Classical Financial Mathematics: Basic Ideas, Central Formulas and Terms at a Glance (essentials) [1st ed. 2021] 3658320370, 9783658320379

Citation preview

Bernd Luderer

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

essentials

Springer essentials

Springer essentials provide up-to-date knowledge in a concentrated form. They aim to deliver the essence of what counts as "state-of-the-art" in the current academic discussion or in practice. With their quick, uncomplicated and comprehensible information, essentials provide: • an introduction to a current issue within your field of expertis • an introduction to a new topic of interest • an insight, in order to be able to join in the discussion on a particular topic Available in electronic and printed format, the books present expert knowledge from Springer specialist authors in a compact form. They are particularly suitable for use as eBooks on tablet PCs, eBook readers and smartphones. Springer essentials form modules of knowledge from the areas economics, social sciences and humanities, technology and natural sciences, as well as from medicine, psychology and health professions, written by renowned Springer-authors across many disciplines.

More information about this subseries at http://www.springer.com/series/16761

Bernd Luderer

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

Bernd Luderer Chemnitz, Germany

ISSN 2197-6708 ISSN 2197-6716  (electronic) essentials ISSN 2731-3107 ISSN 2731-3115 (electronic) Springer essentials ISBN 978-3-658-32037-9 ISBN 978-3-658-32038-6  (eBook) https://doi.org/10.1007/978-3-658-32038-6 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 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. 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, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Responsible Editor: Iris Ruhmann This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

What You Can Find in This essential

• An introduction to classical financial mathematics. • The description of crucial requirements and underlying mechanisms. • The presentation of the most important formulas for calculating interest and capital. • Discussion of the terms present value and final value of capital. • The equivalence principle as the key to calculating yield or effective interest rate. • The solution of basic problems of financial mathematics. • A description of aspects that go beyond classical financial mathematics.

v

Preface

This text is addressed to everyone who is interested in financial mathematics and wants to get acquainted with financial mathematics, be it pupils, students or practitioners. It focuses on the basic concepts and ideas, while strictly mathematical derivations as well as banking and tax-related details play a secondary role. At first the basic ideas are explained rather casually and without many formulas. Only later on the most important models and formulas are developed and given. For a better understanding it is strongly recommended to recalculate the examples contained in the text, whereby the highest possible accuracy is not important. Without a calculator, which should at least have the key x y, this is of course not possible. The literature references refer to books from which one can obtain more detailed information and which contain numerous examples and exercises, often with detailed solutions. Thanks are due to the publisher for including this work in the “essentials” series. For helpful comments and critical remarks I would like to thank especially Mrs. Iris Ruhmann. Any comments and remarks on the present text are always welcome. Chemnitz July 2020

Bernd Luderer

vii

Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Time is Decisive. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 How to Pay Interest?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Reaching Your Goal Through Regular Saving. . . . . . . . . . . . . . . . . . . 5 Annuity Calculation—Not Only for Pensioners. . . . . . . . . . . . . . . . . . 6 Would You Rather Be a Debtor or Creditor?. . . . . . . . . . . . . . . . . . . . 7 Short and Long Periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Bonds, Coupons, and Yields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Cherry on the Cake is the Rate of Return. . . . . . . . . . . . . . . . . . . 10 What Else is Important? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5 7 15 19 27 33 37 39 45

Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

ix

1

Introduction

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.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_1

1

2

1 Introduction

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 results. The framework of classical financial mathematics is formed from very few basic formulas, which are clearly listed in the chapter Basic formulas. From this handful of formulas, even complicated relationships can be modelled by assembling the basic formulas like building blocks. When analyzing a financial agreement, it is advisable to have an overview of all inpayments and payouts, together with the times at which they are made (the particular role of time is discussed in detail in the next chapter). This can be done by using the general scheme of payments, called cash flow, as shown in Fig. 1.1, which must be specified in each specific case. The scheme can easily be generalized to non-integer points in time. Such an overview of all payments helps to translate the usually verbally formulated problem into the language of mathematics, that is, to model it. This means assigning the given data to the appropriate variables and selecting or combining the correct basic formulas.

1 Introduction

3

Fig. 1.1   General scheme of inpayments and payouts

Unfortunately, it is not always done by inserting values into a formula or rearranging an equation. Some equation relationships cannot be explicitly resolved for a variable that occurs in them. In this case, the value of this quantity can only be calculated approximately with suitable numerical methods (trial and error methods), but with any degree of accuracy. As a rule, this involves the calculation of zeros of polynomial equations (see Luderer 2015, Point 2.3). Typically, such questions arise in the calculation of returns.

2

Time is Decisive

In classical financial mathematics, the time factor is often mentioned. In fact, the (current) value of a payment is always dependent on the time at which it is due. This is one of the basic ideas underlying financial mathematics, and this is what it thrives on. Even if “Joe Average Consumer” does not always pay attention to this thesis in his daily life by far or does not consider it to be so essential, it can be understood immediately: If one compares, for example, a payment of, say, 100 €, which one either receives today or can only expect to receive in five years, everyone would probably prefer to receive this payment today. And not only because nobody knows what will be in five years’ time, whether the payer will still exist then, no, even if one leaves all emotions aside: to receive 100 € today, so to speak “cash in hand”, is more advantageous than a payment of the same amount in the future, because one can invest the money received today, so that its value will have increased after five years. Conversely, it is immediately clear that a payment of, for example, 100 € to be made today is less favorable than a payment of this amount in the future. As a consequence, all calculations in financial mathematics are based on the corresponding point in time or term of financial investments. The corresponding point in time can generally be chosen arbitrarily; once chosen, however, it must remain fixed. The point in time t = 0, often referred to as today, plays a prominent role; it usually represents the beginning of a financial agreement. The corresponding value is called present value. But also the end of a contract is predestined to be chosen as a point in time, whereby the final value is then spoken of. In general one speaks of the time value of a capital. From this point of view, information on total payments, for example in a savings agreement or when repaying a loan, does not say much, as the time factor is completely disregarded.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_2

5

6

2  Time is Decisive

Fig. 2.1   Accrued interest and discounting of a capital

On the other hand, payments due at different points in time can only be compared if they are related to a uniform, fixed point in time. If an (initial) capital  bears interest at a positive interest rate (as is generally assumed) it increases and is referred to as compounding (or adding interest). Conversely, the present value K0 of a future payment or of the capital Kt considered at time t results from discounting. This value is always smaller than the time value Kt. Figure 2.1 illustrates the procedure for discounting and adding interest.

3

How to Pay Interest?

The following terms are used in the following: Capital

Sums of money left to a third party

Duration

Duration of the surrender (as part or multiple of the interest period)

Interest

Remuneration for the surrender of capital within an interest period

Interest period

The time frame underlying an agreement; often one year, sometimes shorter (month, quarter, half-year), rarely longer

Interest rate

A number expressing the ratio of the interest amount in monetary units (MU) payable on a principal of 100 MU in an interest period; usually expressed as a percentage

Time value

The time-dependent value of a capital

Present value

Time value to be attributed to t = 0

Furthermore, the following notation is used in this and the following chapters: t

Moment, duration; part or multiple of the interest period

i

Interest rate

Zt

Interest for the time period t

K,Kt

Capital; capital at time t (time value)

K0

Initial capital; present value; time value for t = 0

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_3

7

8

3  How to Pay Interest?

Linear Interest Calculation It has already been mentioned that the larger the capital K, the longer the duration (term) t and the higher the interest rate i, the higher the interest. However, this requirement alone does not lead to a uniquely defined formula, there are (in principle infinitely) many possible. Two types of interest calculation are common in practice. The most obvious formula is obtained when all three of the above-mentioned variables are entered proportionally. In this case, one speaks of linear or simple interest calculation:

Zt = K · i · t.

(3.1)

The term t plays a special role. Are the exact calendar days counted? Does the year have 360, 365 or 366 days? How are the first and the last day of the investment calculated? In Germany the year is often counted as 360 days and each month as 30 interest days. Numerous other definitions that differ from this can be found, for example, in Grundmann and Luderer 2009, p. 24 f. The quantity i is the interest rate payable for a full interest period, for example, i = 3% = 0.03. It is a rational number that has no unit of measurement. For t = 1 what corresponds to a full interest period, the relationship (3.1) yields a value of Z1 = K · i, for half an interest period Z1/2 = K · 21 · i, etc. If we assume that at t = 0 the initial capital K0 is available, the final value at time t is therefore Kt = K0 + Zt = K0 + K0 · t · i. Thus we get the basic formula

Kt = K0 · (1 + it) ,

(G1)

which is called the final value formula for linear interest. Example

A capital of 100 € is to be invested for one year or four months, respectively at an interest rate of 3% p. a. It then  grows to 100 · (1 + 0.03) = 103 € after one year and to 100 · 1 + 31 · 0.03 = 101 € after four months. ◄ The basic formula (G1) contains the four quantities Kt, K0, t, and i. It can be resolved for each of these quantities by simple formula conversion, which we leave to the reader. However, one of these conversions, which leads to the present value formula for linear interest, is particularly important:

K0 =

Kt . 1 + it

(G2)

Linear Interest Calculation

9

The basic formula (G2) is used to express the present value, that is, the today’s value of a future payment. To be more precise, if the (positive) interest rate i is given, the value K0 is the equivalent of the payment Kt due at the later date t; it is obviously smaller than Kt. Or to put it another way: if the amount K0 is applied over a period of time t at interest rate i, it increases to the value Kt. Thus, for a given i, the amounts K0 today and Kt at t are equivalent. This is the simplest form of the equivalence principle, which appears repeatedly later on, and which can be used to calculate the (effective) interest rate or yield i (or also other quantities, such as the term). Example

What is the value of a payment of 100 € due in one year, assuming an interest rate of 4%, and what is the value if the payment is due in three months? According to the  (G2), this results in K0 = 100/1.04 = 96.15 €  relationship and K0 = 100/ 1 + 41 · 0.04 = 99.01 €, respectively. ◄ If the interest rate i is therefore given, it does not matter whether the amount K0 is paid today or Kt at time t – neither of the two variants is better or worse. As a rule, the inequality 0 < t ≤ 1 in formula (G1) applies, that is, the period under consideration is shorter than an interest period. However, if the interest is paid out and not reinvested by the investor,1 the relation t > 1 can also apply. This is e. g., provided for in the German Civil Code (BGB §§ 248 and 289). As already mentioned, interest is generally paid at the end of the interest period, that is, in arrears. However, exceptions confirm the rule, because occasionally, although relatively seldom, interest is paid in advance, for example on bills of exchange. This means that the interest is due immediately and the bill debtor does not receive the full amount, but the amount reduced by the interest. Bills of exchange used to be common instruments for financing small and medium-sized enterprises, whereas nowadays they have become considerably less important. If you go back a few centuries in history, you will find this type of financing at every turn in the novels of Balzac, for example. And even earlier,

1This

actually contradicts the assumption that money is always invested; but one can simply separate capital and interest in thought and consider them independently of each other.

10

3  How to Pay Interest?

among the merchants of the Middle Ages, the interest paid in advance was found in the so-called commercial discounting. Here, the interest was deducted from the final value instead of performing a division as in the basic formula (G2):

K0 = Kt · (1 − it).

(3.2)

From a mathematical point of view, the reason for this is that on the one hand 1 ≈ 1 − x applies to small x-values, but on the other hand it is the relationship 1+x much easier to form a difference than to divide. Example

If repayment of 100 thalers in half a year at 4% annual interest was agreed  for  a loan, the amount to be paid out in cash was K0 = 100 · 1 − 0.04 · 21 = 98 thalers according to the relationship (3.2), as can easily be done by mental 100 arithmetic, whereas 1.02 = 98.39 thalers would have had to be paid out if the loan had been discounted linearly. ◄

Geometric Interest Calculation If a capital is invested with interest over several interest periods and the interest is not paid out but accumulated (this is also called capitalized), it is called compound interest (i.e., the interest on the interest) or also geometric or exponential interest, since the value of the capital develops from period to period like the terms of a geometric numerical sequence. In contrast to linear interest calculation, several interest periods are typically considered. Thus, at first the term t is an integer. However, legal regulations, such as the German Price Indication Ordinance (Preisangabenverordnung, PAngV), or procedures customary in the financial markets may also prescribe geometric interest rates for shorter (so-called intra-year) periods. This form of interest rate is also preferred internationally. Now the well-known compound interest formula is to be derived. If a capital is invested over several years (more generally: interest periods) and – as already mentioned above – the interest due at the end of each year is accumulated and consequently also accrued in the following years, compound interest is generated.

Geometric Interest Calculation

11

Using the final value formula (G1) with t = 1 as well as the fact that the capital at the end of one year is equal to the initial capital in the next year, it is now possible to calculate step by step the capital available at the end of each year if the capital at the beginning of the first year amounts to K0. Capital at the end of the first year: K1 = K0 · (1 + i)

Capital at the end of the second year: K2 = K1 · (1 + i) = K0 · (1 + i)2 .. .

Capital at the end of the n-th year:

Kn = K0 · (1 + i)n .

(3.3)

The latter formula is known as the final value formula for geometric interest or as Leibniz’s compound interest formula. Sometimes the final value is also called accumulated value. The quantities Kn and K0, respectively, appearing in it denote the capital at the end of the n-th year and the initial capital, while the compounding (or accumulation) factor (1 + i)n indicates the amount to which a capital of one monetary unit grows at an interest rate i and reinvestment of the interest after n years. The quantity n here is initially an integer. In many cases, however, the formula (3.3) is generalized to an arbitrary time period, so that it is converted into the form

Kt = K0 · (1 + i)t

(G3)

in which t represents an arbitrary rational number. When using the pocket calculator, nothing changes, because the key x y allows any kind of number as input. Example

What is the return on an investment of 100 € over five years, where the annual interest is reinvested (so-called accumulation of interest), if the interest is 6% p. a.? According to formula (G3), with the quantities K0 = 100, t = 5, and i = 0.06, the final value is K5 = 100 · 1.065 = 133.82 €. ◄ As also in the case of the linear interest calculation, the question of how great the value of a future payment is today, that is, at the time t = 0, is of interest. In other

12

3  How to Pay Interest?

words: What is the present value of a payment Kt due at time t? By rearranging the basic formula (G3), the answer is quickly found:

K0 =

Kt . (1 + i)t

(G4)

Analogous to the remarks made above regarding linear interest, the basic formula (G4) expresses the present (or discounted) value of a future payment, so that at a given interest rate i, the value K0 represents the equivalent of the payment Kt due at the time t > 0. It is always smaller than Kt. Conversely, if the amount K0 is invested to the interest rate i in the period [0, t], it increases to the value Kt. If the interest rate i is thus given, it does not matter whether the payment K0 is made today or the amount Kt at time t – both variants are equivalent, because all payments are regarded as secure as agreed upon. Example

A firmly committed payment of 100 €, which will be made in five years, is to be passed on to a third party who is willing to pay a certain amount for it. How much is this if one expects an interest rate of 6% p. a.? The amount in question should, in fairness, be just as high as the present value of the expected payment, so that using the basic formula (G4) the amount K0 = 100/(1 + 0.06)5 = 74.73 € is obtained. ◄ We now want to compare linear and geometric interest rates from different points of view. First we will take a closer look at growth. Figure 3.1 shows, on the one hand, how linear and geometric interest calculation behaves within an interest rate period (left figure: linear interest yields more interest, at the time t = 1 the interest amounts are equal). On the other hand, it shows how a capital with a geometric interest rate develops over time; the higher the interest rate, the faster and stronger the capital grows (the variable p describes the interest rate in percent, so that the relationship p = 100 · i applies). The difference in the final values of the two types of interest calculation is particularly significant for longer maturities. The standard in classical financial mathematics is usually defined as follows: for a period of time 0 < t ≤ 1 linear interest is applied, for t ≥ 1 geometric interest (for t = 1 the results are the same). However, it has already been pointed out that there are also situations where linear interest calculation is applied for t > 1, and geometric interest calculation for 0 < t < 1 (see the German Price Indica-

13

Geometric Interest Calculation

Fig. 3.1   Comparison of linear and geometric interest calculation

tion Ordinance (Preisangabenverordnung, PangV) mentioned above as well as in Chap. 9). A mixed interest rate occurs when linear and geometric interest rates meet, for example, when money is invested in a savings account that begins in the middle of the year and ends at some point after several years. Then, with calendar-based interest, there are three sections in which linear, geometric and again linear interest is calculated. The corresponding formula is relatively complicated (see Luderer 2015, Point 4.2). Therefore, in financial mathematics it is often replaced by the basic formula (G3). The deviation is small, which is due to the fact that for small i the relationship (1 + i)t ≈ 1 + it holds. The solution of a mathematical problem should be independent of the solution path. For financial mathematics this means: when calculating the time value of a capital, the result should not depend on the calculation path taken. To be more precise: Let there be given the capital K0 at the time t = 0 and the two points in time t1 and t2 with 0 < t1 < t2. We are looking for the time value Kt1. Then it should be indifferent whether you accumulate interest directly from t = 0 to t1 or first accumulate interest to t2 and then discount from t2 to t1 (with the difference time t2 − t1); the latter is easier in some situations. This is in fact true for geometric interest calculation, because

Kt1 = K0 · (1 + i)t1 = K0 · (1 + i)t2 ·

1 . (1 + i)t2 −t1

14

3  How to Pay Interest?

However, this does not generally apply to linear interest calculation, as the value K0 · (1 + it1 ) usually differs from the value K0 · (1 + it2 ) · 1+i(t12 −t1 ), which is in some respects inconvenient (see the following counterexample). Example

The amount of 100 € is to be compounded at an interest rate of 5% and linear interest on the one hand by eight months, on the other hand by one year and then discounted by four months. In the first case, this   8 = 103.33 €, in the second in results in K8/12 = 100 · 1 + 0.05 · 12 1 K8/12 = 100 · (1 + 0.05) · 1+0.05· 4 = 103.28 €. The two calculated time val12 ues differ. ◄

4

Reaching Your Goal Through Regular Saving

In this chapter, multiple constant payments made at regular intervals within an interest period are considered and combined into a single payment at the end of the interest period. We will initially focus on the year as the interest period and monthly payments. Such situations arise, for example, in savings plans, but also in the repayment of loans when monthly payments and annual interest calculation have to be brought into line. The question of interest to us is: What is the final amount R at the end of the year if at the beginning of each month (i.e., in the case of payments in advance) an amount r is invested and the underlying interest rate is i? Figure 4.1 illustrates the situation mentioned above. The January payment is subject to interest for a whole year (so that t = 1 = 12 12 applies) and therefore grows up according to the basic formula (G1) to   r · (1 + i) = r · 1 + i · 12 . According to the same formula, the Februarydeposit 12 11 accrues interest until the end of the year to an amount of r · 1 + i · 12 and so 1 on. The December payment finally produces a final amount of r · 1 + i · 12 , as it earns interest for one month. Therefore, the total amount at the end of the year (using the sum formula of the arithmetic series) is   12 11 1 R=r 1+i· +1+i· + ... + 1 + i · 12 12 12   i = r 12 + · [12 + 11 + . . . + 1] 12   i 13 · 12 , · = r 12 + 12 2

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_4

15

16

4  Reaching Your Goal Through Regular Saving

Fig. 4.1   Regular monthly payments (in advance)

and finally

 R = r · (12 + 6.5 · i).

(G5)

In this context, the quantity R is referred to as the annual replacement installment, as it can serve as a substitute for the twelve individual monthly payments. So it does not matter whether someone pays monthly the amount r or once at the end of the year the amount R. We will come back to this later. Example

Mrs. X. regularly saves 100 € at the beginning of each month. How much can she dispose of at the end of the year if the interest rate is 6% p. a.? The basic formula (G5) gives for the concretely selected values r = 100 and i = 6% = 0.06 immediately the result R = 100(12 + 6.5 · 0.06) = 1239. Mrs. X. therefore has an amount of 1239 € available at the end of the year. ◄ It should be noted that the annual replacement rate refers to the end of the year, although the monthly payments are made at the beginning of each year, that is, in advance. If the monthly payments are made at the end of each month, the total amount is   R = r · (12 + 5.5 · i) (G6) and one speaks of monthly payments in arrears. The derivation can be made analogously to the above; however, it is also easy to convince oneself that the number 6.5 must be reduced by 1 to 5.5, since the January payment is omitted and the last payment at the end of December does not bear any interest at all. Now the above problem is to be generalized somewhat by choosing an arbitrary period (e.g., a quarter) and dividing it into m shorter periods of length m1

4  Reaching Your Goal Through Regular Saving

17

(e.g., a quarter is to be divided into three months). At any point in time mk , k = 0.1, . . . , m − 1, that is, at the beginning of each short period, a payment of r is made. The following two formulas are obtained for the annual replacement rate (payments in advance and in arrears):   m+1 R=r· m+ ·i , (4.1) 2

  m−1 R=r· m+ ·i . 2

(4.2)

Conclusion: The payments made at the beginning or end of each short period were converted into an equivalent one-off payment due at the end of the interest period (long period).

5

Annuity Calculation—Not Only for Pensioners

The annuity calculation deals with the question of combining several regularly recurring payments into one value (taking into account the accrued interest) or, conversely, dividing a given value into a certain number of payments (annuitization of a capital sum), taking into account accrued interest. The old age pension is only one example of such regular payments; the repayment of a loan, savings and payout plans and much more also fit into this scheme. Since—as has been emphasized several times—the value of a payment depends on its due date, a distinction is made between the annuity present value and the annuity final value (start and end of the annuity). In the further course, it is assumed that the following applies:

Installment period = Interest period Thus, for example, if interest is paid annually, payments should also be made annually. According to the time at which the annuities are paid, a distinction is made between payments paid in advance (ordinary annuity) and such paid in arrears (annuity due), with payments being made at the beginning and end of the period. Advance annuities often occur in connection with regular savings (savings plans) or rent payments, while payments in arrears are typical for the repayment of loans or for salary payments. A further distinction is made between temporary annuities and perpetual annuities (of unlimited duration), with temporary annuities (of limited duration) being the core of financial mathematics, while perpetuals are a more or less

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_5

19

20

5  Annuity Calculation—Not Only for Pensioners

theoretical but often useful construction for simplifying calculations. The payments in each of the periods are supposed to be constant; this is called a rigid annuity. A generalization is made by dynamic annuities, which can be increasing or decreasing. Life annuities which are paid for as long as the beneficiary lives and which are therefore dependent on the average life expectancy of the policyholders are not dealt with in classical financial mathematics; however, they play an important role in actuarial mathematics. Important variables in the annuity calculation are: n

Number of installments or interest periods

i

Interest rate

q =1+i

Accumulation factor for one year

qn

Accumulation factor for n years

R

Installment

E

Capital at the end of the n-th interest period; final value of annuity

B

Capital at the time t = 0; present value of annuity

Annuities in Advance As already mentioned, the basic problem of the annuity calculation is the aggregation of the n individual payments into one total payment. Since the amount of the latter depends on the time at which this payment is made or at which the settlement is carried out, the point in time t under consideration is therefore important. Two points in time are of particular importance: t = n which corresponds to the final annuity value and the point t = 0 in time which belongs to the present value of the annuity. If the installments are paid at the beginning of each period, this is called an annuity in advance. Figure 5.1 shows the (constant) payments together with their payment dates and below the final values of the individual payments; these are obtained by applying the basic formula (G3) to each individual payment (with q = 1 + i). First of all, the final annuity value E adv is to be calculated, that is, the amount which at the time t = n represents an equivalent for the n installments of amount R to be paid. To calculate it, we consider the final values of the individual payments according to the final value formula (G3) with K0 = R using geometric

Annuities in Advance

21

Fig. 5.1   Final values of individual advance payments

interest calculation, whereby it should be noted that the individual installments must be compounded over a different number of periods according to the different payment dates. Using the well-known formula of the geometric series, all individual values are now added up (see for example Grundmann and Luderer 2009, p. 13):

E adv = R · q + R · q2 + . . . + R · qn−1 + R · qn = R · q · (1 + q + . . . + qn−2 + qn−1 ) =R·q·

qn −1 . q−1

If you now replace again q by 1 + i, the final value formula of the advance annuity calculation is finally obtained: (G7) Example

At the beginning of each year, a grandmother pays 100 € into a savings account for her granddaughter. To what amount do the deposits increase after 18 years at 5% interest p. a.? According to the formula (G7), the 18 payments along with interest pay18 −1 ments result in a final value of E adv = 100 · 1.05 · 1.05 = 2953.90 €. ◄ 0.05 To determine the present value of the annuity, the present values of all individual payments could be calculated by discounting using the present value formula (G4) and adding them together. However, it is simpler to use the result just obtained and calculate the present value by discounting the value E adv over n years:

Badv =

1 · E adv . qn

(5.1)

22

5  Annuity Calculation—Not Only for Pensioners

If the expression from (G7) is used, (5.1) results in the present value formula of the advance annuity calculation: (G8) The factor with R indicates the value of an annuity of amount 1 payable in advance for n periods at the time t = 0 or, in other words, the number of years over which an annuity of amount 1 can be paid (taking into account the accrued interest) if the amount available at t = 0 is Badv. Example

How much should a pensioner have at his disposal if he goes on pension so that he could receive an advance payment of 10,000 € a year for twenty years at 6% interest per year? What is needed here is the present value of an advance annuity of the amount R = 10,000. According to the present value formula of the advance 1.0620 −1 annuity calculation, this is Badv = 10,000 · 1.06 19 ·0.06 = 121,581.16 €. ◄

Annuities in Arrears Here, the installments are paid at the end of each interest period. This is illustrated in Fig. 5.2. By adding the n individual final values, the final value of the annuity in arrears is calculated as a geometric series with the initial value R, the constant quotient q and the number of elements n:

E arr = R + R · q + . . . + R · qn−1 = R · (1 + q + . . . + qn−1 ) =R· Fig. 5.2   Final values of individual payments in arrears

qn −1 . q−1

Annuities in Arrears

23

If q is again replaced by 1 + i, one obtains the following basic formula (G9) for the final value of an annuity in arrears: (G9) If one compares the final value formulas of the annuity calculation in advance and in arrears, one can see that the factor 1 + i is missing in the case of arrears. This is due to the fact that each payment is made one period later and is therefore compounded once less. Logically, the final value of an advance annuity payment (with all other parameters remaining the same) is therefore greater than the final value of a payment in arrears. Example

(See the example on p. 21) For her granddaughter, a grandmother pays 100 € into a savings account at the end of each year. To what amount do the payments increase after 18 years at 5% interest p. a.? According to the basic formula (G9), the 18 payments including interest 18 −1 result in a final value of E arr = 100 · 1.05 = 2813.24 €. The final amount 0.05 achieved is therefore about 140 € lower for payments of the same amount but made later. If one would make a one-off payment today, which, with the same interest rate, leads to the same final value as in the example just considered, this would 1 arr have to be a value of Barr = (1+i) = 2813.24 = 1168.96 according to the n · E 1.0518 basic formula (G4). For comparison: the total sum G of all deposits (which of course does not take into account the payment dates!) is 1800 €. In general, the following inequality holds for i > 0 and t > 0: K0 < G < Kt. ◄ The present value of an annuity in arrears is calculated most simply by discounting the final annuity value E arr from the basic formula (G9) over n years, that is,

Barr =

1 · E arr qn

(5.2)

or by using (G9)

B

arr

 (1 + i)n − 1  =R· . (1 + i)n · i 

(G10)

24

5  Annuity Calculation—Not Only for Pensioners

Example

What amount of money should a pensioner have at his disposal if he retires so that he could receive an annual payment of 10,000 € in arrears for twenty years at 6% interest p. a.? What we are looking for is the present value of an annuity in arrears of the amount R = 10,000 €. According to the present value formula of the annuity calculation for arrears (G10), the present value is 1.0620 −1 Badv = 10,000 · 1.06 20 ·0.06 = 114,699.20 €. Compared to the payment method in advance (see p. 22), about 7000 € less initial capital is required here, because the payments are made one year later, so that the remaining capital earns interest for longer. ◄ If you compare the two formulas (G8) and (G10) with each other, you will see that the following applies: Badv = (1 + i) · Barr . The reader can easily check this for the examples above. In addition, one can also give an interpretation of the present value: If the agreed installment of 10,000 € is taken annually from the initial sum (i. e., the present value), then after 20 years the account is just empty.

Perpetual Annuities (Perpetuities) If the annuity payments are made for an unlimited period of time, one speaks of a perpetuity. This seems unrealistic at first, but on the one hand it simplifies the calculation for a large number of periods n. On the other hand, there are real situations in which perpetual annuity is appropriate (mortgage loans with no repayment of principal or foundations where only the interest income is paid out and the actual capital remains untouched, i.e., no capital consumption). The question of the final value of a perpetual annuity is not meaningful, so that only the present value is of interest. This is determined by converting the present value formula and then changing to the limit value (let again q = 1 + i). One gets adv = lim B∞

n→∞

R qn−1

·

1 q − qn−1 qn − 1 = lim R · . n→∞ q−1 q−1

(5.3)

Due to i > 0 we have q > 1. Therefore the relations lim qn−1 = ∞ as well as n→∞ 1 = 0 hold true, so that the relationship (5.3) for the present value of the lim n−1 n→∞ q perpetual annuity in advance finally results in this formula:

Perpetual Annuities (Perpetuities)

adv B∞ =R·

25

q 1+i =R· . q−1 i

(5.4)

Similarly, the present value of the perpetual annuity payable in arrears is adv B∞ =R·

R 1 = . q−1 i

(5.5)

This formula has a simple interpretation when slightly reshaped: from B = Ri it follows R = B · i. The right side represents the interest due, so the installment adv must be equal to the interest in order that the capital B∞ remains unchanged.

6

Would You Rather Be a Debtor or Creditor?

Closely related to the annuity calculation is the redemption (or amortization) calculation (or amortization of a debt), which is used when a creditor lends money to a debtor, which the latter then pays back in (usually equal) installments. It is therefore a matter of determining the repayment installments for interest and redemption of a borrowed capital sum. However, you may also be looking for other determinants such as the term to full repayment or the effective interest rate. In principle, the creditor expects the debtor to pay interest on his debt and repay it as agreed. The repayments, consisting of redemption and interest, are called annuities (this term comes from annus: Latin for “year”; however, it can also refer more generally to any interest or payment period). Furthermore, the following general agreements are always assumed to hold. Thus the formulas for the calculation of annuities in arrears have to be applied: • Installment period = interest period = 1  year • The number of repayment periods is n years • The annuity payment is made at the end of the period There are two main forms of amortization, depending on the method of repayment: • Repayment by installments (constant redemption installments) • Annuity amortization (constant annuities)

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_6

27

28

6  Would You Rather Be a Debtor or Creditor?

The following notation will be used: S0

Credit amount, initial debt

Sk

Residual debt (outstanding principal) at the end of the k-th period, k = 1, . . . , n

Tk

Redemption in the k-th period, k = 1, . . . , n

Zk

Interest in the k-th period, k = 1, . . . , n

Ak

Annuity in the k-th period: Ak = Tk + Zk

i

Underlying (nominal) interest rate

Repayment by Installments (Annual Payments) Here the annual repayment installments are constant:

Tk = T = const =

S0 , n

k = 1, . . . , n.

For the other quantities only the corresponding formulas are given; their derivation can be found, for example, in Luderer 2015, Chapter 6. The outstanding debt Sk after k periods constitutes an arithmetic sequence with the initial element S0 and the difference d = −T = − Sn0 :   k Sk = S0 1 − , k = 1, . . . , n. n The interest to be paid on the residual debt Sk−1 at the end of the previous period also forms an arithmetically decreasing numerical sequence, with the difference d = − Sn0 · i between successive elements:   k−1 Zk = Sk−1 · i = S0 · 1 − · i. n Since the interest payments decrease over time, but the redemption installments remain constant, due to the relationship Ak = Tk + Zk the annuities are falling, as shown for n = 5 schematically in Fig. 6.1. It is customary to present all interrelationships in a clear and concise manner in the form of a repayment plan. It consists of a tabular list of the planned repayment of a borrowed amount of capital within a certain term. For each repay-

Annuity Amortization (Annual Agreements)

29

Fig. 6.1   Development of annuities in the case of repayment by installments

ment period, it contains the residual debt at the start and end of the period, interest, redemption, annuity and any other necessary information (such as agios). A repayment plan is based on the following rules: Zk = Sk−1 · i

Interest is paid on the remaining debt

Ak = Tk + Zk

Annuity = sum of redemption plus interest

Sk = Sk−1 − Tk

Residual debt at the end of a period = Residual debt at the start of this period minus redemption installment

With the help of these rules, the values in the repayment plan can be calculated successively one after the other, although once an error has been made, it will affect the entire calculation. As an alternative, the formulas derived above can be used for direct calculation of all occurring quantities or for checking the calculation.

Annuity Amortization (Annual Agreements) As explained above, the annuities are constant with this form of repayment:

Ak = Tk + Zk = A = const. Due to the annual redemption payments, the residual debt decreases from year to year, so that the interest to be paid decreases and a constantly increasing share of the annuity is available for redemption, as illustrated in Fig. 6.2 (exemplary for n = 5). To calculate the annuity, the formulas of the annuity calculation in arrears can be used, where the equivalence principle is used (for more details, see Chap. 9).

30

6  Would You Rather Be a Debtor or Creditor?

Fig. 6.2   Development of interest and repayment amounts for annuity amortization

In the context of the annuity amortization, this compares the creditor’s payments with the debtor’s payments, whereby, for the sake of comparability, one refers to a uniform date. This is often the point in time t = 0, so that the present values of creditor and debtor payments are compared with each other (present value comparison). This approach can then be used to determine each of the variables that occur, provided that all the others are given. The performance of the creditor (bank, lender) consists in the provision of the credit sum S0 at time zero; this sum therefore corresponds to its present value. (In practice, however, there can also be several partial payments, for example, depending on the progress of the construction work; these are then discounted to t = 0). The present value of all the debtor’s payments is (due to the usual method of payment of annuities in arrears) equal to the present value of an annuity in arrears with constant installments R in the amount of the annuity A being sought, which, according to the present value formula of the annuity calculation in arrears (G10), results in the relationship

S0 = A ·

(1 + i)n − 1 . (1 + i)n · i

(6.1)

By transforming this expression one finally obtains the following formula for the calculation of the annuity:

A = S0 ·

(1 + i)n · i . (1 + i)n − 1

(6.2)

The factor standing with S0 in the above expression is called the annuity factor or capital recovery factor. It indicates the amount to be paid annually in arrears

Annuity Amortization (Annual Agreements)

31

in order to fully pay off a debt of one monetary unit in n years, with the remaining debt bearing interest at interest rate i. Example

A loan of 100,000 € is to be repaid in full within twenty years at an interest rate of 4% per annum. How much does the borrower have to pay to the lender at the end of each year? The question is about the annuity A at given values n = 20, i = 0.04, and 20 ·0.04 S0 = 100,000. From Eq. (6.2), the relation A = 100,000 · (1.04) = 7,358.18 (1.04)20 −1 results. So 7,358.18 € are to be paid annually. An interesting question (although from a financial mathematical point of view rather insignificant, as payment dates are not taken into account) is how much the borrower has to pay in total. The answer is very simple— the total amount is G = A · n; in the case under consideration we obtain G = 7358.18 · 20 = 147,163.60 €, that is, almost one and a half times the amount of the loan itself. On other dates (lower interest rate, longer term), it may well be double or triple. ◄ Again, of interest are formulas for the redemption installments Tk, residual debts Sk, and the interest payments Zk, k = 1, . . . , n. These will be given below (in some cases in several versions), but are not derived; for details see Luderer 2015, Chapter 6:

Tk = T1 · (1 + i)k−1

with

T1 = A − S0 · i,



 Zk = Z1 − T1 · (1 + i)k−1 − 1 = A − T1 · (1 + i)k−1 , Sk = S0 − T1 ·

(1 + i)k − 1 (1 + i)k−1 = S0 · (1 + i)k − A · . i i

(6.3) (6.4) (6.5)

The formula (6.1) can be resolved not only for A, but also to n. If, for example, the loan amount S0, the interest rate i and the annuity A are given, the duration until complete repayment can be determined by rearranging the relation (6.2) to n, where n need not necessarily be an integer:

n=

1 A · ln . ln(1 + i) A − S0 · i

(6.6)

The situation where maturity n is sought occurs, for example, with the so-called percentage annuity, which in the case of a loan is characterized by the fact that the repayment in the first year is predetermined (by the bank, which often insists

32

6  Would You Rather Be a Debtor or Creditor?

on a minimum repayment, or by the customer, who can choose a higher initial redemption rate); the nominal interest rate is in any case predetermined by the bank at normal market conditions. Example

In a loan agreement, there is the following passage: “The loan shall bear interest at 8% per annum and shall be repaid at the rate of 1% of the original principal plus the interest saved by the repayment.” Then the question arises after how many years the loan will be completely repaid. The annuity, which is constant during the annuity amortization, can be easily determined by using the annuity in the first year: A = const = A1 = Z1 + T1 = 0.08S0 + 0.01S0 = 0.09S0. In concrete terms, let € and therefore S0 = 100,000 A = 0.09 · 100,000 = 9000 €. The formula (6.6) initially yields 9000 1 · ln 9000−100,000·0.08 = ln 9/ ln 1.08 = 28.55 [years]. According to n = ln 1.08 the relationship (6.5) the remaining debt (outstanding principal) after 28 years 28 −1 amounts to S28 = 100,000 · 1.0828 − 9000 · 1.08 = 4661.16 €. For the 0.08 given interest rate of 8% p. a. this results in an interest of 372.89 € in the last year, so that the annuity in the 29th year is 5,034.05 €. ◄ If, as in the example just considered, there is no integer solution value for the number of repayment periods, a lower annuity occurs in the last year of repayment.

7

Short and Long Periods

Previously, it was always assumed that the payment period and the interest period were the same. But what should be done if the payments or the interest calculation do not correspond to the originally defined interest period? In practice, a common situation is that interest is assumed to be payable annually, but payments are made monthly. If, on the other hand, a monthly interest rate is agreed upon, the question of the interest rate per year arises, because only for this one has a “feeling” or only this serves in most cases as a basis for comparison. For the sake of simplicity, the (original) interest rate period should be one year; this is called the long period. However, shorter periods often occur in contracts in financial practice. For example, six-monthly, quarterly or monthly interest or installment payments may be agreed. This is referred to as interest or payment during the year or short periods.

Payments during the year, linear interest In Chap. 4, twelve monthly (or more generally: m during the year) payments were converted into one payment per year (in arrears), with linear interest being assumed during the year. This can be used in the annuity calculation by assuming only one payment of the amount R = r · (12 + 5.5i) and R = r · (12 + 6.5i) respectively, instead of twelve monthly payments in arrears (in advance, resp.) of the amount r. Note that the replacement installment R is payable in arrears in both cases.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_7

33

34

7  Short and Long Periods

Conversely, in the amortization calculation presented in Chap. 6, an annuity A payable annually in arrears can be converted into monthly annuities a (payable in arrears), where the following applies:

a=

A . 12 + 5.5i

(7.1)

Interest during the year In the following, an interest period of length one (long period, e. g., one year) is to be divided into m sub-year (short) interest periods of length m1 . An amount of K0 is to be invested. The connection between the interest rates in the long and short periods is important, whereby two cases are of particular interest. 1 Let be given the nominal interest rate i for the long period. Then it is obvious to assign a pro rata interest rate of i/m to the short period (linear interest), which is called the relative interest rate during the year. So far so good. However, since interest is paid m times during the original interest period, the final value formula for geometric interest (G3) yields a value of   i m (m) K1 = K0 · 1 + . (7.2) m This value is greater than the final value in the case of a one-off interest payment at the nominal interest rate i, i.e., the inequality K1(m) > K0 · (1 + i) applies, which is due to the fact that in the case of interest paid during the year, the paid interest bears interest again. This leads to the compound interest effect. Therefore, if one asks for the effective interest rate related to the original interest period, the effective annual interest rate ieff , one must use this approach:   i m ! (m) = K1 = K0 · (1 + ieff ) K1 = K0 · 1 + . m After cancelling K0 and rearranging, this results in   i m ieff = 1 + − 1. m

(7.3)

2 Let again be given the nominal interest rate i for the original (long) interest period. In this case, the interest rate ˆim for the (short) interest period of length m1 can be derived from the ansatz

Continuous compounding

35

 m i = 1 + ˆim − 1,

(7.4)

from which the relationship

ˆim = (1 + i) m1 − 1 =

√ m

(7.5)

1+i−1

results. This quantity is called the equivalent intra-year interest rate. Similar considerations as above can be made if one starts from the interest rate in the short period and asks for the corresponding interest rate in the long period. Example

A capital of 1000 € is invested over one year at 6% interest p. a. From the relationships (7.2) and (7.3) the following results are obtained for different values of m: m

Interest calculation

Final value K1(m) 1000 · 1.06  1000 · 1 +  1000 · 1 +  1000 · 1 +  1000 · 1 +

1

Yearly

2

Half-yearly

4

Quarterly

12

Monthly

360

Daily

 0.06 2 2  0.06 4 4  0.06 12 12  0.06 360 360

ieff (%) = 1060.00

6.00

= 1060.90

6.09

= 1061.36

6.14

= 1061.68

6.17

= 1061.83

6.18

◄ This concrete example shows that the final value after one year (and consequently the effective interest rate) increases the more often interest is paid or the shorter the short period. This also applies in general, as can be proven. This raises the question of whether the ever-increasing final capital is tending toward a limit and, if so, to which value if the periods become shorter and shorter (i.e., m1 → 0 or m → ∞). This means that interest is being paid every hour, every minute, every second or at every moment.

Continuous compounding We come to the problem of the continuous interest rate, also called instantaneous interest. Using the known limit value

36

7  Short and Long Periods

  i m lim 1 + = ei , m→∞ m

(7.6)

where e = 2.718 281 828 459 . . . is Euler’s number, the calculation rule for the final capital after the time t is

Kt = K0 · eit .

(7.7)

In this context, the quantity i is called interest intensity. In order to distinguish it from the nominal interest rate i, it is denoted by i∗. Continuous interest means that interest is paid at every moment proportionally to the current capital. The model of continuous interest is a useful theoretical construction, but it is also of great interest, for example, in calculating the value of options and in other financial market models, not least because the exponential function f (x) = ex has many good properties. The interest intensity i∗ and the corresponding effective interest rate ieff , which is to be applied for a one-off annual interest payment, are related through these two relationships: ∗

ieff = ei − 1,

(7.8)

i∗ = ln(1 + i).

(7.9)

Example

To what amount does a capital of 1000 € under continuous compounding with an interest intensity of 0.06 grow within one year? What is the effective interest rate? Relationship (7.7) directly gives the final amount K1∞ = 1000 · e0.06·1 = 1061.84, corresponding to an effective interest rate of 6.18%; see relationship (7.8). ◄

8

Bonds, Coupons, and Yields

Bonds are fixed-interest securities. They function as follows: When you buy them, you pay a price P, which is called the market value. If the security has the nominal value (par or face value) N = 100 monetary units, the market value corresponds to the price. Interest is paid annually (or at shorter intervals). At the end of the term, interest is paid plus a repayment R. In most cases, R corresponds to the nominal value, but sometimes there is also a repayment of a different amount. The amount of the interest rate, called coupon, is p (measured in percent, e.g., p = 5). We want to limit ourselves here to integer residual terms, so that the bond is purchased precisely on the interest date (but after interest payment).1 Furthermore, let N = R = 100, and the (remaining) term should be n. This leads to the cash flow shown in Fig. 8.1. A distinction must be made between the coupon and the market interest rate i. The latter is the interest rate currently customary on the markets for fixed-interest securities with a term of approximately n. Now, using the basic formula (G9), all coupon payments can be summarized, and using the basic formula (G4) all payments can be discounted to the point in time t = 0. Thus a present value comparison can be made (the price P is payable in t = 0, so it does not have to be discounted):

  1 (1 + i)n − 1 P= · p· + 100 (1 + i) i

1If

(G11)

the bond is bought or sold on any date, accrued interest must be taken into account.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_8

37

38

8  Bonds, Coupons, and Yields

Fig. 8.1   Inpayments and payouts of a standard bond

A more realistic, but also more complicated task is to calculate the yield rate i = i eff achieved from the bond at a given price. This effective interest rate usup ally differs from the coupon, so that in general the relation i = 100 holds. Unfortunately, it is not possible to resolve the relationship (G11) explicitly to i, so that numerical approximation methods must be used (see Chapter 9). p It should be noted that for P = 100 just i = 100 applies; one then speaks of papers traded at par. For P > 100 (above par) the rate of return is lower than the coupon, because the security is “expensive”; for P < 100 (below par) we have p , the paper is “cheap,” but promises a high coupon. i > 100 Example

For a bond with a term of n = 10 years and a coupon of p = 4, the fair value (i.e., the theoretical price) should be calculated at a market interest rate i = 2.25 %. Preliminary considerations: Since the coupon is higher than the market rate, the bond is worth more than a current investment would yield. Therefore its price must be higher than 100 (above par). If the given values are inserted into the formula (G11), the result is P = 115.52, a value that is significantly higher than 100 (especially because of the long maturity). ◄

9

The Cherry on the Cake is the Rate of Return

The rate of return or effective interest rate (two terms that are more or less synonymous) is the actual, uniform, average and, unless otherwise explicitly agreed, one-year interest rate underlying a financial agreement or investment or borrowing. It is this quantity in particular that serves to compare different payment plans, offers etc. and is therefore extremely important. It is not without reason that there is a legal obligation to always show the effective interest rate in financial contracts. Reasons why the rate of return (yield rate) or effective interest rate differs from the nominal interest rate may include fees, bonuses, and discounts on the disbursement of a loan, postponements of payments or a method of payment during the year. In practice, financial transactions such as loan agreements, payment plans or financing usually have a large number of the special conditions mentioned above, which means that a direct comparison is usually not possible. The only way is to calculate the rate of return. Unfortunately, with the exception of some special cases, these calculations belong to a relatively complicated type of tasks, because polynomial equations resulting from the equivalence principle (see below on p. 44) have to be solved, which is generally possible only approximately, but always with any degree of accuracy. For the mathematical layman: Test procedures like (systematic) trial and error are to be applied.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_9

39

40

9  The Cherry on the Cake is the Rate of Return

Trial and Error Procedures Typical for financial mathematics is the appearance of higher order polynomial equations. These have the following form: !

f (x) = an x n + an−1 x n−1 + . . . + a2 x 2 + a1 x + a0 = 0.

(9.1)

The function f on the left is called polynomial of degree n, because the highest power is n. The coefficients an , an−1 , . . . ,a2 , a1 , a0 are given real numbers, where an = 0. We are looking for a value (x0 or several values) with the property that, if we plug them into the function f, the function value is just zero:

f (x0 ) = 0.

(9.2)

The quantity x0 is therefore called zero. (It should be noted that if any value x˜ is plugged in (9.1), the function value f (˜x ) will normally be different from zero.) How can you find such a value x0? Most readers will remember at least two cases—the linear and the quadratic equation. In the first case a1 x + a0 = 0, the solution can be found by simple conversion: x0 = −a0 /a1 In the second case, the corresponding solution formula (p, q-formula) is certainly still well-known. There are some more cases where a direct solution is possible, but in general the value x0 (if such a number exists at all) can only be obtained by systematic trial and error (see Luderer 2015, Chap. 2). Of course, nowadays zeros are mostly calculated with the help of programmable pocket calculators or suitable computer programs, so we will not go into this in detail here.

The Equivalence Principle This could be, for example, “the debtor’s payments are equal to the creditor’s payments” or “the value of all inpayments is equal to the value of all payouts” or, slightly modified, “different payment methods (e.g., cash and financing for car purchase) are equally favorable”. This is of course based on a certain interest rate, which is either known or to be calculated. Since, as we know, the value of payments is always time-dependent, the date t = 0 is often chosen as the point of comparison, which leads to the present value comparison. Sometimes, however, it is easier to carry out a final value comparison. The equivalence principle is one of the most important tools for performing calculations and is the key to determining yield rates (effective interest rates). It leads in each case to a conditional equation, from which—depending on the values given—the remaining variables can be determined.

Examples for the Determination of Returns

41

Examples for the Determination of Returns 1. The simplest example of a return calculation is the basic formula (G3), i.e., Kt = K0 · (1 + i)t, if Kt, K0, and t are given. This situation occurs, for example, if we consider a zero bond (zero-coupon bond) where no interest payments are made in the meantime, but interest is accounted. In this case, we have Kt = 100, K0 = P is the price of the zero bond and t is its term. A simple conversion of (G3) then provides  100 t (9.3) ieff = i = − 1. P Example

A zero bond with a term of six years is sold at the price P = 92.18. Then its  100 yield rate amounts to ieff = 6 92.18 − 1 = 0.013664 = 1.37% ◄ 2. A financial investment runs over n years, whereby it is subject to different interest rates, or more precisely: in the k-th year it is subject to the interest rate ik. What is the return on this investment ieff if it is held for the full term of n years? From the comparison of the final values, using the basic formula (G3) the following result is obtained: K0 · (1 + i1 ) · (1 + i2 ) · . . . · (1 + in ) = K0 · (1 + ieff )n , (9.4) √ n which after a short transformation yields ieff = (1 + i1 ) · . . . · (1 + in ) − 1. Example

A savings bank offers an investment with a maximum term of five years, with dynamic interest rates that increase over time: the first year i1 = 1%, the second year i2 = 1.5%, the third i3 = 2%, then i4 = 2.5%, and finally√i5 = 3%. What is the return? Formula (9.4) gives the relationship ieff = 5 (1.01 · 1.015 · 1.02 · 1.025 · 1.03 − 1 = 0.019975 ≈ 2%. The obvious desire to simply take the arithmetic mean of all interest rates leads here (after rounding) to the same result, but is wrong from a financial mathematical point of view. ◄

42

9  The Cherry on the Cake is the Rate of Return

3. It is not uncommon to find savings plans that grant a bonus B at the end of the agreed term n in addition to the fixed nominal interest rate i. This naturally increases the effective interest rate. You can calculate this by equating the final value actually achieved with the final value that would be achieved if the same inpayments were paid, but the interest rate is the rate ieff to be calculated. Let R be the amount of the yearly deposits. Using now the basic formula (G7), we get:

R · (1 + i) ·

(1 + i)n − 1 (1 + ieff )n − 1 . + B = R · (1 + ieff ) · i ieff

(9.5)

Example

Mrs. A. concludes a savings plan which provides for her to pay in 1000 € at the beginning of each year for six years, with an interest rate of 2%. After six years, a 4% bonus is paid on all deposits. By using the relationship (9.5), we n 6 −1 + 0.04 · 6000 = 1000 · (1 + ieff ) · (1+iieffeff) −1, get the Eq.  1000·1.02 · 1.02 0.02 from which we obtain ieff = 3.05% by systematic trial and error (inserting confirms that the solution is correct). ◄ 4. In practice, it often happens that in loan agreements it is agreed that the annual annuity, specified by the initial redemption and the nominal interest rate, is divided by 12 and monthly payments in just this amount should be made. This of course has an impact on the annual effective interest rate, which is increased due to the fact that payments are made earlier. To determine it, one has first to calculate the monthly effective interest rate i12 from the ansatz

S0 =

A (1 + i12 )12n − 1 · 12 (1 + i12 )12n · i12

(9.6)

by trial and error, where A is the annual annuity calculated using formula (6.2). The annual effective interest rate is then calculated according to formula (7.3): ieff = (1 + i12 )12 − 1. Example

For S0 = 100, a nominal interest rate of i = 5% as well as the term n = 5, one calculates initially an annual annuity of A = 23.017451 and from this the

Examples for the Determination of Returns

43

monthly annuity A/12 = 4.6194902, from which one determines i12 = 0.0114 and ieff = 0.058314 = 5.83% (inserting the calculated values confirms their correctness). ◄ 5. In price calculation, the corresponding return i = ieff must often be calculated from the basic formula (G11) for a given price observed on the stock exchange, which is only possible with the help of a trial and error procedure. In other words, in the relationship (G11), the quantities P, n and p are now given, while i = ieff is searched. Example

We modify the example on p. 42 and increase the price from 115.52 to P = 120. Then we can determine i = ieff from the relationship (1+i)10 −1 1 + 100]. 120 = (1+i) 10 · [4 · i Preliminary considerations: Since the bond price is now higher than that calculated on p. 42, the return will be lower than 2.25%. This consideration is useful for choosing the starting value. Systematic trial and error leads to i = 0.1797 ≈ 1.80% ◄ 6. The effective interest rate calculation of loans is regulated by law in accordance with the new version of the German Price Indication Ordinance (Preisangabenverordnung, PangV) of July 28, 2000 (BGBl. I p. 1244). The following parameters play a role in this context: m, n

Number of individual payments of the loan or redemption payments

tk , tj′

Intervals, expressed in years or fractions of a year, between the date of the first loan disbursement and the date of disbursement of loan with the number k and repayment of principal number j, resp., where t1 = 0

Ak , A′j

Amount of the disbursement with the number k and the repayment of the principal with the number j, resp.

Approach for calculating the i = ieff annual effective interest rate: m  k=1

n  A′j Ak = ′ . t (1 + i) k (1 + i)tj j=1

(9.7)

The annual effective interest rate is calculated either algebraically (if possible) or by a trial and error method.

44

9  The Cherry on the Cake is the Rate of Return

Example

The loan amount is 4000 €, but the lender retains 80 € for processing costs, so that the payout amount is 3920 €. The loan will be disbursed on February 28, 2021. The borrower has to repay the following installments: 30 € on March 30, 2021, 1360 € on March 30, 2022, 1270 € on March 30, 2023, 1180 € on March 30, 2024, 1082.50 € on February 28, 2025. From the equation

3920 =

30 (1 + i)

1 12

+

1360 (1 + i)

13 12

+

1270 (1 + i)

25 12

+

1180 (1 + i)

37 12

+

1082.50 48

(1 + i) 12

the solution i = 0.09958 . . . ≈ 9.96% is determined with the help of a suitable numerical test procedure. ◄

What Else is Important?

10

In this text, the key points of classical financial mathematics were presented in an overview. Of course, there are many other topics which are particularly important in practical application. At least some of them will be briefly discussed here: • The reader could rightly ask the question why one should even bother with financial mathematics if there has been almost no interest for several years, and even negative interest (penalty interest) has to be paid (especially in payment transactions between banks and central bank). But that is only half the truth, because firstly, interest rates are—historically speaking—always subject to fluctuations, there are low and high interest rate phases. Many loans, such as municipal loans, have a very long maturity. A (high) interest rate that was fixed, say, 30 years ago is still valid today. Secondly, debit interest (e.g., for an overdraft facility or a line of credit) is always significantly higher than earned interest. Thirdly, there are market segments (such as corporate bonds) where interest rates of up to 10% are by no means uncommon. And fourthly and finally, one should not only look at United Kingdom or Europe. • In financial mathematics, a fixed interest rate is always assumed. In practice, the interest rate usually depends on the term. In this context, the terms yield curve and spot rates play an important role. • In classical financial mathematics, interest is quite naturally regarded as the remuneration for the provision of capital. Historically, however, there have been prohibitions on interest at various times and in various regions of the world. Almost all religions know such regulations. Admittedly, at all times a variety of tricks were used to circumvent such prohibitions. • In classical financial mathematics, the calculation of final values is always based on an underlying nominal interest rate; inflation, that is, the reduction

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6_10

45

46

10  What Else is Important?

in the purchasing power of money, does not exist. Therefore, when making practical considerations, especially of a long-term nature, one should also take inflation into account. The easiest way to do this is to use the real interest rate, that is, the difference between the nominal interest rate and the inflation rate, instead of the nominal interest rate in the calculations. However, if the term is very long, more precise methods should be used (see e.g., Korn and Luderer 2019, p. 47 f.). • In classical financial mathematics, all payments are always regarded as secure. Anyone who once owned Greek government bonds or bonds issued by a company that had to file for bankruptcy knows that this assumption is deceptive. • Multiple payments in the annuity calculation are classically assumed to be constant. In practice (e.g., in actuarial mathematics), however, dynamic (i.e., increasing or decreasing) payments often occur. • In today’s (scientific) language use, the term financial mathematics is usually understood in a completely different, more general way than described here. In modern financial mathematics, stochastic models are at the center of attention, be it in the evaluation of uncertain events, in interest or stock price forecasts or in the calculation of the fair value of various financial products, called derivatives (options, certificates, swaps, futures etc.). Without classical financial mathematics, however, all the models examined there would be inconceivable.

What You Learned From This essential

• The time-dependence of the value of a payment is the linchpin of all classical financial mathematics, basically all formulas are based on it. • The calculation of interest is based on relationships which, from a mathematical point of view, can be assigned to arithmetic or geometric sequences of numbers and number series. • The equivalence principle serves as a starting point for calculating the rate of return or effective interest rate, resp., of a financial agreement. This involves comparing different payment schedules or all inpayments and all payouts, or comparing the creditor’s payments with those of the debtor. In order to apply the equivalence principle correctly, it is useful to make a graphical representation of all inpayments and payouts to illustrate when payments are due. • Classical financial mathematics is the starting point for a variety of problems of modern stochastic financial mathematics and actuarial mathematics. Aspects beyond classical financial mathematics are legal regulations, inflation, uncertain payments, supply and demand on the financial markets, chance, and numerous others.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6

47

Basic Formulas

Final value formula for linear interest:

Kt = K0 · (1 + i · t)

Present value formula for linear interest:

K0 =

(G1) (G2)

Kt 1+i·t

Final value formula for geometric interest (Lei- Kt = K0 · (1 + i)t bniz’s compound interest formula):

(G3)

Present value formula for geometric interest:

K0 =

(G4)

Annual replacement installment (payments in advance):

R = r · (12 + 6.5 · i)

(G5)

Annual replacement installment (payments in arrears):

R = r · (12 + 5.5 · i)

(G6)

Final value formula of the advance annuity calculation:

E adv = R · (1 + i) ·

Present value formula of the advance annuity calculation:

Badv = R ·

(1+i)n −1 (1+i)n−1 ·i

(G8)

Final value formula of an annuity in arrears:

E arr = R ·

(1+i)n −1 i

(G9)

Present value formula of an annuity in arrears:

Barr = R ·

Price formula:

P=

Kt (1+i)t

1 (1+i)n

(G7)

(1 + i)n −1 i

(G10)

(1+i)n −1 (1+i)n ·i



· p·

(1+i)n −1 i

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6

+ 100



(G11)

49

References

Cissell R, Cissell H, Flaspohler DC (1990) Mathematics of finance. Houghton Mifflin, Boston Brown R, Zima P (2011) Mathematics of finance (Schaum’s Outlines), 2nd edn. McGrawHill Education, New York Garrett S (2013) An introduction to the mathematics of finance: a deterministic approach, 2nd edn. Butterworth-Heinemann, Oxford Grundmann W, Luderer B (2009) Finanzmathematik, Versicherungsmathematik, Wertpapieranalyse. Formeln und Begriffe, 3. Aufl. Vieweg + Teubner, Wiesbaden Heidorn T, Schäffler C (2017) Finanzmathematik in der Bankpraxis: Vom Zins zur Option, 7th edn. Springer Gabler, Wiesbaden Hull JC (2018) Options, futures, and other derivatives, 10th edn. Pearson Education India, Chennai Jacques I (2018) Mathematics for economics and business, 9th edn. Pearson, London Korn R, Luderer B (2019) Mathe, Märkte und Millionen: Plaudereien über Finanzmathematik zum Mitdenken und Mitrechnen, 2nd edn. Springer, Wiesbaden Luderer B (2015) Starthilfe Finanzmathematik. Zinsen – Kurse – Renditen, 4. Aufl. Springer Spektrum, Wiesbaden Luderer B (2017) Mathematik-Formeln kompakt für BWL-Bachelor. Springer Gabler, Wiesbaden Luderer B (2017) Facetten der Wirtschaftsmathematik. Eine unterhaltsame Einführung ganz ohne Formeln, Springer Spektrum, Wiesbaden Ortmann KM (2017) Praktische Finanzmathematik: Zinsrechnung – Zinsanleihen – Zinsmodelle. Springer Spektrum, Wiesbaden Pfeifer A (2015) Finanzmathematik: Das große Aufgabenbuch (mit herausnehmbarer Formelsammlung). Europa-Lehrmittel, Haan-Gruiten Pfeifer A (2016) Finanzmathematik: Lehrbuch für Studium und Praxis. Mit Futures, Optionen, Swaps und anderen Derivaten, 6. Aufl. Europa-Lehrmittel, Haan-Gruiten Sydsaeter K, Hammond P et al (2016) Essential mathematics for economic analysis, 5th edn. Pearson, London Tietze J (2014) Einführung in die Finanzmathematik. Klassische Verfahren und neuere Entwicklungen: Effektivzins- und Renditeberechnung, Investitionsrechnung, Derivative Finanzinstrumente, 12. Aufl. Springer Spektrum, Wiesbaden © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 B. Luderer, Classical Financial Mathematics, Springer essentials, https://doi.org/10.1007/978-3-658-32038-6

51