Classical Financial Mathematics: Basic Ideas, Central Formulas and Terms at a Glance

By Bernd Luderer

This essential teaches basic formulas, methods and ideas of classical financial mathematics. Since classical financial mathematics makes do with elementary mathematical tools, any interested reader with average mathematical school knowledge can easily follow this text. The core of the text is the calculation of interest and compound interest, annuity calculation, amortization calculation and price calculation. A large number of practical examples illustrate the mathematical questions.

This Springer essential is a translation of the original German 1st edition essentials, Klassische Finanzmathematik by Bernd Luderer, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2019. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

• Language: English
• Publisher: Springer
• Release date: May 25, 2021
• ISBN: 9783658320386

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© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021

B. LudererClassical Financial Mathematicsessentialshttps://doi.org/10.1007/978-3-658-32038-6_1

1. Introduction

Bernd Luderer¹  

(1)

Chemnitz, Germany

Bernd Luderer

Email: [e-mail]

Classical financial mathematics considers the development of a capital over time as well as the calculation of interest on borrowed capital. Interest represents remuneration for the surrender of capital; it always refers to an agreed interest period. By far the most common interest period in practice is the year. One often writes p. a. then (per annum, per year). As a rule, one also has a good feeling for this period: 1% is quite little (although still better than nothing), 3% or 5% is quite common, 10% or even 20% is quite a lot. However, it always depends on various factors (see also the considerations in Chap. 10).

In the following, these conditions will be assumed:

There is always money available in any amount. (Isn’t that a wonderful assumption?)

The agreed interest rate is positive and constant, that is, independent of the term. Unless otherwise agreed, it should refer to one year.

Interest payments are always made at the end of the interest period.

Existing money that exceeds the consumption share is always invested.

All future payments are secure (considerations and emotions of the kind What I have now, I have, who knows what will happen in the future should therefore not play a role; risks and uncertainties of all kinds are also not considered).

There is no inflation (but see Chap. 10).

Anyone who has objections to these conditions is not necessarily wrong, because in practice, of course, not all of the abovementioned points need to be fulfilled. However, within the framework of classical financial mathematics, one limits oneself to precisely these assumptions in order to clarify the essential ideas and develop the basic formulas with the help of this somewhat limited model. However, no account is taken of banking details (the small print), legal regulations, emotions, and tax aspects.

Why is the title classical financial mathematics? Because only those areas are considered here that can be worked on with the help of comparatively simple models and thus on the basis of elementary mathematics: calculation of interest and compound interest, annuity computation, amortization calculus (repayment of principal), and price calculation. From a practical point of view, this includes, for example, savings plans, loans, and bonds, that is, fixed-interest securities. There are close links to the business management issues of depreciation and investment analysis. The calculation of effective interest rates (= rates of return) of financial products or investments runs like a golden thread through all areas of financial mathematics.

At the same time, classical financial mathematics is the starting point for a variety of problems in both actuarial mathematics and modern financial mathematics. The latter has developed rapidly in the past decades. Numerous independent and mathematically sophisticated disciplines have emerged: methods for the valuation of shares and for forecasting share prices, pricing of derivatives (options, futures, swaps, reverse convertible bonds, certificates, etc.), which play an important role especially in investment banking. Furthermore, interest rate models, probabilities of default and much more are examined, which usually requires in-depth stochastic