Pnce level increase

with the losing phase, there would be a profit of 50° for the first year and 100° for each subsequent year.

Many traders would not net 100°o each year from a sjStem that performs as the one in the example. As their profits increased, positions would be added so that at the time the total equity was worth $20.000, the maigin requirements would also be $20,000. The 50°o loss is then applied to the total equity:

Tradmg futures would be a great deal of effort for no retum.

Holding the investment constant as shown can also be viewed by studjing the growth and decline of the accouni excess, called the reserve. The size of the reserve relative to total equity is the key to successful management. Starting with maigin and reserves equal, reserves increase during profitable periods and decrease during losing ones. Proportionately more of the total equity is traded during losing phases. This pattem can be used to improve results safely, as follows

Using the natural equity cycles, hold the number of positions the same and allow the reserve to increase during profitable periods; maintain the same position size through the beginning of the next losing period. \ien the equitj drop has slowed or stabilized, the total equity can be redishibuted into maigin and reserve according to the original 50°o formula. In the next example, the total equity of $25,000 is dishibuted 40<fe to maigin and 60°o to reserve at the end of the first cycle. It is redistributed so that the next profit phase will be entered with a larger base than the previous losing cycle. The result is a gradual increase in profits:

Tracing on Equity Cjcles

If a moving average technique is traded, the equity resulting from this sfrategj will fluctuate with the trending nature of the maiket. By applying a moving average analysis to the equity itself, the frending and nontrending periods are identified by buy and sell signals just as though the equity series was a price series.

An equity buj means that the maiket has begun trending; the length of this period depends on the calculation period of the trend. A sell signal means that the maiket is no longer trending. These signals can be taken as "buy the sjStem" or Short the sjStem"; that is, enter all positions that the sjStem currently holds or liquidate the entire portfolio and hold cadi. An equity buy could also be taken as the point to redisfribute the equity into the original ratio of margin to reserves. If the normal profile of price movement is to apend a large proportion in a sideways pattern, then there should be sustained periods of downward equity trends., trading equity cycles can improve performance.

INVESTING ) REimTSTING: OPTIMAL f

Ctimal f is the optimal fixed fraction of an account that should be invested at any one time, or the size of the bet to place on any one trade. The amount to be risked is measured as a percentage of the portfolio size. The objective is to maximize the amount invested (put at rid;), yet avoid the possibility of total loss. Trading a very small part of assets can be a poor use of capital, while trading too much guarantees bankruptcj or ruin. Ctimal f is the ideal portion of an inveshnent that should be placed at rid; at any one time. If you risk less than the ctimal f then you are noi generating the peak profits; however, if you trade more than ctimal f you assure eventual ruin.

Investing generally has a two-level ctimal f the part of the total portfolio put at rid; compared with that part

held in cadi equivalents, and the individual size of the commitment to each stock or futures martlet within that portfolio. This is particulariy important for futures, where the high leverage of individual martlets makes it very easj to rid; too much on each trade.

Fixed Initial Invedments

Then, how much should be invested? The cptimal amount is difficult to pinpoint because you would have to know what risks lie shead, and of course, thats not possible. However, based on a very small likelihood of losing 50°o of the portfolio when 50°o is invested, one might say that a portion under a 50°o investment is best. To account for greater uncertainties in the future, you could increase your confidence by investing only ISo of the initial portfolio. This approach has negative effects, because a smaller relative investment reduces both rid; and retums. When the invedment becomes too small, the returns are no longer attractive.

The chance of a catasfrophic risk is an important concem for any investor or portfolio manager. For practical pU oses this is always figured on the historic profile of the data or trading results. This still leaves uncertainty in the final values. Nevertheless, the most exfreme situation is often found by using the calculation for risk of ruin (in a previous section of this chapter), most often applied to gambling situations in which the bet sizes, pay out, and odds are well defined. \ien there are enough test data and trades, this technique has been applied to frading sjStems (see the section "Wins Not Equal to Losses").

Some analysts have tried to deal with the uncertainties of price movements by using a Monte Cario technique in teding, which shifts the sequence of blod;s of data, or profit and loss results, so that they occur in random order. The worst results are considered the greatest risk. This approach may be unrealistically severe, yet even the real performance is not likely to reflect the size of the risk in the future. For those traders applying a long-term

frend-following technique to capture moves that are based on economic or government policy, sustained profits are most often followed by a reversal before the trade is ended. In fact, the ending change of direction is directly related to the sustained move. To move this data around so that the loss comes at a different time may create a laige loss without the offsetting profit, which is a situation that is unfair to the trading drategj. Before applying aMonte Cario analysis, il is first necessaij to identify the dominant period of sequential correlation to avoid segmenting the series incorrectly.

For an initial investment, the optimal f is simply the maximum part of that portfolio that can safely be traded without any significant risk of ruin. For those investors who take out profits when they occur and continue to trade based on the same assumed initial investment, nothing need be changed unless exceptionally high risk causes a reassessment and decrease in the amount of leverage. However, it is more common and more complicated for the investor to varj the amount committed to the mari;et by either increasing or decreasing leverage. This involves (1) determinmg the right time to change the leverage, (2) calculating the amount to increase the invesbnent when there are profits, and (3) figuring the size of the reduction when there are losses exceeding some designated amount. These are issues that are addressed by cptimal f

Finding Ctimal f

Ralph Vince, in his popular book. Portfolio Management Formulas, focuses on cptimal f, rid; of ruin, and other practical items. Optimal f is the ideal amount of an invesbnent that should be put at risk at any one time. As background for this, we need to formulate the relationship where the percentage gain needed to recover a loss is laiger than the percentage lost.

Inquired gain =------1

1 - percent loss

That is, a 50°o loss requires a 100°o gain to restore the original value. Because the amount rid;ed on each trade depends on our expectations of loss, the results obtained from the cptimal f calculation will be the size of the bet, invested amount, or the number of contracts to be fraded, as a percentage of the maximum loss. As we alreadj know, the value used as a maximum loss will be an estimate, because losses can alwajS be greater than only those experienced in the maiket, whether theoretical or real. In addition, the cptimal f will be different for each sjdem, depending upon its performance profile.

The mathematics needed to determine cptimal f is based on the Kelly Betting System Kelly dates that the cptimum bet is the one that maximizes the growth function Gif):