The Kelly Criterion revisited

In my previous post about the Kelly Criterion for gambling and investing, I went through the mathematics to derive the Criterion. One of the models we discussed was this:

You are going to play an infinite series of games starting with a bankroll of B dollars. In each game, for each dollar bet, with probability p you will win W dollars and, with probability q (equal to 1-p) you will lose L dollars. In each game, you bet some fraction f of your current bankroll at the start of that game. The question is what should f be?

The solution given by the Kelly Criterion is the value of f that maximizes the geometric mean of your final bankroll (or equivalently the arithmetic mean of the logarithm of your final bankroll). For the model just given, that value of f is

f = (pW – qL) / WL

In the long term (an infinite series of bets), with probability one, this value of f minimizes the time required to reach any specified goal for the bankroll and will result in a larger final bankroll than any other strategy.

As we discussed, there is no controversy about the math underlying the Kelly Criterion, but there is a significant controversy about whether the f specified by the Kelly Criterion should actually be used in real life gambling or investing situations. In those situations, the number of games is finite, and the properties of infinite sequences might not be an appropriate guide to what you should do.

Some people are concerned that when the value of f corresponding to the Kelly Criterion is used, there is a large amount of volatility in the bankroll as the sequence of games progresses. For example, starting at any point in the infinite sequence, the probability is 1/3 that the bankroll will halve before it doubles. Many of these people would recommend betting Half Kelly (half the value corresponding to the Kelly Criterion), which gives about 75% of the growth rate but with significantly less volatility. (For example, the probability is 1/9 that the bankroll will half before it doubles.)

Other people would recommend betting more than the Kelly Criterion, which increases the volatility, but allows a larger possible gain (but with a low probability). For example, a value of f equal to one and a half times the Kelly Criterion also give about 75% of the growth rate, but with greater volatility, and a value of f equal to twice the Kelly Criterion gives a growth rate of 0 and even greater volatility.

To gain some insight into this situation, I made up a simple example, with

p = ½, W = 1, and L = .5

For these values

Kelly f = 0.5 and gain G = 1.06

Assume we are going to play only four games, and the initial bankroll B = 100.

We consider using four different values of f

f = 0.25, f = 0.5, f = 0.75, and f = 1.

For each of these values of f and each of the possibilities after four games, the final bankroll is given by the table.

Final Bankroll After Four Games

	Probability	1/2 Kelly	Kelly	3/2 Kelly	2 Kelly
		f = .25	f = 0.5	f = .75	f = 1.0
4 Wins, 0 Losses	0.06	244	506	937	1600
3 Wins, 1 Loss	0.25	171	235	335	400
2 Wins, 2 Losses	0.38	120	127	120	100
1 Win, 3 Losses	0.25	84	63	43	25
0 Wins, 4 Losses	0.06	59	32	15	6.25
Arithmetic Mean		128	160	194	281
Geometric Mean		105	106	105	100

The first column in the table enumerates the possible number of wins and losses after four games. The second column gives the probabilities of each of these outcomes.

The remaining columns give the final bankroll for each outcome for each value of f.

The last two rows of the table give the arithmetic and geometric means of the outcomes for each value of f.

The median and mode of the distributions for each value of f occurs when there are 2 wins and 2 losses. The median and modes are both largest (as is the geometric mean) for the value of f corresponding to the Kelly Criterion, (f = 0.5), but the arithmetic mean is largest for the value of f corresponding to twice the Kelly Criterion.

But suppose we removed all references to the Kelly Criterion from the table and just looked at the four possible ways to gamble. It certainly isn’t clear (to me at least) which is best.

For example, some people might want to bet their entire bankroll on each game (twice the Kelly Criterion) even though the most likely outcome is that they just break even, because there is some probability that they win a large amount of money (1600 dollars). Real people often make “bad” bets because of the small possibility of making a large amount of money. Think of state lotteries. (Insurance is also a bad bet because the insurance companies set the premiums so that they have an edge, but people nevertheless buy insurance because when they need it, they need it.)

I copied this quotation from the Web (http://www.bjmath.com/bjmath/kelly/mandk.htm)

If I maximize the expected square-root of wealth and you maximize expected log of wealth, then after 10 years you will be richer 90% of the time. But so what, because I will be much richer the remaining 10% of the time. After 20 years, you will be richer 99% of the time, but I will be fantastically richer the remaining 1% of the time.

That person is willing to take a large risk if there is a possibility of making a large amount of money

Some other people are not willing to take a significant risk and might want to bet the Half Kelly because it has the smallest possible loss of the four alternatives even though it has the smallest arithmetic mean and the smallest maximum gain.

Again, looking at the table, it seems to me that there are some people who would want to use each of the four ways to gamble (as well as possibly other ways).

I am not sure about what all of this says about the Kelly Criterion, except that it is mathematically interesting and a good benchmark to use to begin thinking about how to manage your money when you have an edge in gambling or investing.

Women Think Differently Than Men

Girls think differently than boys. Women think differently than men.

• Anyone who is married knows that those statements are true
• Anyone who has a sibling of the opposite sex knows that those statements are true
• Anyone who has ever had a date with a person of the opposite sex knows that these statements are true
• Anyone who has attended a coed school knows that these statements are true
• Anyone who has a mother and a father knows that those statements are true
• Anyone who has children of both sexes knows that these statements are true
• Anyone who has lived in the world knows that those statements are true

Can we just agree that those statements are true?

Let’s not debate:

• Why women think differently than men
• In what ways women think differently than men
• What implications this difference in thinking has on education and on careers

There are so many issues of political correctness involved in such debates that they hide the obvious fact that women do think differently than men.

Let’s just agree then on the one obvious fact:

Women think differently than men.

I Hate the Term "IT Professional"

These days almost everyone who is paid to work with computers is referred to as an IT professional. I find that terminology very confusing.

So what is an IT professional? There are many occupations involved with computing: for example, Computer Scientist, Information Systems Professional, Software Engineer (all three of which are accredited professions; their educational programs are accredited by ABET, the Accreditation Board of Engineering and Technology), System Support Person, Database Administrator, Network Administrator, Project Manager, Web Designer, Help Desk Person, etc. People working in each of those occupations have different skills and the abilities to accomplish different kinds of tasks. Calling them all IT professionals blurs these differences and can cause serious problems.

As an example outside of the computing field consider airplanes. We do not call all the people who are paid to work with airplanes APs (Airplane Professionals). We call those people who design airplanes, Aeronautical Engineers; those who service them, Airplane Mechanics; those who fly them, Pilots. We might also talk about Stewardesses, Baggage Handlers, Ticket Agents, etc.

Continuing with the airplane example, if American Airlines wanted a new airplane built, they would not think of asking the airplane mechanics who service their planes to design the new plane; they would, for example, ask the aeronautical engineers who work for Boeing. And Boeing would not think of designing and building that plane without using engineers from accredited engineering programs.

But in the computing field, we do that kind of thing all the time. Since we call everyone in the field an IT professional, people often assume that all IT professionals have the same skills. Companies who want a new software system built often ask the “IT professionals” on their system support staff, who might have excellent skills in maintaining their existing system, to design and build their new system. That often leads to projects that are later cancelled, or, if they are completed, are late, over-budget, incorrect, unreliable, insecure, hard to use, etc. Supporting a large heterogeneous system is a difficult skill, but it is a very different skill than specifying, designing, and building a high quality system, which is also a difficult skill.

Even asking the system support staff to oversee a project performed by outside consultants can be a problem, because the skills required to oversee such a project (as well as to evaluate the abilities of proposed consultants) are surprisingly difficult and again different from those required to support and maintain systems.

The same kind of reasoning often occurs when companies select project managers for their software projects. Many people think that software project managers do not need professional skills in software design, in addition to the obviously-needed skills in project management. However, no one would think that the manager of an airplane design project does not need such skills. Not long ago, I copied the qualifications for a particular project manager position from the Boeing Job Search Web site:

Bachelor’s degree in a technical field with thirteen years experience in an engineering classification. Master’s degree desirable. Training in systems engineering and project management.

No one can even apply for this job without a technical degree and at least thirteen years of engineering experience. Compare that with the current practice in the software area. No wonder so many software projects fail while so many airplane design projects are successful.

Many books and papers have been written on the subject of why so many software projects fail. (Some people would say it is because software design and implementation is so complicated, but few would argue that it is more complicated that airplane design and implementation.) Whenever anyone asks me why software projects fail, I always give the same answer: “incompetent project manager.” I firmly believe in the statement “The buck stops here.” If the software project succeeds, give the project manager a raise; if it fails, fire him.

At a still higher level, many companies have a CFO (Chief Financial Officer) with a professional accounting degree and a CTO (Chief Technical Officer) with some professional technical degree, but the CIOs (Chief Information Officers) of most companies do not have any professional degree in the computing field. The companies would say that it is more important that a CIO understand the business needs of the company than that he have technical skills in the IT area. If that is true, which is doubtful, then why do they insist that their CTO have a professional degree in some technical field? Why is it not more important that the CTO understand the business needs of the company than that he have technical skills. One reason the companies believe that their CIO does not need to have a technical degree is that by calling everyone an IT professional, we have deemphasized the importance of professional degrees in the computing field.

Another related issue is outsourcing. The statement that “X hundred thousand IT jobs have been outsourced” is very confusing, particularly to potential students. Again this is partly the fault of the words “IT professional.” Since we have blurred the distinctions between the various jobs within IT, it is not usually made clear which kinds of IT jobs are being outsourced and which are likely to remain. Thus many potential students have been scared away from Computer Science programs. (See my post dated 12/07/06, Will My Computer Science Job Be Outsourced?)

One source of confusion is that many of the jobs that are called professional within the IT community are not really professions in the usual sense of the word. For example, someone who has a Computer Science degree is obviously a professional, but do we want to use that term to refer to someone who has a Cisco CCNP (Cisco Certified Network Professional) certification? How about someone who has Help Desk Professionals certification? Calling all of these people IT professionals diminishes the reputation of the real professions within the IT community and confuses the companies for which they work and the public in general.

To summarize, the title IT professional can be very confusing at a number of levels. Different people working in the computing field have very different skills. Giving the same title to all of those people, many of whom are not professionals in the usual sense of the word, is a source of confusion and often a recipe for serious failure.

Will My Computer Science Job Be Outsourced?

Question: I am a student thinking of majoring in Computer Science. If I do, will my job be outsourced?

Answer: You might have been warned not to become a computer scientist because so many “IT jobs” are being outsourced to India and China. Is that a good reason to decide not to become a computer scientist?

First of all, the words “IT” (Information Technology) refer to many types of computer-related jobs other than computer scientist, for example a help desk person or a low-level programming job. And, indeed, many of these lower-level jobs are being outsourced to countries where people are willing to work for a much lower salary than would be required in this country.

But yes, many high-level design and programming jobs are also being outsourced. And several companies have established research and development laboratories in such countries as India, China, Ireland, and Israel to perform research in various areas of Computer Science.

However, even with the reality of outsourcing, computer-related jobs are still predicted to be among the fastest growing occupations over the next decade and beyond. For example:

· The College Board, the organization that administers the SAT examinations, has a Web site with a page, 10 Hottest Careers for College Graduates (2002-2012), and on that page is a list of the “Occupations with the Most New Jobs: Bachelor's Degrees.” Six of these ten occupations are Computer Science related:

o Computer systems analyst

o Computer software engineer (applications)

o Computer software engineer (system analyst)

o Network systems and data communications analyst

o Network and computer systems administrator

o Computer programmer.

· The U.S. Department of Labor says that while jobs for computer programmers will grow about as fast as average through 2012, other Computer Science related jobs (software engineers, support specialists, system administrators, systems analysts, database administrators, computer scientists) will grow faster than average, and jobs for software engineers are projected to increase faster than almost any other occupation.

· The National Association of Colleges and Employers in its 2005 salary survey identified ``Software Design and Development'' as one of the top 10 jobs for students with new Bachelor's degrees, and the job with the highest starting salary among the top 10. Computer Science majors with new Bachelor's degrees earn the second highest average starting salary overall.

· In 2006, Money magazine published a list of the 50 “Best Jobs in America.” Number 1 on the list was software engineer, and number 7 was computer/IT analyst. One measure they used in making up this list was their “10 year job growth forecast.” That forecast for software engineers was 46.07% and for computer/IT analysts was 36.10%.

The technology is changing so fast and the need for new computer professionals is growing so rapidly that there will be plenty of jobs for computer professionals in the United States as well as in all the other countries of the world. In fact, in 2004 Bill Gates predicted that there would be a serious shortage of computer professionals in the United States by the year 2012.

The Real Issue

Outsourcing is not the real issue. The issue is what country will be the world leader in conceiving and developing the coming generations of exciting new applications of computers and in starting the new companies that will produce and sell those applications.

Instead of using outsourcing as a reason not to become a computer scientist, we should instead view it as a challenge. These other countries are challenging our technical leadership in the computer area. That leadership has been an important factor in our economic growth and well being over the past several decades. For example, companies such as Google, Microsoft, eBay, Cisco, and Dell were formed by American entrepreneurs, many of them while they were still in college.

Do we want to retain that leadership? If so, we need to attract more, not fewer computer scientists who will develop the new concepts, technologies, and products that will certainly be developed elsewhere if not here.

If the word “outsourcing” scares off too many young people, and not enough smart, creative, entrepreneurial people in the United States decide to become computer scientists, we might find ourselves giving up our leadership to India, China, or some other country. That would almost certainly be bad for our economic well-being and that of our children.

You are the generation that will determine whether or not we retain our position as leaders in computer technology.

Web Sites Containing the Data Mentioned Above:

http://stats.bls.gov/oco/oco1002.htm

http://www.jobweb.com/SalaryInfo/default.htm

http://money.cnn.com/magazines/moneymag/bestjobs/top50/index.html

ftp://ftp.bls.gov/pub/special.requests/lf/aat9.txt

http://www.collegeboard.com/article/0,3868,4-24-0-236,00.html#table%203

http://stats.bls.gov/oco/oco1002.htm

http://www.jobweb.com/SalaryInfo/default.htm

The Kelly Criterion

When I read William Poundstone’s Fortunes Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street, I was intrigued by his discussion of the Kelly Criterion for gambling and investing. But since the book was written for a general audience, I was not satisfied with the mathematical level of his description of the Kelly Criterion. So I decided to read more about it to see if I could understand the math a bit better. Below is the result of my endeavors. I think I understand the math now, but that still does not answer the question of whether the Kelly Criterion is appropriate for the average person to use for gambling or investing. But at least it helps to understand the math. (To understand the math in this post, you need a first course in calculus.)

The Kelly Criterion

Suppose you are gambling. You are going to play a sequence of games. If you win a game, you win W dollars for each dollar bet. If you lose, you lose your bet. For each game, the probability is p that you will win and q (equal to 1 – p) that you will lose. You start out with a bankroll of B dollars. You bet some percentage, f, of your bankroll on the first game, that is you bet fB dollars. After the first game, your new bankroll is B’ dollars depending on whether you win or lose. You then bet the same percentage, f, of your new bankroll on the second game, that is you bet fB’ dollars. And so on. For each game, you bet the same percentage, f, of your current bankroll on that game. The problem is: what value of f should you chose. The value of f that maximizes your expected gain per game is called the Kelly Criterion.

To see why this problem is not as easy as it might sound, suppose you can play a game where you flip a perfect coin, with probability of 1/2 for heads and 1/2 for tails.

If you bet on heads and it comes up heads, you win $100 for each dollar bet; if it comes up tails, you lose your bet. What fraction of your bankroll should you bet? Since the odds are overwhelmingly in your favor, you might first think you should bet your entire bankroll on each game. However, if you do, you will eventually lose a game and thus lose all your money. So what fraction of your money should you bet? If you bet too much, you will go bankrupt. If you bet too little you will not make as much money as you could from this excellent opportunity. We will give the answer later.

Winning W Dollars Or Losing Your Bet

To see how the betting works in the more general case where you win W dollars for each dollar bet with probability p, and lose your bet with probability q, suppose your initial bankroll is B. In the first game you bet fB. If you win, your new bankroll is

B’ = B + WfB

Therefore on the second game you will bet

fB’ = f(1 + fW)B

If you are lucky enough to win again, your new bankroll will be

B” = (1 + fW)B’ = (1 + fW)^2

If you then lose the third game, your bankroll will be

B’’’ = (1 – f)B” = (1 + fW)^2 * (1 – f) B

and so on. Suppose that after n games, you have won w times and lost l times. Then your bankroll will be

(1 + fW)^w * (1 – f)^l * B

Note that the final value of your bankroll does not depend on the order in which you win and lose games, just on how many games you win and lose.

Thus the gain in your bankroll after n games is

(1 + fW)^w * (1 – f)^l

The expected value of this gain per game, G, is the limit as n approaches infinity of the nth root of this gain (the geometric mean),

G = lim (1 + fW)^(w/n) * (1 – f)^(l/n))

Since the limit as n approaches infinity of w/n is p and the limit of l/n is q.

G = (1 + fW)^p * (1 – f)^q

Note that in our original example where you can lose your entire bet in one game, if you bet your entire bankroll on each game (that is, you select f = 1), then G = 0; the expected gain per game is 0.

The expected value of your bankroll after n games is

G^n * B

Because this gain is exponential, the expected rate of gain per game, R, (sometimes called the growth rate) is the logarithm of the expected gain per game.

R = log G = p * log(1 + fW) + q * log (1 – f)

(All logarithms in this paper are natural logarithms to the base e.)

We want to find the value of f that maximizes this expected rate of gain per game.

(Note that maximizing the expected rate of gain also maximizes the expected final value of your bankroll.)

Taking the derivative with respect to f and setting that derivative equal to 0 gives

pW / (1 + fW) – q / (1 – f) = 0

Solving for f gives

f = (pW – q) / W

This value of f is called the Kelly Criterion. People sometimes say that this equation for the Kelly Criterion can be interpreted as Edge/Odds. (pW - q) is your edge (how much you expect to make per game), and W is the odds.

Note that

f = (pW – q) / W = p – q / W

so that f is never more than p, no matter how favorable the odds.

In our first example, where W = 100 and p = 1/2 the value of f is

f = (pW – q) / W = (50 - .5) / 100 = .495

Your expected gain per game is then

G = (1 + .495* 100) ^(1/2) * (1 – .495) ^(1/2) = sqrt(50.5 * .505) = 5.05

One other interesting special case is when. You either win 1 or lose 1.

f = p – q = 2p - 1

For example, if p = 1/2 (when W = 1), then f = 0. You have no advantage and you shouldn’t bet anything.

Winning W Dollars or Losing L Dollars on Each Bet (More Like Investing)

Now consider a more general case, closer to investing, in which on each bet (investment) you might win W dollars or lose L dollars for each dollar bet. (If you assume you cannot lose more than you bet, then) As before the probability that you will win is p and that you will lose is q (equal to 1 - p). Also as before, you always bet some percentage f of your current bankroll.

Now the expected gain per game is

G = (1 + fW)^p * (1 – fL)^q

Again to find the value of f that maximizes the expected rate of gain per game we first take the log of G and then find the value of f that maximizes that log

R = log G = p * log(1 + fW) + q * log (1 – f L)

Taking the derivative with respect to f and setting that derivative equal to 0

pW / (1 + fW) = qL / (1 – fL) = 0

Solving for f gives

f = (pW – qL) / WL

which is the Kelly Criterion for this case.

As a simple example, suppose you play a game in which you have a probability of 1/2 of doubling your bet and a probability of 1/2 of losing 60% of your bet. In this case

f = (.5 * 1 – .5 *6) / 1 * .6 = 1/3

For this value of f, the expected gain per game is

G = (1 + .33)^(1/2) * (1 – (.33 * .60))^(1/2) = sqrt(1.33 * .08) = 1.0328

To see how this works out in a specific situation, suppose your bankroll is 100. You play two games using this value of f, winning the first and losing the second. On the first game you bet 33.33, and after winning, your bankroll is 133.33. On the second game you bet 1/3 of your bankroll, which is 44.44. You lose 60% of that, which is 26.67. Your final bankroll is

133.33 – 26.67 = 106.67

which could have been calculated directly as

(1 + .33)*(1 – (.33*.60)*100 = 106.67

If you had lost the first game and won the second, the final result would have been the same.

By contrast, if you had bet your entire bankroll on each game, after the first (winning) game your bankroll would be 200, and after the second (losing) game your bankroll would be 80. Your expected gain per game would be negative.

N Possible Outcomes

Now consider a still more general case, in which you make a bet and there are n possible outcomes, xi each with probability pi (For example, you might be buying a stock and keeping it for some period of time, and at the end of that period of time, there might be n possible values that that stock might have: x1, x2, …,xn)) In this case

G = Prod((1 + f xi)^pi)

Some values of xi might be positive and other values might be negative.

Taking the log of G gives

R = log G = Sum(pi * log(1 + f xi))

The math would now get a bit complicated. However, if f xi <<>

log(1 + z) = z – z^2 / 2 + z^3 / 3 – z^4 / 4 + ….

to obtain

R = log G = f * Sum(pi xi) – f^2 * Sum(pi xi^2 / 2)

Taking the derivative with respect to f and setting that derivative equal to 0 gives

Sum(pi xi) – f * Sum(pi xi^2) = 0

Solving for f gives

f = Sum(pi xi) / Sum (pi xi^2)

The numerator is the mean of the outcomes, and the denominator is close to the variance of the outcomes (actually slightly larger than the variance), which is

Sum(pi xi^2) – (Sum(pi xi))^2

Varying the Kelly Criterion

The value of f corresponding to the Kelly Criterion often leads to a large amount of volatility in the bankroll. For example, the probability of the bankroll dropping to 1/n of its initial value at some time in an infinite sequence of bets is 1/n. Thus, for example, there is a 50% chance that the bankroll will drop to half of its value at some time. As another example, there is a 1/3 chance that the portfolio will half before it doubles.

Therefore many people propose using a value of f corresponding to one half of the Kelly Criterion (sometimes called a half Kelly), thus obtaining somewhat slower growth but with less volatility. Half Kelly betting yields about 75% of the return of full Kelly. Another reason for using half Kelly is that people often overestimate their edge.

You should also note that if you bet more than the Kelly Criterion, the return is less and the volatility is even greater. If you bet twice the Kelly Criterion, the growth is zero. If you bet more than twice the Kelly Criterion, the growth is negative.

Some Final Comments

The Kelly Criterion is actually quite controversial, as you can see by just Googling the words “Kelly Criterion.” You should realize, however, that the mathematics in this paper is not controversial. What is controversial is whether you should actually use the Kelly Criterion when you gamble (or invest) because you are going to make only a relatively short sequence of bets, compared with the infinite sequence used in the math, and there might be a significant risk because there is a wide variability in the final bankroll for such a relatively short sequence of bets. We do not make any comments on that controversy

References

http://www.geocities.com/gbosmis/MM1.htm

http://en.wikipedia.org/wiki/Kelly_criterion

http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/Context/Kelly%20Resources/

KellyIndex.htm

Poundstone, William, “Fortunes Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street,” Hill and Wang, New York, NY, 2005

Kelly, John L, Jr., A New Interpretation of Information Rate, Bell Systems Technical Journal, Vol. 35, pp917-926, 1956.

The Estate T ax

When I was younger, I was taught that one reason for the estate tax is that if we don’t take money away from the rich every once in a while, they will invest it and, due to the “magic of compound interest,” they will soon have all the money in the world. Now that the government is thinking of abolishing the estate tax, I thought I would look at this “magic of compound interest” situation.

I looked at two situations:

• A rich person invests $1 million (once) in some stocks where he gets 6% gain per year after taxes. (The capital gains tax is 15 %, so his gain before taxes would have to be 7.06%)
• An average person invests $4,000 every year (in a Roth IRA) in some stocks where he gets 7% gain per year (with no income or capital gains taxes).

We assume the investments continue to grow at those rates for 100 and 200 years in the absence of any estate tax.

The math is simple. After n years,

• The rich person’s investment is worth (1 + .06)^n
• The average person’s investment is worth ((1 + .07)^n – 1) / .07

The math is even easier than that because there are many books that contain tables of how the investments grow. I used an ancient book “The Dow Jones-Irwin Guide to Interest” with a copyright date of 1981. (I assume the math hasn’t changed since then.) I have rounded off the numbers somewhat.

The rich person

• After 100 years would have $339 million
• After 200 years would have $115 billion

The average person

• After 100 years would have $49 million
• After 200 years would have $43 billion

By comparison, in 2005 the net worth of Bill Gates, the richest man on earth, was about $47 billion and the U.S. Gross Domestic Product was about $13 trillion. Of course no one knows what those numbers will be 100 or 200 years from now.

It is interesting to note that if Bill Gates invested $10 billion of his money and got 6% return per year after taxes,

• After 100 years his descendents would have $3.39 trillion
• After 200 years his descendents would have $1.15 quadrillion.

A quadrillion is a thousand trillion, a trillion is a thousand billion, and a billion is a thousand million. These definitions of quadrillion, trillion, and billion are the U.S definitions, not the U.K. definitions.

Since the expected life span in the United States in 2004 was 77.9 years (http://www.cdc.gov/nchs/fastats/lifexpec.htm), an investment period that lasts 100 years would escape one round of estate tax and an investment period that lasts 200 years would escape two (maybe three) rounds of estate taxes.

So do we need an estate tax or not?

My Prius

I own a 2006 Prius. In my actual usage of the car, it gets just about 45 miles per gallon. We are going to estimate how much money I save (if any).

But before we do that, note that actual usage can be different than the published EPA mileage numbers. For example, for the 2006 Prius, the EPA mileage for city driving is 60 miles per gallon, for highway driving is 51 miles per gallon, and for combined driving is 55 miles per gallon. In the computations below, we will use the actual miles per gallon that I get, which is 45 miles per gallon.

So how much money do I save on gas? Let’s compare my car with

1. Car A that (actually) gets 30 miles to the gallon and

2. Car B that (actually) gets 15 miles to the gallon.

Assume first that all three cars go 10,000 miles and gasoline costs $3.00 a gallon.

(Since I wrote the first draft of this post, the price of gasoline has started to come down. So I am going to include the results for $2.00 gallon gasoline in parentheses after the results for $3.00 per gallon gasoline.)

1. At 45 miles per gallon, my Prius uses 10,000 / 45 = 222.22 gallons. Since gas costs $3 per gallon, my cost is 222.22 * 3 = $666.66 (at $2.00 per gallon my cost is $444.44)

2. At 30 miles per gallon, Car A uses 10,000 / 30 = 333.33 gallons at a cost of $1,000 (a cost of $666.66)

3. At 15 miles per gallon Car B uses 666.67 gallons at a cost of $2,000 (a cost of $1,333.33)

So when I drive 10,000 miles,

1. Compared to Car A, I save $333.33 (I save $222.22)

2. Compared to Car B, I save $1,333.33 (I save $888.89)

Stated differently

1. Compared to Car A, I save 3 1/3 cents per mile (I save 2.22 cents per mile)

2. Compared to Car B, I save 13 1/3 cents per mile (I save 8.88 cents per mile)

Suppose I keep my Prius for 150,000 miles, which is the mileage my last Toyota had when I donated it to a charitable organization. My total savings over the life of my car would be

1. Compared with Car A, I would save $0.03333 * 150,000 = $5,000 (I would save $3,333)

2. Compared with Car B, I would save $0.13333 * 150,000 = $20.000 (I would save $13,333)

But did I really save anything when considering the extra cost of my car. After all, I certainly paid something extra for the hybrid mechanism. Taking that into account, did I save any money? It’s hard to determine how much I paid for the hybrid mechanism because Toyota doesn’t make a non-hybrid Prius. But they do make both a hybrid and a non-hybrid Camry. The base price for the 2007 hybrid Camry is $25,900, and the base price for the 2007 non-hybrid Camry is $18,445. Thus the base price for the non-hybrid Camry is 71.2% of the base price of the hybrid Camry. Using that same ratio and the base price of the (hybrid) Prius, which is $22,175, the base price of a non-hybrid Prius would be $15,788. Thus the extra cost I paid for the hybrid mechanism would be $6,387.

Hence, over the 150,000 mile estimated life of my Prius,

1. Compared with Car A (with mileage of 30 miles to the gallon), I would lose $1,387 (I would lose $3,054)

2. Compared with Car B (with mileage of 15 miles per gallon), I would gain $13,613 (I would gain $6,946)

However, I got a $3,145 income tax rebate for buying a 2006 Prius. Factoring that in

1. Compared with Car A, I would gain $1,758 (I would gain $91)

2. Compared with Car B, I would gain $16,758 (I would gain $10,071)

Suppose we consider a different car, Car C. How many miles per gallon would Car C have to get so that I would exactly break even after going 150,000 miles considering the cost of the hybrid mechanism (and ignoring my tax rebate)? A little math shows the answer to be about 27.5 miles per gallon (18.3 miles per gallon).

Now let’s consider a different question: what would gasoline have to cost so that the drivers of Car A and Car B would spend the same amount of money on gasoline to go 10,000 miles as I do in my Prius, $666.66 ($444.44). (Of course, the same price of gasoline would apply no matter how many miles we used in the computation.)

1. Since Car A used 333.33 gallons, gasoline would have to cost 666.66 / 333.33 = $2 per gallon ($1.33 per gallon)

2. Since Car B used 666.67 gallons, gasoline would have to cost 666.66 / 666.67 = $1 per gallon ($0.67 per gallon)

Unfortunately, I can remember when gasoline could be bought for all of these prices (and even cheaper).