Mathcad and financial pyramids
Valery
Ochkov
On first page of the Russian newspaper “Informationexpertise”
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 123, 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
children’s 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
joinstock companies, Author remembered children’s dream, school’s 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
lowlevel 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 stockjobbing
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 Dth day (rubles)
Rate of
purchase of the shares for the Dth day (rubles)
The
development of the pyramid by the days for the year
NK_{D} –
number of the shares that were bought per the (D+1)th day
SNK_{D}
– 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 dayX: maximum income (rubles) for the181th 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) 
1 
2 
3 
... 
51 
... 
365 
... 
Sale (rubles) 
105 
107 
109 
... 
205 
... 
833 
... 
Purchase (rubles) 
100 
102 
104 
... 
200 
... 
828 
... 
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
bywords.
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
stockjobbing fly. This stockjobbing preheats with help of the abovestated
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 (NSNK). 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 Â ñëó÷àå ôèíàíñîâîé ïèðàìèäû ýòîò êîýôôèöèåíò (we call it conditionally constant
of stockjobbing – 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 (Ëåîíàðäî Ïèçàíñêèé) (11801240) 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
Generation 
1 
2 
3 
4 
5 
6 
7 
... 
27 
... 
Number of rabbits 
1 
1 
2 
3 
5 
8 
13 
... 
196418 
... 
Reader, of course, has already guessed, that we speak
about Fibonacci numbers: the name Ëåîíàðäî Ïèçàíñêèé is well know as Fibonacci (Fibonacci – is short form from filius and
Bonacci (Bonacci’s 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 – http://www.sdstate.edu/~wcsc/http/fibhome.html).
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
Day 
Income (Up to rubles are rounded) 
Remark 
1 
70 000 000 
Beginning of the pyramid 
2 
72 100 000 

3 
74 128 022 

4 
76 086 203 

... 
... 

180 
311 700 019 

181 
311 780 298 
The dayX 
182 
311 569 396 

... 
... 

228 
2 990 028 
Last money 
229 
 6 984 025 
The debt hole 
... 
... 

We call numbers of the second column of the table 3 the Ìàâðîäè numbers. We’ll hope that in due one will forget about financial pyramids. But the
Mavrody (Ìàâðîäè) numbers will enter the history. Especially, Sergej Mavrody (Ñåðãåé Ìàâðîäè) is a mathematician
on formation.
In points 46 we map graphically the development of
the pyramid. One can calculate “the dayX ”, when the income of an organizer is
maximum (in our case it is the 181th 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.
That’s great you have enough money for it (about three hundred millions at an only threepercentage 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 we’ll next money in some doubtful firm,
at first we’ll 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.