You though you could break even?
Risk can be put down to a formula. This is based on random selections, not including any empirical data or fundamentals.
R{2K,2n} = (2k/k) X ((2n-2k)/(n-k)) X 2^-2n
Here R is ones money, N is the number of investments
"The implications of this theorem are rather straightforward. Those who fall on the negative side (lose money on a stock for a period) tend to stay on the same side and therefore have less chance to recover their losses, even if they continue to play the game for a substantially long period of time. Specifically, the only practical solution for those who have huge losses is to leave the game as soon as possible, if feasible."
"With these differences in mind, imagine the following situation. An individual is involved in a coin-tossing game. If he gets heads, he adds + 1 to his score, and if he gets tails, he adds – 1 to his score. Now the question is, how likely it is that his score reverts to zero, namely that the number of heads and tails up to the time of counting is identical, as he continuously plays the game over a long period of time. The intuitive answer to this question is, because the game itself is fair in the sense that both heads and tails appear with the same probability (i.e. probability 0.5), the resulting score tends to stay around zero. The arc-sine theorem, however, states that this is not the case. The startling message of the theorem is, unlike basic intuition, that the score breaking even is the least likely outcome. "
This is why I say that a company offering the least likelihood of losing, i.e. a production oriented company is the seeming safest investment. It actually seems to be to most people, as people do not invest due to expected gain, but expected marginal utility of gain, even if the driller offers to you the expectation of the possibility of enormous gain, with littled downside risk, if you keep playing.
To suppose that safety-first consists in having a small gamble in a large number of different [companies] where I have no information to reach a good judgment, as compared with a substantial stake in a company where one’s information is adequate, strikes me as a travesty of investment policy.
John Maynard Keynes
Letter to F.C. Scott, February 6, 1942
A simple example illustrates the point. Assume you can participate in a coin toss game where heads pays $2 and tails costs $1. You start with a $100 bankroll and can play for 40 rounds. What betting strategy will allow you to achieve the greatest probability of the most money at the end of the fortieth round?
We’ll get to the answer in a moment, but let’s consider the obvious extremes: if you bet too little, you won’t take advantage of a clearly positive expected-value opportunity. On the other hand, if you bet everything, you risk losing all of your money. Money management is all about determining the right amount of capital to allocate to an investment opportunity, given the edge and the frequency of such opportunities. Position size is extremely important in determining equity portfolio returns. Two portfolio managers with the same list and number of stocks can generate meaningfully different results based on how they allocate the capital among the stocks. Great investors don’t stop with finding attractive investment opportunities; they know how to take maximum advantage of the opportunities. As Charlie Munger says, good investing combines patience and aggressive opportunism.
The Mean/Variance Way
So how best to allocate capital, either across asset classes or within an asset class? The classic answer comes from the concept of mean/variance efficiency, first formalized by Harry Markowitz in 1952. 3 The premise is that risk and reward are related linearly (see Exhibit 1). The mean is the average arithmetic return from an asset or portfolio. Variance measures how spread distribution points are from the average.
A risk averse investor seeks the highest return for a given level of risk. For all portfolios with a given level of risk, the investor will select the one with the highest return. And for an assumed level of return, the investor prefers the one with the least risk. No optimal portfolio exists since different individuals have different risk preferences, but portfolios away from the efficient frontier—the best reward for a given level of risk—are suboptimal. Mean/variance is powerful because if you specify the function that accurately expresses your utility, you can find a portfolio that’s right for you. But what if you ask the asset allocation question a different way: How do you maximize the likelihood that you’ll have the most money at the end of a particular period? As it turns out, mean/variance doesn’t answer that question.
Shannon, Chance, and The Kelly Criterion
Bell Labs scientist Claude Shannon is well known for developing information theory—essentially, the necessary properties and systems for transmitting intelligence. Before Shannon, most engineers tried to understand the information problem by focusing on a message’s meaning. Shannon’s insight was that information is related to chance. As author William Poundstone notes, “Information exists only when the sender is saying something that the recipient doesn’t already know and can’t predict. Because true information is unpredictable, it is essentially a series of random events like spins of a roulette wheel or rolls of a dice.” 4 As an example, Poundstone points to a television commercial depicting a wife asking her husband to bring home “shampoo.” The husband, misunderstanding her, shows up with “Shamu,” the killer whale. Neither the wife’s request nor the husband’s misunderstanding is surprising. The Risk (variance) Return (arithmetic mean)
Legg Mason Capital Management
The commercial captures our attention because the husband acts on a highly improbable and unpredictable request without further information.
For Shannon, the incompressible part of a message relates to its unpredictability. The less probable a message, the more bandwidth it requires. A request to bring home Shamu undoubtedly demands more bandwidth than a routine demand for shampoo. Shannon’s theory also considers equivocation—the chance the message is wrong—and shows you must subtract equivocation from the channel capacity to determine the information rate. More reliable information leads to a higher information rate for a given channel capacity. Most of the information channels we use today, including phones, television, the Internet, operate using Shannon’s ideas. What does any of this have to do with optimal bet size? Shannon’s colleague at Bell Labs, John Kelly, recognized another application for information theory’s ideas: gambling.
Information in a betting setting is something the market does not already know. Consistent with the idea of equivocation, true information is also probabilistic. Kelly imagined a system where you have an edge; a set of expectations that differs from those of the market. He then developed a formula, based on Shannon’s work, showing the exact amount of your bankroll you should bet in order to maximize your capital over the long term. Consistent with the theory, the maximum rate of return comes when you know something the market doesn’t. We can express the Kelly formula a number of ways. We’ll follow Poundstone’s exposition:
"edge/odds = f"
Here, edge is the expected value of the financial proposition, odds reflect the market’s expectation for how much you win if you win, and "f" represents the percentage of your bankroll
you should bet. Note that in an efficient market, there is no edge because the odds accurately represent the probabilities of success. Hence, bets based on the market’s information have zero expected value (this before the costs associated with betting) and an f of zero.
Let’s go back and answer our opening coin-toss question using the Kelly formula. The payoff scheme, a $2 win for a heads and a $1 loss for a tails, suggests 2-to-1 odds. Since we’re dealing with a fair coin, we know the tosses will be 1-to-1. 7 So we recognize something the market doesn’t: heads will show up more often than the payoff scheme suggests.
Solving the formula, edge is $0.50 (expected value, or 50 percent x $2 + 50 percent x -$1) and odds are $2 (the amount you win if you win). The optimal amount to bet is 25 percent of your bankroll in each round. Said differently, betting 25 percent will lead to a greater accumulation of wealth, on average, than any other betting strategy.
0.50/2.00 = f = 25%
Exhibit 2 shows wealth outcomes based on a range of f values for 40 rounds. Betting too little leaves a substantial amount of money on the table, while betting too much leads to near-certain ruin. The latter point bears emphasis: if there is a probability of loss, even with a positive expected value economic proposition, betting too much reduces your expected wealth. Such overbetting may have been the source of demise for a number of high-profile hedge funds.
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