I have recently been in the process of educating my 30-year old nephew about finances and investing. Towards that end, I have posed the following problem, which I call the Magic Black Box Problem.

Late one night, your geeky neighbour comes banging on your door excited about his latest invention. Reluctantly, you allow him to enter and he excitedly plants himself on your living room sofa with a big smile on his face.

“Well?”, you say.

He proudly produces a small black box from his pocket and plants it on the table.

“What is that?”, you say.

He explains as follows: this is a black box.

“Yet another black box!”, you retort.

He responds as follows: not just a black box, but a magic black box. He continues as follows: you purchase this box from me; simply allow it to sit on the table for ten years; during the course of each year, it extracts particles from the air; then on each anniversary of the day that you purchased the magic black box, it produces a crisp, genuine, one-dollar bill. He finishes by stating that the box will produce precisely 10 of these one-dollar bills and will then self destruct.

The question is: how much would you pay for this box and why?

My nephew answered as follows: that he would pay no more than 92¢ for the magic black box. I then posed the question: what if this box were to be offered on the open market and multiple individuals were bidding to purchase the box? Then, I asked, what would be a reasonable price for the box on the market? He has been stumped for three-quarters of a year. So I thought that I would help him reach the answer to the question by teaching the underlying concepts and methods necessary to reach a reasonable answer.

I have become convinced that the understanding obtained by deriving the answer to this question will position one to understand how to manage just about every aspect of life.

The first step towards answering this question is to obtain some information. It will be necessary to determine the present value of monies received in the future. The present value of future revenue is the inverse problem to the future value of current revenue. That is, if we receive one dollar today, what will that dollar be worth in the future. Once one knows how to compute the future value of the dollar, one can readily compute the present value of a dollar received at some time in the future. Of course, it is still one dollar; however, the question is relative to the purchasing power of that dollar today, what will be the purchasing power of the dollar at the end of the first year. To obtain that information requires the Consumer Price Index. The official, and most reasonable, data may be obtained from the website of the bureau of labor statistics. The data may be readily loaded into a spreadsheet and analysed. The annual inflation rate may be computed as the year over year change in the consumer price index as a ratio of the start of the year. I used the monthly data starting in 1950 and determined that the average rate of inflation was 3.78% with a standard deviation of 2.93%.

It is also necessary to account for state and federal income taxes as well as long-term capital gains taxes. Since each dollar bill issued by the magic black box each year is counted as income, then it must be taxed as income. I shall leave until later the discussion of the need to account for long-term capital gains taxes. It is now time to create a spreadsheet, enter the long-term rate of inflation, the federal and state income taxes, and the federal and state long-term capital gains tax rate.

I am a Macintosh user, so my spreadsheet program is Numbers. Type the following words into the first few cells of your spreadsheet: “Inflation” into cell $a$1; “Interest” into cell $a$2; “Treasury” into cell $a$3; “Excess” into cell $a$4; and “Payment” into cell $a$5. Of course, omit the quotation marks. Also, type the following into the adjacent cells as follows: “0.0374” into cell $b$1; “0.02” into cell $b$2; “0.0314” into cell $b$3; “0.0300” into cell $b$4; and “1” into cell $b$5. Again, omit the quotation marks.

Also, enter the following in columns $d and $e: the phrase “State Tax” into $d$1; the phrase “Federal Tax” into $d$2; “0.079” into $e$1; and “0.25” into cell $e$2.

By now, your spreadsheet should look something like the following.

Inflation | 3.74% | State Tax | 7.9% | |

Interest | 2.00% | Federal Tax | 25.0% | |

Treasury | 3.14% | |||

Excess | 3.00% | |||

Payment | $1.00 |

The state tax rate is fixed and I am using the income tax rate for Minnesota. The federal tax rate that you should use is not the marginal tax rate; rather, you should use the average tax rate that you pay.

The cell adjacent to the word “Payment” will contain the amount that you would be willing to pay for the “Magic Black Box.” Since this is the answer, which is currently not known, we use $1.00, as a space holder. Also, having $1.00 in cell $a$5 will allow our computations to proceed without confusing the spreadsheet program.

In the analysis of our “Magic Black Box” problem, we are going to allow for the possibility that the crisp new one-dollar bill that it issues on each anniversary of purchase is invested into some sort of a vehicle. To align with standard financial and investment concepts, let us call the crisp new one-dollar bill that the “Magic Black Box” issues on each anniversary of purchase the “dividend.”

Presume for a moment that the “Payment” had been placed into an interest bearing bank account. Of course, these days you would be luck to receive an interest of 0.015% on a standard savings account. Each month, the bank will send you a statement indicating that some interest had been paid for the month and that that interest payment had been credited to your account. I ask forgiveness from the accountants if the interest is “debited” rather than “credited.” I have never been able to remember the difference.

Nevertheless, on the following month, you will see listed the principle that you had at the beginning of the first month, the interest that you were paid at the end of that first month, and the interest that you were paid at the end of the second month. The a fraction of the interest paid during the second month is actually interest paid on the interest that was paid during the first month. Thus, at the end of the second month, you will have an amount in your bank account that can be readily computed using the following formulation:

Total = Principle x (1 + i) x (1 + i)

That formula represents accrual of compound interest.

However, in our “Magic Black Box” problem, we cannot accrue compound interest from the device until we have had a sufficient number of dividend payments, net of taxes, to slightly exceed the amount that the market is demanding for the box. In the case of shares in companies, many companies have dividend reinvestment programs, which are commonly known as “Drips”, that allow the dividend to be used to purchase fractional shares. However, with equity, it is not at all clear that the next dividend payment is made on the fractional share rather than on the integer number of shares.

Now, once we have sufficient dividends accumulated from the “Magic Black Box” net of taxes, we can use the cash to purchase a new box. Until that happens, we should invest the dividends in an alternative vehicle that accrues some interest. There are a wide variety of possibilities. For the time being, let us presume that we have found a bank that is willing to allow us to purchase a $1.00 one-year certificate of deposit that pays an annual interest rate of 2.0%. Thus, the number in cell $b$2.