set . The algorithm for the optimal set of outcomes consists of four steps: Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes: Reorder the outcomes so that the new sequence is non-increasing. Thus will be the best bet. Set (the empty set), , . Thus the best bet will be considered first. Repeat: If then insert -th outcome into the set: , recalculate according to the formula: and then set , Otherwise, set and stop the repetition. If the optimal set is empty then do not bet at all. If the set of optimal outcomes is not empty, then the optimal fraction to bet on -th outcome may be calculated from this formula: One may prove that where the right hand-side is the reserve rate. Therefore, the requirement may be interpreted as follows: -th outcome is included in the set of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction may be interpreted as the excess of the expected revenue rate of -th horse over the reserve rate divided by the revenue after deduction of the track take when -th horse wins or as the excess of the probability of -th horse winning over the reserve rate divided by revenue after deduction of the track take when -th horse wins. The binary growth exponent is and the doubling time is This method of selection of optimal bets may be applied also when probabilities are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that and . Stock investments The second-order Taylor polynomial can be used as a good approximation of the main criterion. Primarily, it is useful for stock investment, where the fraction devoted to investment is based on simple characteristics that can be easily estimated from existing historical data – expected value and variance. This approximation leads to results that are robust and offer similar results as the original criterion. For single assets(stock, index fund, etc.), and a risk-free rate, it is easy to obtain the optimal fraction to invest through geometric Brownian motion. The value of a lognormally distributed asset at time () is from the solution of the geometric Brownian motion where is a Wiener process, and (percentage drift) and (the percentage