Risk Management

(c) Ed Seykota, 2003

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).

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
 

 Fixed Bet

$10
 Fixed-Fraction Bet

1%
 
Start
 1000
 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/font>
 1050
 1049.97
 
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.
 
% 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
 
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.
Notice that the expected value of the system rises from $1000.00 with increasing bet fraction to a maximum value of about $1,800 at a 25% bet fraction. Thereafter, with increasing bet fraction, the profitability declines. This curve expresses two fundamental principles of risk management: (1) The Timid Trader Rule: if you don't bet very much, you don't make very much, and (2) The Bold Trader Rule: If you bet too much, you go broke. In portfolios that maintain multiple positions and multiple bets, we refer to the total risk as the portfolio heat.
Note: Note the chart illustrates the Expected Value / Bet Fraction relationship for a 2:1 payoff game. For a graph of this relationship at varying payoffs, see Figure 8.
 
The Kelly Formula
 
K = W - (1-W)/R
 
K = Fraction of Capital for Next Trade

W = Historical Win Ratio (Wins/Total Trials)

R = Winning Payoff Rate

-------

For example, say a coin pays 2:1 with 50-50 chance of heads or tails. Then ...

K = .5 - (1 - .5)/2 = .5 - .25 = .25.

Kelly indicates the optimal fixed-fraction bet is 25%.
 
This graph shows the optimal bet fraction for various values of luck (Y) and payoff (X). Optimal bet fraction increases with increasing payoff. For very high payoffs, optimal bet size equals luck. For example, for a 5:1 payoff on a 50-50 coin, the optimal bet approaches about 50% of your stake.
This graph shows optimal expected value for various values of luck and payoff, given betting at the optimal bet fraction. The higher the payoff (X: 1:1 to 5:1) and the higher the luck (Y: .20 to .70), the higher the expected value. For example, the highest expected value is for a 70% winning coin that pays 5:1. The lowest expected value is for a coin that pays 1:1 (even bet).
 
This graph shows the expected value of a 50% lucky (balanced) coin for various levels of bet fraction and payoff. The expected value has an optimal bet fraction point for each level of payoff. In this case, the optimal bet fraction for a 1.5:1 payoff is about 15%; at a 2:1 payoff the optimal bet fraction is about 25%; at a 5:1 payoff, the optimal bet fraction is about 45%. Note: Figure 4 above is the cross section of figure 8, at the 2:1 payoff level.
 
Stock Price/Share Shares Value
A
B $100
C $200  $50,000
Stock Price/Share Risk/Share Shares Risk Value
A
B $100 $10
C $200 $5 1000 $5,000 $200,000

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.