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 8½ 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-à-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.

Endnotes:

- 12+ years for men, 15+ for women.
- 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.
- Even theoretical particle physicists!
- The author connects this very counter-intuitive result to the law of large numbers.
- It is the compound effect, for example, which makes trees grow so fast.
- 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
8½ 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 8½ 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 8½ 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 - 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.
- They are due to the individual's reluctance to admit that he/she has made a mistake.
- 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.
- But don't
*serve*mammon (cf. KJV, Matthew vi. 24).