Money and mathematics

Springer Texts in Business and Economics

Ralf Korn
Bernd Luderer

Money and
Mathematics
A Conversational Approach
to Modern Financial Mathematics
and Insurance


Springer Texts in Business and Economics


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Ralf Korn • Bernd Luderer

Money and Mathematics
A Conversational Approach to Modern
Financial Mathematics and Insurance


Ralf Korn

Fachbereich Mathematik
TU Kaiserslautern
Kaiserslautern, Germany

Bernd Luderer
Fakultăat făur Mathematik
TU Chemnitz
Chemnitz, Germany

ISSN 2192-4333
ISSN 2192-4341 (electronic)
Springer Texts in Business and Economics
ISBN 978-3-658-34676-8
ISBN 978-3-658-34677-5 (eBook)
/>Original German edition has been published with title Mathe, Märkte und Millionen with Springer
Fachmedien Wiesbaden GmbH in 2019.
© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019, 2021
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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To the two most critical readers,
our wives, gratefully dedicated


Preface

Mathematics—abstract, dust-dry, unrealistic, incomprehensible, and not for you?
Wrong, all wrong! There is really no truth to this stereotype. In particular, applied
mathematics deals with many extremely exciting and practical questions. It is the
intention of this book to show that it is worth studying them. In more than five
dozen stories, you, dear reader, will get a relaxed but nevertheless mathematically
exact introduction into the colorful world of financial and insurance mathematics
and into the financial markets.
Mathematical knowledge is often only required at school level; however, for
some stories, basic knowledge of differential calculus and probability theory is
an advantage. For those readers who want to learn more about the mathematical
background, the last chapter describes the associated theory.
The stories are mostly independent of each other, but are grouped by content.
Frequently used formulas, a glossary, and introductory literature are compiled at the
end of the book, while more specific sources can be found in the individual stories.
The book is based on an earlier German edition, but has been expanded and adapted
for the English-speaking reader.
We are indebted to Elke Korn for introducing us to many colorful expressions
that enhanced our text. We are also grateful to Springer Verlag for the inclusion of
our work in the publishing program.

Kaiserslautern, Germany
Chemnitz, Germany
May 2021

Ralf Korn
Bernd Luderer

vii


Contents

Part I

Income Taxes, Lottery, and Lion Hunting—Elementary
Mathematics

1

“We Take over Your VAT!” How Big Is the Actual Discount? . . . . . . . .

3

2

Millions Every Week, but Not for Me. Six Numbers in the
Lottery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5

Where Did My Money Go? Loss
Compensation After a Price Drop . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9

How Do You Catch a Lion? Finding a Zero by Halving
the Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13
16

Upside Down and Up Again. How Many
Zeros Does a Polynomial Have? .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17

“According to Adam Ries, that Comes to . . . ” About Fusti,
Freight, and Cartage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21
24

7

How to Invest? The Cost-Average Effect .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25

27

8

§ 32a, the Politician and the Coaster. Calculating the Income
Tax of a Person .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

29

This Makes the Taxpayer Shudder. What Does “Cold
Progression” Actually Mean? . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

33
37

3
4

5
6

9

Part II

Interest Rates, Prices, Yields—Classical Financial
Mathematics

10 A Fair Deal? Or: There’s Nothing Like Starting Young . . . . . . . . . . . . . . .

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

41
42
ix


x

Contents

11 Should I Pay the Bill Quickly? Cash Discount . . . . . .. . . . . . . . . . . . . . . . . . . .

43

12 The Children of the Interest Rate Are the Grand-Children of
the Capital. Compound Interest . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

45
48

13 When Will Scrooge McDuck Be Satisfied? The Doubling Problem .. .

49

14 How Real Is Nominal? The Actual Rate of Return on a Principal .. . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53

56

15 “Have I Learned to Calculate Correctly?” Why Dr. X. from
Gifhorn Was Wrong .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

57

16 “What, I Have to Pay that Long?” Full Repayment of a Loan .. . . . . . .

59

17 The Widow of the General and the Painter. A Loan à la Chekhov.. . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

63
66

18 Why Does Nominal not Equal Effective? The Effective Interest
Rate of an Immediate Loan. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

67

19 Sandwich with a Car Inside. Financing with Hooks and Eyes . . . . . . . .

69

20 The Assiduous Clerk. Capital Certificates and Federal Bonds . . . . . . . .

73

21 7500 Euros Monthly: A Lifetime. Or Better Yet, Two Millions
Right Away? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

77

22 Financing a Car with Zero Percent: A Bargain? .. . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

81
82

23 Interest Payments Anytime: Isn’t That Wonderful?
Continuous Compounding of Interest . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83

24 Bearer Bonds and Coupons. Bond Prices and Returns of Bonds . . . . .

87

25 Oops! A Law Containing Formulas and Numerical Methods?
The Calculation of the Effective Interest Rate According to
the German Price Indication Ordinance .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

91
94

Part III

Financial Products and Strategies—Modern Financial
Mathematics

26 Fair Prices and Market Prices . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

97

27 The Short End and the Long End. Yield Curves, Spot Rates,
and Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 104
28 Simple as Vanilla Ice Cream. On Standard Financial Products .. . . . . . 105


Contents

xi

29 Exchanges for Mutual Benefit. Swaps . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
30 The Telescope That Has Been Pushed Together. How to
Calculate a Swap Rate? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111
31 Pull out Yourself of the Swamp by Your Own Hair. The
Bootstrapping Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 115
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 118
32 No Risk, No Fun! Risk Indicators of Fixed-Income Securities . . . . . . . . 119
33 Sleep Well Despite Turbulent Markets? The Immunization
Property of the Duration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 125
34 Rising Like a Phoenix from the Ashes. New Shine
for Your Depot? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 132

35 The Crop of Standing Corn. Are Speculators Really Bad People?.. . . 133
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 134
36 Orange Juice and Pork Bellies. Forward Transactions.. . . . . . . . . . . . . . . . 135
37 Empty Pockets and No Money. About Short Sales and
No-Arbitrage Portfolios .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137
38 Earning Money Without Capital and Risk. Arbitrage
Transactions and Fair Prices . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 141
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145
39 Fibonacci and His Rabbits. A Few Words About
Technical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
Part IV

Only Rights, No Obligations—Options

40 A Trip Around the World: Different Types of Options.. . . . . . . . . . . . . . . . 153
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 155
41 Two Triumvirates: From Arbitrage to Speculation .. . . . . . . . . . . . . . . . . . . . 157
42 Nothing Is for Free: The Arbitrage Principle . . . . . . .. . . . . . . . . . . . . . . . . . . . 159
43 How Much Do I Have to Pay for My Right? Option Pricing
According to Black and Scholes. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 161
44 It Takes Two: Option Pricing in the Binomial Model . . . . . . . . . . . . . . . . . . 165
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 172
45 Safe Behind the Hedge: Hedging of Stock Positions .. . . . . . . . . . . . . . . . . . . 173


xii

Contents

46 Wrong Calculation—Right Result: Can This Really Be? The

Correct Derivation of the Risk Measure Delta . . . . . .. . . . . . . . . . . . . . . . . . . . 177
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 180
47 The Greeks and the Risk: About Risk Indicators
for Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181
48 “In, At and Out of the Money”: The Language of the Actors at
the Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185
49 Volatility Determines the Option Price—Really? . . .. . . . . . . . . . . . . . . . . . . . 187
50 Speculating with Options: Rich by Using Leverage? .. . . . . . . . . . . . . . . . . . 191
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 194
Part V

It Is All in the Mix—Portfolio Theory

51 A Portfolio of Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197
52 Risky Investments: Everything Under Control . . . . .. . . . . . . . . . . . . . . . . . . . 201
53 Negative with a Positive Impact: Risk Reduction
Using Correlation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 209
54 Above Your Needs and Maybe Even More? The CPPI Strategy .. . . . . 215
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 220
55 High Risk Pays Off!? Sometimes: On Strategies in Stock
Market Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221
Part VI

The Collective Against Risks—Insurance

56 A Duo Taming Uncertainty: The Law of Large Numbers and
the Central Limit Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 229
57 Do You like Classics? A German Life Insurance Concept . . . . . . . . . . . . . 231
58 More Opportunities: Dynamic Hybrid

Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 239
59 A Million Dollar Roulette in the Financial and Insurance
Market? The Monte Carlo Method. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 245
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 248
60 Insurance for Millions–Billions for the Insurer . . . . .. . . . . . . . . . . . . . . . . . . . 249
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 251
61 CRK: One Number for Risk and Return. Classification of
Pension Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253


Contents

xiii

62 Living with the Mortality Table . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 257
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 260
63 What Relates Honoré de Balzac and 30 Young Geneva Girls
with Life Annuities and Life Tables? . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 261
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 265
64 Sometimes It Clicks and Sometimes Not. A Riester Pension
Product with Index Participation .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 267
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 270
Part VII

Theoretical Foundations—Classical and Stochastic
Financial Mathematics

65 Classical Financial Mathematics .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.1 Linear Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.1.1 Basic Notions and Definitions . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

65.1.2 Interest Formula . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.1.3 Time Values .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.1.4 Final Value with Linear Interest . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.1.5 Present Value with Linear Interest . . . . .. . . . . . . . . . . . . . . . . . . .
65.1.6 Multiple Constant Payments . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.2.1 Interest Formula . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.2.2 Present Value with Compound Interest .. . . . . . . . . . . . . . . . . . .
65.2.3 Sub-annual and Continuously Compounded Interest . . . . .
65.3 Annuity Calculation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.3.1 Annuities in Arrears . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.3.2 Annuities in Advance.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.3.3 Formula Conversion . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.3.4 Perpetual Annuity . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.4 Amortization Calculation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.4.1 Basic Notions and Forms of Repayment . . . . . . . . . . . . . . . . . .
65.4.2 Annuity Amortization . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.4.3 Percent Annuity.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.5 Price Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.5.1 Price of a General Cash Flow . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.5.2 Price of a Bond . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
65.5.3 Price of a Zero Bond . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

273
273
273
274
275
275

276
276
278
278
279
281
282
283
284
285
286
288
288
289
291
291
291
292
292
293

66 Stochastic Financial Mathematics . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
66.1 Basic Notions from Probability Theory . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
66.1.1 Discrete Probability Distributions . . . . .. . . . . . . . . . . . . . . . . . . .
66.1.2 Probability Distributions with Density . . . . . . . . . . . . . . . . . . . .

295
295
296
297

xiv

Contents

66.2 Stochastic Modeling of Stock Prices . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 302
66.3 Option Pricing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 305
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 308
Glossary . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 309
Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 313
Literature . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 317
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 319


Part I
Income Taxes, Lottery, and Lion
Hunting—Elementary Mathematics


1

“We Take over Your VAT!” How Big
Is the Actual Discount?

The Taylor couple is looking around for new living room furniture. The furniture
shop in Manchester has a huge advertising banner:

We take over the value-added tax on your furniture purchase!

“That comes just in time,” Mrs. Taylor is pleased, “we can save 20%. Excellent!”
“There must be a trick to it.” Mr. Taylor is skeptical. “They don’t just take 20% off
the sale price. I’ll do the math at home.”
Suppose a piece of furniture costs P pounds. Then its price minus 20% VAT is
P
only P1 = 1.20
, because if you add the value-added tax on this lower amount, the
20
· P1 = 1.20 · P1 = P . If you now want to calculate the discount r,
result is P1 + 100
which is behind the advertising campaign, you have to start with P1 = P · (1 − r),
P
which leads to the relationship P · (1 − r) = 1.20
. After dividing both sides by P ,
1
one gets 1 − r = 1.20 and finally
r =1−

1.20 − 1
0.20
1
=
=
= 0.16666 . . . = 16.67%.
1.20
1.20
1.20

“I knew it,” Mr. Taylor triumphs, “there’s a trick to it, 20% is not the truth, it’s barely
17%.”

“But 17% are not bad either,” his wife calms him down.
“Yeah, I know. But it’s not 20%, as is suggested by the furniture store,” grumbles
Mr. Taylor. Somehow, he is still unhappy.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021
R. Korn, B. Luderer, Money and Mathematics, Springer Texts in Business
and Economics, />
3


2

Millions Every Week, but Not for Me. Six
Numbers in the Lottery

If wishes were horses, beggars would ride.
English proverb

“I don’t believe it! You are interested in lottery numbers?” My friend Peter is usually
very passionate about the stock market and very keen on making systematic gains.
The fact that of all people he told me something about the latest Saturday draw
seemed as unlikely to me as, uh, winning the lottery.
“Well, the jackpot was hit. And that with the numbers 34, 35, 36, 40, 41, 42.
Unbelievable!”
“Hitting the jackpot is no big deal. And besides, any combination of numbers for
the grand prize is equally likely,” I needled Peter.
In fact, for example, the combination 1, 2, 3, 4, 5, 6 is just as likely as any other
fixed number combination. In the game “6 from 59” there are 59 possibilities for
the first ball to be drawn. As the drawn ball is not returned, there are 58 possibilities
for the second ball and 57 possibilities for the third ball, 56 for the fourth, 55 for the

fifth, and, finally, 54 for the sixth. As they can be freely combined, there are after all
59 · 58 · 57 · 56 · 55 · 54 = 32,441,381,280
possibilities that we can observe during the draw of the lottery numbers.
“Wait a minute, I always thought there were only about 45 million possibilities,”
Peter interjected.

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and Economics, />
5


6

2 Millions Every Week, but Not for Me. Six Numbers in the Lottery

“That’s right, because above I counted cases as different although practically in
the end they are the same. For example, it doesn’t matter if 34 was drawn first and
then 40, or vice versa. So, you have to divide by the number of possible assignments
of the six numbers to the six places in the draw. This results in
32,441,381,280
= 45,057,474
6·5·4·3·2·1
possible combinations of the numbers 1 to 59 that can be drawn.1 The chance of
matching exactly the right numbers has therefore only the tiny value of 0.0000022%.
This means that every week something extremely unlikely happens,” I explained.
Peter protested, “You don’t want to understand me. I meant something else. Two
times three numbers in a row. That’s really quite rare, isn’t it?”
“Well, certainly it appears more often than any particular combination. However,
I know what you mean. The likelihood is small, since among the almost 45 million

possibilities, there are only 1485 that have this property. Among those, I also
counted the 54 possibilities, where all six numbers are directly one after the other.”2
“I knew it was rare!” Peter was pleased.
“Correct. And there are three more exciting things to mention about the lottery.
First, note there’s actually at least one jackpot winner almost every week. This is
because there are so many people playing that game that quite a substantial fraction
of the 45 million possibilities have been chosen. Secondly, I should say that the
lottery is an unfair game in general, because significantly less money is paid out
than paid in. Even if all participants win, the individual profit amounts would be
reduced accordingly. And thirdly, while you cannot increase your probability of
winning, you can indeed increase your expected gains.”
“Whoops! That sounds like a strategy. That’s what I want to hear!” Peter was
wide awake again!

The reader who is familiar with binomial coefficients notices that this is just the expression nk ,
pronounced “n over k,” with n = 59 and k = 6.
2 Counting the possibilities, there are 54 further possible triples for (1; 2; 3), while for the
combination (23, 24, 25), for example, there are only the triples (26, 27, 28), (27, 28, 29), . . . ,(57,
58, 59) left we have not counted yet. In total, we arrive at 54 + 53 + . . . + 2 + 1 = 1485 possible
combinations of different triples.
1


2 Millions Every Week, but Not for Me. Six Numbers in the Lottery

7

© Nina Garman/www.
pixabay.com

“Many gamblers have their own, especially chosen lucky numbers. Usually, they
include their date of birth. But this also means that the numbers from 1 to 31 are
picked more frequently. In a draw with relatively small numbers, you must therefore
share your prize with many other players. Consequently, you won’t receive several
millions for matching the six correct numbers.”3

3

In January 1995, 133 winners had to share a single jackpot that left each player winning only
122,510 £.


3

Where Did My Money Go? Loss
Compensation After a Price Drop

Just because a stock has already fallen does not mean that it
cannot fall further.
Market wisdom

Mr. Scott rubs his eyes in disbelief: “Jeepers!” He had followed the advice of stock
exchange guru Kostolany “Buy shares, leave them there, sleep peacefully,” and
bought shares of a solar panel producer, then went on a trip around the world and no
longer cared about the share prices. And now this—his shares are just worth 12% of
the original purchase price. What a mess! He had paid 200 euro each, now the price
is only 24 euro.
The words of his pal Harrison still seem to echo in his ear: “The stock markets
breathe. So don’t worry if a stock falls by 3%. At some point it rises again by 3%
and everything balances out.” Easy for him to say, 3%. The solar stock was down

88%. Who knows if they will ever go back up 88%? Wait a minute, wait a minute.
What happens if they go up 88%? Mr. Scott does the math: Something can’t be
right. Did his friend tell him the truth? If the price of 24 euro rises 88%, the new
price is just about
24 · 1 +

88
100

= 24 · 1.88 = 45.12 [euros].

That is miles away from 200 euros! That is what happens when you take advice
from friends or “experts.”

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and Economics, />
9


10

3 Where Did My Money Go? Loss Compensation After a Price Drop

Mr. Scott sighs and starts calculating. Suppose the purchase price is P and then
S
the share price falls by S percent. If you set s = 100
, the new rate is P1 = (1 − s)P .
What is the required growth G percentage-wise to get back to P ? Using the notation
G

g = 100
it leads to the equation
!

P2 = (1 + g)P1 = (1 + g)(1 − s)P = P .
After division by P and a short transformation you get
g=

s
1−s

G=

or

100S
.
100 − S

This results in the following values, for example:
S

5

10

20 50

G 5.26 11.11 25 100.
By what percentage must the price of my shares rise, after they have fallen by

88%? Mr. Scott considers, and after a short calculation receives . . . 733.33%. Oh
dear! Will this really happen at some point?1
Although his specific question has now been answered, Mr. Scott continues to
brood. Harrison is no fool. Is he perhaps right after all that a price loss and increase
by the same percentage cancel each other out, but only if the fluctuation is small?
He makes an attempt: Let the price be 100 and let it drop by 1%. Then it is 99.
Now, if it goes back up 1%, it amounts to 99.99, which is almost 100. If it falls
by 3% and then rises again, it first slides to 97 and then grows up to 99.91. Yes,
Harrison seems to be right. He is indeed a smart guy.
The reason behind that for small x values is indeed the relationship
(∗)

1
≈ 1 − x.
1+x

1
Thus, at least for values of s close to zero, the quotient 1+s
can be replaced by
2
the difference 1 − s. Therefore, the division, which can practically only be done in
writing or with a pocket calculator, can be replaced by subtraction. Long live mental
arithmetic!

1

Even if the share price rises back to its original level, Mr. Scott suffers a loss due to lost interest
from a risk-free fixed interest investment that he could have made alternatively.
2 This results from the Taylor expansion of the function f (x) = 1 in a neighborhood of x = 0
0

1+x
or equivalently from the equation of the tangent to the graph of the function f in (0, f (0)). Both
are: l(x) = f (0) + f (0) · (x − 0) = 1 − x.


3 Where Did My Money Go? Loss Compensation After a Price Drop

11

P
Back to the slump. If the stock price drops from P to 1+s
, it approximately equals
P (1 − s). If this value rises again by S percent, then we obtain

Pnew = P (1 − s)(1 + s) = P (1 − s 2 ) ≈ P ,
because s 2 is a very small number.
By the way, in the so-called commercial discounting the merchants in the Middle
Ages used exactly the relation (∗) by deducting the interest from the final value
instead of dividing it (see the basic formula (2)):
K0 = Kt · (1 − it) ≈

Kt
.
1 + it

If, for example, a repayment of 1000 thaler in half a year at 4% annual interest
was agreed, the sum to be paid out in cash was then B = 1000 · (1 − 0.04 · 12 ) = 980
thaler.3

3

With linear discounting,

1000
1.02

= 980.39 thaler should have been paid out.


4

How Do You Catch a Lion? Finding a Zero
by Halving the Interval

The desert is cut in half in a north-south direction by a fence.
Then the lion sits either in the western or eastern half of the
desert. Assume he is in the western half. Now the western half is
cut in half by a fence in an east-west direction. Then the lion is
either in the northern or southern part. If you continue in this
way, the side lengths of the parts that are created in this halving
process will tend towards zero. In this way the lion is finally
enclosed by a fence of an arbitrarily small length.
Bolzano-Weierstrass method to catch a lion, Math joke

Oops, the reader may wonder. What does a lion have to do with mathematics?1
If we take the lion as a synonym for the zero point of a continuous function, for
example a polynomial of a higher degree,2 which very often occurs in the calculation
of returns in financial mathematics, we get much closer to the point. Indeed, the
situation is as follows: Contrary to the widespread impression that with simple
transformations or formulas everything can be solved, this is by far not the case.

Often one has to resort to numerical methods, i.e. methods of “smart trying.”
In many situations you need the zeros of a continuous function f , i.e. values x
with f (x) = 0. These are exactly the points where the graph of the function f
intersects the x-axis. As already mentioned, in financial mathematics the function f
is usually a polynomial.
Let us start with polynomials of first degree, i.e. with linear functions y =
f (x) = a0 + a1 x, which admit straight lines as their graphical representation. To

1

Bernardus Placidus Johann Nepomuk Bolzano (1781–1848), Bohemian Catholic priest, philosopher, and mathematician; Karl Theodor Wilhelm Weierstrass (1815–1897), German mathematician.
2 The reader can imagine f (x) = x 11 − 25x 10 + 3x 2 − 7x + 5 as an example.
© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021
R. Korn, B. Luderer, Money and Mathematics, Springer Texts in Business
and Economics, />
13


14

4 How Do You Catch a Lion? Finding a Zero by Halving the Interval

determine the zero point (which is unique for a1 = 0), you have to set a0 + a1 x = 0
and to solve this equation for x, which is possible and easy for a1 = 0: x = − aa01 .
Also for the quadratic function y = f (x) = a0 + a1 x + a2 x 2 with a2 = 0 the
equation f (x) = 0 or (after a short transformation) the relation3
x 2 + px + q = 0
can be solved. The at most two real zeros are determined using the solution formula
x1,2 = −

p
±
2

p2
−q
4
2

that is well-known from school. If the radicand D = p4 − q is equal to zero, there
is only one zero, and for D < 0 there exists no root.
There are also solution formulas for polynomials of 3rd and 4th degree, but
they are so complicated that practically nobody actually uses them. For polynomial
functions of fifth and higher degrees, there are in principle no solution formulas,
as the Norwegian mathematician Niels Henrik Abel showed at the beginning of the
nineteenth century. Therefore, you have to rely on numerical methods such as the
interval bisection method.
Now we are approaching our lion again, but very carefully, he might attack. We
go on a stalk to encircle the lion (i.e. the zero). The situation now is easier than
described above, because our search is only along a straight line and therefore only
one-dimensional. First of all, we set up a table of values: For selected x values the
corresponding y-values are calculated. Sometimes we have a guess where the lion,
pardon the zero, could be. In financial mathematics, for example, we know that
the so-called accumulation factor q = 1 + i with the interest rate i is just over 1,
i.e. between 1 and 2, because interest rates usually have orders of magnitude of 1,
2, 5, or 8%, i.e. 0.01 or perhaps 0.08.
Now let us assume that our search was successful and we found a value xL with
f (xL ) < 0 and another value xR with f (xR ) > 0. Since polynomials have no jumps,
the graph of the function f between xL and xR must intersect the x-axis at least
once.4 We have surrounded the lion. He is trapped!

Now we want to corner him. But softly, softly, catchee monkey, pardon—the
lion. For this purpose we calculate the function value in the center xM of the interval
[xL , xR ]:
xM =

a1
a2

and q =

1
(xL + xR ).
2

a0
a2 .

3

Here p =

4

This is not only valid for polynomials, but for any continuous function.


4 How Do You Catch a Lion? Finding a Zero by Halving the Interval

15

If f (xM ) = 0, then we have caught the lion: xM is a zero. If, however, f (xM ) > 0,
then the lion lurks in [xL , xM ], which is now half the length of [xL , xM ], whereas
for f (xM ) < 0 it lurks in [xM , xR ]. In both cases we know exactly in which half of
the interval the lion is sitting. If we repeat this process, the length of the interval in
which the zero is located is halved in each step. Thus the lion is encircled closer and
closer. The longer you calculate, the more precisely the zero can be determined and
consequently any desired accuracy can be achieved.
The described “hunting method” should now be explained by the example of the
function
f (i) = 96 −

1
(1 + i)8 − 1
+ 100 ,
·
4
·
(1 + i)8
i

whose zero i we are looking for. This function arises after a short transformation
from the price formula
96 =

1
(1 + i)8 − 1
+ 100
·
4
·

i
(1 + i)8

of a bond (see basic formula (17)), which has a maturity of n = 8 years, a coupon
of 4% and a (below par) price of 96. Thus the financial market expert knows
immediately that the yield of the bond must be higher than 4%. Maybe 5%?
We set xL = 0.04 and xR = 0.06. That yields
f (xL ) = −4.000 < 0

and

f (xR ) = 8.4196 > 0

and the lion has already been located (see Fig. 4.1). He “roars” between 0.04 and
0.06. In the center xM = 0.05 the inequality f (xM ) = 2.4632 > 0 holds. So the
zero is located in the left sub-interval [0.04, 0.05]. The new center is xM = 0.045
and has a function value of f (xM ) = −0.7021, so the zero is to the right of it. In
the next step we get the central point xM = 12 · (0.045 + 0.05) = 0.0475.

0 04

0 045

Fig. 4.1 The “lion” is trapped

0 0475

0 05

0 06

16

4 How Do You Catch a Lion? Finding a Zero by Halving the Interval

In this point we get the function value of f (xM ) = 0.8678. After calculating the
function value in the new center point
xM =

1
· (0.045 + 0.0475) = 0.04625,
2

which is 0.1015 and is already close to zero, we stop the process: The zero is about
4.6%. If you need a higher accuracy, you have to calculate a few more steps.5
The reader can practice in “lion catching” by calculating the discount to be
obtained when Kevin buys a car and if he has invested his money at 5% (see p. 71).
The result can then be compared with the exact result given there. The only thing
the reader needs is diligence, endurance, and a little bit of courage. Not like the
sluggard who is quoted in the Holy Bible with the words, “There is a lion outside! I
shall be killed in the streets!” (Proverbs 22:13)
In conclusion, it should be noted that interval bisection is only one of many
numerical methods for determining zeros. There are certainly more elaborated ones,
which are also usually faster. For example, the secant method or multiple linear
interpolation should be mentioned. The described methods belong to the derivativefree procedures. Other methods, however, use the first or higher derivatives, for
example the tangent method (also called Newton’s method) or the Quasi-Newton
method. Convergence analysis is needed to be able to make statements if such
a procedure actually succeeds in delivering a zero, while the convergence speed
says something about how fast a zero is found. In Schwarz and Koeckler (2011) or

Zeidler (2013) you can find more information about this.

Literature
Schwarz HR, Koeckler N (2011) Numerische Mathematik (Numerical Mathematics), 8th edn.
Vieweg, Teubner, Wiesbaden
Zeidler E (ed.) (2013) Springer-Taschenbuch der Mathematik (Springer paperback of mathematics), 3rd edn. Springer Spektrum, Wiesbaden

5

If you want to know exactly: For a calculation with two decimal places, the zero is 0.0461 =
4.61%.