 # Money Management*

An important part of Money Management is changing the number of contracts you trade as your account size increases or decreases. There are several ways to mathematically define money management strategies. Here are a few of the more commonly accepted approaches.

Percent risked

LotSize = RiskFraction * Equity / TradeRisk
This model can skip trades or stop trading if the risk fraction of equity shrinks smaller than the risk or initial stop loss one must endure to enter a trade. If your risk input contains a constant value for risk (as you would input if you didn’t have risk data on a per-trade basis), then this model becomes what’s called the fixed fractional model. Its power derives from having the risk, or initial stop loss size, of each individual trade.

Percent volatility Adjust the lot size so that the market volatility in dollars per lot, often measured as the average true range of the last 10 to 20 bars, is no more than a fixed fraction of your equity.
LotSize = VolatilityFraction * Equity / Volatility
This model can skip trades or stop trading if the volatility fraction of equity shrinks smaller than the market’s volatility. This model also converts to a fixedfractional model if you have a constant value in the volatility input.

Optimal f — An overview Optimal f is a fixed factional money management method. In 1956, J.L. Kelly Jr. published a paper called “A New Interpretation of Information Rate.” Professional blackjack players realized the application of this work and began using it in their gaming efforts. The basic concept was to use the probability of winning and the ratio of wins to losses to calculate the optimal bet size.
Larry Williams popularized this concept for traders in 1987 during the Robbins’ World Cup trading competition. Money management is a powerful tool when an individual has an edge. Roulette will not work with money management because you cannot get a theoretical edge in that game. However, in backgammon or blackjack an expert player can get a small edge on the casino and use Kelly’s formulas to supercharge their returns. The Kelly formula is:
F = ((B + 1) * P - 1) / B
Where:
P is the probability of a winning bet
B is the ratio of the amount won vs. the amount loss
If there is a 60% chance of winning \$1.50 or a 40% chance of losing \$1.00, the
optimal bet size can be calculated as:
f = (1.5 + 1) * 0.60 - 1) / 1.5 F = 0.33
We would conclude that betting 33% of our stake on each bet would produce the best or optimal results.

Another researcher, Ralph Vince, discovered the problem with the Kelly formula in 1987 while working with Larry Williams. He found that the formula was not valid if the amount won or lost on each event was different. Vince developed his own set of equations to solve this problem based on the concept of a Holding Period Return (HPR). The Holding Period Return is the rate of return on any given trade plus 1.00. So, a 10% return equals 1.10 and a 25% loss equals 0.75. Because percentage returns are being calculated based on a fixed fraction of the account size, we can define HPR as:
HPR = 1 + f * (-T / BL)
Where:
f is the fixed fraction of the account to trade
T is the profit/loss of an individual trade
BL is the largest losing trade of an entire sequence of trades
The HPR formula is applied to each trade. By multiplying HPR for each trade, we can obtain a multiple of our original stake, the Terminal Wealth Relative (TWR):
TWR = Product (1 + f * (-T / BL))
We maximize the TWR function by changing the values of “f” to find the value that produces the highest TWR, which is called optimal f. After calculating optimal f and TWR, we need to calculate how much equity is required to trade one unit:
U = (ML / - f)
Where:
U is the trading units in dollar
ML is the maximum loss in dollars
f is the optimal value for f
Using the trading units in dollars, starting account size and trade history, we can run a simulation of the equity curve for any trading system using optimal f.

* Extracted from an article in Technology & Trading written by Murray A. Ruggiero Jnr.

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