A risk premium is the minimum difference between the expected value of an uncertain bet that a person is willing to take and the certain value that he is indifferent to.

Example

Suppose a game show participant may choose one of two doors, one that hides $1,000 and one that hides $0. Further suppose that the host also allows the contestant to take $500 instead of choosing a door. All three options (door 1, door 2, or take $500) have the same expected value of $500, so there is no risk premium for choosing the doors over the guaranteed $500.

A contestant unconcerned about risk is indifferent to these choices. However, a risk averse contestant may be more likely to choose no door and accept the guaranteed $500.

If too many contestants are risk averse, the game show may encourage selection of the riskier choices (door 1 or door 2) by creating a risk premium. If the game show offers $2,000 behind the good door, increasing to $1,000 the expected value of choosing doors 1 or 2, the risk premium becomes $500 (i.e., $1,000 expected value – $500 guaranteed amount). Contestants with a minimum acceptable rate of return of $500 or more will likely choose a door instead of accepting the guaranteed $500.

Finance

In finance, the risk premium can be the expected rate of return above the risk-free interest rate.

• Debt: In terms of bonds it usually refers to the credit spread (the difference between the bond interest rate and the risk-free rate).
• Equity: In the equity market it is the returns of a company stock, a group of company stock, or all stock market company stock, minus the risk-free rate. The return from equity is the dividend yield and capital gains. The risk premium for equities is also called the equity premium.

The white paper Equity Risk Premium: Expectations Great and Small notes that “it is dangerous to engage in simplistic analyses of historical ERPs to generate ex ante forecasts that differ from the realized mean.” Standard & Poor’s states “the most correct method is to use an arithmetic average of historical returns.”

Links

• Hussman Funds – Estimating the Long-Term Return on Stocks – June 1998
• equity risk premium and information quality

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Originally, martingale referred to a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth is guaranteed to eventually flip heads, the martingale betting strategy was seen as a sure thing by those who practiced it. Unfortunately, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would quickly bankrupt those foolish enough to use the martingale after even a moderately long run of bad luck.

Analysis

Suppose that someone applies the martingale betting system at an American roulette table, with 0 and 00 values; a bet on either red or black will win 18 times out of each 38. If the player’s initial bankroll is $160 and the betting unit is $10, the player will make a win in approximately 96% of sessions, gaining an average of $4.30 from each winning session. In the remaining 4% of sessions, the player will “bust”, exhausting his bankroll, for a loss of $160. It follows then that the average session losses of a gambler employing this strategy are $2.27. Given a larger bankroll, the odds of making a win before running out of cash increase; however, the average winnings grow more slowly than the average losses, so the game remains a losing proposition.

Modern casinos generally have table minimums and maximums to prevent players from doubling their bets more than five or six times, rendering the martingale system obsolete.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Kelly gambling is an application of information theory to gambling and (with some ethical and legal reservations) investing. An important but simple relation exists between the amount of side information a gambler obtains and the expected exponential growth of his capital (Kelly). The so-called equation of ill-gotten gains can be expressed in logarithmic form as

for an optimal betting strategy, where K₀ is the initial capital, K_t is the capital after the tth bet, and H_i is the amount of side information obtained concerning the ith bet (in particular, the mutual information relative to the outcome of each betable event). This equation applies in the absence of any transaction costs or minimum bets. When these constraints apply (as they invariably do in real life), another important gambling concept comes into play: the gambler (or unscrupulous investor) must face a certain probability of ultimate ruin, which is known as the gambler’s ruin scenario. Note that even food, clothing, and shelter can be considered fixed transaction costs and thus contribute to the gambler’s probability of ultimate ruin.

This equation was the first application of Shannon’s theory of information outside its prevailing paradigm of data communications (Pierce).

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

The Kelly Criterion or as it is sometimes referred to as the Kelly formula is a formula used to maximize the long-term growth rate of repeated plays of a given gamble that has positive expected value. The formula specifies the percentage of the current bankroll to be bet at each iteration of the game. In addition to maximizing the growth rate in the long run, the formula has the added benefit of having zero risk of ruin, as the formula will never allow a loss of 100% of the bankroll on any bet. An assumption of the formula is that currency and bets are infinitely divisible, though this is met for practical purposes if the bankroll is large enough.The most general statement of the Kelly criterion is that long-term growth rate is maximized by finding the fraction f* of the bankroll that maximizes the expectation of the logarithm of the results. For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the following formula can be derived from the general statement:

   f* = (bp - q) / b
   where
   f* = percentage of current bankroll to wager;
   b = odds received on the wager;
   p = probability of winning;
   q = probability of losing = 1 - p.

As an example, if a gamble has a 40% chance of winning (p = 0.40), but the gambler receives 2:1 odds on a winning bet, the gambler should bet 10% of her bankroll at each opportunity, in order to maximize the long-run growth rate of the bankroll.

For even-money bets (i.e. when b = 1), the formula can be simplified to:

   f* = 2p - 1

The Kelly Criterion was originally developed by AT&T Bell Laboratories physicist John Larry Kelly, Jr, based on the work of his colleague Claude Shannon, which applied to noise issues arising over long distance telephone lines. Kelly showed how Shannon’s information theory could be applied to the problem of a gambler who has inside information about a horse race, trying to determine the optimum bet size. The gambler’s inside information need not be perfect (noise-free) in order for him to exploit his edge. Kelly’s formula was later applied by another colleague of Shannon’s, Edward O. Thorp, both in blackjack and in the stock market.

Cited References

1. American Scientist online: Bettor Math, article and book review by Elwyn Berlekamp

Link

• Original Kelly paper

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

Video: Understanding Kelly Criterion

Betting strategies or betting systems are approaches to gambling intended to increase the odds of winning.

Independent Events

The following betting strategies have been recorded as being applied to games which operate on independent events. For such games, the odds of a particular outcome are identical for every bet played. No such strategy can beat the house edge (if any) in the long run, and all of them trade off many small wins for a big loss or vice versa.

• Martingale – doubling bet after each loss until a win is achieved (or fails when the amount of the bet becomes excessive).
• Kelly criterion;

Video: Pferdewetten Horse Betting Strategy SelMcKenzie Selzer-McKen