Understanding the Mathematics of Personal Finance
From Chapter 13, Investing: Risk versus Reward, Ponzi Schemes and Other Scams: A Hypothetical Ponzi Scheme
........ A more sophisticated scheme of this sort is called the Ponzi scheme, named after Charles Ponzi, an Italian immigrant to the United States in the early twentieth century. Ponzi schemes are devilishly clever in their ability to fool a lot of people for a long time. I’m going to work through a hypothetical example of a Ponzi scheme. It’s a bit messy but worth your while to follow the money flow(s). Just in case you think this couldn’t happen to you, note that in early 2009, a New York fund manager named Bernard Madoff was arrested for running a Ponzi scheme that fooled the most sophisticated investors and even fooled the Securities and Exchange Commission examiners for quite a while. Right now nobody is quite sure of the numbers, but it’s estimated that he stole somewhere between $20 and $50 billion from his investors. (This is not a typo; he made off with billions of dollars.)
I’m making my example somewhat rigid in that everybody invests the same amount of money and each investment can start and end only on January 1. I’m doing this so that following the logic of what’s happening doesn’t get so entangled with the details of calculations that are certain to give you a headache. Keep in mind, however, that it’s precisely these entanglements that help make a Ponzi scheme very difficult to spot. Even with this very constrained example, it’s tricky to follow what’s happening. As Mr. Madoff recently showed, a real working complex Ponzi scheme can be so intricate and entangled that even experienced professionals can’t figure out what’s going on.
Assume that I’m an experienced stock advisor and money fund manager with some level of credibility. This is necessary just to get peoples’ attention.
I’m announcing a new investment fund. Each year, on January 1, I will accept 100 investors, each investing $100,000. I am promising a 25% annual return on invested money. An investor can pull out and withdraw all of his or her money on January 1, but only on January 1, of any year.
I know that I can get a 5% return on money quite safely.
I will assume that 10 people from each year’s starting group of 100 will want to pull out each year, taking all of their money with them.
Something I’m not telling my clients is that, when they sign up and give me $100,000, I immediately take $25,000 and put it into my personal account. This account is hidden offshore somewhere—for reasons that will become clear soon.
Table 13.5 is very busy. That’s because, since I’m setting up a scam, it’s necessary to keep several sets of books. I need a book that shows my profits, a book that honestly tracks clients’ money, and a book that I publish showing “how well” my fund is doing.
The first line in the table shows all my books at the beginning of year 1, the very start of the fund. My first group of 100 people has signed up. They each gave me $100,000, so I have $10,000,000 in my hands. Taking $25,000 from each of them gives me $2,500,000 in my own personal account and leaves $7,500,000 of clients’ funds. I will spread this money over several bank accounts so nobody really knows how much money there is in the clients’ fund accounts. Externally, each client believes that his or her account is now worth $100,000 and that the total fund is worth $10,000,000.
At the end of the first year, I do my own accounting. I have only one group of 100 people. My personal account has grown at 5% to $2,625,000 and my internal (honest) record of clients’ funds has grown, also at 5%, to $7,875,000. Externally, my clients each believe that their funds have grown to $125,000 each.
The next day is the beginning of year 2. Ten people pull out of group 1, leaving it with 90 people. My personal account doesn’t change. My internal clients’ funds account for group 1 drops by 10($125,000) = $1,250,000 to $6,625,000 because I returned the promised $125,000 to each of the 10 people who pulled out. At the same time, group 2 now joins the fund. I repeat everything I did with the money of group 1 at the beginning of the first year. That is, I take $2,500,000 for my personal account, which is now worth $5,125,000. I have $7,500,000 new clients’ funds, bringing my clients’ funds account up to $13,125,000. The group 2 people each have an account worth, so they believe, $100,000. I publish my funds report showing how much I paid out and what’s left in the fund: $11,250,000 of first year investors’ money plus $10,000,000 second year investors’ money, totaling to $21,250,000.
The scam is evolving. Internally, I actually have only $13,125,000 in clients’ funds, but I’m publishing a report showing $21,250,000 of clients’ funds.
Following through the next few years, you can see the mess evolving. My personal account is growing healthily, the clients’ funds account is growing slowly, and the published external funds total value is really looking fabulous! At the beginning of each year, 10 people from each group leave, and they each walk away with their original $100,000 compounded annually at 25%.
At the beginning of year 6, you can see the beginning of the end. The clients’ fund value for group 1 (with only 50 people remaining in it) has gone negative. There isn’t enough money to fund the necessary payouts. No one except me knows this. Because there is still plenty of money in the total clients’ funds accounts, I just pay the missing piece out of other clients’ money.
By the beginning of year 8, there isn’t enough money in groups 1, 2, or 3 internal funds to pay the people pulling out. Since the total clients’ fund account still has money, however, I can still pay them.
Let’s take a minute to look at how things appear to the outside world at the beginning of year 8. The people who stuck with my fund for 8 years are each pulling out with $476,837. This is a 25% APR compounded for 8 years. The people who stuck with it for 7 years are pulling out with $381,470 and so on. These are very happy people. The published value of my fund going into year 8 is over $100,000,000.
Figure 13.10 shows the history (and the future) of my fund. To the outside world, it looks great. Internally, however, the end is in sight. There will not be enough money to pay all the people who will be pulling out at the beginning of year 9.
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Table 13.5 (Excerpted) Workings of a Hypothetical Ponzi Scheme | ||||||
|
Time |
Group |
Number of people |
My Personal Account ($) |
Internal Books ($) |
External per person ($) |
External Total ($) |
|
Beginning, year 1 |
1 |
100 |
2,500,000 |
7,500,000 |
100,000 |
10,000,000 |
|
End, year 1 |
1 |
100 |
2,625,000 |
7,875,000 |
125,000 |
|
|
Beginning, year 2 |
1 |
90 |
2,625,000 |
6,625,000 |
125,000 |
11,250,000 |
|
|
2 |
100 |
2,500,000 |
7,500,000 |
100,000 |
10,000,000 |
|
|
Total |
190 |
5,125,000 |
13,125,000 |
|
21,250,000 |
|
End, year 2 |
1 |
90 |
2,756,250 |
6,956,250 |
156,250 |
|
|
|
2 |
100 |
2,625,000 |
7,875,000 |
125,000 |
|
|
|
Total |
190 |
5,381,250 |
13,831,250 |
|
|
|
..... |
|
|
|
|
|
|
|
Beginning, year 6 |
1 |
50 |
3,190,704 |
- |
305,176 |
15,258,789 |
|
|
2 |
60 |
3,038,766 |
1,454,422 |
244,131 |
13,647,438 |
|
|
3 |
70 |
2,894,063 |
3,710,313 |
195,313 |
13,671,875 |
|
|
4 |
80 |
2,756,250 |
5,393,750 |
156,250 |
12,500,000 |
|
|
5 |
90 |
2,625,000 |
6,625,000 |
125,000 |
11,250,000 |
|
|
6 |
100 |
2,500,000 |
7,500,000 |
100,000 |
10,000,000 |
|
|
Total |
450 |
17,004,782 |
23,158,870 |
|
77,329,102 |
|
..... |
|
|
|
|
|
|
running the fund. I’d like to let a lot of people, whenever they show up, invest in the scheme. Not exactly 10 people will pull out from each group every year, and they certainly won’t all want to pull out on exactly January 1. In a real fund, people will come and go at random times throughout the years. However, the workings of the scheme (scam?) are accurately portrayed. A lot of people made a lot of money (exactly what they were promised). They, unknowingly, helped the scam to succeed by building other investors’ confidence that the fund was real.
A most unbelievable fact is that many people, including some professional fund managers, will watch their apparent fund value grow, year after year, at an incredible rate, without asking for a full annual report to see just what was happening. Bernie Madoff showed that they really will. Maybe many of them suspected what was going on but hoped that they had gotten in early enough to be one of the lucky winners. I’ll never know.
I think the old adage that “something that seems too good to be true probably is” covers the ground very well here.
