Q uantitative T rading, E RNEST P. C HAN : Chp6

Money and Risk Management

essential to manage your risks : limits your drawdowns to a tolerable level and yet be positioned to use optimal leverage of your equity to achieve maximum possible growth of
your wealth. Furthermore, if you have more than one strategy, you will also need to find a way to optimally allocate capital among them so as to maximize overall risk-adjusted return.
central tool: Kelly formula

OPTIMAL CAPITAL ALLOCATION AND LEVERAGE

excellent expository article on this subject in oneof his papers (Thorp, 1997) :

• Thorp, Edward. 1997. “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market.” Handbook of Asset and Liability Management, Volume I, Zenios and Ziemba (eds.). Elsevier 2006. Available at: www.EdwardOThorp.com.
• http://www.eecs.harvard.edu/cs286r/courses/fall12/papers/Thorpe_KellyCriterion2007.pdf

Our objective here is to maximize our long-term wealth. Equivalent to maximizing the long-term compounded growth rate g of your portfolio. Implicitly means that ruin (i.e., equity’s going to zero or less because of a loss) must be avoided.
approximation: probability distribution of the returns of each of the trading strategy i is Gaussian, with a fixed mean m_i and standard deviation s_i.
Calculating the optimal fractions:

Let’s denote the optimal fractions of your equity that you should allocate to each of your n strategies by a column vector F^∗ = (f^∗₁ ,f^∗₂ , . . . , f^∗_n )^T. Here, T means transpose. optimal allocation is given by:
F^∗ = C⁻¹M

• C is the covariance matrix such that matrix element C_ij is the covariance of the returns of the ith and jth strategies
• −1 indicates matrix inverse
• M = (m₁,m₂, . . . ,m_n)^T is the column vector of mean returns of the strategies. These returns are one-period, simple (uncompounded), unlevered returns. For example, if the strategy is long $1 of stock A and short $1 of stock B and made $0.10 profit in a period, m is 0.05, no matter what the equity in the account is.

If we assume that the strategies are all statistically independent, the covariance matrix becomes a diagonal matrix (and the C^-1 matrix is also diagonal with 1/a_ii in the diagonal)
so we get the simple formula:
f_i = m_i/s_i²

average compounded rate of return (in the simple case of only 1 strategy)

• the average compounded rate of return is:
• g =m − s²/2. (follows from the general formula for compounded growth g(f ) given in the appendix to this chapter, with the leverage f set to 1 and risk-free rate r set to 0.)  [The take-away lesson here is that risk always decreases long-term growth rate]
• The appendix says: the general formula: g( f ) = r + fm− s² f²/2  : where
• f is the leverage,
• r is the risk-free rate;
• m is the average simple, uncompounded one-period excess return; 
• s is the standard deviation of those uncompounded returns.
• optimal f which maximizes g : f=m/s², the Kelly formula for one strategy or security under the Gaussian assumption.

“half-Kelly” : Often, because of uncertainties in parameter estimations, and
also because return distributions are not really Gaussian, traders prefer to cut this recommended leverage in half for safety. This is called “half-Kelly” betting.
For retail trading account, your maximum overall leverage l will be restricted to either 2 or 4, depending on whether you hold the positions overnight or just intraday. In this situation, you would have to reduce each f_i by the same factor l/(| f₁| + | f₂| +
· · ·+| f_n |), where | f₁| + | f₂|+· · ·+| f_n| is the total unrestricted leverage of the portfolio.

what is this maximum compounded growth rate?
g = r + S²/2
where the S is none other than the Sharpe ratio of your portfolio!
the higher the Sharpe ratio of your portfolio (or strategy), the higher the maximum growth rate of your equity (or wealth), provided you use the optimal leverage recommended
by the Kelly formula.
Note that following the Kelly formula requires you to continuously adjust your capital allocation as your equity changes so that it remains optimal.
As a practical procedure, this continuous updating of the capital allocation should occur at least once at the end of each trading day. One should also periodically update F* itself by recalculating the most recent trailing mean return and standard deviation.

What should the lookback period be and how often do you need to update these inputs to the Kelly formula? These depend on the average holding period of your strategy. If you hold your positions for only one day or so, then as a rule of thumb, I would advise using a lookback period of six months. Using a relatively short lookback period has the advantage of allowing you to gradually reduce your exposure to strategies that have
been losing their performance. As for the frequency of update, it should not be a burden to update F* daily once you have written a program to do so.

RISK MANAGEMENT

Kelly formula is for

1. optimal allocation of capital
2. determination of the optimal leverage
3. risk management

selling at a loss is the frequent result of risk management, whether or not the risk management scheme is based on Kelly’s formula. Risk management always dictates that you should reduce your position size whenever there is a loss, even when it means realizing
those losses. (The other face of the coin is that optimal leverage dictates that you should increase your position size when your strategy generates profits.)

Handle fat tails – max tolerable one-period drawdown /max historical loss:  Because the returns are not really Gaussian: there are “fat tails”: large losses occur at far higher frequencies than would be predicted by Gaus. Dist. To handle the extreme events:  use our simple backtest technique to roughly estimate what the maximum one-period loss was historically. (The period may be one week, one day, or one hour. The only criterion to
use is that you should be ready to rebalance your portfolio according to the Kelly formula at the end of every period.) You should also have in mind what is the maximum one-period drawdown on your equity that you are willing to suffer. Dividing the maximum tolerable
one-period drawdown on equity by the maximum historical loss will tell you whether even half-Kelly leverage is too large for your comfort. The leverage to use is always the smaller of the half-Kelly leverage and the maximum leverage obtained using the worst historical
loss. (e.g.: for S&P500, max hist one-day loss = 20.47% (Oct 19 1987, black monday)

The truly scary scenario in risk management is the one that has not occurred in history before.

IS THE USE OF STOP LOSS A GOOD RISK MANAGEMENT PRACTICE ?

It is a common fallacy to believe that imposing stop loss will prevent the portfolio from suffering catastrophic losses. When a catastrophic event occurs, securities prices will drop discontinuously, so the stop loss orders to exit the positions will only be filled at prices much worse than those before the event. So, by exiting the positions, we are actually realizing the catastrophic loss and not avoiding it.

For stop loss to be beneficial, we must believe that we
are in a momentum (=trending) regime. In other words, we must believe that
the prices will get worse within the expected lifetime of our trade. Otherwise, if
the market is mean reverting within that lifetime, we will eventually recoup our
losses if we didn’t exit the position too quickly.

It is not easy to tell whether one is in a momentum regime (when stop loss is beneficial) or in a mean-reverting regime (when stop loss is harmful). My own observation is that when the movement of prices is due to news or other fundamental reasons (such as a company’s deteriorating revenue), one is likely to be in a momentum regime, and one should not “stand in front of a freight train,” in traders’ vernacular.

 

Types of risks

Types of risks: position risk (which is comprised of both market risk and specific risk), model risk, software risk, and natural disaster risk, in decreasing order of likelihood.

1. Model risk: the trading model is wrong. Why? a lot of reasons, among them:
   • data-snooping bias, survivorship bias, and so on.
     • To eliminate errors: it is extremely helpful to have a collaborator or consultant to duplicate your backtest results independently to ensure their validity.
   • Model risk can also come from increased competition from other institutional traders all running the same strategy as
     you; or it could be a result of some fundamental change in market structure (=regime shift).
     • not much you can do to alleviate these sources of model
       risk, except to gradually lower the leverage of the model as it racks up losses, up to the point where the leverage is zero. This can be accomplished in a systematic way if you constantly update the leverage according to the Kelly formula based on the trailing mean return and standard deviation. This
       is preferable to abruptly shutting down a model because of a large drawdown (see my discussion of the psychological  pressure to shut down models prematurely
2. Software risk : The ATS does not faithfully reflect your backtest model. because of bugs.
   • way to elminate (see Ch5): compare the trades generated by your automated trading system with the theoretical trades generated by your backtest system to ensure that they are the same.
3. physical or natural disasters: (not dramatic as earthquakes or tsunami):
   • Internet connection went down before you could enter a hedging position
   • your power went down in the middle of transmitting a trade?
     • methods for preventing physical disasters: see Chp4

 

PSYCHOLOGICAL PREPAREDNESS

human traders who are not psychologically prepared will often override their automated trading systems’ decisions, especially when there is a position or day with abnormal profit or loss.

when one has entered a position by mistake, the rational step to take is to exit the position
immediately upon discovery of the error.

“representativeness bias”—people tend to put too much weight on recent experience and underweight long-term average: After a big loss, traders—even quantitative traders—tend to immediately modify certain parameters of their strategies so that they would have avoided the big loss if they were to trade this modified system.

two major psychological weaknesses: despair and greed.

• Despair occurs when a trading model is in a major, prolonged drawdown. You will be under great pressure under this circumstance to shut down the model completely. Other overly self-confident traders with a reckless bent will do the opposite: They will double their bets on their losing models, hoping to recoup their losses eventually, if and when the models rebound. Neither behavior is rational: if you have been managing your capital allocation and leverage by the Kelly formula, you would lower the capital allocation for the losing model gradually.
• Greed is the more usual emotion when the model is having a
  good run and is generating a lot of profits. The temptation now is to
  increase its leverage quickly. a well-disciplined quantitative trader will keep the leverage below the dictates of the Kelly formula as well as the caution imposed by
  the possibility of fat-tail events.
• How should we train ourselves to overcome these psychological weaknesses and learn not to override the models manually and to remedy trading errors correctly and expeditiously? start with a small portfolio and gradually gain psychological preparedness, discipline, and confidence in your models.

SUMMARY

Risk management is a crucial discipline in trading.

The danger:

• The trading world is littered with numerous examples of giant hedge funds and investment banks laid low by enormous losses due to a single trade or in a very short period of time.
• WHY? overleveraging positions and not to an inherently erroneous model.
• Typically, traders will not overleverage a model that has not worked very well. It is a hitherto superbly performing model that is at the greatest risk of huge loss due to overconfidence and overleverage.
• => Here is an important tool for risk management: the determination
  of the optimal leverage using the Kelly formula.
• useful side of Kelly:  determines the optimal allocation of capital among different strategies, based on the covariance of their returns.
• Be psychologically prepared for ups and downs. Do not succumb to either despair or greed.
• To gain practice in this psychological discipline, one must proceed slowly with small position size, and thoroughly test various aspects of the trading business (model, software, operational procedure, money and risk management) before scaling up according to the Kelly  formula.
• I have found that in order to proceed slowly and cautiously, it
  is helpful to have other sources of income or other businesses to
  help sustain yourself either financially or emotionally (to avoid the
  boredom associated with slow progress). It is indeed possible that
  finding a diversion, whether income producing or not, may actually
  help improve the long-term growth of your wealth.