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In Marc Artzrouni’s paper « the mathematics of ponzi scheme” (2009), I’m having some difficulty understanding the logic of an equation. It is assumed by the author that the continuous cash inflow is that of exponential growth: $$S(t)=S_0e^{r_it}$$ Where S0 is the initial density of the deposits and ri is the investment rate. Subsequently, he uses the data available on Ponzi’s original scheme to crudely fit the model. The parameters S0 et ri for the density of new investments S0 exp(rit) are estimated on the basis of information on deposits made between the first day (December 26,1919, t=0) and last day (July 26, 1920, t* = 217/365 = 0.58) of the scheme’s history. We also know that on the last day Ponzi collected 200,000 dollars for a total of 10 million deposited over the seven-month period. Under the assumption of exponential growth, and with the year and a million dollars as the time and monetary units, the parameters S0 and ri must satisfy: $$S_0e^{r_i0.58}=0.2\times365,$$ $$S_0\frac{e^{r_i0.58}-1}{r_i}=10$$ Which yields S0 = 1.130 and ri = 7.187. This value of So translates into an initial flow S0/365 of $ 3095 a day, says the author. My problem is that I have no idea why the right term in the above equation is multiplied by 365. Could someone explain it to me, please?

Lio
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Aug 15 '22 at 10:05
  • For me, the first equation in the system of the two equations to solve should be: S0e(ri0.58) = 0.2 not S0e(ri0.58) = 0.2 times 365. Where does the "times 365" part come from? How to justify this? I don't know precisely. – Lio Aug 15 '22 at 12:40

1 Answers1

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It explains why it's multiplied by 365 in the sentence before the equation.

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It is because $s_0$ is in units of million-dollar years, and the $\$200,000$ dollars is the amount accumulated in one day. In other words $\$200,000$ dollars a day is equivalent to $365\times\$200,000$ dollars a year.

Suzu Hirose
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  • Why not just use the equation : Soexp(0.58ri) as defined previously? Why he adds "times 365"? How does he get that? And why it disappears when he integrates (second equation)? – Lio Aug 15 '22 at 12:30
  • Why do you think I know the answers to those questions? – Suzu Hirose Aug 15 '22 at 12:36
  • You don't have to answer if you don't know. I didn't see the end of your answer. It helps me to understand. Thank you. – Lio Aug 15 '22 at 12:45