A Man for All Markets: From Las Vegas to Wall Street, How I Beat the Dealer and the Market

by Edward O. Thorp

It is vastly less stressful to be independent—and one is never independent when involved in a large structure with powerful clients.

True success is exiting some rat race to modulate one’s activities for peace of mind.

trait that showed up at about this time was my tendency not to accept anything I was told until I had checked it for myself.

Understanding and dealing correctly with the trade-off between risk and return is a fundamental, but poorly understood, challenge faced by all gamblers and investors.

In the abstract, life is a mixture of chance and choice. Chance can be thought of as the cards you are dealt in life. Choice is how you play them.

I felt then, as I do now, that what matters is what you do and how you do it, the quality of the time you spend, and the people you share it with.

This plan, of betting only at a level at which I was emotionally comfortable and not advancing until I was ready, enabled me to play my system with a calm and disciplined accuracy. This lesson from the blackjack tables would prove invaluable throughout my investment lifetime as the stakes grew ever larger.

What I had done was focus on a price that was of unique historical significance to me, only me, namely, my purchase price. Behavioral finance theorists, who have in recent decades begun to analyze the psychological errors in thinking that persistently bedevil most investors, call this anchoring (of yourself to a price that has meaning to you but not to the market). Since I really had no predictive power, any exit strategy was as good or bad as any other. Like my first mistake, this error was in the way I thought about the problem of when to sell, choosing an irrelevant criterion—the price I paid—rather than focusing on economic fundamentals like whether cash or alternative investments would serve better.

Lesson: Do not assume that what investors call momentum, a long streak of either rising or falling prices, will continue unless you can make a sound case that it will.

I learned from this that even though I was right in my economic analysis I hadn’t properly evaluated the risk of too much leverage.

I also learned from my losing silver investment that when the interests of the salesmen and promoters differ from those of the client, the client had better look out for himself. This is the well-known agency problem in economics, where the interest of the agents or managers don’t coincide with those of the principals, or owners. Shareholders of companies that have been pillaged by self-serving CEOs and boards of directors are painfully familiar with this.

The pamphlet explained that a common stock purchase warrant is a security issued by a company that gives the owner the right to buy stock at a specified price, known as the exercise price, on or before a stated expiration date. For instance, in 1964 a Sperry Rand warrant entitled the holder to purchase one share of common stock for $28 until September 15, 1967. On this final day, if the stock trades above that price, you can use one warrant plus $28 to buy one share of stock. This means the warrant is worth the amount by which the stock price exceeds $28. However, if the stock price is below $28, it is cheaper to buy the stock outright, in which case the warrant is worthless.

To form a hedge, take two securities whose prices tend to move together, such as a warrant and the common stock it can be used to purchase, but which are comparatively mispriced. Buy the relatively underpriced security and sell short the relatively overpriced security. If the proportions in the position are chosen well, then even though prices fluctuate, the gains and losses on the two sides will approximately offset or hedge each other. If the relative mispricing between the two securities disappears as expected, close the position in both and collect a profit.

The way to profit was to sell them short. To sell a security short you borrow the desired quantity through your broker from someone who owns it, sell it in the marketplace, and collect the proceeds. Later you have to repurchase it at whatever price then prevails to meet your contractual obligation to return what you borrowed. If your buy-back price is below your earlier sale price, you win. If it is higher, you lose.

“Be fearful when others are greedy and greedy when others are fearful.”

find pairs of closely related securities that were priced inconsistently with respect to each other, and use them to construct investments that reduced risk. To form these hedges, we simultaneously bought the relatively underpriced security while offsetting the risk from adverse changes in its price by selling short the comparatively overpriced security.

preferred stock’s dividend is paid first—in preference—before any payments due to the common stock. In the typical case, where the dividend amount is fixed, the preferred is like a bond but more risky because the dividend payments and the claim on assets upon liquidation are only paid after the corresponding bond payments. A so-called convertible preferred is one that can be exchanged for a specified number of shares of the common. So a convertible preferred is like a convertible bond but less secure, as it is paid only if there is enough money to do so after the bondholders receive their interest. At that time they gave us numerous investment possibilities.

The diagrams were prepared with curves that were drawn by a computer from my formula and showed the “fair price” of the convertible. The beauty of this was that I could immediately see from the picture whether we had a profitable trading opportunity.

The slope of the curve near the data point on my diagram gave me the hedge ratio, which is the number of shares of common stock to use versus each convertible bond, share of preferred, warrant, or option.

Besides the new hedges we wanted to add, the tables told us how to adjust existing positions that needed the hedge ratio changed (so-called dynamic hedging) because the stock price had moved or that should be closed because we reached our objective.

Unknown to Einstein, his equations describing the Brownian motion of pollen particles were essentially the same as the equations that Bachelier had used for his thesis five years earlier to describe a very different phenomenon, the ceaseless, irregular motion of stock prices.

Bachelier’s paper appeared in 1964 in The Random Character of Stock Market Prices, edited by Paul Cootner and published by the MIT Press.

Bachelier had assumed that changes in stock prices followed a bell-shaped curve, known as a normal or Gaussian distribution. This didn’t match real prices well, especially for periods longer than a few days.

Back in 1967, I had taken a further step in figuring out how much a warrant was worth. Using plausible and intuitive reasoning, I supposed that both the unknown growth rate and the discount factor in the existing warrant valuation formula could be replaced by the so-called riskless interest rate, namely that which was paid by a US Treasury bill maturing at the warrant expiration date. This converted an unusable formula with unknown quantities into a simple practical trading tool.

This launched the development and widespread use of so-called derivative securities throughout the financial world.

We modified our performance fee of 20 percent of the profits, billed annually, by including a “new high water” provision. This meant that if we had a losing year, we carried forward the losses and used them to offset future profits before we were paid more fees. This helped align our economic interests with those of the limited partners. As it happened, we never had a losing year, or even a losing quarter, and this calculation was never invoked.

In order to attract and keep superior staff, I paid wages and bonuses well above the market rate. This actually saved money because my employees were far more productive than average. The higher compensation limited turnover, which saved time and money otherwise used to teach my one-of-a-kind investment methodology. At the higher levels, it kept people from breaking off and going into business for themselves.

realized that I could use a computer and my undisclosed “integral method” for valuing options to get numerical results to any desired degree of accuracy for this as-yet-unsolved “American put problem.” In a productive hour in the fall of 1973 I outlined the solution, from which my staff programmed a computer to produce precise calculated values. My integral method also had another advantage over the Black-Scholes approach. Whereas the latter was based on one specific model for stock prices, one with limited accuracy, my technique could value options for a wide range of assumed distributions of stock prices.

Convertible bonds today may have complex terms and conditions. However, the basic idea is simple. Consider the hypothetical XYZ 6s of 2020. Each bond was originally sold for approximately $1,000 on July 1, 2005, to be redeemed by the company for exactly $1,000, the “face amount,” on July 1, 2020. The bond promises to pay 6 percent of the face amount in interest for each year of its life, in two semiannual installments of 3 percent, or $30, payable to holders of record on January 1 and on July 1. So far these are like the terms of a typical ordinary bond. However, the convertible has one more feature. At the option of the owner, it can be converted into twenty shares of XYZ common stock anytime until bond maturity on July 1, 2020. So this bond combines the features of both an ordinary bond and an option. The market price of the bond can be thought of as the sum of two parts. The first is the value of a comparable bond without the conversion feature, which will fluctuate with the level of interest rates and the financial soundness of the company. This sets a “floor” to the price.

The second part is the option value of the conversion feature. In our example, if the stock is at $50, the bond can be exchanged for twenty shares of stock, worth $1,000, which the bond is worth anyhow when it matures so there is no benefit from the conversion feature. However, if the stock were to rise at any point to $75, twenty shares of stock would be worth $1,500. The bond, which can be exchanged immediately for this amount of stock, should trade in the market then for at least that amount.

From the 1980s on, some of these techniques came into usage by modern investment banks and hedge funds. They also adopted a notion we rejected, called VaR or “value at risk,” where they estimated the damage to their portfolio for, say, the worst events among the most likely 95 percent of future outcomes, neglecting the extreme 5 percent “tails,” then acted to reduce any unacceptably large risks. The defect of VaR alone is that it doesn’t fully account for the worst 5 percent of expected cases. But these extreme events are where ruin is to be found. It’s also true that extreme changes in securities prices may be much greater than you would expect from the Gaussian or normal statistics commonly used. When the S&P 500 Index fell 23 percent on October 19, 1987, a leading academic finance professor said that if the market had traded every day for the thirteen-billion-year life of the universe, the chance of this happening even once was negligible.

Another tool used today is to “stress-test” a portfolio by simulating the impact of major calamitous events of the past on the portfolio. In 2008, a multibillion-dollar hedge fund managed by a leading quant used ten-day windows from the crash of 1987, the First Gulf War, Hurricane Katrina, the 1998 Long-Term Capital Management crisis, the tech-induced market drop in 2000–02, the Iraq War, and so forth. All this data was applied to the fund’s 2008 portfolio and showed that these events would have led to losses of at most $500 million on a $13 billion portfolio, a risk of loss of no more than 4 percent. But they actually lost over 50 percent at their low in 2009, brought to the brink of ruin before finally recovering their losses in 2012. The credit collapse of 2008 was different in kind from the past worst cases for which they tested, and their near-extinction reflects the inadequacy of simply replaying the past.

We took a more comprehensive view. We analyzed and incorporated tail risk, and considered extreme questions such as, “What if the market fell 25 percent in one day?” More than a decade later it did exactly that and our portfolio was barely affected. When, with our expanding range and size of trades, we moved our account to Goldman Sachs as our prime broker, one of the questions I asked was: “What happens to our account if Goldman Sachs New York is destroyed by a terrorist nuclear bo...

The prototype was Value Line, an investment service that launched a program in 1965 using information such as surprise earnings announcements, price-to-earnings ratios, and momentum to rank stocks into groups from I (best) to V (worst). A stock is said to have positive momentum if its price has recently been trending strongly up, and negative momentum if strongly down.

The Compustat database provided historical balance sheet and income information. Of the scores of indicators we systematically analyzed, several correlated strongly with past performance. Among them were earnings yield (annual earnings divided by price), dividend yield, book value divided by price, momentum, short interest (the number of shares of a company currently sold short), earnings surprise (an earnings announcement that is significantly and unexpectedly different from the analysts’ consensus), purchases and sales by company officers, directors, and large shareholders, and the ratio of total company sales to the market price of the company. We studied each of these separately, then worked out how to combine them. When the historical patterns persisted as prices unfolded into the future, we created a trading system called MIDAS (multiple indicator diversified asset system) and used it to run a separate long/short hedge fund (long the “good” stocks, short the “bad” ones). The power of MIDAS was that it applied to the entire multitrillion-dollar stock market, with the possibility of investing very large sums.

Meanwhile, an army of PhDs, following our path, greatly expanded the theory of derivatives and implemented the revolution in quantitative finance on Wall Street. They helped direct investing at hedge funds, investment banks, and other institutions. Driven in part by the sell side—the sales force that finds and sells new products—these quants invented new derivative securities that the salespeople then pushed. These products undermined the world financial system in a series of increasingly grave crises. The first of these surprised almost everyone.

Though there was no major outside event to explain this one-day collapse, when I thought it through that evening I asked myself, Why did this happen? Will the disaster continue tomorrow? Will there be profit opportunities created by the chaos? I believe the cause was a new financial product called portfolio insurance. Had I paid closer attention earlier to the vast expansion in its use, I might have foreseen the disaster. This investment technique was largely created and marketed by the quant firm of Leland, O’Brien and Rubinstein. Suppose a company pension and profit-sharing plan with a broad portfolio of equities wants to protect itself against a sharp drop in the market. It sets up a program, either in-house or by a portfolio insurance specialist, to switch from equities into US Treasury bills as the market falls. This is done in stages: For each drop of, say, a couple of percent, part of the stock portfolio is sold and the proceeds used to buy T-bills. If the market later rises, the process is reversed until, once again, the portfolio is fully invested in equities.

At the time of the crash $60 billion or so of equities were insured by this technique and implemented largely by computers. When the market fell 4 percent on Friday the insurance programs placed orders, to be executed at Monday’s opening, to sell stock and buy Treasury bills. When trading began on Monday, these sales drove stock prices down further, triggering more selling from portfolio insurance programs. As prices continued to plummet, investors panicked and added their selling to the deluge. This “feedback loop” continued throughout the day, building to a devastating climax. Portfolio insurance was designed to protect investors from large market declines. Ironically, the cure became the cause.

The rule was part of the Securities Exchange Act of 1934 (rule 10a-1). It specified that, with certain exceptions, short-sale transactions are allowed only at a price higher than the last previous different price (an “uptick”). This rule was supposed to prevent short sellers from deliberately driving down the price of a stock. Seeing an enormous profit potential from capturing the unprecedented spread between the futures and the index, I wanted to sell stocks short and buy index futures to capture the excess spread. The index was selling at 15 percent, or 30 points, over the futures. The potential profit in an arbitrage was 15 percent in a few days. But with prices collapsing, upticks were scarce.

We added extraordinary investment products that could expand our capital base to billions. They included: 1. State-of-the-art convertible, warrant, and option computerized analytic models and trading systems. With this we had already become the biggest player in the Japanese warrant market. 2. Statistical arbitrage, which was a computerized analytic model and trading system for common stocks using a real-time feed of the ticker into our $2 million computer center, where we generated automated electronic orders and sent them to the floor. From one eight-by-eight cubicle we traded between one and two million shares a day, which was then 1 or 2 percent of NYSE daily volume. 3. A group of experts on interest rates joined us from Salomon Brothers. While there, they made $50 million for that firm in just eighteen months. 4. MIDAS: This indicator-driven stock prediction system was to be our entry into the broader money management business. 5. OSM Partners: a “fund of hedge funds” that invested in other hedge funds.

Knowing when to haggle and when not to is valuable for traders. In the days of Princeton Newport our head trader used to crow about how, by regularly holding out for an extra eighth or quarter, he saved us large amounts of money. Here’s the idea. Suppose we want to buy 10,000 shares of Microsoft (MSFT), currently trading at, say, 71 bid for 50,000 shares, and 71¼ asked for 10,000 shares. We can pay 71¼ now and buy our 10,000 shares. Or, as our trader would do, we can offer to buy our 10,000 shares at 711⁄8 and see if we have any takers. If this works—and it does most of the time—we’ll save $1⁄8 × 10,000 or $1,250. This sounds good. Is there any risk? Yes. By trying to save $1⁄8 per share we may miss a big winner if the stock always trades at 71¼ or higher for however long we’re trying to buy. Those stocks we miss, which run away to the upside, would have given us windfall profits. Put simply, you might scalp $1⁄8 twenty times but lose $10 once. Do you like that arithmetic? I don’t. I asked our trader how he could tell whether his repeated scalping for an eighth of a point offset his losses from missed opportunities. He could not make a case for what he was doing. I asked other traders around the street the same question and didn’t find anyone who could clearly show that they gained more than they lost by doing this.

What the hagglers and the traders do reminds me of the behavioral psychology distinction between two extremes on a continuum of types: satisficers and maximizers. When a maximizer goes shopping, looks for a handyman, buys gas, or plans a trip, he searches for the best (maximum) possible deal. Time and effort don’t matter much. Missing the very best deal leads to regret and stress. On the other hand, the satisficer, so-called because he is satisfied with a result that is close to the best, factors in the costs of searching and decision making, as well as the risk of losing a near-optimal opportunity and perhaps never finding anything as good again.

Though many details of these schemes are either complex or not yet public knowledge, one of the mechanisms is. Some exchanges, such as NASDAQ, let HF traders peek at customer orders ahead of everyone else for thirty milliseconds before the order goes to the exchange. Seeing an order to buy, for instance, the HF traders can buy first, pushing the stock price up, then resell to the customer at a profit. Seeing someone’s order to sell, the HF trader sells first, causing the stock to fall, and then buys it back at the lower price. How is this different from the crime of front-running, described in Wikipedia as “the illegal practice of a stock broker executing orders on a security for its own account while taking advantage of advance knowledge of pending orders from its customers”?

Since the more the rest of us trade the more we as a group lose to the computers, here’s one more reason to buy and hold rather than trade, unless you have a big enough edge. Although it’s politically not likely, a small federal tax, averaging a few cents a share on every purchase, could eliminate these traders and their profits, possibly saving more for investors than the extra tax, and adding cash to the US Treasury. If this cut a trading rate of about $30 trillion a year for equities by half, a 0.1 percent tax (3 cents a share on a $30 stock) would still raise about $15 billion.

Our computer simulations and experience show that this portfolio is close to market-neutral, which means that the fluctuations in the value of the portfolio have little correlation to the overall average price changes in the market. Our level of market neutrality, measured by what financial theorists call beta, has averaged 0.06. When beta is zero for a portfolio, its price movements have no correlation with those of the market, and it is called market-neutral. Portfolios with positive beta tend to move up and down with the market, more so for larger beta. The beta of the market itself is chosen to be 1.0. Negative beta portfolios tend to fluctuate oppositely to the market. Our risk-adjusted excess return, the amount by which our annualized return has exceeded that from investments of comparable risk and called alpha by finance theorists, has averaged about 20 percent per year. This means that our past annual rate of return (before fees) of 26 percent can be thought of as the sum of three parts: 5 percent from Treasury bills with no risk, about 1 percent due to our slight correlation to the market, plus the remaining 20 percent, the amount by which our return exceeds investments with comparable risk.

Market professionals describe stocks with large trading volume as “liquid”; they have the advantage of being easier to trade without moving the price up or down as much in the process.

To control risk, we limit the dollar value we hold in the stock of any one company. Our caution and our risk-control measures seem to work. Our daily, weekly, and monthly results are “positively skewed,” meaning that we have substantially more large winning days, weeks, and months than losing ones, and the gainers tend to be bigger than the losers.

When he finds events such as the unexpected announcement of a merger, takeover, spin-off, or reorganization, he tells the computer to put the stock on the restricted list: Don’t initiate a new position and close out what we have.

Our entire holding of stocks, long and short, turns over about once every two weeks, or twenty-five times per year.

Why is statistical arbitrage so-called? Arbitrage originally meant a pair of offsetting positions that lock in a sure profit. An example might be selling gold in London at $300 an ounce while at the same time buying it at $290 in New York for a $10 gain. If the total cost to finance the deal and to insure and deliver the New York gold to London were $5, it would leave a $5 sure profit. That’s an arbitrage in its original usage.