Money management is changing the number of contracts you trade as youraccount 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
Adjust the lot size so the total amount risked (stop loss) on each trade is a fixed
fraction of your trading equity.
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. These
simulations often yield astronomical results after 50 or 100 trades. The problem is
optimal f quickly can require trading more contracts than is realistic for a given market.
Another problem is that optimal f returns are also based on trading fractional
contracts. For example, if U is $4,000 and our account equity is $10,000, optimal f
would call for 2.5 contracts to be traded. In real life, we would round the number of
contracts down to the nearest whole number, which would be two contracts.
Because the largest losing trade is used to calculate TWR, it has a major effect
on optimal f. This is not a problem when working with historical simulations, but
when we are using optimal f on a real system where protective stops are based on
volatility or channel size, we cannot define optimal f.
The distribution of trades greatly affects the value of optimal f.
We can have two trading systems that make $100,000 on 1,000 trades for an
average profit of $1,000 per trade. The optimal f values for these two systems can
vary widely based on the distribution of the returns on the trades. It is dangerous
to trade anywhere near optimal f because the distribution of trades in real time
can change. For illustration’s sake, say, in testing you had 500 straight winners and
500 straight losers, while real life may deliver any mix of winners and losers to
achieve the same results.
The problem is that when an account is in a run up, the number of contracts
being traded can increase rapidly and when the system goes into a drawdown, the
account takes a hit that takes it below the level before the run up. This happens
because lot numbers can double within a few trades.
Professional money managers also trade a fixed percentage of an account on a
given trade. The standard for professional money managers is to risk 1% to 3% of
trading capital on a given trade. We will call this term RiskFraction, so it is between
0.01 and 0.03. The number of units to be traded can depend on market conditions
as well as the system.
define money management strategies. Here are a few of the more commonly
accepted approaches.
Percent risked
Adjust the lot size so the total amount risked (stop loss) on each trade is a fixed
fraction of your trading equity.
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. These
simulations often yield astronomical results after 50 or 100 trades. The problem is
optimal f quickly can require trading more contracts than is realistic for a given market.
Another problem is that optimal f returns are also based on trading fractional
contracts. For example, if U is $4,000 and our account equity is $10,000, optimal f
would call for 2.5 contracts to be traded. In real life, we would round the number of
contracts down to the nearest whole number, which would be two contracts.
Because the largest losing trade is used to calculate TWR, it has a major effect
on optimal f. This is not a problem when working with historical simulations, but
when we are using optimal f on a real system where protective stops are based on
volatility or channel size, we cannot define optimal f.
The distribution of trades greatly affects the value of optimal f.
We can have two trading systems that make $100,000 on 1,000 trades for an
average profit of $1,000 per trade. The optimal f values for these two systems can
vary widely based on the distribution of the returns on the trades. It is dangerous
to trade anywhere near optimal f because the distribution of trades in real time
can change. For illustration’s sake, say, in testing you had 500 straight winners and
500 straight losers, while real life may deliver any mix of winners and losers to
achieve the same results.
The problem is that when an account is in a run up, the number of contracts
being traded can increase rapidly and when the system goes into a drawdown, the
account takes a hit that takes it below the level before the run up. This happens
because lot numbers can double within a few trades.
Professional money managers also trade a fixed percentage of an account on a
given trade. The standard for professional money managers is to risk 1% to 3% of
trading capital on a given trade. We will call this term RiskFraction, so it is between
0.01 and 0.03. The number of units to be traded can depend on market conditions
as well as the system.
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