Download:Ponzi Financial Dynamics
The Financial Dynamics of Unsustainable Fraudulent Investment Schemes
Carlos A. Abadi
June 3rd, 2019
We use a first order differential linear equation to describe the dynamics of an investment scheme that promises more than it can deliver (a Ponzi scheme). The model is based on (i) a promised, unrealistic, rate of return, (ii) the rate at which new investments are gathered, and (iii) the withdrawal rate. We establish the parameters that lead to the scheme’s collapse.
The principle of Ponzi schemes is simple: they entice potential investors by promising high returns, which they cannot possibly deliver. The only was to pay the promised returns is by attracting new investors whose money is used to subsidize the returns of those already in the fund.
In this article we posit a simple model that attempts answer the basic questions: How fast must new investments come in? How long can the scheme last? What parameters drive the dynamics of the scheme?
The model is described in Section 2, with detailed results on the behavior of the scheme as a function of seven parameters.
Section 3 highlights the main results and concludes.
We assume that the fund starts at time with an initial investment , followed by a continuous cash inflow . We next assume a promised rate of return, and a nominal rate at which the money is actually invetsed. If then the fund is legitimate and has a profit rate . If the fund is promising more than it can deliver. We call the promised rate, , the “Ponzi rate”.
We need to model the fact that investors withdraw at least part of their money along the way. The simplest way of doing this is to assume a constant withdrawal rate, , applied at every time to the promised accumulated capital. The withdrawal rate at time by those who invested the initial amount is . If is less that the promised return , then these withdrawals increase exponentially; can also be larger than , in which case withdrawals decrease exponentially, as these investors are eating into the capital .
In order to calculate the withdrawals at time from those who added to the fund between times 0 and , we note that those who invested at time will want to withdraw a quantity at time . Integrating these withdrawals from 0 to and adding the previously calculated withdrawals from the initial deposit , results in the following withdrawals at time :
We note that the nominal interest rate, , does not appear in : withdrawals are a function of only the promised rate of return, .
The differential Equation
If S(t) is the amount in the fund at time , then is obtained by adding to , the nominal return , the inflow of fresh money , and subtracting the withdrawals :
For , the amount is the solution to the first-order linear differential equation:
We let be the initial condition which may or may not be equal to , the initial investment made by customers. The fund managers can make an initial “in-house” investment , which will also be invested at the nominal rate . In this case, the initial value is greater than . An initial condition formally corresponds to the cae where, for some reason (theft or other), a fraction of the initial investment is not available. We will see later that the solution to the differential equation with an initial condition , other than , will be used when there is, at some subsequent time , a sudden change in parameter values (for example, if the cash inflow or withdrawal rate changes at ).
One often made assumption in the literature on the subject is that the cash inflow grows at an exponential rate:
were is the initial density of the cash inflows and will be called the investment growth rate. Then, Eq.(1) becomes:
where the fraction should equal when .
The solution, , to the differential equation (3) has a closed-form solution, which will be formulated using the function:
Based on this notation, becomes:
The solution to Eq.(7) is a linear combination of three exponentials which we are not able to tackle directly by elementary methods. Instead, the zeros of and of its derivative can be calculated numerically but providing no insights into the function’s behavior.
On the other hand, if we know the number of positive zeros of we can shed light on the conditions under which the fund is solvent ( remains positive). We will see below that, depending on parameter values, of Eq. (7) has 0, 1, or 2 positive zeros. When there is no positive zero then remains positive and the fuind is solvent. One positive zero means that becomes negative and the scheme has collapsed. Two positive zeros mean that becomes negative, reaches a negative minimum, then becomes positive again. The fund has collapsed but could recover with a bailout equal to the absolute value of the negative minimum. To simplify, we will say that, in this case, the scheme has collapsed and then recovered.
Analytical results on the number of positive zeros will result from noting that the zeros of are also those of of Eq. (6). This function is a linear combination of only two exponentials plus a constant but the zeros still cannot be found in closed form by elementary method. However, the derivative of is a linear combination of two exponentials with no constant, which can be analytically studied. The following proposition provides results on the number of positive zeros of .\\[12pt]
Proposition 1. We consider the function of Eq. (6) in the non-trivial case where and . We also assume that . We further consider the following set of two conditions:
The function then has an extremum:
at the positive value:
if and only if Condition (13) is satisfied.
Depending on the number of positive zeros of , we get results for four different cases :
: Condition (13) is satisfied and . If then has no positive zero (and, therefore, remains positive). If and , then the function has exactly one positive zero at a value smaller than . In all other cases, with , the function has one positive zero on each side of .
: Condition (13) is satisfied and . If , then the function has no positive zero. In all other cases there is one positive zero.
: Condition (13) is not satisfied and . If , the function has no positive zero. In all other cases there is one positive zero.
: Condition (13) is not satisfied and . There is no positive zero.
Proof. The proof is elementary and hinges on the following facts:
- The derivative equals 0 at and is the value of at ; is positive if and only if Condition (13) is satisfied. The derivative at 0 is equal to .
- If and are negative, the function tends to for . If or is positive, the function tends to (depending on the signs of ).
- If Condition (13) is not satisfied, then either has an extremum for a negative value of or no extremum at all. In both cases, the function for is monotone increasing if and monotone decreasing otherwise.
In order to apply Proposition 1 to the parameters of Eqs. (8)-(12), we first define:
We will need the function:
which is the critical value of above which of Eq. (12) is positive.
We then define the function:
and note that if and only if (i.e., ).
The extremum of Eq. (14) and the corresponding of Eq. (15) are:
We also define the function as:
The quantity of Eq. (21) is the critical,value of above which the extremum of Eq. (19) is positive.
With these notations, we have the following result on the number of positive zeros of of Eq. (7).
Theorem 1. We consider the solution to Eq. (7) defined by the non-negative parameters and . The number of positive zeros of is given as a function of the sign of :
Case : .
Sub-Case : . has no positive zero.
Sub-Case : . We first consider the case .
If (which includes the case ), then has no positive zero for and, therefore, remains positive for all . For the function has one positive zero on each side of . For the function has one positive zero. When the function has one positive zero for (which includes the case ) and none if . When the function has one positive zero for (which includes the case if is larger than the fixed point of ) and none if (which includes the case if is smaller than the fixed point ).
Sub-Case : or . The function has one positive zero.
Sub-Case : and . For (which includes the case if the function has no positive zero. For (which includes the case if ), then has one positive zero.
Proof. The application of Proposition 1 hinges on the following observations:
- The parameter of Eq. (12) is positive if and only if .
- The extremum of Eq. (14) is positive if and only if .
- The difference has the same sign as .
- For the function is a non-decreasing function of that has no positive fixed point if and one positive fixed point if .
- The parameter and the derivative
of at 0 have opposite signs.
With of Eqs. (8)-(12) the quantities and of (13) are:
and are both negative if and only if and have the same sign (because ).
In the case () we considered two sub-cases:
Sub-Case : . Condition (13) is not satisfied and the derivative of at 0 is positive. This result follows from of Proposition 1.
Sub-Case : . Condition (13) is satisfied and the derivative of at 0 is positive. The sub-cases and correspond to or positive and to and negative, respectively. The results follow from of proposition 1. When Condition (13) is not satisfied and the derivative of at 0 is negative. These results follow from of Proposition 1.
Interpretation of Results
Theorem 1 breaks down the results depending on whether the rate of growth of new investments is larger or smaller than the promised return .
We first consider the case when is larger than (legitimate fund). In the case the fund is solvent () regardless of the initial condition . In the sub-case the fund remains solvent when remains above (which includes the case ). For between and the scheme collapses () as soon as drops below . For larger than the fund collapses but recovers () if does not fall too much below . If is too small() then the scheme collapses ().
Case illustrates what happens to a Ponzi scheme ) even if the rate of growth of new investments is larger than . The fund will remain solvent for only if is not too large ( less than the fixed point ). If and is larger than the fixed point , then the combined withdrawals by the initial and subsequent investors eventually cause the scheme to collapse.
For Case (the rate of growth of new investments is smaller than the promised return ) we first consider sub-case with (legitimate fund). The fund remains solvent for , which included the case . In the Ponzi sub-case , with and the fund does not grow too fast and is solvent if is larger than , which is itself larger than . This means that, despite an and an smaller than , the Ponzi scheme is solvent if the manager can add to an “in-house” investment at least equal to . This type of scheme is unprofitable of the manager and is often called a “philanthropic Ponzi scheme”. This scenario relies on remaining smaller than the nominal return . If the manager does not invest enough () the scheme collapses.
The Ponzi sub-case consists of the values and of the values for which and is between and . In this sub-case the fund grows too fast and collapses ().
This analysis shows that the role of is ambiguous when is between and . Although a small may seem desirable, the fund will grow more in the long run and eventually collapses. A large may seem dangerous but depletes the fund and means smaller withdrawals in the long run. The fund is ultimately solvent if is large enough to absorb the large early withdrawals (“philanthropic Ponzi scheme”).
Actual and Promised Amounts in the Fund
In order to describe the dynamics of a fund that includes a suden parameter change at some time , we need to make explicit the role of the parameters by denoting the solution of differential equation (7).
We introduce the actual and promised amounts and . The actual amounts in the fund (resulting from the nominal rate of return and the initial condition ), is the one given in Eq. (7) and rewritten explicitly as:
The value is the amount that is promised to and belongs to investors; is obtained by setting in Eq. (25) the parameter equal to and the initial condition equal to . Under these conditions, the third term in Eq. (25) becomes zero and:
Contrary to the actual amount in the fund, the promised amount is positive regardless of the parameter values.
Change in Parameter Values
The assumption of an exponentially increasing density of new investments is not realistic in the long run and we may wish to examine what happens in the particular case where the irate of growth of new investments suddenly drops to 0. This means that the flow of new investments becomes a constant. More generally, it would be useful to describe the future dynamics of the fund if at a point the parameters experience a sudden (discontinuous) change of values and become .
We call and the actual and promised amounts in the fund at time , respectively:
These values will be the initial condition and initial investment starting at time . The actual and promised amounts at any time then become:
Several discontinuous parameter changes at different times can be dealt with in this fashion.
As we expected, the fund is always solvent with in the case of a legitimate fund characterized by . In a Ponzi scheme (), the fund can remain solvent depending on the values of the rate of growth of new investments and the withdrawal rate . Our model sheds light on the ambiguous role played by these two parameters. If is too large or too small, the fund grows fast and can be in jeopardy as withdrawals increase. If is too small or too large, the fund may not keep up with withdrawals.
Our model yields a variety of increasing trajectories that may look alike initially, but are fundamentally different in their long-run behavior. Some will continue to increase as long as new investments come in — others will increase possibly for a long time before they collapse. This happens when parameter values in the phase spaces of solutions are close to border regions between different qualitative behaviors (for example, between no zero and one zero for the function ). In some cases initially decreases, reaches a positive or negative minimum, and then recovers.