The Math of Money

Morton G. Davis

Copernicus Books, 2001. 199 pages.

I well remember the good old days when faculty and other scholarly types could devote all their mental powers to their research and other intellectual activities without worrying about retirement. They could depend on TIAA or some other company-sponsored retirement plan to look after them when they shut up shop at age 70.

Nowadays many of the company retirement plans are defunct, or at least have changed from "defined benefit" plans to plans, such as 401k or 403b, where the retirement benefit derives from the participant's (voluntary) contributions. In these, as in the more traditional plans still in existence, the participant must decide in which vehicle(s) his/her money should be invested, i.e. growth stocks, value funds, bonds, money markets, etc. This situation is exacerbated by the fact that advances in medical technology as well as lifestyle changes (use of seat belts, general disapproval of smoking, predilection to exercise, etc.) have considerably extended the number of years that the retired scientist or engineer may expect to survive.1

This is a major reason why these days everybody should have some expertise in finance.2 And that is what the reader can hope to attain from this easy-to-read book, written by this professor emeritus of mathematics at CCNY. It is written at the level of high-school algebra, however, which means it should be accessible to all.3 There are nine chapters, each preceded by a mini-quiz testing the readers' intuition in matters monetary (the answers are provided in the Appendix). Below I describe, chapter-by-chapter; some of the most interesting material I found most interesting, but the list is by no means inclusive.

The first chapter "Investment Strategies" is really about betting (showing us what the author thinks of the stock market!) Inter alia, a formula is presented, but not derived, for the probability of losing all of your money in a casino. For example, if you come to the table with $90 and make $1 bets and if the probability of winning any bet is .49, then 34% of the time you will lose all of your money before winning $10.4 Obviously, such a formula is easier to apply in Las Vegas, where the probabilities are all known, than on Wall Street, where they are not.

Chapter 2 is entitled "Interest." While most of the material in this chapter, dealing with the effects of compounding, for example, should be well known to readers of this journal, some other material is perhaps less well recognized.5 One example is the very useful formula, used throughout the rest of the book, for calculating the present worth of an annuity. It is used to compute the "lump sum" payoff for lottery winners (cf. below under Chapter 5) as well as by states suffering financial crises (such as Virginia) who want to sell their annuitized "tobacco settlement" payments for a lump sum in order to meet current expenses. (In using the same formula to compute the annual annuity payment based on one's equity in an investment, the length of the annuity period is taken from the so-called "mortality tables," available on the internet.)

Another example is the convenient formula for computing the doubling time of an investment (also useful for engineers in energy-use calculations). The doubling time is equal to 69 divided by the interest rate, so if you can invest your money at 7 percent it will double in less than ten years (and octuple in less than 30 years!) Incidentally, only annual compounding is considered in most of the formulas presented in the book. Daily compounding can produce significantly more growth. For example at an interest rate of 8 per cent for a period of forty years, annual compounding will multiply the original investment by a factor of 26 [(1.085)40] whereas daily compounding will multiply it be a factor of 30 [exp(.085x40)]. (The rule of 69 assumes daily compounding.)

Chapter 3, "Bonds" uses the present value formula again, and explains how interest rates, duration and years to maturity affect bond prices. Various types of bonds are described, including zero-coupon bonds and "junk" bonds (which the author doesn't consider completely junky).

Chapter 4 is on "Mortgages." This is a short chapter, since mortgages are merely mirror images of bonds. However one very worthwhile idea suggested is to increase mortgage payments through time, at a set rate matching inflation, say. For example (Table 4.5) for a $100,000, 8 percent mortgage with no annual increase, the monthly payments are $714. If the payments were increased 3 percent per year, the initial payment would be only $522. Of course, the final payment would be much larger, $1284, but the combined effects of inflation and increased salary (hopefully) over the years would make the larger payment affordable. I am not aware that lenders who allow this sort of loan, but what one can do is to begin with a 30-year loan and periodically increase the principal payment. This reduces the term of the loan and, I might add, saves a considerable amount in interest.

Chapter 5 is entitled "retirement." It tells you how to calculate how much capital you will need for a specific desired income, including the effects of inflation. It also presents a formula for how long your money will last (important if you decide to live too long). Then there is a nice section advising how to accumulate the necessary capital during your working years.6 (Note a misprint on page 100. The first equality sign in the equation displayed there should be a multiplication symbol.) An interesting computation which might have presented, but is not, would compare the merits of lump sum payoffs vis--vis annuitization upon retirement. This question is the same as the decision one must make upon winning a $25 million lottery, whether to take the lump sum payment or an annuity of $1 million per year for 25 years. Most winners, according to anecdotal newspaper evidence, take the lump sum payoff, but the mathematics strongly favors the annuitization. The basic reason is that first the lump sum is taxed and then the earnings on it are taxed again.7 Presumably people choose the lump sum because they think they can get a better return for their money than the discount rate (5 per cent) on which the lump sum has been calculated. Suppose they can get 10 percent. Using formulas presented in the text plus some tax assumptions (45 per cent rate, including state and federal) I calculate that after 25 years you would be almost $6 million better off with the annuity than with the lump sum!

Chapter 6, "The Psychology of Investing," had a lot of material which was new to me (although I consider myself a pretty sophisticated investor), especially the Dow theory. This technical idea explains "resistance levels" which are more important in the market than a mathematically trained person would intuitively imagine.8

Chapter 7 is entitled "A Mathematical Miscellany," and discusses, for example, game theory and its applications to business and auctions.

Chapter 8, "Statistics," introduces concepts such as variance and correlations which should be well known to our readership, but there are intriguing examples which are worth studying. These involve both investing (how much you can trust your broker) and legal cases (how much you can trust a witness. How much you can trust your lawyer is another question entirely!)

The final chapter talks about "options." It explains what puts and calls are, how to value options and how to use them to decrease risk in investing. They are, of course, used more often for leverage than for risk-avoidance, and that brings many investors to grief. The formula for evaluating an option is known as the Black-Scholes (or Black-Scholes-Merton) equation for which Myron Scholes shared the 1997 Nobel Prize in economics with Robert Merton.9 (Fischer Black died in 1996). A couple of years or so after the Nobel Prize was awarded, Black-Scholes became "black holes" into which Scholes' company, formed to apply his theories to the trading of options, managed to dump an enormous amount of money. That, along with earlier options fiasco in Orange County, California, have given options a bad name, but still this is a subject which every savvy investor should understand.

I really enjoyed reading this book, and recommend it heartily to everyone who is interested in mammon.10 But be sure to supplement it with a book which discusses taxes.


    1. 12+  years for men, 15+ for women.

    2. Another is the incredible complexity of today's tax laws, which affect everything from investing and borrowing money to providing for one's children's education. Regrettably, Davis, in the interest of simplicity, has not considered taxes in his book.

    3. Even theoretical particle physicists!

    4. The author connects this very counter-intuitive result to the law of large numbers.

    5. It is the compound effect, for example, which makes trees grow so fast.

    6. Suppose you are 25 years old and decide that you will need $2,000,000 when you retire at age 65. If you can invest your money at 8 percent, then the "rule of 69" gives a doubling time of 8 years, so in 40 years your investment will increase by a factor of 32. You will need to come up with an initial investment of only $62,500 (a more refined calculation gives $66,6667.) You can expect to realize 8.5 percent from the stock market if past history is any guide. (In fact many stockbrokers suggest 10 or even 12 per cent as a realistic figure, but let's stick to 8.5.) Investing in mutual funds involves paying capital gains taxes every year as well as fees called "expense ratios," both of which will stunt the growth of your money. (Unmanaged index funds are an exception, but there you run into the problem that they are unmanaged.) And if you invest in a tax-deferred annuity, you have to pay ordinary income tax, rather than capital gains, on your profits. Your best bet might be to buy a portfolio of ordinary stocks, which will subject you only to capital gains taxes of about $386, 667 if you sell them all at age 65. So to realize a full $2,000,000 you would have to invest $83,333 rather than $62,500. If you don't have a lump sum to contribute, you can realize the same $2 million after taxes by investing $7000 annually. (Incidentally, I have ignored dividend distributions in these calculations, a factor that further complicates the issue since they are taxed every year as ordinary income.) This little exercise illustrates the danger of ignoring taxes when carrying out financial calculations; cf. Note 2, supra. To illustrate the sensitivity of growth to interest rates suppose that instead of 8 percent you realize only an 8 percent return. With the same $83,333 initial investment your nest egg would grow (after taxes) to about $1,635,000 rather than $2,000,000. This is an after tax difference of $364,500. A simple generalization to mutual funds (assuming you decide to go that route) demonstrates the importance of considering the expense ratios charged by the various funds. I will leave it as an exercise to the reader to calculate that a fund with a 6 percent up-front load and an expense ratio of one point will outperform a no-load fund with a 1.5 percent expense ratio by $203,000 assuming the same parameters as above ($ 83,333 at 8 percent before expenses, for 40 years. Year-by-year capital gain taxes have been ignored. An "expense ratio" of q points, incidentally, simply means that the fund extracts q per cent of the value of the fund every year.) I am indebted to Ms. Phyllis Albritton of First Union Securities for a fruitful discussion of the material in this endnote

    7. If taxes are not taken into account, then the lump sum beats the annuity if the recipient invests at the discount rate, or even slightly below. Presumably, the people who choose lump-sum payoffs don't plan to pay taxes. The discount rate d is related  to the rate available for a "risk free" investment i, usually taken to be a U.S. Treasury bond, by the formula  d = i/i+1.  For the time intervals we are talking about here it would be natural to take the interest on the long bond which, alas, no longer exists. But looking at prevailing rates on the 10 year bond and highly-rated corporate bonds it seems reasonable to take a discount rate of  about 5 per cent, and that is what our calculations have been based upon. I don't know what the lottery people choose for a discount rate, probably more since the higher the rate the lower the lump sum payoff.

    8. They are due to the individual's reluctance to admit that he/she has made a mistake.

    9. This equation is derived pretty much from the single assumption that "arbitrage" is impossible. Arbitrage is discussed in various places in the book; it refers to the ability to make an instantaneous risk-free profit, for example by buying currency in London at one price and selling it at a higher price in New York (instantaneously, by computer). So the prices in London and New York must be equal.

    10. But don't serve mammon (cf. KJV, Matthew vi. 24).