Assumptions, mathmatical basics and goals

The process of modelling usually includes the translation of a real world problem to a mathematical problem. We start the modelling cycle by giving assumptions:
  • We are credit-worthy for all appearing credits.
  •  There exist only four credits and one key interest rate $p$ (think of Euribor)
  •  Credit 1: The credit interest rate $p_1$ equals the key interest rate $p$ in every interest rate period.
     Credit 2: The credit interest rate $p_2$ is 1\% in the first period, 2\% in the second period and in the following periods $p_2$ equals $p$.       Credit 3: The credit interest rate $p_3$ takes only values in a certain interval $\left[2;7\right]$. If the key interest rate is below 2, then $p_3$ equals $2$ and if the key interest rate is above 7, then $p_3$ equals 7, otherwise $p_3$ equals $p$.             Credit 4: This credit is a foreign currency loan, particularly in Swiss franc. We assume that this credit depends on the key interest rate $p$ as well. In the real world this is not the case, because one takes his debt in Swiss franc, so the credit interest rate has to depend on the Swiss-Libor. As expected, the exchange rate is important on such forms of credits. We denote $er$ as the exchange rate which says how many Swiss franc we get for one Euro. %Then $\frac{1}{wk}$ denotes how many Euros we get for one Swiss franc. We don't distinguish between bid and ask price and assume $er$ to be constant over the lent term.  
  • For convenience the interest rate period is one year.
  • The key interest rate is variable, but constant over the whole credit term.
  • The value of the instalments, the annuities, is constant over the whole credit term and accounts for 8 400 Euro. One pays such amounts in arrears. In other words, at first the debt level will be raised by interest rate and then it will be reduced by paying the annuity amount.
  • We neglect all kind of charges and taxes related to the borrowing.
Now, we shortly introduce the most important mathematical tool for this article the "repayment-equation" and assume that the reader knows the following equation very well. We identify important variables and denote $S$ for the start level of debt, $S_n$ for the debt level after $n$ years, $R$ for the yearly instalments and $p$ for the value of the key interest rate in percent. The recurrence relation for $S_n$ is: \begin{equation} \end{equation} One finds an explicit version for $S_n$ below: \begin{align}      S_n=S\cdot \left(1+\frac{p}{100}\right)^n-R\cdot \frac{\left(1+\frac{p}{100}\right)^n-1}{\frac{p}{100}} . \end{align} Which criteria are needed to mark the best credit? One assumption assures the constancy of the instalments over the whole credit term. Then it's almost obvious, the best credit is the one with the shortest lent term. In the following we are going to simulate the process of the four different credits. An interest rate of a credit depends on the key interest rate $p$, so the value of $p$ represents the essential feature and is the one to be modelled. The modelling takes place in a GeoGebra applet. We start with deterministic and end up with probabilistic considerations.