2.1.1 Madoff’s Ponzi scheme

We now describe Madoff’s Ponzi scheme. Madoff began his investment operations in his early twenties when he established the firm of Bernard L. Madoff Investment Securities (BLM). Anticipating the potential of automatic trading, he became an early adopter of this technology, establishing BLM as an innovative leader in the over-the-counter market as a broker-dealer. Battalio (Reference Battalio1997) noted in the Journal of Finance that after BLM entered the market as a broker-dealer, “the quoted bid-ask spreads tightens,” indicating increased market efficiency.

Madoff became recognized as a pioneer in the industry, and he began to play an active and increasingly important role in the securities industry. He served on various committees of the National Association of Securities Dealers (NASD), rose to become a member of the NASD board, and was later elected Chairman of NASDAQ. These activities enabled him to establish and maintain good relations with the Securities and Exchange Commission (SEC). From this vantage point, Madoff was able to anticipate, influence, and exploit regulatory changes.

While establishing his broker-dealer business, Madoff also provided investment advice to his family and friends. This advisory business, which started as a side activity, eventually grew into the largest Ponzi scheme in history. His early investors included his father-in-law, Saul Alpern, and Alpern’s clients. By enabling four of his early Jewish clients to become very rich, Madoff’s reputation spread by word of mouth, expanding his client base to Jewish groups, wealthy individuals, and charities. Madoff claimed he was able to make profitable trades because his trading experience as a broker-dealer gave him an edge.^{Footnote 3}

In the early 1990s, Madoff claimed to use a new trading strategy that enabled him to deliver stable returns. The claim of steady returns was used by financial institutions as a marketing tool to impress clients and potential investors. Thus, it helped raise vast amounts for Madoff’s Ponzi scheme. The strategy, known as the split-strike conversion strategy, involved sophisticated option trades. It consists of a long position in a stock portfolio, a long put option on the Index, and a short call of similar maturity, also on the Index. It was advertised that, with the help of the two options, the two extreme tails of the stock returns would be trimmed. Bernard & Boyle (Reference Bernard and Boyle2009) and Clauss et al. (Reference Clauss, Roncalli and Weisang2009) provided an in-depth analysis of this strategy and showed that the returns reported by Madoff’s split-strike portfolios were implausible.^{Footnote 4}

The split-strike strategy turned out to be beneficial to each of the key parties involved. It was appealing to the hedge funds who collected funds from the investors. These so-called feeder funds were handsomely compensated through the fees they charged. The fees consisted of a fixed percentage of the asset value plus a performance fee. By the time Madoff’s scheme collapsed in 2008, there were 15 feeder funds. The investors were attracted to the strong, steady stream of returns, which the feeder funds used as a powerful marketing gimmick. As for Madoff himself, even though he appeared to charge very little, he was presumably very satisfied having access to an increasing flow of new funds, as long as he could meet the outflow.

Madoff’s Ponzi scheme collapsed in 2008 as a result of the financial crisis. The turmoil, which caused panic selling, led to massive stock price markdowns and huge waves of redemptions that exhausted the cash in the scheme. On December 11, 2008, Madoff confessed to the massive fraud and was later sentenced to 150 years in prison.^{Footnote 5} After the scam was exposed, Irving Picard was appointed the trustee under the Securities Investment Protection Act. From an investor’s viewpoint, this final stage tended to be a bitter experience. For those who lost money with Madoff, their dream of outsized investment returns was dashed. For those who withdrew more than the amount invested, the excess was clawed back and used to pay those who lost money. Many of Madoff’s victims felt a deep sense of shame, embarrassment, guilt, and remorse from their unfortunate experience. As of this writing, March 2025, the trustee has recovered $\$14.7$ billion for the victims, and the Department of Justice recovered an additional $\$4.3$ billion, with the recovery process still ongoing.^{Footnote 6} The average recovery over all investors is about 75% on the dollar.

In the Madoff case, the SEC has been deservedly criticized for its regulatory negligence. The SEC completely failed to uncover the scam despite receiving various warnings, including six substantive complaints.^{Footnote 7} In retrospect, the SEC investigation team should have engaged experts with strong quantitative skills, such as financial economists, financial engineers, or actuaries. The team also lacked seasoned equity and option traders with market nous who would have been able to analyze and dissect Madoff’s trades. The lawyers on the exam team, to compound matters even more, were too young and inexperienced to detect such a sophisticated scam and tended to be overawed and intimidated by Madoff’s reputation.

2.1.2 Stanford’s Ponzi scheme

Robert Allen Stanford’s scheme affected 280,000 investors and incurred a loss of around USD $8 billion. Compared with Madoff’s case, it has received less attention in the economics and finance literature. Nevertheless, there is considerable documentation available on this scheme, much of which is contained in various SEC reports (Kotz, Reference Kotz2014).^{Footnote 8}

Stanford grew up in Texas and obtained a finance degree from Baylor University. His foray into business began with the purchase of fitness clubs in Texas. These clubs flourished for a few years but were in Chapter 11 following the economic collapse in 1982. In November 1984, Stanford managed to get a bankruptcy court to clear his debts so he could start afresh. In 1985, he founded a bank on the Caribbean island of Monserrat, offering certificates of deposit (CDs) that paid 200 basis points above the market rates. As many Monserrat banks were involved in money laundering for drug companies, Stanford’s operation also came under scrutiny. His Monserrat banking license was revoked in May 1991.

In 1990, Stanford moved his operations to Antigua, setting up the Stanford International Bank (SIB). Antigua proved to be a hospitable habitat for the continued operation of his Ponzi scheme. Stanford’s bank in Antigua also offered CDs at unusually high yields. He marketed these investments aggressively through affiliate companies in the US and across Central and Latin America. Stanford and his team assured investors that the deposits were secure by falsely claiming that (i) the bank invested the funds in liquid, secure assets; (ii) the portfolio was monitored on an ongoing basis by a team of 20 analysts; and (iii) the investments were subject to annual audits by Antiguan regulators.

Stanford’s scheme managed to survive for almost twenty years, and the reasons for its remarkable duration, as well as why Stanford was not indicted, are threefold. First, Antigua’s financial regulation was weak and amenable to manipulation. While his bank was subject to local regulations, Stanford was well-positioned to outmaneuver the regulatory system. Not only was he on the regulatory body, but he was also in charge of overseeing Antigua’s banking laws. He gamed the enforcement system by bribing the main officials. One of the bribees, Leroy King, was the administrator and chief executive officer of Antigua’s Financial Services Regulatory Commission (FSRC). King accepted thousands of dollars per month as bribes and turned a blind eye to Stanford’s Ponzi scheme. Moreover, King supplied Stanford with confidential information about the SEC’s investigation. In February 2021, King was sentenced to ten years in prison by a Texas court. He admitted to receiving some $500,000 in cash as well as other benefits in kind from Stanford.

Second, the island’s political leadership was corrupt. For example, Stanford made it his business to establish good connections with Vere Bird, who was prime minister of Antigua and Barbados from 1981 to 1994, and also with his son Lester Bird, who led the country from 1994 to 2004. In 1995, Stanford lent the government millions of dollars to pay salaries and pensions.

Thirdly, Stanford worked hard to create and maintain an image of integrity and success to ensure the continuation of his fraud. Stanford promoted himself as a prominent businessman and philanthropist in Antigua and, at one time, was the island’s largest employer.^{Footnote 9} In addition to his proactive endeavors to maintain his reputation, Stanford also employed strong-arm tactics. He was known to sue journalists and others who impugned his reputation and paid a security firm to vigorously and aggressively defend his good name.

On the US side, inter-agency conflict, incompetence, and borderline corruption prevented an early shutdown of Stanford’s scheme. In fact, at least three US agencies – the FBI, the SEC, and FINRA – had been monitoring Stanford for several years.^{Footnote 10} The SEC suspected that he was running a Ponzi scheme as early as 1997. In particular, the SEC’s Fort Worth office conducted four reviews of Stanford’s investment operations and concluded each time that his returns were highly unlikely. Regarding the Fort Worth inaction, Spencer Barasch, one of the senior actors who decided not to pursue Stanford, eventually ended up working for Stanford. As the head of the enforcement group, Barasch claimed that the case was complicated, noting that the group would be rewarded based on the number of cases they resolved, not the complexity thereof. Later, in September 2006, Stanford hired Barasch – who at that time had left the SEC – to seek advice on how to deal with regulators. Barasch was later fined $50,000 for breaching conflict of interest guidelines.

2.1.3 WinCapita (Finland)

WinCapita was started in 2003 by Hannu Kailajärvi, who had a background in information technology but none in finance. The scheme was operated through a shell company domiciled in Panama and became the largest investment fraud in Finnish history. At first, it was claimed that the profits were generated by sports betting, but later, foreign currency was advertised as the purported investment medium. By promising investors that they would get rich quickly, the scheme swindled more than 10,000 victims between 2003 and 2008, representing 0.2% of the entire Finnish population (Rantala, Reference Rantala2019).

To create a sense of exclusivity, investors in WinCapita could only join the scheme via invitation from a sponsor. This created an affinity network that resembled the transmission of an epidemic (Huebscher, Reference Huebscher2016). After the scheme’s demise, Rantala (Reference Rantala2019) obtained detailed information on the scheme’s investors. Using this dataset, he showed that the investor engagement with the scheme was not transmitted evenly from one investor to a fixed number of peers; instead, a small fraction of investors (i.e., sponsors) at the hub of the system were responsible for most of the observed transmission. Rantala (Reference Rantala2019) found that investors put in more funds if their sponsor had higher income, was older, or had higher education. Presumably, these attributes made the information more credible. He also found that sponsors typically invested more than non-sponsors.

The Finnish regulator, Oikeusrekisterikeskus (ORK), started investigating the scheme in 2008. The investigation turned out to be a time-consuming legal process. Investors who made profits and withdrew their earnings from the scheme were ordered to pay back their gains. Vironen (Reference Vironen2019) reported that 626 former investors were ordered to pay EUR 64 million in illegal benefits to the Finnish state. By the autumn of 2020, ORK claimed it had received a total of EUR 34.2 million in illegal proceeds from the scheme.

According to Huebscher (Reference Huebscher2016), the promoter, Hannu Kailajärvi, received only a five-year sentence, the maximum duration under Finnish laws. Nevertheless, even this short sentence was later cut to two and a half years. Given the size of the scheme and the seriousness of the crime, the punishment seems very mild compared to comparable sentences in the US and China.

2.2.1 Ponzi schemes in Jamaica

Ponzi schemes were very popular in Jamaica in the early years of the twenty-first century. Tennant (Reference Tennant2011) provides an accessible description of these schemes and conducts an empirical analysis based on a sample of the participants. The data examined includes the personal profiles of 402 respondents who invested in 17 Ponzi schemes in Jamaica. The main dependent variable is exposure, defined as the money invested in any Ponzi scheme divided by personal income. Tennant (Reference Tennant2011) found that those who are relatively well-off treat their investment in Ponzi as experiments. The motivation is consistent with the case of MMM in Africa: it is those individuals who want to improve their economic situations that have the largest exposure. Moreover, people who got referrals from friends and received high payments from Ponzi schemes tend to increase their exposure. This finding alludes to repeated investing behavior in Ponzi schemes.

2.2.2 Ponzi schemes in Columbia

Cortés et al. (Reference Cortés, Santamaría and Vargas2016) explored the consequences of the crackdown of 12 Ponzi schemes, which caused the Colombian government to declare a state of emergency. The serial insolvencies of these scams, which posed a huge disruption to the economy, started with the collapse of DRFE and DMG on November 12 and 15 of 2008. DMG sold prepaid cards with a 6-month maturity that promised yields ranging from 50% to 300%. DRFE (Dinero Rápido, Fácil y Efectivo – Money in Cash, Fast and Easy) promised monthly returns between 80% and 150%. While an overall estimate of the total damage was not available, the reported losses are already significant. According to Hofstetter et al. (Reference Hofstetter, Rosas and Urrutia2018), DFRE attracted USD $\$865.6$ million as deposits, victimizing 153,878 investors, and DMG attracted USD $\$1,191.3$ million as deposits, victimizing 356,631 investors.

Hofstetter et al. (Reference Hofstetter, Rosas and Urrutia2018) found that prior to (subsequent to) the crackdown of DRFE and DMG, victims invested in DMG and DFRE enjoyed more (fewer) loans and better (worse) credit standings. Municipalities that were more affected by the collapse saw a larger decrease of deposits placed in the financial sector. Using data from all 50 municipalities in Columbia, Cortés et al. (Reference Cortés, Santamaría and Vargas2016) found that the crime rates of shoplifting and robbery increased significantly following the crackdown of the 12 schemes, especially in 2009Q4 and in municipalities with weak law enforcement and limited access to credit.

2.2.3 Other US Ponzi schemes

Stephen et al. (Reference Stephen, Waymire and White2021) analyze 376 Ponzi schemes in the US that were prosecuted by the SEC from 1988Q1 to 2012Q4. The study found that the median size of these collapsed schemes was USD $14.7 million, with the median proportion purloined by the promoter being 13.2%. The median duration of these schemes was 3.1 years, implying that most scams are short-lived. Similar to Tennant (Reference Tennant2011), trust built by affinity – either through kinship, friendship, shared religious belief, or ethnic origin – was very important for the promoter to ensnare his victims, evinced by 74% of the schemes being perpetrated by local promoters. Meanwhile, trust built by advertising on mass media had a mixed effect – it was negatively related to the duration but positively related to the size of the scheme. However, the analysis did not show how the promised interest rate affects the duration or the size of the scheme.

Jordan Maglich presents data on 1108 schemes covering the period 2008–2023, of which the average size was over USD $\$70$ million.^{Footnote 11} For schemes that ended between 2008 and 2022, there is a full record of the total losses. Based on available news reports and the regulator’s documents, 768 perpetrators were sentenced to a fixed term of prison time. The average (median) length of their sentence was 130 months (96 months). The ratio of male versus female is 8.5:1, implying that nearly 90% of the schemes were initiated by men.

2.2.4 Ezubao and other P2P lending scams in China

We next turn to a class of Ponzi schemes that swept through China during the 2011–2019 period. Unlike the heterogeneous schemes discussed in Section 2.1, these scams appeared in clusters and closely resembled one another. They flourished because of a gap in the country’s regulations and were shut down when the gap was closed.

Thanks to China’s long period of economic growth, there were millions of households with excess funds to invest in the early years of this century. However, with the majority of the population possessing limited financial knowledge, individuals preferred simple fixed-income investments. This preference stemmed from their everyday experience with bank deposits, which used to be the only investment vehicle available. After years of credit expansion in the post-crisis period, various bond-like Ponzi schemes appeared, catering to people’s low risk appetite.

Many of the investment schemes emerged as peer-to-peer (P2P) lending platforms. A P2P platform puts borrowers directly in contact with lenders without involving a formal financial institution (e.g., a bank). Since P2P lending was a new type of financial arrangement in China, the prevailing regulations were ineffective (Wei, Reference Wei2015). In principle, P2P platforms were only allowed to play the role of an information agent, matching up lenders with eligible borrowers; see Li et al. (Reference Li, Hsu, Chen and Chen2016) for details. However, these online platforms became very aggressive in seizing lenders’ funds.

We provide summary statistics of the Chinese P2P industry from 2011 to 2019 in Table 1. Panel A provides the numbers of these schemes, and the last two rows show the number of failures. These numbers, extracted from two sources, did not agree with each other due to the self-reported nature of the statistics. According to the website www.wdzj.com , by the end of 2019, only 343 platforms had survived, suggesting a high rate of failure. Panel B of the table displays several key characteristics, including the average duration of the schemes, the average term of investment (i.e., maturity), the average coupon rate, and the average number of lenders and borrowers. On average, the failed platforms have a short duration. As the first row of Panel B indicates, the mean duration equals 5.19 months in 2013 and 16.37 months in 2016. In 2016, the median duration was 15 months, suggesting that most platforms typically did not survive two years after establishment. From 2013 to 2019, the average promised return decreased from 21.25% to 9.46%.

Table 1.

Summary statistics of the P2P industry in China (2011–2019)

Source: The statistics were obtained from www.wdzj.com , www.p2peye.com , and www.yingcanzixun.com in 2020, all of which have been closed down around 2021. The original sources include the Bluebook for China’s Online Lending Industry published in 2013, Annual Report on China’s Online Lending Industry (wdzj, 2014, 2015, 2016); P2P Industry Annual Report (p2peye, 2016). Notes: 1. Statistics on failed platforms are based on victims’ reports. The actual number of failures may be larger than reported. 2. Durations and investment terms are measured in months. Platforms that last less than a month are assigned a duration of one month. 3. The promised rate of return is annualized.

Researchers have documented a negative relationship between the probability of failure and the promised interest rate. Li et al. (Reference Li, Hsu, Chen and Chen2016) examined 308 P2P lending platforms during 2011–2014, which included 104 failures and 204 non-failures from the website www.wdzj.com . Using a Cox proportional hazard model, they find that the risk of failure is positively related to the promised interest rate but negatively related to the registered capital. A similar survival analysis by Wang et al. (Reference Wang, Shen and Huang2016) studied a larger sample of 3407 P2P platforms, among which 1048 failed. They found that failed platforms were more likely to promise returns exceeding 20%. Also, platforms that do not make key information available, such as the interest rate and registered capital, are less likely to survive.

Among all P2P scams, Ezubao became the most well-known. Launched by the Yucheng Group in July 2014, Ezubao underwent a period of rapid growth (Albrecht et al., Reference Albrecht, Morales, Baldwin and Scott2017). It offered six products, each of which had a promised return rate exceeding 9%. To attract small investors, each product only required a minimum investment of one RMB (approximately 1/7 USD). Ezubao claimed that it put investors’ money into a subsidiary of its parent group, which professed to operate a series of financial leasing projects. The subsidiary was supposed to purchase pieces of equipment and lease them to several lessee companies. According to Ezubao, when the lessees paid their rent, the subsidiary would transfer the money back to investors registered on the platform.

To make it appear more credible, Ezubao worked diligently to bolster its reputation and create a positive image. For example, it spent a huge amount of money placing advertisements in various media outlets. In particular, it used prime time commercials on national television, China Central Television (CCTV), as well as several provincial channels (Li, Reference Li2016). Furthermore, Ezubao rented luxurious offices at the Shanghai Pilot Free Trade Zone and the Beijing Central Business District. The company also held its annual meeting in the Great Hall of the People in Beijing. These trust-building activities were designed to instill confidence in the investors and disguise the underlying sham. In reality, all of Ezubao’s investment projects were fictitious.

In December 2015, Ezubao eventually attracted the attention of the regulators. By then, the platform had swindled RMB 58 billion (USD $\$$ 8.2 billion) from 900,000 investors across China (Bai & Chen, Reference Bai and Chen2016). The scheme collapsed in early 2016. This scandal made investors more suspicious of P2P lending platforms, causing regulators to increase surveillance of suspect companies. The demise of Ezubao and the trial of its management foreshadowed a large-scale crackdown on the P2P sector from 2018 to 2020. By the end of 2019, there were just 343 active platforms compared to 3,500 four years earlier. And this number fell to zero in November 2020.

4.2.1 How Ponzi scheme models differ from the economic model

A Ponzi scheme model differs from Equation (1) in three major ways: capital accumulation, withdrawal processes, and uncertainty in the duration. First, the cash accumulation in the resource constraint, $(1+r_t)W_{t}$ , is incorrect, as the promised investment opportunity does not actually exist. This can be accounted for by setting $r_t=0$ , assuming no real investment occurs, or $r_t=r_f$ , assuming the previous cash balance has been deposited at the risk-free rate.

Second, in standard economic models, the entire amount withdrawn, $C_t$ , represents the agent’s consumption. In a Ponzi scheme, only a portion of this amount goes to the promoter, and the rest goes to early investors who redeem from the scheme.^{Footnote 16} We denote the two streams of withdrawals, attributable to the promoter and investors, by $C_t$ and $O_t$ , respectively. Projecting the cash inflow, $I_t$ , the outflows $O_t$ , and the promoter’s takeout, $C_t$ , is key to capturing the evolution of $W_t$ .

Third, the duration or decision horizon, $T$ ,, is not predetermined at the outset but depends largely on the evolution of the scheme. The finite duration property of a Ponzi scheme implies that the total withdrawal, $O_t+C_t$ , eventually outpaces the inflow, $I_t$ , exhausting all the cash balance. The duration can be manipulated somewhat using the marketing strategies described in Section 3.6. Due to this interdependence of $T$ and other variables, to simplify the model, the utility maximization component is often dispensed with in Ponzi models.^{Footnote 17}

4.2.2 A simple, stylistic Ponzi model

Before delving into more advanced Ponzi models, we first discuss a simple discrete-time version. This model will highlight the basic differences mentioned above and provide some intuition. Suppose each new investment has a maturity of one period, with the promised rate of return being $r$ . Due to the one-period lock-up, investors cannot withdraw any funds in period zero, i.e., $O_0 = 0$ ; the initial investment is $I_0=K$ , with $K$ being a fixed number. Starting from $t=1$ , the cash balance is

(2) \begin{equation} W_{t+1} = W_t + I_t - O_t - C_t. \end{equation}

This last equation shows that the cash balance in period $(t+1)$ , $W_{t+1}$ , can be broken down into four parts: (i) the previous balance, $W_t$ ; (ii) new investment, $I_t$ ; (iii) the withdrawal by investors, $O_t$ ; and (iv) the takeout by the promoter, $C_t$ .

The amount invested, $I_t$ , is assumed to grow at a constant rate $g$ , so that after $t$ periods, the amount of new investment becomes $I_t = K (1 + g)^t$ . At the end of the $t$ -th period, the total amount to be paid out equals $O_t = I_{t-1}(1+r)$ , where $I_{t-1}$ is the investment in period $t-1$ . Since there is no real investment activity, the cash inflow and outflow attributable to investors evolve according to the following rules:

(3) \begin{align} I_t & = I_{t-1} (1+g), \end{align}

(4) \begin{align} O_t & = I_{t-1} (1+r), \end{align}

where the growth rate of cash inflow, $g$ , and the promised rate, $r$ , are assumed constant over time. These are, of course, strong assumptions made to simplify the analysis. In reality, both $g$ and $r$ can vary over time and need not be independent of one another.

Suppose the promoter has quadratic utility; that is, his single-period utility function can be written as $U(C_t) = C_t - b C_t^2$ , with $b\gt 0$ and $C_t \leq 1/(2b)$ .^{Footnote 18} In addition, the promoter is assumed to have zero time preference, so that $\rho = 1$ . It is convenient to assume that his takeout each period corresponds to the satiation point, so that $C_t = 1/(2b)$ for $1 \leq t \leq (T-1)$ . Adding Equations (2), (3), and (4) recursively, the cash balance of the scheme at the beginning of period $t$ is

(5) \begin{equation} W_t = \frac {K}{g} \left [(1+g)^{t-1} (g-r) + r \right ] - \frac {t-1}{2b}. \end{equation}

It is easy to show that $W_t$ is a decreasing function of $r$ .

If the scheme lasts $T$ periods, we must have $W_{T-1}\gt 0$ and $W_T\leq 0$ . This requires the first-order derivative of $W_t$ with respect to $t$ to be negative at $t=T$ ; that is,

(6) \begin{equation} \left. \frac {\partial W_t}{\partial t} \right|_T= \frac {K}{g}\left [ (g-r)(1+g)^{T-1}\ln (1+g)\right ] - \frac {1}{2b} \lt 0. \end{equation}

This inequality holds for $g\lt r$ , as both the first and second terms are negative. When $g =r$ , the system corresponds to the balanced path where the cash inflow offsets the cash outflow, with $I_t = O_t$ . A more interesting case is when $g\gt r$ . With certain parameter combinations, $\partial W_t/\partial t \lt 0$ is attainable for small $t$ , resulting in a scheme that wraps up quickly due to large, early outflows. When $g \gg r$ , $W_t$ can be explosive, such that the scheme can last forever. However, the assumption that the inflow can grow perpetually is unrealistic.

Numerically, the duration of the scheme can be obtained by solving the two inequalities, $W_{T-1}\gt 0$ and $W_T\leq 0$ . Table 2 illustrates how different combinations of parameter values affect the duration, $T$ . For the sake of simplicity, we standardize $K$ to 1 and set $b=4$ . We can see that the duration of the scheme declines as $r$ increases, holding $g$ fixed. The duration of the scheme increases with $g$ , holding $r$ fixed. Both results are intuitive. When $r=g$ , $T=(2b+1)=9$ . It is also noteworthy that even if $g\gt r$ , we can obtain a finite duration in some cases. For example, when $r=0.06$ and $g=0.08$ , the duration is 12.

Table 2.

Duration $T$ assuming different $g$ and $r$ pairs in the stylistic model ( $b=4$ )

5.1.1 A too-big-to-fail Ponzi scheme: Bhattacharya (Reference Bhattacharya2003)

Bhattacharya (Reference Bhattacharya2003) constructs a model of a large Ponzi scam in a small economy. He explains how to design a Ponzi scheme that will attract investors, even though they know it will collapse: the likelihood of a bailout provides the incentive. Bhattacharya (Reference Bhattacharya2003) makes assumptions regarding the shared information in the economy, the nature of the population, the type of investments, the decisions of the promoter, the design of the bailout, and the behavior of the regulator. His model is rather complicated, and we will only be able to cover the main ideas.

We now provide a brief review of his model. The promoter expects the scheme to last for $T$ periods before it winds up. In each period, he sells a one-period investment product priced at $P$ and promises a rate of return $r$ throughout. Formally, denote $n_t \in [0,\,1]$ as the mass of individuals that join the scheme in the $t$ -th period; for each additional period, this mass grows at rate $g$ , i.e., $n_{t+1} = (1+g) n_t$ . The gross growth rate $(1+g)$ is bounded above by $D$ , the natural speed of information transmission.

The model rests on two strong assumptions, symmetric information and government bailout, which Bhattacharya admits are severe assumptions. First, under symmetric information, as soon as individuals realize it is a Ponzi scheme, they know when it began, which stage it is at, and when it will end. Second, since the scheme is considered too big to fail, it will end up with a government bailout.^{Footnote 19} Consequently, an individual in the economy has two choices: either they invest in the scheme and end up with some compensation when it collapses, or they do not invest in the scheme and end up defraying the cost of the bailout.

The period zero cash flows are complicated. To attract an initial mass of $n_0$ investors and receive $n_0P$ , the promoter incurs a marketing expense of $ f_0 + f_1(n_0)n_{0}P$ and retains $f_2$ to pay for a regulatory penalty when the scheme collapses. The promoter’s takeout, therefore, equals $C_0 = n_0P - \big ( f_0 + f_1(n_0)n_{0}P + f_2)$ . Here, $f_0$ and $f_2$ are fixed, while $f_1(n_0)$ is increasing and convex in $n_0$ , the initial investor mass.

The cash flows from period one onwards follow a balanced path. That is, in period $t=1,\, \ldots , \, T$ , the cash inflow, $I_t = n_{t}P$ , is split between (i) investor’s withdrawal, $O_t = (1+r) n_{t-1} P$ , and (ii) the promoter’s takeout, $C_t = cn_tP$ . This balanced path implies $ n_t P = (1+r) n_{t-1} P + c n_{t} P$ , such that the proportion of promoter takeout equals $c=(g-r)/(1+g)$ , implying that $g\gt r$ .

The regulator can detect the Ponzi game and close it down with probability $\theta$ , implying that the expected promoter takeout is ${\textrm E}[C_t]=(1-\theta ) c n_t P$ . Once detected, the promoter’s takeout in that period and onward is confiscated and becomes zero. Therefore, the utility maximization problem facing the promoter reduces to find the promised return ( $r$ ), growth rate of the investor population ( $g$ ), price of the investment ( $P$ ), initial population mass ( $n_0$ ), and duration of the scheme ( $T$ ) to maximize his expected utility:

(8) \begin{equation} {\textrm E} \Bigg [\sum _{t=0}^T \rho ^t U(C_t) \Bigg ] = n_0P - \Big ( f_0 + f_1(n_0)n_0 P + f_2 \Big ) + \sum _{t=1}^T (1-\theta )^t c n_t P, \end{equation}

where $\rho =1$ and $U(C_t) = C_t$ are chosen to characterize a patient, risk-neutral promoter.

Bhattacharya (Reference Bhattacharya2003) uses a series of steps to characterize the optimal solution. It can be shown that the expression in Equation (8) is decreasing in $r$ and $n_T$ and increasing in $g$ and $P$ . Consequently, the optimal solution reduces to finding the lower bounds of $r$ and $n_T$ but the upper bound of $g$ and $P$ . To induce participation in period $T-1$ , the expected gain of participation in per capitaterms must be greater than that of non-participation, such that $(1-\theta )Pr + \theta (-P) \geq 0$ , which yields $r\geq \theta /(1-\theta )$ . The optimal $r$ is the lower bound, $\theta /(1-\theta )$ . Moreover, $n_T$ must exceed $n^*$ , a threshold that triggers a bailout. The optimal $n_T$ , therefore, is $n^*$ . Since $(1+g)$ is bounded by $D$ , the optimal $g$ equals $1-D$ . Let the per capitaloss for non-participation be $\alpha$ , then the total gain for a bailout – given only to the mass of $n_T$ who invested – is $\alpha (1-n_T)$ . In period $T$ , suppose the probability of the bailout is $\beta$ , the expected total gain for a bailout is $\beta \cdot \alpha (1-n_T)$ . To induce participation in period $T$ , the per capita loss in the Ponzi scheme plus the bailout gain must be greater than the loss from non-participation, i.e., $-P+ \beta \cdot \alpha (1 - n_T)/n_T \geq \beta \cdot (-\alpha )$ . This implies $P\leq \alpha \beta /n_T$ .

Combining these results, the optimal end-period investor mass, investment price $P$ , and the duration of the scheme are

(9) \begin{equation} n_T = n^*, \quad P = \frac {\alpha \beta }{n^*}, \quad T = \frac {\ln n^* - \ln n_0}{\ln D}, \end{equation}

where $n_0$ solves the first-order condition of Equation (8).^{Footnote 20}

As already noted, this model rests on assumptions about the wealth redistribution effect of the bailouts. The reason an individual joins the scheme is because if they do not and the scheme fails, they will be forced to contribute to the bailout. In the case of contemporary Ponzi schemes, the bailout provisions envisaged in this model appear highly unusual. Currently, compensation to victims is normally provided by clawing back funds from investors who withdrew more than they invested in the scheme; see, for example, the Madoff case.

5.1.2 A balanced path pay-as-you-go pension system

Pay-as-you-o (PAYG) pension plans have sometimes been described by their critics as Ponzi schemes (Boccia, Reference Boccia2024). In both PAYG plans and Ponzi schemes, the benefits payable to successive cohorts are funded by the contributions of the previous cohort, essentially implying a balanced path of cash flows.

It is of interest to consider a simple model of a PAYG system. For the sake of convenience, we tweak the notation in Morsomme et al. (Reference Morsomme, Alonso-Garcia and Devolder2025). The current contributions to a PAYG system can be considered as the inflow, $I_t$ , expressed as

(10) \begin{equation} I_t = \pi _t \cdot S_t \cdot N_t^I, \end{equation}

where $\pi _t$ is the contribution rate, $S_t$ is the average salary, and $N_t^I$ is the number of workers who contribute to the pension plan at time $t$ . The total pension payments at time $t$ , which resemble the cash outflows of a Ponzi scheme, are given by

(11) \begin{equation} O_t = P_t \cdot N_t^R, \end{equation}

where $P_t$ is the average pension and $N_t^R$ is the number of retirees.

In the ideal case, the total contributions are set equal to total benefits, such that $I_t = O_t$ . This balanced path implies that the contribution rate equals

(12) \begin{equation} \pi _t = \frac {P_t}{S_t } \cdot \frac {N_t^R}{N_t^I} = \delta _t \cdot d_t, \end{equation}

where $\delta _t = P_t /S_t$ is the replacement rate and $d_t = N_t^R / N_t^I$ is the dependency ratio, which equals the number of retirees divided by the number of active workers.

Equation (12) provides a convenient tool for examining the dynamics of a PAYG system. We can see that, ceteris paribus, if the relative number of retirees increases, the contribution rate will have to rise; and if the life expectancy increases or the birth rate falls, the dependency rate will increase. PAYG systems can offset these adverse effects by reducing the benefit level, raising the contribution ratio, increasing the retirement age, or implementing a combination of these changes (Boccia, Reference Boccia2024).

Despite the similarity in balanced cash flows, there are profound differences between PAYG plans and Ponzi schemes: transparency and the tools available to manage the cash flows. First and most significantly, the operations of a PAYG system are completely transparent. Under a state-sponsored PAYG scheme, the rules can generally be changed by a democratic process. On the contrary, a Ponzi scheme is inherently fraudulent: only the promoter knows what is going on, and he lies to his clients and steals their money. The sustainability of the arrangement depends critically on having an adequate number of new entrants at each stage to fund the benefit payments.

Second, the administrator of a PAYG system has more tools in its arsenal to tackle cash flow problems than a Ponzi promoter. For example, in China, to ensure the solvency of the pension system, the government has made several fundamental changes to the population policy to influence people’s fertility and retirement decisions, thereby dynamically adjusting the replacement rate and the dependency ratio.^{Footnote 21} In contrast, in a Ponzi scheme, if the net cash flows become negative and remain so, then as soon as any buffer of cash reserves is depleted, the scheme will collapse. Given the nature of a Ponzi scheme, it will be very difficult for the promoter to make any changes that will reverse this trend. For example, if he reduces the promised return, the number of new investors will decline even further.

Nevertheless, even with all its policy tools, a PAYG plan can still encounter liquidity issues when demographic changes become too extreme or adverse economic conditions result in a sharp cut in salary levels. In recent history, PAYG schemes have run into difficulties in several countries; see, for example, the cases of Italy and Russia (Forni & Giordano, Reference Forni and Giordano2001; Jensen & Richter, Reference Jensen and Richter2004). When this happens, the payments will be halted and the scheme restructured, thus preventing it from collapsing.

5.2.1 A Ponzi scheme with a fictitious balance and no redemption withdrawal: Zhu (Reference Zhu2011)

In her doctoral thesis, Zhu (Reference Zhu2011) proposed a Ponzi model with two balances: the actual balance, $W_t$ ; and the fictitious balance, $F_t$ , which the investors (wrongly) perceive as real. Denote the promised rate of return as $r_t$ , the risk-free rate as $r_f$ , and $C_t$ as the takeout of the promoter.^{Footnote 22} Then, the two balances, $W_t$ and $F_t$ , evolve according to the following equations:

(17) \begin{align} W_{t+1} &= (1+r_f)W_t + I_t - C_t, \end{align}

(18) \begin{align} F_{t+1} & = (1+r_t) F_t + I_t, \end{align}

(19) \begin{align} I_t & = g(r_t) F_t , \end{align}

where the cash inflow, $I_t$ , grows in proportion to the fictitious balance, $F_t$ ; investors do not redeem their money until the terminal time, thus $O_t=0$ ; the proportion of takeout shown on the fictitious balance is $c_t= 0$ . The proportion of fictitious balance invested, $g$ , is defined as

(20) \begin{equation} g(r_t) = g_0 \Big[ 1- \exp \big(-a(r_t-r_f)\big) \Big]. \end{equation}

As $r_t \to r_f$ , the growth rate tends to zero; and as $r_t \to \infty$ , the growth rate approaches the upper limit, $g_0$ .

In each period, the regulator observes $r_t$ and sets an arbitrary detection threshold $\tilde {r}$ , which is a random draw from the normal distribution, with $\tilde{r} \sim N(\bar {r},\,\sigma )$ . If the promised return $r_t$ exceeds $\tilde {r}$ , the regulator detects the Ponzi scheme and will force it to close. This would drive both the fictitious and actual balances to zero. Otherwise, if there is no regulatory detection, the Ponzi scheme continues.

Assume that the duration of the Ponzi game, $T$ , is known to the promoter, and at the terminal time $T$ , there is a bequest $B(W_T,\,T)$ . Denote $U(C_t, \, t)$ as the promoter’s utility function, and assume that the promoter chooses a consumption and promised rate pair $(C_t, \, r_t)$ in each period to maximize his expected utility. His optimal behavior can be obtained by solving the following Bellman equation:

(21) \begin{equation} J(W_t, \, F_t, \, t) = \max _{\{C_t,\, r_t\}} \, {\mathrm E}_t\Big [ U(C_t, \, t) + \rho \cdot J(W_{t+1}, \, F_{t+1}, \, t+1) \Big ], \end{equation}

subject to the resource constraints in Equations (17)–(19) and a non-detection condition $r_t\lt \tilde {r}$ . For simplicity, the discount factor $\rho$ can be set to 1. At the final period $T$ , the value function reduces to the bequest function, i.e., $J(W_T, \, F_T, \, T) = B(W_T,\, T)$ .

The model rests on a strong assumption: there is no withdrawal by the investors from the fund during its life, or $O_t=0$ . That is, all promised benefits stay in the fund. Hence, the fictional account $F_t$ grows geometrically at a rate of $ r_t + g(r_t)$ . This condition, as well as the assumption of known duration, is unrealistic.

5.2.2 Continuous time models: Clauss et al. (Reference Clauss, Roncalli and Weisang2009)

Up to now, we have dealt with discrete-time models. The model proposed by Clauss et al. (Reference Clauss, Roncalli and Weisang2009) is formulated in continuous time. One advantage of continuous-time models is that obtaining solutions is generally easier.

Similar to Zhu (Reference Zhu2011), Clauss et al. (Reference Clauss, Roncalli and Weisang2009) consider the evolution of two fund balances. As before, $W(t)$ is the real cash balance, which grows at a return of $r(t)$ . Let $F(t)$ be the fictitious balance, and $\mu (t)$ be the gross return. Unlike Zhu (Reference Zhu2011), Clauss et al. (Reference Clauss, Roncalli and Weisang2009) specifically model cash outflows related to fund redemptions and assume the scheme ends when the cash runs out. In their model, the promoter’s takeout takes the form of a management fee, which equals the unit rate, $m(t)$ , times the fictitious balance, $F(t)$ . Note that this management fee is assumed to be disclosed to the investors and is publicly known.

Both the cash inflow and outflow are proportional to $F(t)$ , with an instantaneous intensity of $\lambda ^+(t)$ and $\lambda ^-(t)$ , respectively. Given the above assumptions, the fund dynamics are given by

(22) \begin{align} \dot {W}(t) & = r(t) W(t) + I(t) - O(t) - C(t), \end{align}

(23) \begin{align} \dot {F}(t) & = \mu (t) F(t) + I(t) - O(t) - C(t), \end{align}

(24) \begin{align} I(t) & = \lambda ^+(t) F(t), \end{align}

(25) \begin{align} O(t) & = \lambda ^-(t) F(t), \end{align}

(26) \begin{align} C(t) & = m(t) F(t), \end{align}

with the initial condition being $W_0 = F_0$ . Apparently, the proportion of takeout shown on the fictitious balance is $c(t)=1$ . In other words, the management fee is an actual cash withdraw, of which the full amount is deducted from both the actual wealth in Equation (22) and fictitious wealth in Equation (23). The scheme collapses if $W(t)\lt 0$ . One solution to the system is

(27) \begin{equation} \begin{pmatrix} W(t) \\ F(t) \end{pmatrix} = W_0 \cdot \exp \bigg (\int _0^t \mathbf{A}(s) {\textrm d}s \bigg ) \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad \text{where} \,\, \mathbf{A}_t = \begin{pmatrix} r(t) & \lambda ^+(t) - \lambda ^-(t) - m(t) \\ 0 & \lambda ^+(t) - \lambda ^-(t) + \mu (t) - m(t) \end{pmatrix} \end{equation}

However, it is difficult to set the value of $\lambda ^+(t)$ and $\lambda ^-(t)$ , as either may evolve over time; in particular, the sudden release of negative news concerning a liquidity crisis may lead to a jump in $\lambda ^-(t)$ .

5.2.3 Auxiliary fictitious balance: Artzrouni (Reference Artzrouni2009)

Artzrouni (Reference Artzrouni2009) proposes a model based on a set of four differential equations to capture the four processes underlying a Ponzi scheme.^{Footnote 23} These equations specify the dynamics of (i) $W(t)$ , the actual cash balance; (ii) $F(t)$ , the fictitious cash balance, which only serves an illustrative purpose for investors; (iii) $I(t)$ , the cash inflow process, driven by new investors; and (iv) $O(t)$ , the cash outflow, attributable to previous investors who redeem their investment. The promoter’s takeout, $C(t)$ , is not modeled and taken to be zero. This assumption seems unrealistic. As before, $r$ denotes the promised rate of return, only that it is now expressed as continuous.

Suppose the initial real cash balance is $W(0)= W_0 = F_0$ . The four cash processes evolve according to

(28) \begin{align} \dot {W}(t) & = r_f W(t) + I(t) - O(t), \end{align}

(29) \begin{align} \dot {F}(t) & = (r - \gamma ) F(t) + I(t), \end{align}

(30) \begin{align} \dot {I}(t) & = \beta I(t), \end{align}

(31) \begin{align} \dot {O}(t) & = (r - \gamma ) O(t) + \gamma I(t), \end{align}

where $r$ $(r-\gamma )$ is the net return to the investor; and there is no promoter takeout, i.e., $C(t)=0$ . The inflow, $I(t)$ , brought in by new investors, grows at an exponential rate of $\beta$ , with the initial condition $I(0)=I_0$ . The outflow, $O(t)$ , is subject to an initial condition, $O(0)=\gamma \big ( K + I_0 \big )$ . Artzrouni (Reference Artzrouni2009) was able to derive explicit solutions for $W(t)$ and $F(t)$ .

It is debatable whether Artzrouni’s model can be used as a candidate tool to study the dynamics of a social security system as the author claimed. The assumption that $I(t)$ follows exponential growth is not attainable in the long run, as it is equivalent to assuming an ever-increasing pool of new investors. As a result, without specifying a finite horizon, the real balance in Equation (28) can be explosive or follow a balanced path for some parameter values.^{Footnote 24}

Interestingly, the model suggests an auxiliary role of the fictitious balance, $F(t)$ , as it does not depend on the actual outflow, $O(t)$ . In other words, the promoter does not equate the withdrawal from the fictitious balance, $\gamma F(t)$ , with the actual investor withdrawal, $O(t)$ . This specification illustrates the fact that $F(t)$ only serves to show the rosy yet bogus fortune promised by a Ponzi scheme and thus can be dispensed with in modeling.

It is worth noting that the dynamic of $O_t$ , while exponential, can be either explosive or contractual. This model is prescient of Ponzi models based on the a class models that integrate investor demographic dynamics. These models, by imposing hump-shaped dynamics on the inflow and outflows, are able to eliminate the unrealistic assumption of perpetual growth and focus on the actual cash balance, $W(t)$ . We discuss them in the next section.

5.3.1 SIR model with add-on assumptions

To get a serviceable model, we adopt a gradual approach and start with some refinements. These refinements help simplify the model structure while retaining the link with the model’s economic lineage.

First, we impose restrictions on investor behavior. In many Ponzi schemes, an investor may redeem part of her promised benefits, add more funds, or re-enter the scheme after her previous redemption. Therefore, the boundaries between “invested” ( $I$ ) and “redeemed” ( $R$ ) may be blurred. To exclude this complication, we assume that once an investor redeems, she withdraws all her principal and accrued interest. She takes no further part in the scheme.

Second, we impose a lock-up period on the investment, which is an analog of the single-period investment products in the discrete model introduced in Section 4.2. The lock-up period also links with the recovery rate in the SIR model, $\gamma$ , which indicates that the average duration of an epidemic is $\gamma ^{-1}$ . In Ponzi schemes, we translate $\gamma ^{-1}$ as the lock-up period of the scheme. This assumption makes the investment product in a Ponzi scheme similar to a fixed-income instrument with a predetermined investment term.

A direct benefit of the two restrictions is that they significantly simplify the real cash balance process. Otherwise, investors could enter and leave the scheme in a random fashion, hence leading to overly complicated inflow and outflow dynamics, and examples of such complications can be found in Mayorga-Zambrano (Reference Mayorga-Zambrano2011) and Zhu, et al. (Reference Zhu, Fu, Zhang and Chen2017).^{Footnote 26}

5.3.2 The basic model

Our basic model involves two structural parameters: the transmission rate $\beta$ and the recovery rate $\gamma$ , both of which are assumed constant. Moreover, we assume the scheme only offers one product, promising a continuously compounded return of $r$ . The risk-free rate, or any rate used as the benchmark, is $r_f$ . We disregard any exogenous shocks, such as the possibility that the Ponzi scheme collapses due to forced liquidation after regulatory detection. The existence of a lock-up period, which is of length $\gamma ^{-1}$ , divides the Ponzi scheme into two stages.

Stage 1.

Before the lock-up period expires for the first batch of investors, the recovery or redemption rate $\gamma$ is equal to zero. The dynamics of the three groups of investors are then characterized by a susceptible-infected (SI) model:

\begin{align*} \dot {N}^S(t) & = - \beta \cdot \tfrac {N^I(t)}{N} \cdot N^S(t), \\ \dot {N}^I(t) & = \beta \cdot \tfrac {N^I(t)}{N} \cdot N^S(t), \\ \dot {N}^R(t) & = 0. \end{align*}

These specifications are simply Equations (32) and (33) without the redemption term $\gamma \cdot N^I$ . As before, $\beta$ is the transmission rate. The actual wealth or balance of the scheme, $W$ , has the following dynamics:

(35) \begin{equation} \dot {W}(t) = - c_0 - K \cdot \dot {N}^S(t), \end{equation}

where $K$ is the average amount invested in the scheme, and the initial wealth level is $W(0) = K \cdot N^I(0)$ ; $c_0$ is a fixed rate of takeout by the promoter. To set the Ponzi game in motion, the basic reproduction rate, $\mathcal{R}_0 = \beta /\gamma$ , must be greater than one.

Equation (35) shows that the changes in wealth are composed of two terms: $-c_0$ , the promoter’s withdrawal, which is fixed; and $-K \cdot {\textrm d}N^S/{\textrm d}t$ , the money brought by new investors, whose population is the negative of the change in the number of susceptible investors, $\dot {N}^S$ . This means that $N^S$ decreases with time so that $\dot {N}^S$ is negative and the last term in Equation (35) is positive.

Stage 2.

After the lock-up period ( $t \geq \gamma ^{-1}$ ), earlier investors begin to redeem. The demographic dynamics are restored to the following SIR model:

\begin{align*} \dot {N}^S(t) & =- \beta \cdot \tfrac {N^I(t)}{N} \cdot N^S(t), \\ \dot {N}^I(t) & = \beta \cdot \tfrac {N^I(t)}{N} \cdot N^S(t) - \gamma \cdot \tfrac {N^I (t-\gamma ^{-1})}{N} \cdot N^S\big ( t-\gamma ^{-1} \big), \\ \dot {N}^R(t) & = \gamma \cdot \frac {N^I (t-\gamma ^{-1})}{N} \cdot N^S \big ( t-\gamma ^{-1} \big). \end{align*}

Here, the second and third equations become delay differential equations (DDE) that must be solved numerically.^{Footnote 27} Compared with the standard SIR model in Equation (33), the dynamics of $N^I$ now depend on the current $N^I$ and $N^I$ with a delay. With a delay equating the lock-up period ( $\gamma ^{-1}$ ), the promoter’s actual wealth evolves according to

(36) \begin{equation} \dot {W}(t) = - c_0 - K \cdot \dot {N}^S(t) - {\textrm e}^{r / \gamma } \cdot K \cdot \dot {N}^R \big ( t-\gamma ^{-1} \big ), \end{equation}

where the first two terms are the same as in Equation (35). The last term is the amount withdrawn, in which ${\textrm e}^{r/\gamma } \cdot K$ is the future value of an investment $K$ made $\gamma ^{-1}$ unit of time before $t$ . As long as $W(t)$ remains non-negative, it evolves according to Equation (35) within the lock-up period and Equation (36) after the period.

Due to the takeout by the promoter and withdrawal by investors, the cash balance will eventually approach zero, leading to the scheme’s demise. As before, the duration of the scheme is the maximum time length that the wealth $W(t)$ stays non-negative, i.e., $T = {\textrm {argmax}}_t \, \{ W(t) \geq 0 \}$ .

The demographic dynamics and the wealth process of our bond-like Ponzi scheme are illustrated in Panels (a) and (b) of Fig. 1. We can see that the actual wealth declines to zero in the 53rd month. This timing is earlier than that predicted by the standard SIR model, in which the wealth process is disregarded, such that the process ends when the number of susceptible investors drops to zero.

Figure 1

An illustration of the basic SIR-type Ponzi model.

Note: In this graph, we set $N=100,000$ , $\beta =0.2$ , $\gamma =1/12$ , $r=10\%$ , $K=5$ , $c_0=0$ . We implicitly assume that the promoter is risk-neutral. The Ponzi scheme collapsed in month 53 when the cash balance $W$ dropped below zero.

5.3.3 Extensions: functional forms of structural parameters

There are many possible extensions to the baseline model. For example, we can also add $-c_1 W(t)$ to Equations (35) and (36), where $c_1\gt 0$ is a fixed constant. This addition entails that the promoter’s withdrawal has a component that increases with the real cash balance.

Additionally, it is possible to adopt other plausible forms for the structural parameters. As before, we denote the promised rate of return by $r$ and the risk-free by $r_f$ . We can assume that the target population depends on $r$ and satisfies the following equation:

(37) \begin{equation} N = N_0 \cdot \exp \big ( -a (r-r_f) \big ), \quad \text{for } r\gt r_f \end{equation}

where $N_0$ is the total population in the area, and $a$ is a positive constant. This implies that the higher the promised rate, the smaller the pool of susceptible investors.^{Footnote 28}

Alternatively, the two key parameters – the transmission rate, $\beta$ , and the recovery rate, $\gamma$ – can also depend on the promised interest rate, $r$ . For example, Amona & Oduro (Reference Amona and Oduro2019) assume that $\beta$ and $\gamma$ follow the following rules

(38) \begin{equation} \beta =\lambda {\textrm e}^{-\lambda r}, \quad \gamma = k(1-\beta ) = k(1-\lambda {\textrm e}^{-\lambda r}). \end{equation}

Using this specification, the transmission rate peaks at $r=0$ , which is inconsistent with empirical observations.^{Footnote 29}

By using the same structural parameters, the basic model assumes homogeneous characteristics across members of the three groups. There is evidence that this may not apply. Rantala (Reference Rantala2019) documented that in the WinCapita scheme, a subset of influential investors, known as “highly connected hubs” in network parlance, had a much higher impact on the spread of the scam than average investors. Such influential investors correspond to the so-called super spreader in an epidemic, who exhibits an above-average risk of becoming infected and can infect a large number of susceptibles (Wong et al., Reference Wong, Liu, Liu, Zhou, Bi and Gao2015). In a recent paper, Szapudi (Reference Szapudi2020) replaces $\beta$ , the transmission rate, by $p_k$ , the probability that an individual with $k$ connections will be infected, and explains how to modify the SIR model to include super spreaders.