Money and Mathematics

If a retailer takes over the VAT, then the actual discount granted is different, where the discount is smaller.

Like winning the lottery is synonymous with improbability, and yet there are winners almost every week. We explain why and propose a strategy for improving the chances of better gains.

By what percentage must a share price rise again to return to its old level after a price slump?

In many problems of financial mathematics, the solution cannot be achieved by formula conversion, but only by numerical solution methods. A simple method of this kind is clearly presented.

To calculate the yield of bonds or other financial products, polynomial equations usually have to be solved. It is important to know how many (positive) zeros they have.

Even 500 year old tasks of the German reckoner Adam Ries are still of great interest today. They show how practice-oriented this medieval master of arithmetic was, who can quite rightly be called a business mathematician.

What is the best way to invest in fund savings plans in order to achieve a high return?

Law texts can also contain mathematics. In this story, the not so simple calculation of income tax serves as an example.

The phenomenon of cold progression is studied in detail on a simplified tax scale.

In many everyday situations one cannot avoid interest calculations. This story is about linear interest.

An interesting, practice-relevant example of linear interest is the discount on an invoice amount.

The most common form of interest in daily life is the compound (geometric) interest. An easy way to understand introduction is given.

A frequently asked question is that of doubling a capital at a given interest rate.

In the real development of a capital one has to take into account not only the interest rate but also inflation, which is usually forgotten or simply not considered.

A very important factor in financial mathematics is time. The value of a deposit or a withdrawal always depends on the point of time when it is made. Not taking into account or not correctly including the date inevitably leads to wrong results.

A descriptive introduction to redemption calculation using the annuity amortization is given. In practical problems, one often encounters results, which are surprising for a layperson, e.g. concerning the time until full repayment or the total sum of all payments.

Also in novels one occasionally founds extremely interesting financial mathematical problems. For example, from a narrative written by the famous Russian novelist Chekhov one can deduce several problems of annuity calculation.

Why do nominal interest rates usually differ from effective interest rates? Sub-annual interest calculation or payments during year are possible reasons.

Financing offers from car or furniture dealers often sound very attractive. But are they really always beneficial?

The analysis of cash flows is well suited for the calculation of yields. However, the calculation is not always easy.

In financial mathematics, the present value concept plays a central role. It is often not easy to understand it correctly because the time factor is underestimated and the timing of payments is not taken into account.

Does a zero-percent financing really always mean that the underlying interest rate is zero? An answer can be found in the calculation of yields.

We give a vivid introduction to continuous interest, which is often used in financial market models.

The financial product bond is examined in detail. Important terms are explained in a clear way.

In the legal text of the German Price Indication Ordinance, the mathematically interested person will find detailed formulas and calculation rules for determining the effective interest rate of loans. Central terms here are the equivalence principle and the comparison of present values.

What is the difference between fair (i. e. theoretical) and practical prices of financial products?

Interest rates generally depend on the maturity of the considered products. While in classical financial mathematics the interest rate is usually assumed to be a fixed value, one encounters numerous situations where more detailed considerations are needed. The concept of the yield curve plays an important role here.

The actors at the financial markets often speak of plain-vanilla products. This notion refers to products that are as simple as possible and do not have any special features.

A swap is an interesting and often used financial instrument, which consists in the exchange of different cash flows for mutual benefit.

In connection with swaps, the swap rate is important as the interest rate to be paid. The calculation of the swap rate leads to an interesting mathematical phenomenon—the telescope sum.

In order to construct yield curves, one needs the corresponding spot rates. In this story we describe how spot rates can be obtained from swap rates observable at the financial markets.

The value of a bond or an other fixed-interest security changes when market interest rates change. So-called risk indicators are used to describe these changes.

A frequently used risk measure is duration. This quantity has several interesting features such as the immunization property.

Certificates are financial products that are often not easy to understand. The mathematical calculation of key parameters are complicated, too.

Forward transactions serve both hedging and speculation purposes. The sale of crop of standing corn is one of the historically first commodity futures transactions. In his novel “Buddenbrooks” Thomas Mann impressively describes how such transactions can end.

Not everyone who enters into commodity futures transactions actually wants to purchase the goods. Often, buying and selling (closing out) are merely for speculation.

An arbitrage possibility describes the situation if one can profit from (usually very small) price differences at different markets. In theoretical studies it is usually assumed that such possibilities do not exist. Under this assumption, fair (theoretical) prices of complicated financial products can be determined.

In practice, price differences can occur at different markets, at least for a short time. By taking advantage of arbitrage, one may be able to benefit from this phenomenon.

Technical Analysis is a methodology for predicting price trends through the analysis of previous market behaviour. Fibonacci numbers play an important role in this.

Options are conditional forward transactions. Different types of options are presented in this story.

The terms of yield, risk and liquidity as well as arbitrage, hedging and speculation are closely related to each other.

Does the financial market live only on gut decisions and speculation? We explain that simple principles can even lead to the Nobel Prize.

The Black–Scholes formula is a famous relation for calculating option prices. This is explained in this story using a simple example.

We present principles of stock-price modelling with the help of the binomial model as the simplest but popular stock price model used in practice.

In addition to speculation, the strategy of hedging is a common method of investing money.

The fact that it is sometimes possible to arrive at the right result even with a wrong calculation is shown using the example of the risk indicator delta for stock options.

The price of options changes when certain influencing factors change. So-called risk indicators are used to describe these price changes.

Each professional discipline has its own language. Thus, the actors at the financial markets often use terms that are unusual for the layman.

The term of volatility is one of the most commonly used mathematical terms in options trading. The fact that it is sometimes used ambiguously leads to confusion, which we clarify in this story.

“Only with options you can make fast money, and mainly when you buy options with a big leverage.” What sounds like a stock market wisdom is examined in detail in this story. And a big lever can also lead to really fast losses.

A brief introduction to portfolio optimization and the Markowitz model is given.

“You once wanted to explain to me how to get as rich as possible with mathematics. Or did you want to keep it to yourself?,” is a question that a financial mathematician is often asked. And of course, the financial mathematician’s answer will be more detailed, referring to profits and losses and pointing to balancing opportunities and risks, e. g. with the help of the Markowitz approach.

Why do investors look for shares that develop in opposite directions? We explain the attractiveness of negative correlation and also why Markowitz received the Nobel Prize.

Beat the market and still risk nothing is an unattainable goal. But staying above a given asset level and still participating in equity gains is the most attractive feature of the CPPI strategy explained here.

Why do teams with extremely risky strategies almost always win at stock market games? Why is it not worth it for yourself? We explain why a cautious approach to risk is usually more recommendable when playing with real, own money.

Probability theory is a young branch of mathematics and was only accepted in Europe as a fully-fledged mathematical subject from the second half of the last century onwards. Conversely, entire branches of the finance and insurance industry would be unthinkable without two of the most prominent results of probability theory—the Law of Large Numbers and the Central Limit Theorem.

“Not life insurance! It’s absolutely outdated and it doesn’t give you a good return either.” That is a common saying these days. We show that the product of a classic life insurance is not at all that simple and has some features that are attractive to customers, especially if you can take advantage of the German Riester premium.

Modern Riester products often contain strategies with which one can participate in the stock market but still have the guarantee of having all paid-in premiums and received subsidies available at the beginning of the pension phase. We describe dynamic three-pot hybrids.

Often one reads or hears about the application of so-called Monte Carlo methods in extensive calculations in the finance and insurance industry. Should it really be the case that money is put at risk by a kind of roulette in the truest sense of the word?

Everyone has an insurance, be it only health insurance, liability insurance, or automobile insurance. If the premiums paid in by the insured in Germany are added up, it is easy to see that the sum reaches the billion euro range. We present some impressive figures.

Opportunities and risks play a decisive role in old-age provision products. With the help of the opportunity-risk-class, a number is presented that should allow the end consumers to decide about the pension product that fits their needs best.

The most important variable for the calculations of a life insurer is the expected (remaining) life time of a policyholder. The more accurately the insurer can predict it, the better it can estimate its financial risks in terms of its future payment obligations. A suitable life table is used for this purpose.

Sometimes one can find interesting financial mathematical problems in the literature. The great French novelist Balzac, for example, had an excellent knowledge of bills of exchange or life annuities, while finance ministers and bankers at that time were very creative in generating money.

Index participations are a modern Riester pension product. We explain how the product works and describe the role of the cliquet option, which is typically purchased once a year.

A detailed description of the most important parts of classical financial mathematics is given. For linear and compound interest, special attention is paid to the final and the present value. Moreover, sub-annual and continuously compounded interest are described. In annuity calculation we consider annuities in arrears and in advance as well as the percent annuity. Finally, in price calculation the price of a general cashflow as well as the price of a (zero) bond are derived.

We give an overview of basic notions from probability theory including discrete probability distribution and probability distributions with density. We explain the Law of Large Numbers, the central Limit Theorem as well as the Theorem of de Moivre-Laplace. Moreover, stochastic modeling of stock prices is described with a special focus on geometric Brownian motion. Further, we give an insight into mathematical foundations of option pricing (e.g. the risk-neutral market model and the Black-Scholes formula).