult analytical tools such as the price-to-earnings ratio without realizing the intuitive and obvious way that this relates to interest rates. And when it comes to haggling over your salary, you may or may not know how game theory affects your chances of an increase.

Along the way we’ll be meeting a strange assortment of problems that are also interesting from a purely mathematical point of view, from the Keynesian beauty contest to the Byzantine Generals Problem, and from the Kelly criterion to Maverick solitaire.

You don’t need to be any kind of maths genius to put mathematical thinking to work in your everyday life. In fact most successful investors and businesspeople do not use complex maths, but instead rely on a clear understanding of how the numbers work, and of the mistakes we tend to make when we analyse data and probability. Avoiding irrational blunders can often be as crucial as making good judgement calls, and being familiar with the common mathematical and statistical errors people make is an enormous help.

It’s not all easy maths – later in the book I’ll discuss the mathematics of the broader financial system and maths prizes and awards, and this can’t be done without an attempt to outline the more complex maths involved. It would take a far more advanced mathematician than me to have a detailed knowledge of every single maths theorem I mention. I’ll be honest and own up where I’m getting out of my depth and will make it clear where the theory is likely to go beyond the reach of an amateur mathematician. But for the most part the maths required in this book is no more complex than you would learn at school.

CHAPTER 1

The Power of Exponential Growth

If a man is proud of his wealth, he should not be praised until it is known how he employs it.

Socrates

If you ask fifty people what money is, you’ll get fifty different answers: it’s a peculiarly hard thing to define, so let’s start by trying to pin that down. That definition will underpin the most basic ways you can make your money grow and help to explain why exponential growth is the key to successful wealth accumulation.

What is Money?

At its most basic, money is just a mathematical tool for counting and measuring value. In pre-monetary societies goods could be traded by barter in which, for instance, a sack of grain might have been swapped directly for pots or beans or for a day’s labour in the fields.

Let’s imagine a transaction where one dairy cow was swapped for three bushels of wheat. We could use a visual equation to express their comparative value (see Figure 1).

But you can only use pure barter if you have exactly the goods the other party wants and vice versa. Otherwise, you can end up in complicated webs of buyers and sellers where person A gives person B a cow, they give their wheat to person C, person C gives person D some beehives, and they give person A their pots and pans. This would be monstrously tricky to choreograph. So, very quickly systems of money and credit were developed. By using tally sticks or other primitive records of trades, people could sell their goods or services and store a credit to be used for purchases at a later time. If we call the unit of currency ‘x’, then we might have market prices of 15x for a cow and 5x for a bushel (see Figures 2 and *).

Figure 2. One cow costs 15x.

Figure 3. One bushel costs 5x.

In algebra, we would represent these as:

We can also manipulate these equations to get valuations for one unit of x:

Note that money can be treated as an additional item in the marketplace, whose own value can be measured in terms of other items. Its main advantage is that you can use it as an intermediary that enables transactions involving other items.

So we immediately have counting as the basis of monetary systems. (In fact, the whole act of counting large numbers may have been inspired by commerce – there is evidence that primitive societies would count ‘one, two, three, many …’ or only up to ten or twenty, based on fingers and toes.) And we also have money being used from the start as a measure of comparative value.

From an early point, debt was also part of monetary systems – many societies had rules against usury (charging interest on lent money) but any system that recognizes a credit owed by one person to another already contains the concept of debt. In fact the concept of negative numbers was initially introduced by Chinese mathematicians specifically to deal with the problem of keeping accounts which recognized both credits and debits – in a ledger, the red debits were subtracted while the black credits were added.

Some people distinguish ‘real money’ from ‘token money’ or ‘fiat money’. By real money they mean objects such as gold, which they see as having a real, intrinsic value, as opposed to tokens such as wooden coins or cowrie shells (which were being used as money tokens three millennia ago on the shores of the Indian Ocean). I would argue that money is always to some degree a token or representation, regardless of its physical form, but I don’t want to get into the complex debate over whether gold money is more real than, say, the US dollar other than to say this: any kind of money, whether it be gold or paper, government-backed or private, digital or a plastic token, can be valued only in relative terms.

What this means is that the value of a unit of money can only ever be measured in terms of the goods and services (or even other currencies) it can be exchanged for.

So there is no such thing as inherent or absolute value: you can measure the current value of gold against wheat, a dollar against gold, or even the value of one yen against the value of one euro. But it is meaningless to describe any of these goods as having value in themselves without referring to who is valuing them and what they might exchange for them. All monetary values are relative and all of them fluctuate over time. And if, for instance, the price of petrol in dollars increases, it is equally valid to say that the price of dollars, as measured in petrol, has fallen.

As well as being relative, monetary value is always subjective. The same bottle of water might be worth nothing to someone who lives by a clean stream, but worth a million dollars to you if you are lost in the middle of a desert and at death’s door.

The art of wealth management is based on identifying differential value and fluctuations in value. This concept is perhaps most easily understood when you consider the idea of ‘net worth’. This is defined as the amount of money you would end up with if you sold all your assets and paid off all your debts at current values.

It can be hard to shake the idea that money does or should have an objective value. But in these days of quantitative easing (and money printing) it should be clearer than ever that money itself can gain or lose value. And it gives us a much more rigorous mathematical basis for thinking about money if we regard it simply as an item that can be exchanged for other goods and services.

Buy Low, Sell High

The next basic thing to bear in mind is that economic transactions generally rely on two individuals or groups placing a different value on the same item and then agreeing a mutually acceptable price. (If the two parties value the item exactly the same, they may agree a deal, but neither will have a strong motivation to do so.) Suppose you go out tomorrow planning to buy a second-hand car, let’s say you are willing to pay up to £3,000, while the seller is willing to sell for at least £2,500. In this case a deal can usually be done somewhere in between the two prices, and this will help to set the market price, which is the theoretical average of many similar transactions.

The supply and demand curves (see Figure 4) that are used in economic theory are just easy ways to show how prices are set in markets. You can use mathematical tools to describe idealized versions of markets, and these are valuable analytical tools so long as you remember that the idealized markets they describe aren’t actually real.

Figure 4. A supply and demand chart: As the price rises, supply tends to rise, meaning more people are willing to produce or sell an item, while demand tends to fall, meaning that fewer people are willing to buy the item. Theoretically the market price, or equilibrium price, will be found where the demand and supply curves meet.

Similarly if you buy a share, then this transaction works because you are assuming that the share is undervalued or valued correctly while the seller is assuming it is overvalued or valued correctly.* There may be rational or irrational reasons for these assumptions, but the key point is that the buyer and seller have different motivations and reasons for valuing these items differently and a compromise is reached. So rather than talk about ‘value’ it is often more useful to look at the market price, which can be measured.

If you want to make money, you have to think about the ways in which you can exchange assets, goods or services of varying price in a way that allows you to accumulate more money or possessions.

There are fundamentally four ways you might approach this task.

The first is to sell your labour for a wage or salary. In other words, get on your bicycle and go out and find some work.

The second is to create a business, large or small, in which you create goods or services. In this process you take the raw materials (whether they be labour, ingredients, materials or ideas) and create something that can be sold at a higher price. For instance, you might buy modelling clay and make brooches that you can sell for a higher price, and advertise them online via social media to keep your costs down. By adding value to the raw materials, you are creating wealth.

The third is to invest in other people’s businesses and wealth creation, either directly (by investing in the business of a friend, for instance or via stocks and shares, which you can buy directly or via a broker).

The fourth is to take advantage of the variation in value of assets, buying at low points and selling at high points – this is the basic activity of any trader who sells goods for a higher price than they pay for them, but it also describes the activities of speculators and gamblers. (It can be hard to pin down the distinction between speculation and investment, but it’s worth thinking about whether the money invested is genuinely helping others to create wealth. If not, it’s probably speculation rather than investment.)

However you aim to make your money, the obvious mathematical rule of ‘buy low, sell high’ is applicable in a world of fluctuating values. Even in the world of work, you can analyse the time and money you spend acquiring particular skills or experience and compare this to how much difference it will make to your pay. But more obviously in business and investment, the more you can take advantage of variations in value the faster your wealth will grow.

However, we shouldn’t only think in terms of buying and selling. The legendary investor John C. Bogle was a great advocate of holding on to assets, writing for instance that ‘the real money in investment will have to be made – as most of it has been made in the past – not out of buying and selling but of owning and holding…’ In this case the relevant question is whether an asset is earning more than it is currently costing you to hold onto it, and how this equation compares to other assets you could swap it for. This is where the idea of comparative value is also crucial, as there is no gain to be made by selling an asset just to swap it for other less profitable assets. And the economic concept of opportunity cost refers to the fact that capital invested in one asset ‘costs’ us the ability to invest that same capital in an alternative asset.

What to Do

One reason to think about value as a purely mathematical equation is that it helps us to avoid some common irrational errors. It is easy to make the mistake of considering irrelevant factors when valuing an asset: for instance, how much you paid or how hard you worked to get this asset, or how much you hoped to sell it for. This leads to errors such as the sunk costs fallacy (in which people find it hard to give up on a lossmaking project because of the money that has already been spent on it).

The only way you should value an asset is by considering its current value and comparing this to your other options: most of what has happened in the past is irrelevant. The trajectory of its past value may, of course, give us some information about the future trajectory, though as the adverts always say, ‘past performance is no guarantee’. While the aim should always be to sell an asset for more than you paid for it, refusing to sell at a loss can be more damaging than accepting the loss and moving on.

The Rule of 72

When considering an investment opportunity or business model, it is often useful to know how long it will take to double your money at a particular rate of growth. (And if you’re not anticipating doubling your money at some point, maybe you should be considering different, more profitable opportunities?)

The Rule of 72 is a quick way to calculate this in your head. It has been used since at least the fifteenth century when Luca Pacioli (1445–1514) included it in his Summa de arithmetica.

The rule is to divide 72 by the rate of growth (or the interest rate, for savings and investments): the result gives you the number of periods it will take for the initial investment to be doubled. For instance, for an interest rate of 9% a year, we divide 72 by 9 and get 8 years. The actual time it would take money to double at 9% is 8.043 years (see Figure 5), so this is reasonably accurate.

Figure 5. 9% growth per period starting with £1,000. It takes approximately eight years to double.

If you want to use this as a rule of thumb, bear in mind it is only a rough approximation, and one that works best for interest rates in the range 5–10%. Also, it’s actually more accurate to use 69 or 70 as the numerator in your fraction. (72 has been used historically because it has so many factors: it can easily be divided by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, or 36.)

If you really want to get geeky about it you can use the more precise 69.3 as the numerator and use what is known as the Eckhart–McHale second order rule, which is this equation:

where t is the number of periods taken to double your money and r is the rate of growth. The second part of this equation helps to improve the accuracy of the estimate for high rates of growth, for which it is otherwise increasingly inaccurate.

But for most standard situations, the basic Rule of 72 is perfectly adequate, as is shown by the fact that it has served so many financiers and investors well over the centuries.

The Easy Way to Make a Million

Now that we know how to quickly work out how long it will take to double your money, let’s look at an extremely simple recipe for turning an initial investment of £1,000 into a million within a year.

Let’s imagine you stumble across a way to buy a supply of magic beans on Mondays. On Friday afternoon you can always sell the beans you bought for twice the price you paid for them. So you spend your start-up money on a supply of magic beans, sell them and double your money, then use the money to stock up on twice as many magic beans the next week. Hey presto, you can then keep on doubling your money – after one week you will have £2,000, after two weeks £4,000 and so on until after 10 weeks you will have £1,024,000.

I’m sure you can spot the flaw in this plan: there is no such thing as magic beans (or, more to the point, any other foolproof way of doubling your money ad infinitum). The mathematics is, however, perfectly sound. As you keep doubling n times, you multiply your original investment by the nth power of 2, so you have 2, 4, 8, 16, 32, 64, 128, 256, 512 then 1,024 (= 210) times your original investment.

So this is just basic maths, but not terribly useful in practice. However, think of it as a thought experiment in how money can grow, given a good, reliable business model. The doubling period may be somewhat longer than a week, you will undoubtedly have to work hard to find your own version of ‘magic beans’, and you are going to have to manage uncertainty rather than being gifted a guaranteed profit. But in the end all business plans and investment strategies rely on finding a way of making your money grow and then repeating the process.

The other thing to bear in mind is that, even if you could find a guaranteed method of doubling small amounts of money, it will become increasingly difficult to scale this up to larger amounts. For instance, if you had a system that allowed you to double your money in a casino, then it would only take a few doublings for the casino either to ban you or go out of business. Even with the magic beans, you would soon have difficulties carrying enough of them away in your wheelbarrow on a Monday. All businesses and investment systems have ceilings, but some are lower than others.

So what we need to do in this book is to focus on the grittier detail of how you might use mathematical skills and rules of thumb in combination with real-world applications to go about taking that first £1,000 and making it into £2,000. And we also need to bear in mind how much a particular strategy can be scaled up before it reaches a natural ceiling.

What to Do

When looking for your own ‘magic beans’, think from the start about how long it will take to double your money. And also think about how quickly this approach will hit a ceiling beyond which it is no longer possible to keep growing at the same rate.

Magic Beans in the Real World

I said there was no such thing as magic beans, and sadly there isn’t. However, it is instructive to compare the magic beans business to the markets in property, land and stocks or shares. Land prices and stock markets can suffer from wild fluctuations in the short term, but in the long term they have grown pretty reliably in real terms – over decades or even centuries. So the investor or land owner who succeeds in buying at the lows and either selling at the highs or securing an income on their asset when prices rise will, in the long term, always make a good profit (as long as the long-term trend in the market continues).

What is the difference between this and the magic beans business? Firstly, there is always a degree of uncertainty about where you are in the market cycle. And secondly the cycle is much slower than the weekly interval over which I imagined magic beans doubling in value. However, there is an underlying similarity: land prices in most economies and the major markets for stocks or shares have tended to increase at about 5–10% over inflation for decades. For instance, investing in index funds (which track the performance of the entire market) will generally give this kind of return, or slightly more if you are able to enter the market in a dip. Not exactly magic beans, but a pretty good substitute for those with sufficient funds. To see how big a difference fairly small variations in annual return can make, look at these changes from 1984 to 2015 in the UK market: £100,000 invested in property over that period would have returned £502,500 (at 5.7% per annum (pa) against an inflation rate of 3.5% retail price index (RPI)), while equities (which rose at the slightly higher rate of 5.9% pa) would have returned £533,000. And constantly reinvesting the dividends from the equities could have turned that into a whopping £1,533,500 (which is equivalent to 9.9% pa). Property and land have grown more than equities since 2000, but this is largely because the property market was at such a low at that stage.

This is one of the reasons why the wealthy tend to stay wealthy (see p. 264 [the Pareto Principle]) – because these kinds of investments are most accessible for those with enough wealth to tie some of it up in long-term assets.

For those with less disposable wealth, long-term investments in property and index funds can still play a significant part in wealth accumulation but it is likely that faster methods will also be required if results are desired over years rather than decades.

The Power of Exponential Growth

The magic beans example demonstrates exponential growth, which means growth that continues at a constant percentage rate. This is an extremely powerful concept when it comes to wealth acquisition, and helps to explain why the wealthiest people tend to have acquired their money through investment or through owning businesses which could be successfully scaled up over time. Figure 6 shows an exponential curve in which money increases at a constant rate against time.

Figure 6. Exponential growth.

By comparison someone on a fairly good salary, which increases over time (but not exponentially), might see a growth in their wealth that looks more like the curve shown in Figure 7 (the vertical lines indicate pay rises).

Figure 7. A slow increase in wealth as a result of pay rises.

This is a crude comparison of course, but it should be obvious that the exponential curve is the one that has the greatest potential in the long run – if you work hard and succeed, your salary might increase by a factor of 2, 10 or even 20 times over the length of your career, but to increase your earnings by 100 times or more you need to be looking for exponential growth.

So when you look at ways of making money, the very first questions you should be asking are: firstly, how long will it take me to double my money? And secondly, can this approach be scaled up, so that the money keeps growing at an exponential rate (at least in the medium term)?

Chapter 1 Summary

1.   Money can be treated as a variable in a comparative value equation.

2.   Use the Rule of 72 for a rough estimate of how rapidly your money will grow.

3.   Exponential growth should be part of your ideal business model.

4.   Unless you can find some magic beans, you will need to learn how to manage risk and uncertainty, and how to make reasonable predictions of future value.

____________

* Alternatively, the seller may be a ‘forced seller’ in which case they may believe the price to be below the fair price but have no option but to sell.

CHAPTER 2

Beating the Casino

The gambling known as business looks with austere disdain on the business known as gambling.

Ambrose Bierce

There can be a fine line between gambling and certain types of business, especially when it comes to speculation and investment. When mathematician Ed Thorp refined the theory of cardcounting (in which a player gains an advantage by keeping track of the remaining cards in blackjack) as a gambling strategy, his book on the subject inspired a generation of quantitative and financial analysts as well as gamblers, and he went on to become a successful hedge fund manager himself (see Chapter 5). Gambling can supply demonstrations of some basic ways of analysing luck and probability, as well as giving us an understanding of the rational fallacies that afflict gamblers. These tools and fallacies are also applicable to other, less risky investment and business options, so using gambling to explore the basic uses of maths in assessing risk and opportunity will give us a solid foundation for looking at how maths can help you to make better decisions about what to do with your money in general.

Gambling Mathematicians

The sixteenth-century polymath Gerolamo Cardano was one of the first mathematicians to lay out the foundations of probability. His book Liber de ludo aleae (Book on Games of Chance) outlined the idea of analysing events by considering all possible outcomes and how many of these were favourable to the gambler. In modern terminology, he described the ‘sample space’ of dice games by observing that there were 36 possible ways in which two dice might fall (see Figure 8), and, for instance, that six of these were cases where the two dice fell on the same number, of which only one was a double six. This allows us to define the probability of a double six as 1 in 36 (which in more rigorous terms means that if we throw two dice repeatedly, the number of double sixes will, over time, tend towards a limit of 1 in 36).