(Trading)SEYKOTA _Risk_Management_(Ed.Seykota,2003,seykota.com)_[pdf].pdf

Risk

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Risk Management

Risk

RISK is the possibility of loss. That is, if we own some stock, and there is a possibility of

a price decline, we are at risk. The stock is not the risk, nor is the loss the risk. The

possibility of loss is the risk. As long as we own the stock, we are at risk. The only way

to control the risk is to buy or sell stock. In the matter of owning stocks, and aiming for

profit, risk is fundamentally unavoidable and the best we can do is to manage the risk.

Risk Management

To manage is to direct and control. Risk management is to direct and control the

possibility of loss. The activities of a risk manager are to measure risk and to increase

and decrease risk by buying and selling stock.

The Coin Toss Example

Let's say we have a coin that we can toss and that it comes up heads or tails with equal

probability. The Coin Toss Example helps to present the concepts of risk management .

The PROBABILITY of an event is the likelihood of that event, expressing as the ratio of

the number of actual occurrences to the number of possible occurrences. So if the coin

comes up heads, 50 times out of 100, then the probability of heads is 50%. Notice that a

probability has to be between zero (0.0 = 0% = impossible) and one (1.0 = 100% =

certain).

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Let's say the rules for the game are: (1) we start with $1,000, (2) we always bet that

heads come up, (3) we can bet any amount that we have left, (4) if tails comes up, we

lose our bet, (5) if heads comes up, we do not lose our bet; instead, we win twice as

much as we bet, and (6) the coin is fair and so the probability of heads is 50%. This

game is similar to some trading methods.

In this case, our LUCK equals the probability of winning, or 50%; we will be lucky 50%

of the time. Our PAYOFF equals 2:1 since we win 2 for every 1 we bet. Our RISK is the

amount of money we wager, and therefore place at risk, on the next toss. In this

example, our luck and our payoff stay constant, and only our bet may change.

In more complicated games, such as actual stock trading, luck and payoff may change

with changing market conditions. Traders seem to spend considerable time and effort

trying to change their luck and their payoff, generally to no avail, since it is not theirs to

change. The risk is the only parameter the risk manager may effectively change to

control risk.

We might also model more complicated games with a matrix of lucks and payoffs, to see

a range of possible outcomes. See figure 1.

Luck Payoff

10%

lose 2

20%

lose 1

30%

break

even

20%

win 1

10%

win 2

10%

win 3

Figure 1: A Luck-Payoff matrix, showing six outcomes.

This matrix might model a put-and-take game

with a six-sided spinning top, or even trading.

For now, however, we return to our basic coin example, since it has enough dimensions

to illustrate many concepts of risk management. We consider more complicated

examples later.

Optimal Betting

In our coin toss example, we have constant luck at 50%, constant payoff at 2:1 and we

always bet on heads. To find a risk management strategy, we have to find a way to

manage the bet. This is similar to the problem confronting a risk manager in the

business of trading stocks. Good managers realize that there is not much they can do

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about luck and payoff and that the essential problem is to determine how much to wager

on the stock. We begin our game with $1,000.

Hunches and Systems

One way to determine a bet size is by HUNCH . We might have a hunch and and bet $

100.

Although hunch-centric betting is certainly popular and likely accounts for an enormous

proportion of actual real world betting, it has several problems: the bets require the

constant attention of an operator to generate hunches, and interpret them into bets, and

the bets are likely to rely as much on moods and feelings as on science.

To improve on hunch-centric betting, we might come up with a betting SYSTEM .A

system is a logical method that defines a series of bets. The advantages of a betting

system, over a hunch method are (1) we don't need an operator, (2) the betting

becomes regular, predictable and consistent and, very importantly, (3) we can perform a

historical simulation, on a computer, to OPTIMIZE the betting system.

Despite almost universal agreement that a system offers clear advantages over hunches,

very few risk managers actually have a definition of their own risk management systems

that is clear enough to allow a computer to back-test it.

Our coin-flip game, however is fairly simple and we can come up with some betting

systems for it. Furthermore, we can test these systems and optimize the system

parameters to find good risk management.

Fixed Bet and Fixed-Fraction Bet

Our betting system must define the bet. One way to define the bet is to make it a

constant fixed amount, say $10 each time, no matter how much we win or lose. This is

a FIXED BET system. In this case, as in fixed-betting systems in general, our $1,000

EQUITY might increase or decrease to the point where the $10 fixed bet becomes

proportionately too large or small to be a good bet.

To remedy this problem of the equity drifting out of proportion to the fixed bet, we

might define the bet as as FIXED-FRACTION of our equity. A 1% fixed-fraction bet

would, on our original $1,000, also lead to a $10 bet. This time, however, as our equity

rises and falls, our fixed-fraction bet stays in proportion to our equity.

One interesting artifact of fixed-fraction betting, is that, since the bet stays proportional

to the equity, it is theoretically impossible to go entirely broke so the official risk of total

ruin is zero. In actual practice, however the disintegration of an enterprise has more to

do with the psychological UNCLE POINT ; see below.

Simulations

In order to test our betting system, we can SIMULATE over a historical record of

outcomes. Let's say we toss the coin ten times and we come up with five heads and five

tails. We can arrange the simulation in a table such as figure 2.

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Fixed Bet

$10

Fixed-Fraction Bet

Start

1000

Heads 1020

1020

Tails

1010

1009.80

Heads 1030

1030

Tails

1020

1019.70

Heads 1040

1040.09

Tails

1030

1029.69

Heads 1050

1050.28

Tails

1040

1039.78

Heads 1060

1060.58

Tails

1050

1049.97

Figure 2: Simulation of Fixed-Bet and Fixed-Fraction Betting Systems.

Notice that both systems make $20.00 (twice the bet) on the first toss, that comes up heads. On the

second toss, the fixed bet system loses $10.00 while the fixed-fraction system loses 1% of $1,020.00

or $10.20, leaving $1,009.80.

Note that the results from both these systems are approximately identical. Over time, however, the

fixed-fraction system grows exponentially and surpasses the fixed-bet system that grows linearly. Also

note that the results depend on the numbers of heads and tails and do not at all depend on the order

of heads and tails. The reader may prove this result by spreadsheet simulation.

Pyramiding and Martingale

In the case of a random process, such as coin tosses, streaks of heads or tails do occur,

since it would be quite improbable to have a regular alternation of heads and tails. There

is, however, no way to exploit this phenomenon, which is, itself random. In non-random

processes, such as secular trends in stock prices, pyramiding and other trend-trading

techniques may be effective.

Pyramiding is a method for increasing a position, as it becomes profitable. While this

technique might be useful as a way for a trader to pyramid up to his optimal position,

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pyramiding on top of an already-optimal position is to invite the disasters of over-

trading. In general, such micro-tinkering with executions is far less important than

sticking to the system. To the extent that tinkering allows a window for further

interpreting trading signals, it can invite hunch trading and weaken the fabric that

supports sticking to the system.

The Martingale system is a method for doubling-up on losing bets. In case the doubled

bet loses, the method re-doubles and so on. This method is like trying to take nickels

from in front of a steam roller. Eventually, one losing streak flattens the account.

Optimizing - Using Simulation

Once we select a betting system, say the fixed-fraction betting system, we can then

optimize the system by finding the PARAMETERS that yield the best EXPECTED

VALUE . In the coin toss case, our only parameter is the fixed-fraction. Again, we can

get our answers by simulation. See figures 3 and 4.

Note: The coin-toss example intends to illuminate some of the elements of risk, and their inter-relationships.

It specifically applies to a coin that pays 2:1 with a 50% chance of either heads or tails, in which an equal

number of heads and tails appears. It does not consider the case in which the numbers of heads and tails

are unequal or in which the heads and tails bunch up to create winning and losing streaks. It does not

suggest any particular risk parameters for trading the markets.

% Bet Start Heads Tails Heads Tails Heads Tails Heads Tails Heads Tails

0 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00 1000.00

5 1000.00 1100.00 1045.00 1149.50 1092.03 1201.23 1141.17 1255.28 1192.52 1311.77 1246.18

10 1000.00 1200.00 1080.00 1296.00 1166.40 1399.68 1259.71 1511.65 1360.49 1632.59 1469.33

15 1000.00 1300.00 1105.00 1436.50 1221.03 1587.33 1349.23 1754.00 1490.90 1938.17 1647.45

20 1000.00 1400.00 1120.00 1568.00 1254.40 1756.16 1404.93 1966.90 1573.52 2202.93 1762.34

25 1000.00 1500.00 1125.00 1687.50 1265.63 1898.44 1423.83 2135.74 1601.81 2402.71 1802.03

30 1000.00 1600.00 1120.00 1792.00 1254.40 2007.04 1404.93 2247.88 1573.52 2517.63 1762.34

35 1000.00 1700.00 1105.00 1878.50 1221.03 2075.74 1349.23 2293.70 1490.90 2534.53 1647.45

40 1000.00 1800.00 1080.00 1944.00 1166.40 2099.52 1259.71 2267.48 1360.49 2448.88 1469.33

45 1000.00 1900.00 1045.00 1985.50 1092.03 2074.85 1141.17 2168.22 1192.52 2265.79 1246.18

50 1000.00 2000.00 1000.00 2000.00 1000.00 2000.00 1000.00 2000.00 1000.00 2000.00 1000.00

55 1000.00 2100.00 945.00 1984.50 893.03 1875.35 843.91 1772.21 797.49 1674.74 753.63

60 1000.00 2200.00 880.00 1936.00 774.40 1703.68 681.47 1499.24 599.70 1319.33 527.73

65 1000.00 2300.00 805.00 1851.50 648.03 1490.46 521.66 1199.82 419.94 965.85 338.05

70 1000.00 2400.00 720.00 1728.00 518.40 1244.16 373.25 895.80 268.74 644.97 193.49

75 1000.00 2500.00 625.00 1562.50 390.63 976.56 244.14 610.35 152.59 381.47 95.37

Figure 3: Simulation of equity from a fixed-fraction betting system.

At a 0% bet there is no change in the equity. At five percent bet size, we bet 5% of $1,000.00 or $50.00 and make

twice that on the first toss (heads) so we have and expected value of $1,100, shown in gray. Then our second bet is

5% of $1,100.00 or $55.00, which we lose, so we then have $1,045.00. Note that we do the best at a 25% bet size,

shown in red. Note also that the winning parameter (25%) becomes evident after just one head-tail cycle. This

allows us to simplify the problem of searching for the optimal parameter to the examination of just one head-tail

cycle.

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