Mathcad and financial pyramids

Valery Ochkov

On first page of the Russian newspaper Information-expertise there is the heading A firm with blotted reputation. On last page of this newspaper new heading Computer a life has just appeared. This article can be typed in both of these headings.

Many problems that we solve on computer are financial ones. These problems are connected with managing the budget of a family, a firm or a plant etc

So, a computer can help managing financial problems, saving them and increasing minimum by three ways:

1.     On computer one sets an electronic version of an account book. In this book one writes all costs and incomes. For these purposes we use the programs Excel, Lotus 1-2-3, QuattroPro and so on (Spreadsheets).

1.     The computer is connected to banking electronic network so that every housekeeper or an accountant can conduct all necessary payments without going out.

1.     On a computer one can simulate and calculate some financial operation (purchase of bonds or shares, open account or take credit in a bank and so on) and look the result of our operations.

Among the examples that Excel includes there is a problem connecting with purchase of securities. It is calculated how many and what shares we have to buy for the sum of future dividends would be maximum one. For buying shares we have limited quantity of free money.

Among the examples that Mathcad includes there is a problem connecting with simulation of development of epidemic. It is known start number of healthy and ill men and then it is calculated how epidemic develops, that is how number of ill men varies every day.

We try to combine these two problems and to get new problem connecting with simulation of development of financial pyramid. This pyramid is similar an epidemic. We do this in Mathcad, that suits for these purposes. Besides in Mathcad we can use electronic textbook Personal Finance. With help of this textbook we can learn to manage finances. And if we need we can transfer formulas and numbers from this textbook to our document.

Author possessed experience of participation in financial pyramid. But it was long ago. That experience was innocent without some consequences. At school someone suggested to give all money (it was 20 kopecks for the dinner, that parents gave their children) every participant of this financial company by turns. One could see in this idea old as the world childrens dream: If every adult had given me only one kopeck I would have had the whole capital. But the adult did not notice that. But in class was next situation: participants, who gave money, left the game. The game slowly finished and someone (among them the author was too) remained with a nose. "One wanted to be better but received as always ".

Now days, hearing about the crash of Russian join-stock companies, Author remembered childrens dream, schools money and not only them.

Banking system of any country consists of three numbers. The first number is N1. It is the payment of a credit. You took in bank 100 rubles, at the end of one year you have to return 100 + N1 rubles. The second number N2. It is money contribution. You put 100 rubles in a bank, at the end of one year you got 100 + N2 rubles. Difference between the first and the second numbers (N1 > N2) makes banks to work profitable. The third number N3, obtained from below two first numbers, makes people to carry money to banks. This number is the value of inflation. In normal economic situation low level of inflation and low-level payment of a credit hold in narrow limits money contribution:

N1 > N2 > N3.

If the inflation makes 1000 and more percents annually, then a lot of people forgetting about abnormality such situation easily believe in 1000 and more money contribution annually (the value N2 has to be more than the value N3) and lay down in base of next pyramid, if the legislation of the country allows to build these pyramids. There are clever people who understand that the pyramids is a kind of a game. The main thing of this game is to run in time .

So, we build the financial pyramid.

1. Initial conditions

Number of people in a town

Daily expenditure (rubles) for building the pyramid

Mean time between purchase and sale a share (days)

Coefficient of stock-jobbing

Norm of profit

2. Position for the first day

Initial capital (rubles)

Number of people who buy the shares for the first day

Common number people who buy the shares for the first day

The profit for the first day (rubles)

3. Simulation of development of the pyramid

Rate of sale of the shares for the D-th day (rubles)

Rate of purchase of the shares for the D-th day (rubles)

The development of the pyramid by the days for the year

NKD number of the shares that were bought per the (D+1)-th day

SNKD common number of the shares that were bought per (D+1) day

Number of the shares that were sold per the(D+1)-th day

Money in cash department per the (D+1)-th day

One earns money for the (D+1)-th day

4. Waves of buyers (solid) and sellers (dotted) of the shares

5. Dynamics of change of quantity of money in cash department (rubles)

6. Dynamics of change of incomes (rubles)

7.Determination of the day-X: maximum income (rubles) for the181-th day

In point 1 of the previously mentioned minutes of work in Mathcad the constants its name, sign of appropriation (:=) and its value are determined. The commenting (blue color) explain them.

In point 2 the condition of the pyramid is determined for the first day. We enter three index variables that are first values of three vectors M, NK, SNK and MMM.

In point 3 dynamics of change of courses of sale and purchase of the shares is written. That are the functions P(D) and K(D). About extension of the shares (tickets) by a rating in 100 rubles with next course of sale P and purchase K is declared:

The table 1

The day that passed after beginning emission of the shares (tickets)









Sale (rubles)









Purchase (rubles)









From the table 1 one can see that purchased shares can give the dividend in 723% annual at the nominal price in 100 rubles. If the rate of inflation is high enough then people believe in reality of such huge dividend and the pyramid grows. But almost all feel danger of crash of this invention that is why they do not give their money for one year. For instance, they can give for 50 days (variable Time is the mean time between sale and purchase of the share. See point 1). During this period for every shares one can weld on magic 100 rubles figuring in many proverbs and by-words.

Then the vectors NK, SNK, NP, M and MMM are filled by simple incremental scheme: it is known the previous value of element of the vector (for the day D). It is calculated its next value (for the day D+1).

In a town where we build the pyramid there million people (N see the point 1). Among these people the certain stock-jobbing fly. This stock-jobbing preheats with help of the above-stated table of courses. With help of the mathematics we can describe it by the formula. This formula connects number of sold shares per concrete day (NK) with common number of sold shares (SNK) and with conditional number of the people who have not purchased while the shares (N-SNK). I want to repeat that the development of the pyramid reminds us the development of an epidemic. Number of falling ill people (people who buy shares) per concrete day is proportional number ill men in a town multiplied on number of healthy men (people who have not bought the shares yet). If we look through the epidemic the constant of proportionality depends on measure of preventive maintenance. If we look though the financial pyramid this constant (we call it conditionally constant of stock-jobbing KA) depends on rate of inflation, advertising, existence other pyramids, period that passed after noisy crash previous pyramid and so on. Many economic phenomenon (crisis, bankruptcy) drive on waves. Period of top of waves of financial pyramids makes by different estimations from 25 till 30 years. It connects: at first, with arrival to fissile life by fresh, not crumpled pyramids of forces, for the second, with short human memory. On such wave financial wreck waits many people. But other people like a brave artful surfer get financial satisfaction.

In our model of the financial pyramid vector formula (two expressions enveloped by brackets) allows to calculate number of the shares (NK), that will be sold tomorrow (D + 1). We can do it with help of today figures (D).

After the wave of people who buy the shares goes the wave of people who want to sell the shares (to turn back their money) and to get their dividend. Here we simplify the model too. We are consider that the wave of people who sell the shares lags for 50 days the wave of people who buy the shares.

Well, now we can calculate our money and roll on waves of the financial pyramid.

It is simple to calculate how much money (M) will be on a score of the organizers of the pyramid tomorrow (D + 1), if we know how much money we have today (D) and if we know the course of the shares and quantity of purchases and soles.

People, who buy the shares, bring money to the cash department. People, who sell the shares, take money from the cash department. But there is one man who gets in the cash department. This is the organizer who has own advantage. He takes from the cash department three percents every day.

Naturally the income is excepted if there are money in the cash department.

In real life, of course, the cash department grows thin for the considerably huge sums taxes, the payment of current expenses, advertising etc. For us it is 300 000 rubles per day.

In 1202 Leonardo from Piza ( ) (1180-1240) one of maiden described the model of development a closed biological system, settled by the conditional rabbits. If in appropriate way to define their fertility and longevity, the number of a population of the rabbits will vary from generation to generation under the stringent law:

The table 2












Number of rabbits











Reader, of course, has already guessed, that we speak about Fibonacci numbers: the name is well know as Fibonacci (Fibonacci is short form from filius and Bonacci (Bonaccis son)). In new generation of rabbits their number will equal the sum of number of rabbits in two previous generations. In due about the Fibonacci rabbits one forgot, but Fibonacci numbers (1, 1, 2, 5, 8, 13 and so on) found application in applied mathematics (see

Our model of development the pyramid also allows to get some numeric series. This numeric series images the condition of the income of the organizers of this financial operation:

The table 3


Income (Up to rubles are rounded)



70 000 000

Beginning of the pyramid


72 100 000



74 128 022



76 086 203






311 700 019



311 780 298

The day-X


311 569 396






2 990 028

Last money


- 6 984 025

The debt hole




We call numbers of the second column of the table 3 the numbers. Well hope that in due one will forget about financial pyramids. But the Mavrody () numbers will enter the history. Especially, Sergej Mavrody ( ) is a mathematician on formation.

In points 4-6 we map graphically the development of the pyramid. One can calculate the day-X , when the income of an organizer is maximum (in our case it is the 181-th day see point 7), and the time, when the pyramid has to be destroyed. That is to leave on floor, to run for in the deputies or to leave abroad. Thats great you have enough money for it (about three hundred millions at an only three-percentage rate of profit). Now one can play the financial pyramid, for example, to change initial conditions and to keep track of the change of dynamics of the incomes.

For the present time we do not go somewhere, but stay with our computer. And before well next money in some doubtful firm, at first well calculate what shall we get in our future. This way we can easily turn back and increase money spent on the acquisition of the computer.