Understanding the Mathematics of Personal Finance

PONZI SCHEMES AND OTHER SCAMS

Suppose that you called two friends tonight and asked them each to drop off \$250 at your house in the morning. Then you would suggest that they recoup their money, and more, by each calling two other friends tomorrow night and repeating the same request to them. You probably wouldn’t get too far with this money-making scheme.

A related scheme that has been used many times starts with an invitation from friends: “Come to a party Friday night and bring \$100.” Your name will be put in position 4 on a list. Then come the following Friday night and bring three friends, each with \$100. They will be given the same instructions. Every week the names on the list are dropped one position and the people whose names are in position 1 split all the money coming in that night and their names get crossed off the list. You attend the party and watch all the people in position 1 receive \$2,700 each. This isn’t bad!

Week 2 you bring three friends with their money. Week 3 you just come to watch the fun. Then, finally, on week 4, you come to collect your \$2,700. Will you really collect all of this money? Maybe.

The biggest problem is this: When you start, you are one of 1 + 3 + 9 + 27 = 40 people in the room. The next week, there are 3 + 9 + 27 + 81 = 120, followed by 360 and then 1,080. Can you picture the madness of showing up as part of a group of 1,080 people, 729 of them bringing money and 27 of them collecting money?

This type of scheme typically works early on, for the first few groups of people, and then falls apart as the number of participants get large. You could find yourself in the position of watching your next-door neighbor collect \$2,700 while you never get to collect anything, or vice versa. In either case, the future of your relationship with this neighbor probably will not be good.

The above scheme is called a pyramid 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 still 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

Table 13.5 Workings of a Hypothetical Ponzi Scheme

 Time Group Number of people My personal account (\$) Internal books (\$) External per person (\$) External total (\$) Beginning, 1 100 2,500,000 7,500,000 100,000 10,000,000 year 1 End, year 1 1 100 2,625,000 7,875,000 125,000 Beginning, 1 90 2,625,000 6,625,000 125,000 11,250,000 year 2 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, 1 80 2,756,250 5,393,750 156,250 12,500,000 year 3 2 90 2,625,000 6,625,000 125,000 11,250,000 3 100 2,500,000 7,500,000 100,000 10,000,000 Total 270 7,881,250 19,518,750 33,750,000 End, year 3 1 80 2,894,063 5,663,438 195,313 2 90 2,756,250 6,956,250 156,250 3 100 2,625,000 7,875,000 125,000 Total 270 8,275,313 20,494,688 Beginning, 1 70 2,894,063 3,710,313 195,313 13,671,875 year 4 2 80 2,756,250 5,393,750 156,250 12,500,000 3 90 2,625,000 6,625,000 125,000 11,250,000 4 100 2,500,000 7,500,000 100,000 10,000,000 Total 340 10,775,313 23,229,063 47,421,875 End, year 4 1 70 3,038,766 3,895,828 244,131 2 80 2,894,063 5,663,438 195,313 3 90 2,756,250 6,956,250 156,250 4 100 2,625,000 7,875,000 125,000 Total 340 11,313,078 24,390,516 Beginning, 1 60 3,038,766 1,454,422 244,131 13,648,438 year 5 2 70 2,894,063 3,710,313 195,313 13,671,875 3 80 2,756,250 5,393,750 156,250 12,500,000 4 90 2,625,000 6,625,000 125,000 11,250,000 5 100 2,500,000 7,500,000 100,000 10,000,000 Total 400 13,813,078 24,683,484 62,070,313 End, year 5 1 60 3,190,704 1,527,133 305,176 2 70 3,038,766 3,895,828 244,131 3 80 2,894,063 5,663,438 195,313 4 90 2,756,250 6,956,250 156,250 5 100 2,625,000 7,875,000 125,000 Total 400 13,504,782 25,917,659

Table 13.5 Continued

 Time Group Number of people My personal account (\$) Internal books (\$) External per person (\$) External total (\$) Beginning, 1 50 3,190,704 -1,524,615 305,176 15,258,789 year 6 2 60 3,038,766 1,454,422 244,131 13,648,438 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 End, year 6 1 50 3,350,239 - 1,600,846 381,470 2 60 3,190,704 1,527,133 305,176 3 70 3,038,766 3,895,828 244,131 4 80 2,894,063 5,663,438 195,313 5 90 2,756,250 6,956,250 156,250 6 100 2,625,000 7,875,000 125,000 Total 450 17,855,021 24,316,813 Beginning, 1 40 3,350,239 -5,415,543 381,470 15,258,789 year 7 2 50 3,190,704 -1,524,615 305,176 15,258,789 3 60 3,038,766 1,454,422 244,131 13,648,438 4 70 2,894,063 3,710,313 195,313 13,671,875 5 80 2,756,250 5,393,750 156,250 12,500,000 6 90 2,625,000 6,625,000 125,000 11,250,000 7 100 2,500,000 7,500,000 100,000 10,000,000 Total 490 20,355,021 17,743,327 92,587,891 End, year 7 1 40 3,517,751 -5,686,320 476,837 2 50 3,350,239 - 1,600,846 381,470 3 60 3,190,704 1,527,133 305,176 4 70 3,038,766 3,895,828 244,131 5 80 2,894,063 5,663,438 195,313 6 90 2,756,250 6,956,250 156,250 7 100 2,625,000 7,875,000 125,000 Total 490 21,372,772 18,630,493 Beginning, 1 30 3,517,751 -10,454,692 476,837 13,305,115 year 8 2 40 3,350,239 -5,415,543 381,470 15,258,789 3 50 3,190,704 -1,524,615 305,176 15,258,789 4 60 3,038,766 1,454,422 244,131 13,648,438 5 70 2,894,063 3,710,313 195,313 13,671,875 6 80 2,756,250 5,393,750 156,250 12,500,000 7 90 2,625,000 6,625,000 125,000 11,250,000 8 100 2,500,000 7,500,000 100,000 10,000,000 Total 520 23,872,772 7,288,635 106,893,005

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

Figure 13.10 Published and internal Ponzi scheme fund values.

enough money to pay all the people who will be pulling out at the beginning of year 9.

At the beginning of year 8, my personal fund has almost \$24 million in it. This isn ’ t a bad retirement package. Since nobody expects to hear from me until the beginning of year 9, I have almost a full year to quietly arrange a new identity in remote but pleasant place in the world. A couple of million dollars goes a long way toward these goals. I might leave the \$7.3 million in clients’ funds, which is still scattered among a variety of banks, alone since I don’t want to arouse any suspicions, or I might withdraw some or all of it. Then I disappear. Nobody notices that I’m gone until the beginning of year 9.

At the end of year 8, there are 540 investors, each believing their fund is worth something between \$125,000 and about \$600,000. These will not be happy people.

The scenario was idealized. I didn ’ t account for my living expenses for the 7 years I was 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 (or 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.

PROBLEMS

1. Suppose that you found three stocks that are ideal examples of prices cycling up and down similar to the dollar cost averaging example above. You are convinced that these three stocks average price at the end of the year will be the same as it was at the beginning of the year. The three variations, however, are significantly different. You wish to invest in each of these stocks, putting a fixed total amount of money in each month for a year. Would you be better off splitting your money equally among the three stocks or putting all your money into one of the three in terms of this idealized return?

2. The tab Sample Data in the spreadsheet Ch13Stocks. xls contains two columns of numbers that represent the prices of two stocks on 25 equally spaced (business) days. Copy these data into the stock price columns in the Correlation tab. Discuss the meaning of the various parameters presented.

Replace the numbers in the stock #1 column with the numbers 1, 2, 3, ... 25. Can you find any meaning in the results?

3. The company you work for has given you some option shares (calls) as a perk with a strike price of \$13 a share. Sketch a graph of your profits and losses and indicate where you would exercise the option.

4. Repeat the above exercise assuming you had purchased these option shares at \$1 a share.

5. Going back to the situation of problem 1, you were given calls with a strike price of \$13 a share. You now sell calls with a strike price of \$18 a share for \$1 a share. Repeat the above, assuming that your option buyer will exercise his or her calls when the stock price exceeds \$18 a share and that both sets of calls expire on the same date.

6. You own both puts and calls for FrisbeesAreUs, Inc. The calls have a strike price of \$90 and the puts have a strike price of \$85. The stock itself is selling for \$88. You paid \$4 for the call and \$5 for the put. Sketch a graph of your profits and losses as a function of the stock price and indicate where you’d exercise the put, the call, both, or neither.

REFERENCE

Dworsky, L. 2008. Probably Not. Hoboken, NJ: John Wiley & Sons.

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