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
[td]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. 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 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.
Figure 2: Simultion of Fixed-Bet and Fixed-Fraction Betting Systems.
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, 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.
Figure 3: Simulation of equity from a fixed-fraction betting system.
Figure 4: Expected value (ending equity) from ten tosses, versus bet fraction, for a constant bet fraction system, for a 2:1 payoff game, from the first and last columns of figure 3.
Optimizing - Using Calculus
Since our coin flip game is relatively simple, we can also find the optimal bet fraction using calculus. Since we know that the best system becomes apparent after only one head-tail cycle, we can simplify the problem to solving for just one of the head-tail pairs.
The stake after one pair of flips:
S = (1 + b*P) * (1 - b) * S0
S - the stake after one pair of flips
b - the bet fraction
P - the payoff from winning - 2:1
S0 - the stake before the pair of flips
(1 + b*P) - the effect of the winning flip
(1 - b) - the effect of the losing flip
So the effective return, R, of one pair of flips is:
R = S / S0
R = (1 + bP) * (1 - b)
R = 1 - b + bP - b2P
R = 1 + b(P-1) - b2P
Note how for small values of b, R increases with b(P-1) and how for large values of b, R decreases with b2P. These are the mathematical formulations of the timid and bold trader rules.
We can plot R versus b to get a graph that looks similar to the one we get by simulation, above, and just pick out the maximum point by inspection. We can also notice that at the maximum, the slope is zero, so we can also solve for the maximum by taking the slope and setting it equal to zero.
Slope = dR/db = (P-1) - 2bP = 0, therefore:
b = (P-1)/2P , and, for P = 2:1,
b = (2 - 1)/(2 * 2) = .25
So the optimal bet, as before, is 25% of equity.
Optimizing - Using The Kelly Formula
J. L. Kelly's seminal paper, A New Interpretation of Information Rate, 1956, examines ways to send data over telephone lines. One part of his work, The Kelly Formula, also applies to trading, to optimize bet size.
Figure 5: The Kelly Formula
Note that the values of W and R are long-term average values, so as time goes by, K might change a little.
Some Graphic Relationships Between Luck, Payoff and Optimal Bet Fraction
The Optimal Bet Fraction Increases with Luck and Payoff
Figure 6: Optimal bet fraction increases linearly with luck, asymptotically to payoff.
The Expected Value of the Process, at the Optimal Bet Fraction
Figure 7: The optimal expected value increases with payoff and luck.
Finding the Optimal Bet Fraction from the Bet Size and Payoff
Figure 8: For high payoff, optimal bet fraction approaches luck.
Non-Balanced Distributions and High Payoffs
So far, we view risk management from the assumption that, over the long run, heads and tails for a 50-50 coin will even out. Occasionally, however, a winning streak does occur. If the payoff is higher than 2:1 for a balanced coin, the expected value, allowing for winning streaks, reaches a maximum for a bet-it-all strategy.
For example, for a 3:1 payoff, each toss yields an expected value of payoff-times-probability or 3/2. Therefore, the expected value for ten tosses is $1,000 x (1.5)10 or about $57,665. This surpasses, by far, the expected value of about $4,200 from optimizing a 3:1 coin to about a 35% bet fraction, with the assumption of an equal distribution of heads and tails.
Almost Certain Death Strategies
Bet-it-all strategies are, by nature, almost-certain-death strategies. Since the chance of survival, for a 50-50 coin equals (.5)N where N is the number of tosses, after ten tosses, the chance of survival is (.5)10, or about one chance in one thousand. Since most traders do not wish to go broke, they are unwilling to adopt such a strategy. Still, the expected value of the process is very attractive, so we would expect to find the system in use in cases where death carries no particular penalty other than loss of assets.
For example, a general, managing dispensable soldiers, might seek to optimize his overall strategy by sending them all over the hill with instructions to charge forward fully, disregarding personal safety. While the general might expect to lose many of his soldiers by this tactic, the probabilities indicate that one or two of them might be able to reach the target and so maximize the overall expected value of the mission.
Likewise, a portfolio manager might divide his equity into various sub-accounts. He might then risk 100% of each sub account, thinking that while he might lose many of them, a few wouldwin enough so the overall expected value would maximize. This, the principle of DIVERSIFICATION, works in cases where the individual payoffs are high.
Diversification
Diversification is a strategy to distribute investments among different securities in order to limit losses in the event of a fall in a particular security. The strategy relies on the average security having a profitable expected value, or luck-payoff product. Diversification also offers some psychological benefits to single-instrument trading since some of the short-term variation in one instrument may cancel out that from another instrument and result in an overall smoothing of short-term portfolio volatility.
The Uncle Point
From the standpoint of a diversified portfolio, the individual component instruments subsume into the overall performance. The performance of the fund, then becomes the focus of attention, for the risk manager and for the customers of the fund. The fund performance, then becomes subject to the same kinds of feelings, attitudes and management approaches that investors apply to individual stocks.
In particular, one of the most important, and perhaps under-acknowledged dimensions of fund management is the UNCLE POINT or the amount of draw down that provokes a loss of confidence in either the investors or the fund management. If either the investors or the managers become demoralized and withdraw from the enterprise, then the fund dies. Since the circumstances surrounding the Uncle Point are generally disheartening, it seems to receive, unfortunately, little attention in the literature.
In particular, at the initial point of sale of the fund, the Uncle Point typically receives little mention, aside from the requisite and rather obscure notice in associated regulatory documentation. This is unfortunate, since a mismatch in the understanding of the Uncle Point between the investors and the management can lead to one or the other giving up, just when the other most needs reassurance and reinforcement of commitment.
In times of stress, investors and managers do not access obscure legal agreements, they access their primal gut feelings. This is particularly important in high-performance, high-volatility trading where draw downs are a frequent aspect of the enterprise.
Without conscious agreement on an Uncle Point, risk managers typically must assume, by default to safety, that the Uncle Point is rather close and so they seek ways to keep the volatility low. As we have seen above, safe, low volatility systems rarely provide the highest returns. Still, the pressures and tensions from the default expectations of low-volatility performance create a demand for measurements to detect and penalize volatility.
Measuring Portfolio Volatility Sharpe, VaR, Lake Ratio and Stress Testing
From the standpoint of the diversified portfolio, the individual components merge and become part of the overall performance. Portfolio managers rely on measurement systems to determine the performance of the aggregate fund, such as the Sharpe Ratio, VaR, Lake Ratio and Stress Testing.
William Sharpe, in 1966, creates his "reward-to-variability ratio." Over time it comes to be known as the "Sharpe Ratio." The Sharpe Ratio, S, provides a way to compare instruments with different performances and different volatilities, by adjusting the performances for volatilities.
S = mean(d)/standard_deviation(d) ... the Sharpe Ratio, where
d = Rf - Rb ... the differential return, and where Rf - return from the fund Rb - return from a benchmark
Various variations of the Sharpe Ratio appear over time. One variation leaves out the benchmark term, or sets it to zero. Another, basically the square of the Sharpe Ratio, includes the variance of the returns, rather than the standard deviation. One of the considerations about using the Sharpe ratio is that it does not distinguish between up-side and down-side volatility, so high-leverage / high-performance systems that seek high upside-volatility do not appear favorably.
VaR, or Value-at-Risk is another currently popular way to determine portfolio risk. Typically, it measures the highest percentage draw down, that is expected to occur over a given time period, with 95% chance. The drawbacks to relying on VaR are that (1) historical computations can produce only rough approximations of forward volatility and (2) there is still a 5% chance that the percentage draw down will still exceed the expectation. Since the most severe draw down problems (loss of confidence by investors and managers) occur during these "outlier" events, VaR does not really address or even predict the very scenarios it purports to remedy.
A rule-of-thumb way to view high volatility accounts, by this author, is the Lake Ratio. If we display performance as a graph over time, with peaks and valleys, we can visualize rain falling on a mountain range, filling in all the valleys. This produces a series of lakes between peaks. In case the portfolio is not at an all-time high, we also erect a dam back up to the all time high, at the far right to collect all the water from the previous high point in a final, artificial lake. The total volume of water represents the integral product of drawdown magnitude and drawdown duration. &nbp; If we divide the total volume of water by the volume of the earth below it, we have the Lake Ratio. The rate of return divided by the Lake Ratio, gives another measure of volatility-normal return. Savings accounts and other instruments that do not present draw downs do not collect lakes so their Lake-adjusted returns can be infinite.
Figure 9: The Lake Ratio = Blue / Yellow
Getting a feel for volatility by inspection.
Reference for Sharpe Ratio: Stress Testing
Stress Testing is a process of subjecting a model of the trading and risk management system to historical data, and noticing the historical performance, with special attention to the draw downs. The difficulty with this approach, is that few risk managers have a conscious model of their systems, so few can translate their actual trading systems to computer code. Where this is possible, however, it provides three substantial benefits (1) a framework within which to determine optimal bet-sizing strategies, (2) a high level of confidence that the systems are logical, stable and efficacious, and (3) an exhibit to support discussions to bring the risk/reward expectations of the fund managers and the investors into alignment.
The length of historical data sample for the test is likely adequate if shortening the length by a third or more has no appreciable effect on the results.
Portfolio Selection
During market cycles, individual stocks exhibit wide variations in behavior. Some rise 100 times while others fall to 1 percent of their peak values. Indicators such as the DJIA, The S&P Index, the NASDAQ and the Russell, have wide variations from each other, further indicating the importance of portfolio selection. A portfolio of the best performing stocks easily outperforms a portfolio of the worst performing stocks. In this regard, the methods for selecting the trading portfolio contribute critically to overall performance and the methodology to select instruments properly belongs in the back-testing methods.
The number of instruments in a portfolio also effects performance. A small number of instruments produces volatile, occasionally very profitable performance while a large number of instruments produces less volatile and more stable, although lower, returns.
Position Sizing
Some position sizing strategies consider value, others risk. Say a million dollar account intends to trade twenty instruments, and that the investor is willing to risk 10% of the account.
Value-Basis position sizing divides the account into twenty equal sub-accounts of $50,000 each, one for each stock. Since stocks have different prices, the number of shares for various stocks varies.
[td][/td] $50 [td][/td]1000[td][/td]$50,000[td][/td] 500 [td][/td]$50,000
Value-Basis Position Sizing Dividing $50,000 by $50/share gives 1000 Shares
Risk-Basis position sizing considers the risk for each stock, where risk is the entry price minus the stop-out point. It divides the total risk allowance, say 10% or $100,000 into twenty sub accounts, each risking $5,000. Dividing the risk allowance, $5,000 by the risk per share, gives the number of shares.
[td][/td]$50[td][/td] $5 [td][/td]1000[td][/td]$5,000[td][/td]$50,000[td][/td] 500 [td][/td]$5,000[td][/td]$50,000
Risk-Basis Position Sizing Dividing $5,000 by $5 risk/share gives 1000 Shares
Note that since risk per share may not be proportional to price per share (compare stocks B & C), the two methods may not indicate the same number of shares. For very close stops, and for a high risk allowance, the number of shares indicating under Risk-Basis sizing may even exceed the purchasing power of the account.
Psychological Considerations
In actual practice, the most important psychological consideration is ability to stick to the system. To achieve this, it is important (1) to fully understand the system rules, (2) to know how the system behaves and (3) to have clear and supportive agreements between all parties that support sticking to the system.
For example, as we noticed earlier, profits and losses do not likely alternate with smooth regularity; they appear, typically, as winning and losing streaks. When the entire investor-manager team realizes this as natural, it are more likely to stay the course during drawdowns, and also to stay appropriately modest during winning streaks.
In addition, seminars, support groups and other forms of attitude maintenance can help keep essential agreements on track, throughout the organization.
Risk Management - Summary
In general, good risk management combines several elements:
1. Clarifying trading and risk management systems until they can translate to computer code. 2. Inclusion of diversification and instrument selection into the back-testing process. 3. Back-testing and stress-testing to determine trading parameter sensitivity and optimal values. 4. Clear agreement of all parties on expectation of volatility and return. 5. Maintenance of supportive relationships between investors and managers. 6. Above all, stick to the system. 7. See #6, above.
翻译:
风险
风险的意义是损失的可能性。也就是说,如果我们拥有一些股票,这些股票价格有下跌的可能性,那么我们就具有风险。股票本身不是风险,损失也不是风险,损失的可能性才是风险。只要我们一天还拥有这些股票,我们就具有风险。控制这些风险的唯一方式就是买进或卖出股票。就拥有股票,想赚取利润这件事来说,风险基本上是无可避免的。我们所能做的,就是管理风险。
风险管理
管理的意思是引导和控制。风险管理在于指引导及控制损失的可能性。风险管理者的任务即在于测量风险,并买进或卖出股票以增加或减少风险。
丢铜板的例子
假设我们丢铜板,正反面出现的机率是相等的。丢铜板的例子能很有效的阐述风险管理的概念。
一件事情发生的机率(Probability),就是这件事情发生的可能性,以百分比来表达。如果丢100次铜板的结果是50次正面,那么机率就是50%。请注意,机率必然介于0和1之间。
再假设丢铜板游戏的规则是:(1)资本$1,000 (2)我们永远都押正面 (3)我们可由剩余资本拿出任何数字来下注 (4)如果是反面,赌注都输掉了 (5)如果是正面,我们可以赢得赌注双倍的彩金 (6)公平的铜板,正反面的机率都是50%。这个游戏跟一些交易方法很相像。
在这样的状况下,我们的胜率(Luck)就是赢的机率,50%;有50%的时候,我们的胜率会不错。我们的报酬(Payoff)是2:1,因为我们每押1元,就能赢2元。在这个例子中,我们的风险(Risk)正是我们下注的金额,也正处于风险中。例子中的胜率和报酬都是固定不变的,只有下注的金额可以改变。
在比较复杂的游戏里,例如股票交易,胜率和报酬会随着状况的不同而改变。交易者花费可观的时间精力来改善胜率和报酬,往往徒劳无功,因为这不是人力所能改变的。要控制风险,“风险”是风险管理者唯一能够有效改变的参数,而非报酬及胜率。
我们也可以为较为复杂的游戏建立模型,改变胜率及报酬,以观察不同的结果。参照图一。
图一:胜率/报酬矩阵,表示六种不同的结果。
这个矩阵甚至可以模拟现实上的交易。
现在我们要回到原来的基本铜板模型了,事实上它的维度就足以说明风险管理中许多的观念。我们等等再来谈比较复杂的例子。
赌注的最佳化
在丢铜板的例子中,我们有一个固定的胜率50%,固定的报酬2:1,而我们永远押正面。要拟定风险管理的策略,我们必须找到管理赌注的方式。这和股票交易里所面临的风险管理是相似的。好的管理者知道,胜率和报酬他们不太插得上手,最根本的问题在于如何找出下注在股票的金额应有多少。我们的游戏以$1,000开始。
直觉和系统
直觉(Hunch)是一种决定赌注的方式。也许我们预感要押$100。
虽然以直觉来决定赌注确实是现实世界里最多人用的方式,它还是有几个问题。它需要一个操作者特续的产生这些预感来决定赌注,把这些预感转为实际的赌注。比较起科学方法来说,这些赌注更仰赖心情和感觉。
要改善以直觉来下注的方式,我们可以使用一套系统。系统的意思是一套逻辑化的方法,来规定一连串的赌注。比较这两种方法,系统的好处在于(1) 我们不需要操作者 (2)赌注变得有规律,可预期,前后一致,而非常重要的是 (3)我们能够在电脑上执行历史资料的模拟,将下注系统最佳化(Optimize)。
虽然一般来说系统的好处很明显,实际上风险管理者却很少清楚定义他们的系统,足以在电脑上进行回溯测试。
我们丢铜板的例子满简单的,我们可以帮它准备一个下注系统。此外,我们可以藉此测试这些系统,找出系统的最佳参数,以便执行最佳化的风险管理。
固定赌注以及固定下注比例
我们的下注系统必须定义赌注。定义赌注的其中一个方法是使用固定金额,例如每次下注$10,不管我们输还是赢。这种就叫做固定赌注(Fixed Bet)。在这个情况下,我们$1,000的资本可能会减少或增加,一直到$10比例上会变得太大或太小,而变成不是最好的赌注了。
要解决固定赌注中资本变动的问题,我们可以定义固定下注比例(Fixed-Fraction)。在我们的资本中,1%的赌注等于$10。这次,不管我们的资本上升或下降,固定下注比例都会和资本成比例。
由固定下注比例我们发现一个有趣的事情,既然赌注和资本保持一定的比例,理论上来说完全破产不可能,形式上毕业出场的风险是零。在实务上,崩溃和心理上的 Uncle Point 比较有关系,参照下文。
模拟测试
我们可以针对历史资料进行模拟测试(Simulate),以便测试我们的下注系统。假设我们丢十次铜板,有五次正面五次反面,我们可以如图二般安排模拟测试。
图二:固定赌注和固定比例赌注的模拟。
请注意,两个系统第一次都赚了$20.00(赌注的两倍),开出来的是正面。第二次,固定赌注的系统输了$10.00,而固定比例系统输了1%,也就是$1,020.00的1%,也就是$10.20,资本剩下 $1,009.80。
两种系统跑出来的结果几乎没什么不同。然而经过长时间后,固定比例系统会以几何级数成长,超越以线性成长的固定赌注系统。另外,系统的结果取决于正反面的个数,至于正反面的顺序并不会影响结果。读者可以自行以试算表进行测试。
金字塔型加码(Pyramiding)以及赌注加倍(Martingale)
如果过程是随机的,像是丢铜板,规律的正反顺序是不可能的,因此会发生一连串的正面或反面的状况。然而,我们无法利用这个现象获利,因为它的本质就是随机的。在非随机的过程中,例如股票价格的趋势,金字塔型加码或是其他趋势追踪技巧都可能有用。
金字塔型加码,是在获利时加码的一种方式。这个技技有助于交易者加码至最佳化部位。在已最佳化的部位之上加码只会引起过度交易的灾难。一般来说,这种系统的小修小补对系统来说,远远不如坚守系统来得重要。事实上,这样的修修补补使交易者对系统的信号产生诠释的空间,可能导致直觉化的交易,徒然削弱坚守系统的努力罢了。
赌注加倍(Martingale)的意思是在赌输时加倍下注。如果又输,则再加倍,如此一直下去。这种方式好比赶在压路机前捡硬币,只要一次失手,资本就完蛋。
最佳化-使用模拟测试
一旦我们选定了一个下注系统,例如固定比例下注系统,我们就能依系统找出最佳化的参数(Parameters),得到最好的期望值(Expected Value)。在丢铜板的例子中,我们唯一的参数就是那个固定比例。再次重申,我们可以经由模拟测试找到答案。参照图三和图四。
请注意,丢铜板的例子的用意在于强调风险的某些元,以及它们之间的关系,特别是我们的例子是报酬2:1,胜率50%。这个例子没有考虑正反面不均匀的情况,也没有考虑一连串的正面或反面。它的用意并非在建议任何市场交易里风险管理的参数。
图三:固定比例下注系统的资本模拟测试
0%时,资本不会改变。在5%时,赌注是资本$1000.00的5%,也就是$50.00。第一次期望值是$1,100,以灰色部份表示。第二次的赌注一样是资本的5%,$55.00,这次我们会输,剩下$1,045.00。请注意,在赌注为25%时,表现最好,以红色部份表示。再请注意,最佳化参数 (25%) 在一次正反面周期后就很明显了。这让我们能够以单一周期求得最佳化参数。
图四:十次铜板后的期望值(期末值)与下注比例的关系,固定比例下注系统,2:1报酬,以图三的第一及最终列作图。
请注意,系统的期望值在25%下注比例时,从$1000.00提高到最大值$1,800。从这之后,随着提高下注比例,获利减少。这条曲线表示了两个表达了两个风险管理的根本法则,(1) 胆小交易者法则:如果你下的注不够大,你的获利也不会大。 (2)鲁莽交易者法则:如果你下的注太大,破产是必然的。在具有多个部位,多个赌注的投资组合中,总风险我们称之为投资组合热度(Heat)。
这个图同时说明了在报酬为2:1的情形下,期望值和下注比例的关系。这样的关系在不同报酬的情况,参照图八。
最佳化-使用微积分
因为我们的丢铜板游戏满简单的,我们也可以用微积分求最佳下注比例。因为我们知道,最佳系统在一次正面和反面的周期后就是显而易见的了,我们也可以用一个正面一个反面的周期,来简化问题。
一正一反的组合后,赌注变成:
S = (1 + b*P) * (1 - b) * S0
S – 一个周期后的赌注
b – 下注比例
P – 报酬2:1
S0 – 一个周期前的赌注
(1 + b*P) - 赢时的影响
(1 - b) – 输时的影响
所以,一个周期后的影响就是:
R = S / S0
R = (1 + bP) * (1 - b)
R = 1 - b + bP - b2P
R = 1 + b(P-1) - b2P
注意,b值很小时,R随着b(P-1)的增加而增加;b值很大时,R随着b2P而减小。这就是胆小交易者、鲁莽交易者法则背后的数学意义。
我们可以画一张图显示R和b之间的关系,这张图看起来会很像我们从模拟的结果,以目测选择最大值。我们也可以观察到,最大值时斜率为零,所以我们也可以令斜率为零,即可求最大值。
Slope = dR/db = (P-1) - 2bP = 0, 于是
b = (P-1)/2P , and, for P = 2:1,
b = (2 - 1)/(2 * 2) = .25
所以最佳化的下注比例就是资金的25%。
最佳化-使用凯利方程式(Kelly Formula)
J. L. Kelly在1965年的论文,A New Interpretation of Information Rate中,讨论使用电话线路来传送资料。论文中的一部份,Kelly Formula,也适用于交易中,求最佳下注比例。
图五:Kelly Formula
注意,W和R都是长期的平均数字,随着时间,K会小小的改变。
胜率、报酬、和最佳下注比例的图形关系
最佳下注比例随胜率和报酬增加而增加
图六:最佳下注比例随着胜率而增加,趋近报酬。
这张图显示在不同的胜率和报酬下的最佳下注比例。最佳下注比例随着酬酬的增加而增加。对于很高的报酬率时,最佳下注比例等于胜率。举例来说,一个5:1报酬的公平铜板,最佳化下注比例趋近于50%。
过程中的期望值和最佳下注比例
图七:最佳化后的期望值随着胜率和报酬增加。
这张图表示,在最佳下注比例时,期望值和不同胜率和报酬的关系。报酬越高,胜率越好,期望值也越高。
由赌注和报酬寻找最佳下注比
图八:对高报酬而言,最佳下注比例趋近胜率。
这张图显示,50%胜率下,期望值和不同下注比例和报酬的关系。对于每一种不同的报酬,都有一个最佳下注比例。在这样的情况,1.5:1的报酬下,最佳下注比例约15%;2:1的报酬下,最佳下注比例约25%;5:1的报酬下,最佳下注比例为45%。请注意,图四是图八在2:1报酬下的一个切面。
非均匀分布以及高报酬
到目前为止,我们看待风险管理,都具于以下的假设:长时间下来,正面和反面出现的次数会相等。然而,偶而也会有一连串的获胜。如果报酬高于2:1,连续的获胜也允许的情况下,期望值会在全押的策略下达到最大值。
例如说,报酬是3:1,每次丢铜板的期望值是报酬乘以机率,也就是3/2。因此,丢十次铜板的期望值变成$1,000 x (1.5)10,将近$57,665。这远 远超出我们原来的期望值$4,200,报酬3:1,下注比例35%,基于正反面平均分配的假设。
几乎确定会毁灭的策略
全押,本质上来说是几乎确定会毁灭的策略。因为对一个公平的铜板来说,存活的机率,变成(.5)N,N表示丢铜板的次数。十次铜板之后,存活的机率大约是千分之一。大部份的交易者当然不想破产,所以就不会采用这样的策略。但是,这种策略的期望值真的很诱人。在毁灭只代表资产的损失时,我们会想要找到这样的系统。
例如,一个将军管理着好多可有可无的士兵。他也许会让士兵全部上场,全面攻坚,不考虑士兵会不会死掉。用这样的战术,将军也许会失去很多士兵,但也许会有一两个士兵攻坚成功。整体上来说,任务成功的机率就大增。
相同的,一个投资组合管理者也许会把资本分散在许多帐户中,然后赌上每个帐户里100%的资本。他想,他也许会输掉很多帐户,但有些帐户的胜利势必可以使整体的期望值最大化。这就是风险分散(Diversification)的原则。当个别的报酬率非常高时适用。
风险分散
风险分散就是把资金分散到很多不同的投资工具,单一投资工具失败时,损失得以限制。这样的策略必须符合“所有部位平均起来有获利期望值”这样的条件。比较起单一投资工具,风险分散也提供心理上的好处。短期内投资的变动,可以由不同投资工具间的成果抵消,而获得较为平滑的投资组合变动率。
The Uncle Point
从分散的投资组合的观点,个别投资工具组合成为总合的绩效。就风险管理者或基金投资人来说,基金的表现就成了注意力的焦点。基金的表现,也会受上述两种人的感觉、态度、还有投资者对个别股票态度上管理者的态度所影响。
基金管理中最重要的,也许也是最不受注意的观点,就是Uncle Point。它的意思是净值水平降低,引发投资者或管理者信心丧失的那个点。如果投资人或管理者失去信心而进行赎回,那基金就宣告死亡。正因Uncle Point发生时的环境通常是很灰心气馁的,很少文献对这个现象有详尽的记载。
尤其是当基金在安全范围里的时候,除了法律文件里必要但却含糊的贴示外,没什么人会注意Uncle Point。在Uncle Point认知上的不协调可以导致其中一方的放弃,说起来也很不幸,明明另一方要的只是再次保证。
当压力真的很大的时候,投资人和管理者不会去看那个看也看不懂的法律文件,他们会看的是自己够不够胆。在净值常常降低,高表现,高变动率的创业界尤其重要。
若双方对Uncle Point没有清楚的共识,风险管理者往往必须假设Uncle Point就在不远处,于是他们寻找降低变动率的方法。如同我们上面所看到的,低变动率系统很少能有最好的获利。然而压力和紧张局势使得对于变动率的侦查和处罚变成必要。
测量投资组合的变动率(Sharpe, VaR, Lake Ratio and Stress Testing)
从分散投资组合的观点,个别投资工具的成败总合成为整体绩效的一部份。投资组合管理者依赖一整套测量基金表现的工具,例如Sharpe Ratio,VaR,Lake Ratio以及Stress Testing。
威廉夏普先生在1966年提出了他的“报酬-变动率比”。经过长时间,它成为我们所熟知的Sharpe Ratio。Sharpe Ratio利用对变动率调整绩效的方法,提供了比较不同绩效不同变动率投资工具间比较的标准。
S = mean(d)/standard_deviation(d) ... the Sharpe Ratio, 而
d = Rf - Rb ... the differential return, 而
Rf - 基金报酬率
Rb - 基准报酬率
夏普指标的变形不断出现。其中一种变形舍弃了基准点,将它设为零。另一个,基本上就只是夏普指数的平方,但它使用获利的变异数,而不是标准差。在使用夏普指标上,一个重要的考量是它并未将上方下方变动率加以区分。高杠杆高绩效的系统,必然有很大的上方变动率,在这标准下也变得不太好了。
VaR,或称风险承担价值,是另一种衡量投资组合风险的方法。基本上它只测量最大净值下降百分比,这种情况很久才会发生一次,机率约95%。VaR的缺点是,(1)历史的计算结果只能提供大概值 (2)还是有5%的机率超过预期。净值下降产生的信心问题多半在非预期中发生,VaR也就无法真正预测它真正想要解决的状况了。
作者的看法是,看待高变动率的一般法则,就看湖泊比。如果我们把绩效画成图,会有山峰和山谷,我们可以想像雨下在山区里,填满山谷。这样会得到山谷和山谷之间有一系列的湖。如果这个投资组合现在不在历史高点,我们也在图形右侧想像一个水坝,和历史高点同高,拦住所有的水。水量代的就是净值降低的幅度,以及降低的时间的积分。
如果我们以水量除以土地量,就得到湖泊比。获利率除以湖泊比就代表另一种测量变动率的方法。一般的储蓄既然没有净值降低的问题,它们以湖泊比调整后的报酬就是无限大。
图九:湖泊比 = 蓝色区域 / 黄色区域
以目测来感受一下波动性
夏普指标的参考文献
压力测试
压力测试是一种针对交易模式以及风险管理系统所做的历史资料测试过程,强调历史绩效中的净值降低。这种做法困难之处在于,很少风险管理者有一套清楚的模型来表示他们的系统,因而无法将他们的交易系统转成程式码。当此法可行时,它具有三个优点:(1) 能作为找出最佳下注策略的基础 (2)对于系统的逻辑性,稳定性和有效性有高度信心 (3)对于投资人和基金管理者达成共识的过程,提供有效的展示。
至于需要多久的历史资料呢?如果测试的期间缩短三分之一或者更多,对结果都没有什么可观的影响,那就差不多了。
选择投资组合
在市场周期中,个别股票的表现很不相同。有的可以涨一百倍,有的可能从高点跌到剩百分之一。像道琼工业指数,S&P,那斯达克这些指数来说,他们的表现也很不同,更加强调的选择投资组合的重要性。表现最好的股票形成的投资组合可以轻易超越表现最差的股票形成的的投资组合。因此,选择投资组合的方法,以及在历史资料测试的投资工具中选择最好的工具,在整体绩效上有决定性的贡献。
投资工具的个数也会影响绩效。由少数投资工具组成的投资组合会产生高变动率,偶尔也会有高获利率。投资工具个数如果很多,变动率就比较小,获利较为稳定,通常也比较低。
部位大小的决定
有些部位大小决定的策略着重价值,有些着重风险。例如一个一百万的户头想用二十种投资工具,而投资人愿意冒10%的风险。
价值取向的部位大小决定策略,将户头分为二十等份,每份五万元,每份交易同一种股票。股票价格不同,买到的股数也会不同。
价值基础的部位大小决定
$50,000 除以 $50/股 得到 1000 股
风险取向的部位大小决定,考虑的是每档股票所承担的风险,这里的风险指的是买进价格减去停损价格。它把总风险的配额,例如10%,或说十万元,分为20个部份,每个部份只能承担五千元的风险。以风险配额五千元,除以每股所承担的风险,就得到买进的股数。
风险基础的部位大小决定
$5,000 除以 $5 风险/股求得 1000 股
请注意每股风险并不会跟每股价格成正比。两种不同的方法也不会表示一样的股数。如果使用很小的停损,很大的风险配额,以风险取向的方式求得应买的股数,有可能超出户头的购买能力。
心理面的考虑
在实际操作上,最重要的心理考量就是坚守系统的能力。要达到这个目标,必须(1)全然了解系统的规则 (2)了解系统行为 (3)在所有参与者中,找到清楚的共识,能够坚守系统的共识。
例如,就我们刚所说的,获利和亏损不见得会平稳的交换出现,通常来说都是一串赢的,一串输的。当一组投资人-管理者团队都了解到这是正常的,在净值降低时坚守系统的可能性就大增,赚大钱的时候也会比较谦虚谨慎。
除此之外,研讨会,心灵支援团体都有助于保持一贯的态度,让组织里上下都能照计划进行。
风险管理总结
一般来说,好的风险管理者包含下列要素:
阐明交易系统和风险管理系统,直到可以转化为程式码为止。
包含风险分散和投资工具选择,再做好历史测试。
历史测试和压力测试决定交易参数敏感性以及最佳化数字。
所有参与者,对于变动率和获利率,有清楚的共识。
投资人和管理者之间,维持具有支持作用的关系。
最重要的是,坚守系统。
参照上面第六条。 |