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Dr. Z’s Mathematics of Gambling
 
by William Ziemba

The return of Gambling Times gives us all a chance to look at strategies for investment in traditional gambling games and in areas such as the stock market. There are two aspects to any repeated risky situation: first can you create an edge for your wager — that is when a dollar is bet do you receive more than a dollar in return and, secondly, how much should you wager?

Winning systems must have positive expected value. It is not possible to devise mathematical betting systems that will provide profits in the long run unless the average return is favorable. Doubling up or similar strategies for losing games will eventually fail in the long run despite possible short-term gains because the eventual large loss will more than wipe out the small gains along the way.

In this column, I will discuss various ways to create such edges and profit from them in horseracing, lotteries, futures and options trading and the stock, bond, and other financial markets.

This first part of the analysis is our strategy development using past data, economic analysis, psychological aspects etc. An important area is behavioral finance, which formalizes people’s behavior into models that try to explain why they do things. Of course, Dr. Z horserace players have known a lot about this for twenty years. This research and that on stock market anomalies answers important questions and explains the behavior of real people. Real people are not as ration-al as the economic man assumed in most analyses. Such concepts are useful for thinking about investment/gambling situations and help create favorable betting environments.

The second key aspect of investing is taking into account the risk and determining proper bets. It is here where the big mistakes in the financial markets occur. Barrings, Orange Country, Long Term Capital Management, the Soros Funds, the Tiger Fund and many others used sophisticated and clever analyses to create winning systems for their investments but ran into trouble when their risk management systems failed. It is not an easy task to devise a betting system that is embedded into a risk management that is fail-safe in all situations. But the careful analysis of this aspect of the investor’s behavior is key to success. There are many situations that we must look at but they have common elements. What is the volatility of the investments? How can you create a truly diversified portfolio? How can you deal if needed with various risks such as:

• Liquidity risk — can the investor sell when he wants to at a decent price?
• Reputation risk — is there an event that occurs extraneous to what the investor is doing that affects the prices of assets held due to the reputation of certain parties has been damaged?
• Credit risk — will those who owe the investor money actually pay this money?
• Operational risk — will something go wrong in the operation of the trading or other systems the investor is using?
• Hedging risk — will neutralizing some risky financial position work, especially in a crisis?

These and other complex risks arise in bond and hedge fund trading departments, which are at the high end of risk taking and profit gathering.

Many aspects of investment/gambling situations are crucial to betting and risk control success. Some of the key elements to keep in mind are what possible scenarios can occur? These are not all the possible events that have occurred in the past but all the events that could conceivably occur. Often we find these rare scenarios from similar economic situations. A crisis in one commodity may have outcomes similar to a crisis in another totally difference asset. The past can guide us into possible future scenarios, the possible ways the world can evolve. Diversification is harder to accomplish than it seems at first blush. It is highly related to overbetting the usual culprit of financial disasters. What appears to be a diversified portfolio may turn into portfolio of many assets moving in the same way. I will investigate various examples in later columns and cover theories and practice of optimal betting, and risk management over time. These we will apply to gambling games and to questions such as how much investors should have in stocks, bonds, and cash over time, and how these allocations should change as new information is received.

Next month I will discuss the most important horse race of 2000 — the Breeders’ Cup Classic and a new strategy for obtaining expected value edges using a novel way of evaluating the performance of race horses throughout various parts of their past races. The analysis will point us towards Tisnow and Giant’s Causeway and away from Fusaichi Pegasus and other horses.

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