“If You Bet Too Much, You’ll Almost Certainly Be Ruined” – Ed Thorp

I follow a few blogs regularly. One is mebfaber.com. Meb uses simple, clear language to share data and insights. His recent podcast covered an interview with Ed Thorp, a pioneer in more ways than I can describe here. Listen to it if you have one hour to spare. If not, just read the summary.

The title of the podcast is also the title of this blog post. The consideration of “bet size” has been on my mind recently. Each year, I place a few bets that have small odds of producing large payouts. “Small odds” means that most of these bets don’t win. This is to be expected, but the process of identifying, placing, and following through on such bets is both mathematically complex and emotionally demanding. Ed Thorp is one of the world’s foremost experts on this kind of betting. And at age 84 he has a boatload of experience to share.

Ed says that if you bet too little, it won’t make an impact on your portfolio, which is obvious, but if you bet too much you’ll almost certainly be ruined. The second part of the statement – the part about losing a lot of money – is interesting for two reasons.

First, for a favorable bet with known odds, the Kelly criterion gives us the optimum bet size. The Kelly criterion is optimum in the sense that if the same bet is placed repeatedly, then the Kelly bet size will grow money faster than any other betting method. To illustrate the problem with large bet sizes, consider the following two bets.

  • In the first bet (Bet 1), we have a 1% chance of increasing the betting money 200-fold, and a 99% chance of losing everything we’ve bet.
  • In the second (Bet 2), we have a 50% chance of quadrupling our money, and a 50% chance of going bust.

Both bets have the same expected return. “On average”, both bets will double our money after a single bet, but stating it in this way is misleading. Here’s why. Kelly says that it is optimum to bet ~0.5% of our wealth if presented with the Bet 1, and ~33.3% of our wealth if presented with Bet 2. Intuitively, this is because the first bet doesn’t win very often, so we bet only small amounts and once in a blue moon we hit the jackpot. With the first bet, the maximum rate of compounding achievable in the long run is less than 0.2%, whereas the with the second bet, we can achieve 14.4%. These numbers can be calculated using some simple formulas that can be found here.

The conclusion from the above is that it is best to bet tiny amounts on profitable bets with rare large payouts (e.g. Bet 1), and one bet of this sort doesn’t contribute very much to long-term growth. This conclusion is perhaps counterintuitive, and in stark contrast to the excitement that the prospect of a 200-fold gain produces. A high-risk high-return bet resembles a lottery, and many people bet on lotteries even though they know that “on average” lotteries lose money. The less obvious point is that even Bet 1 and Bet 2, which have positive expected returns, will lose money if we bet too much.

This brings me to the point about “being ruined” in the title of this post. If, instead of betting the Kelly amount, we bet our favorite amount, say 10% of our wealth, on each bet. The first bet will lose money at a compounded rate of 7.4%. In fact, any bet size larger than 1.26% will lose money on the first bet. Betting less than the Kelly bet size is suboptimal, so it leaves some gains unutilized, but it doesn’t lose money.

There is another consideration here that is perhaps as important. In real life, the odds of profitable bets are seldom known. The odds of some games are known, but the universe is producing an endless stream of opportunities to profit whose odds can only be imperfectly estimated. Human nature messes with our ability to estimate these odds. Placing Kelly bets on mis-estimated odds can also lead to rapid loss of wealth. The fix is simple: be less greedy. By placing smaller bets than we otherwise would, we’ll have a better chance of coming out ahead.

Be happy. Be wealthy.

Happiness is too complex a subject to cover in a blog post. But pursuing wealth without understanding its link to happiness can be harmful. I define happiness as a state of being that every human being aspires to. Since human aspirations are varied, everyone’s happiness is also a little different. Yet, as there are threads that bind humanity, so are there themes that are common to happiness. The question: “How can we be happy?” is so fundamental that it has absorbed the minds of philosophers for ages.

Before I repeat the conclusions of some of my favorite philosophers, let’s consider the four squares below.happiness-four-squares

The squares say ‘Health’, ‘Love’, ‘Respect’ and ‘Wealth’. Each square offers an example of a human need.

  • Can a person who is very sick be happy?
  • How about someone who lacks love, who has no close friends or family?
  • Can a person who does not have the respect of others, and is treated as unimportant, be happy?
  • Finally, can a person in extreme poverty be happy?

I consider Buddha to be one of the greatest philosophers of all time. He lived around 500BC; in short, a long time ago. Buddha was born a prince. He married, had a son, yet the sight of suffering led him to relinquish his princely life and seek higher understanding. His quest brought him to the conclusion that the cause of sorrow is unfulfillment of human desire, and therefore detachment is the way to happiness.

Being an applied mathematician, I think of Buddha’s conclusion as an asymptotic result, meaning something that is true at the extreme limit of self-control. At that limit, excruciating pain of a terminal disease, complete loss of family and friends, and extreme poverty would all fail to lower a person’s spirits. Most of us are not so strong. Buddhism acknowledges this too. Regardless of whether the ultimate goal (of Buddhahood) is actually reachable, one can make progress towards it. This was the state of our understanding of happiness circa 500BC (if not earlier).

The world we live in today is far more comfortable than Buddha’s. His world lacked electricity, running water, climate control, and modern medicine among other things. With time, our understanding of happiness has also evolved. The absence of pain or sorrow is not the same as happiness. The conclusion that we can influence (if not completely control) our happiness by changing our perception is a central idea of Positive Psychology. Evidence to support this idea has been found in psychology experiments as well as in studies on the structure and function of the human brain. The book “The Happiness Advantage” by Shawn Achor talks about some of these ideas.

Clearly, thinkers across thousands of years have concluded that our attitude greatly influences how much fulfilment we experience. Indeed, one of my favorite quotes is this one by John Milton:

The mind is its own place, and in itself can make a heaven of hell, a hell of heaven.

So then, why bother about money? One important reason is that most (if not all) of us are far from attaining the state of Buddhahood. Some desires are needs that are painful not to fulfil. If the cause of sorrow is unfulfilled desire, it helps to satisfy some needs, and have the maturity to understand that the stream of desires is endless; therefore, some of them will never be fulfilled. More importantly, our desires don’t all belong in the same league. While it may be immature to hanker after more money to keep up with one’s neighbors, the pursuit of financial security is often driven by a sense of duty towards one’s children, or a desire to not be a burden on others in old age. In the context of these goals, the pursuit of wealth is an act of responsibility as opposed to something detrimental. 

I will end with a cliché: “Money is not everything.” In this quote, I could replace “money” with any other noun (except ‘everything’), and it would still hold true. But the fact remains that poverty leads to both real and perceived deprivation, and most humans find it hard to stay positive when they are very poor. Being financially independent helps.


Why do we get paid?

Why do we get paid?

Well, you have to do something.

True. Some work has to be done in order to get paid.

Yes, but you could win a lottery too.

Or receive a gift; inherit a fortune perhaps. That’s not quite the same as getting paid though. It’s receiving money by chance, not compensation. So let’s leave aside lotteries and inheritances.

Well then there it is. You do something, and you get paid.

Not quite. I could dig a hole in the middle of nowhere, and I wouldn’t be paid. A lot of effort goes uncompensated.

That’s because it’s useless work. The work has to be useful.

Good point. The work has to have utility. While utility is necessary, as it turns out, it isn’t sufficient for getting paid. A lot of volunteers and interns are unpaid. Sometimes, people devote their time for free to causes they care about. Other times, people would like to be paid, but they work for free anyway hoping that they will gain work experience, add something to their resume, build a good reputation etc. so that they can find work that pays. And all the while, they are working and doing something useful, but not getting paid.

Then what is it? Work experience? A good resume? Trust of others? A brand people recognize? 

It’s simple really. In order to be paid, there has to be someone on the other end who is paying.

So you have to do work that helps someone else.

Yes. When we say “useful work”, it is someone else, effectively a customer or employer of our service, who gets to decide whether the work is worth paying for. It follows that this customer should both have money to pay, and the willingness to pay. Paying customers can be people, businesses, organizations with a budget, governments etc. but they cannot be abstract entities like humanity or society.

But shouldn’t we serve humanity?

Of course we should. A giving nature, a willingness to serve, these are qualities that enrich our lives and the lives of those around us. They also lead to long-term gains in our profession. But one should not expect compensation in the short-term for generously serving abstract entities which don’t have a budget, or for serving those who lack either the means or the willingness to pay for our work. It is vital to understand this.

Okay, so we get paid for work that is useful to someone else who has money and is ready to pay.

Yes, I think that’s the answer to the question: Why do we get paid? This is an important question, one of several important questions whose answers guide us in our search for fulfilment.

Buffett’s journey – a case study

My previous blog post was on the importance of saving in the early part of life. Saving early gives investments the time they need to grow. Now, I use the example of a well-known billionaire – Warren Buffett – to emphasize a few key points.

As of the end of 2015, Buffett’s wealth was estimated at somewhere between $60 and $70 billion. Is Buffett’s example a realistic one to use for the average person? It is. Not in the “everybody can be a billionaire” sense, which is obviously not true. Buffett however, was not born into exceptional wealth, nor did he become a billionaire overnight. He became focused on amassing wealth at an early age, did jobs and earned money for offering his investment skill as a service to others. He was uncommonly frugal, as described in his various biographies. As his earnings grew, so did his savings. At some point, the compounding of his wealth outgrew any salary or fees he would earn. My guess is that his investment return was already more than his salary/fees when he was in his 30s. Buffett is 85 now, and the wealth we see today is the result of over half-a-century of compounding at a rate faster than the US stock market’s. His is an extraordinarily executed example of the wealth growth trajectory I described in my earlier post. Persons who can emulate the things he did right even to a limited extent can find financial prosperity.

Here are a few data points on the growth of his wealth (quoted verbatim from the wikipedia page describing Warren Buffett):

  1. By 1950, at 20, Buffett had made and saved $9,800.
  2. In 1956, … Buffett’s personal savings were over $174,000.
  3. By age thirty-five, he was worth $7 million. – from “Buffett-The making of an American Capitalist” by Roger Lowenstein.
  4. Buffett became a paper billionaire when Berkshire Hathaway began selling class A shares on May 29, 1990.
  5. On August 14, 2014, the price of Berkshire Hathaway’s shares hit US$200,000 a share for the first time. While Buffett had given away much of his stock to charities by this time, he still held 321,000 shares worth US$64.2 billion.

Buffett’s net worth vs. his age is plotted below:

Buffett's net worth over time

And here’s the rate at which his net worth grew over different periods.

Age Rate of growth of net worth
20-26 62%
26-35 51%
35-60 22%
60-84 19%

It is easy to see that Buffett’s wealth grew much more rapidly in his early years. There are two main reasons for this:

  1. When he was young, his wealth was growing from two sources. One was the money he saved from the salary/fees he earned, and the other was his investments. Later on, his savings became insignificant, and the growth of his wealth was due to investments alone.
  2. A skilled investor like Buffett can achieve higher investment return when he is investing only a small amount of money. When investing tens of billions of dollars, one is forced to invest in a lot of different things. The business Buffett likes best might return 50%/yr but the size of such business opportunities is usually small – in the millions, not billions. Buffett commented in an interview: “If I was running $1 million today, or $10 million for that matter, I’d be fully invested. Anyone who says that size does not hurt investment performance is selling. The highest rates of return I’ve ever achieved were in the 1950s. I killed the Dow. You ought to see the numbers. But I was investing peanuts then. It’s a huge structural advantage not to have a lot of money. I think I could make you 50% a year on $1 million. No, I know I could. I guarantee that.” Regardless of whether Buffett can actually achieve 50%, the inverse relationship between investment size and investment return is logical.

My main point here is that early life is a great time to multiply one’s net worth. Even without Buffett’s investment acumen, today a frugal person with a healthy salary can save five figures each year and quickly achieve a net worth of $100,000 in 5-10 years, at which point investment returns start to matter. Buffett went from $9,800 to $174,000 in the six years between age 20 and 26. That was a lot of money in 1956. Today, a person earning $60,000/yr, and saving $18,000 (which happens to be the annual contribution limit for a 401K) can go from $0 to $72,000 in six years by just saving. Saving $18,000 out of a $60,000 salary isn’t easy, but it’s doable. Investing like Buffett is a lot harder.

Now imagine that the saver, with $72,000 saved, decided to invest in the US stock market (S&P500) for the next fifty years, and achieved not Buffett-like returns, but merely stock market returns. Based on the data here, the index was priced at 44.15 in 1956 (when Buffett was 26) and priced at 1278.73 in 2006 fifty years later. If our saver achieved the same growth as the index has done in the past, then the $72,000 would grow into a little over 2 million. The saver wouldn’t be a billionaire – not even close – but I’ve assumed that the saver saved nothing past the first five years, and I’ve also not taken dividends paid by the S&P500 into account. This saver would be one very prosperous retiree by doing two simple things: saving early and staying invested.

Stories of people who earned little but saved, invested, and retired rich are numerous. But in all fairness, I make it sound easy. While the path outlined is real, it is hard to follow because of human nature. I will talk about pitfalls of investing another day.

Saving dominates initially. Investing later.

One of my friends is a retired investment advisor. In a presentation, he emphasized: Start investing early. He illustrated this with an imaginary example of two people, Amy and Bob. They both get jobs at the same time, and both work for 30 years before retiring. Amy saves and invests $10,000 each year for the first 10 years of her life. She then saves no more but keeps her money invested. Bob, on the other hand, saves nothing for the first 10 years, but saves and invests $10,000 each year from year 11 through year 30. At retirement, Amy, who saved for only 10 years, has more money than Bob, who saved for 20 years. Assuming that investments grow at 8% each year (for simplicity) Amy has $675,212 whereas Bob has only $457,620. This is true even though Bob saved for twice as many years. Moral of the story: Start saving and investing early.
But why does Amy have more? There’s an important insight to be gleaned from understanding why the early years are so important. To answer the why, we must observe the how – how money compounds over time. The charts below show the net worths of both individuals over the years. Net worth is divided into two parts: one part that can be attributed to savings accumulated over the years, and another attributed to accumulated investment income. Cumulative savings and investment income are shown as stacked columns for Amy and Bob in the two charts below.
A few basic observations are presented below:
  1. Amy’s savings start from $0 and grow to $100,000 in the first ten years. Her cumulative savings don’t grow after that. However, her investments continue to grow.
  2. Amy’s investment income is relatively modest in the first ten years. Whereas she saves $100,000, her investment income adds up to a total of $44,866 over ten years. Compare this with the last year of her career, when in that one year alone her investment income is $54,017, more than she earns in the first ten years put together.
  3. Bob saves nothing, and consequently has no investment income in the first ten years. His net worth is zero until year 10. From years 11 through 20, his wealth grows exactly as Amy’s grew in years 1 though 10. But then there is a departure. Bob continues to save, whereas Amy stopped saving after the first 10 years.

Sadly for Bob, he still falls short despite saving for twice as many years. Saving is hard. It involves sacrifice. Why does Bob fall short? This question can be largely answered by looking at year 11. In year 11, Bob saves $10,000. Amy saves nothing. But Amy’s investments earn $12,516. Amy’s investments are growing faster than Bob can save! Amy had a huge head start. Even though her investment income seemed small in the first 10 years, the total pile of savings and investments she built in those early years was enough to outpace Bob. In year 30, Bob’s investment income is $36,610, which isn’t small, but it is well short of the $54,017 Amy’s investments earn that year. For savers, investment income becomes sizeable as they approach retirement, as it should be. After all, in retirement, people expect to be able to live off of their investment income without reducing their net worth very much. After all, who knows how long we are to live.

My friend, the retired financial advisor, lamented that this parents hadn’t invested their money. They had government jobs. They didn’t earn much. They had to save painstakingly to retire.  But despite everything, they retired. They did one key thing right. They saved.