GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 109 Sudhir 27A:GK040:Chapter:Chapter-05:
110 TRADING RISK
designed to increase such control considerably and in the process to place you in the best possible position to achieve your performance objectives while minimizing the damage if events play out that you did not anticipate.
DETERMINING THE APPROPRIATE RANGES OF EXPOSURE
Read this next statement very carefully, kids, because it’s important: It is up to the capital provider to determine the appropriate risk ranges for the trading portfolios that it is funding.This means that if you are trad- ing with anything but your own proprietary capital, the risk management call will ultimately belong to someone else. However, this does not imply that the exercise of determining appropriate exposure ranges is one that you can avoid, as I believe that it is an important one for virtually all port- folio managers—whether you are self-funded or otherwise.
This is true because the guidelines that your funding entity is likely to provide will offer little in the way of details beyond setting an upper bound of acceptable portfolio exposure; and these figures may or may not be entirely rational from a risk management perspective. Moreover, while your management will almost certainly be excessively vigilant (if not entirely sensible) in establishing and enforcing risk limits at the upper end of the range, they are likely to provide you with very little guidance as to what an acceptable lower bound for your risk ought to be. This concept is more important than most traders realize, as they are often conditioned to believe that less risk is always better than more risk. However, this is often far from the case. And I wish I had a dime for every trader I know who was profitable on a unit basis but simply did not take on enough risk to gener- ate a revenue stream large enough to cover the fixed costs of his or her accounts or to meet minimally acceptable rates of return on invested cap- ital. Sadly, even tragically, the talents of these traders go fallow, simply because they never feel comfortable enough to take the risks necessary to achieve the absolute revenues that are fully within their power to achieve.
The same dynamic may also apply if you are self-funded; I have encoun- tered many individual traders who make money consistently but at the end of the day aren’t properly compensated for the time and effort they put into trading activities because they simply aren’t taking enough risk.
For all these reasons, I strongly recommend that you take the trouble to establish and routinely review exposure ranges that fit your circum- stances in terms of both minimally acceptable risk levels and the more intu- itive upper bounds. This section will offer guideline on setting these ranges.
In order to arrive at the appropriate figures, let’s draw from that rel- atively small but often untapped resource—common sense. Clearly, we
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 110 Sudhir 27A:GK040:Chapter:Chapter-05:
Setting Appropriate Exposure Levels (Rule 1) 111
want to set our exposure ranges such that when we operate on the lower end of the spectrum, they still provide us with a sportsman’s chance to reach our profitability objectives, while, when operating at the other end of the range, they offer an acceptably low probability of hitting what we have defined as our largest tolerable economic loss (whether stipulated by an external funding entity or self-imposed). In other words, we always want to take enough risk to enable us to achieve our goals while not extending beyond a point where the likelihood of exhausting (or, heaven forbid, exceeding) our risk-taking resources becomes a danger- ous possibility.
While there is no question that setting these parameters is a subjective process, limited in its accuracy by our universal inability to predict the future, our statistical tool kit provides us with the basis for setting these ranges in a manner that should positively impact our ability to achieve our dual objective of reaching our goals without assuming undue exposure.
Specifically, the model features a dual methodology that draws on such concepts as Portfolio Volatility, Sharpe Ratio, Targeted Return, and Maxi- mum Acceptable Loss in order to identify the exposure level sufficient to reach our targets while minimizing both the probability and the magnitude of loss-threshold violations.
Method 1: Inverted Sharpe Ratio
We’ll start the process by focusing on the less intuitive task of developing a methodology for setting a sensible lower bound for portfolio exposure. If we accept the critical notion that risk and return are interrelated, then we ought to be able to express one in units of the other. Indeed, as we have already covered, some of the most important work done in the era of finan- cial engineering is the expression of return in units of risk. Most famously, and most relevantly for our purposes, these concepts are embodied in the Sharpe Ratio calculation, which you may recall is a ratio of returns to their associated volatility.
If we can estimate our Sharpe Ratio with relative accuracy, we have a pretty good notion of the kind of return we can expect for a given level of performance volatility. But what if we are looking to ask the opposite ques- tion? What if instead of wondering what return we can expect, given the riskiness of our portfolio, we want to figure out how much exposure we need in order to generate the level of performance we desire? As it turns out, with a quick sleight of algebraic hand, we can manipulate the Sharpe Ratio equation to determine both what a given level of volatility will pro- duce in terms of returns and (more important for our purposes here) what level of volatility is consistent with a given performance target. In order to understand this critical concept as thoroughly as its importance demands,
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 111 Sudhir 27A:GK040:Chapter:Chapter-05:
112 TRADING RISK
let us take the steps involved one at a time. Begin by resurrecting the basic Sharpe Ratio equation:
Sharpe Ratio (ReturnRisk-Free Rate)Portfolio Volatility The Sharpe Ratio can be thought of as a scorecard of your perform- ance as a portfolio manager from a risk-adjusted-return perspective. It also is a measure of the amount of return you are likely to generate for a given dollar of risk, which, as argued earlier, is a concept that is expressed most succinctly in the portfolio volatility statistic. Thus, once you have carefully analyzed your Sharpe Ratio and are comfortable that the number you have calculated is one that you can sustain, you can invert this equation to determine what level of return you are likely to generate for a given level of risk assumption. Note that for the purposes of this discussion, we will designate this figure to be the “Sustainable Sharpe.” It differs from the
“actual” Sharpe Ratio derived for selected time sequences by containing a qualitative overlay that bases the figure on a conservative assessment of what you are likely to achieve in the future, given the dynamic nature of market conditions. The “actual” Sharpe Ratio is based solely on your trad- ing history. Our “Sustainable Sharpe” is one that uses the historical Sharpe (and perhaps other inputs) to derive a lower bound for what we can con- fidently achieve as a Sharpe Ratio on a going forward basis.
The results of this algebraic maneuvering are as follows:
1. (ReturnRisk-Free Rate) Sustainable Sharpe Portfolio Volatility 2. Portfolio Volatility (ReturnRisk-Free Rate)/Sustainable Sharpe These equations define a concept that I will refer to as the Inverted Sharpe Methodology for setting exposure levels. The first equation is designed to provide you with some idea of the amount of return you can expect given your Sustainable Sharpe and your current level of volatility. The second will tell you what kind of portfolio volatility you should target in order to yield a specific return (again given a Sustainable Sharpe).
Both equations can be extremely useful in the determination of appro- priate target ranges of exposure. To illustrate, consider a portfolio with the following characteristics:
Target Return: 25%
Risk-Free Rate: 5%
ReturnRisk-Free Rate: 20%
Sustainable Sharpe: 2.0
We take the target return, the risk-free rate, and the Sustainable Sharpe as constants in this analysis, meaning in the case of the Sustainable
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 112 Sudhir 27A:GK040:Chapter:Chapter-05:
Setting Appropriate Exposure Levels (Rule 1) 113
Sharpe that the account in question believes it can generate more than $2 in return for $1 of exposure it takes. The variable in these equations is portfolio volatility, which, as you will recall, is our proxy for risk. Now let’s plug various volatility scenarios into the inverted equations in order to deter- mine what they say about performance. In order to do so, however, we must take a careful look at our options for characterizing volatility. In the Sharpe Ratio calculation, volatility is represented as the standard devia- tion of portfolio returns. So, whatever statistic you are currently employ- ing as your benchmark exposure measurement, it must be mapped back into standard deviation units in order to produce meaningful outcomes.
Perhaps the best alternative at your disposal in this regard is the results of a Value at Risk (VaR) calculation, which have the advantage of being based on current portfolio characteristics. If you have access to a VaR calculation, it is therefore possible to substitute this figure into the denomi- nator of the Sharpe Ratio, as long as you take care to scale down the confi- dence interval statistic to the one standard deviation level (for example, if you are using a 95th percentile VaR, you can map it back into a one standard deviation figure by dividing by 1.96). If you choose not to use a VaR approach, the best alternative is simply to calculate a one standard deviation P/L volatility.
Returning to our example, let us first assume that by our best estimate the portfolio is managed such that its projected, annualized volatility is 7%.
Inserting this figure, along with the Sustainable Sharpe and Risk-Free Rate parameters into equation 1, and solving for Return, we find that this port- folio is likely to produce returns of approximately 19%—respectable, but significantly below our target return of 25%.
This analysis begs the question of what portfolio volatility is consistent with a 25% return target, again assuming the portfolio in question is able to sustain a Sharpe Ratio of 2.0. We can arrive at this answer quite simply using the second equation by inputting Return, Risk-Free Rate, and Sus- tainable Sharpe, and solving for Portfolio Volatility. Carrying through these steps with respect to the sample portfolio profile generates a result of 10%, implying that a portfolio with a Sustainable Sharpe of 2.0 should generate about 10% annualized volatility in order to produce returns of 25%.
In considering these concepts, it is important to understand that the algebraic manipulations of the Sharpe Ratio that are embodied in the two equations are nothing more than rough approximations of the amount of volatility that is consistent with a given target return, based on historical risk-adjusted performance. The algebraic results will never be entirely accurate because, in the first place, the figure you select as your Sustain- able Sharpe Ratio will almost certainly deviate from both your historical performance and from your best guess as to what actual Sharpe you can produce in the future.
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 113 Sudhir 27A:GK040:Chapter:Chapter-05:
114 TRADING RISK
Moreover, I must caution against too literal an interpretation of any results you produce in this type of analysis, which, among other things might indicate that those who have lower Sharpes should assume higher risk profiles in order to achieve specified return thresholds (instead, the responsible solution involves targeting a lower return profile). However, when viewed from the proper perspective, the Inverted Sharpe Equations offer useful insights into the appropriate sizing of your risk profile. This is particularly true in cases where (as often occurs in my experience) the volatility characteristics and target returns are entirely inconsistent with one another.
For example, on the one hand, the sample portfolio in our example may target 10% annualized volatility and still come nowhere near the 25%
return target. On the other hand, we can state with a great deal of confi- dence that if this account produces volatility in the low single digits, say, 3%, it has very little chance of ever hitting the 25% objective (indeed, it can only do so if the actual Sharpe Ratio is more than triple the estimated Sustainable Sharpe). By the same token, if this account trades to an annu- alized volatility profile of, say, 30%, the projected returns would be 75%—a figure so far in excess of the targeted objective as to raise questions about either the appropriateness of the target or the level of risk assumption, or some combination of the two. The point here is that by combining infor- mation about your Sharpe Ratio with expectations of return, you begin to create a picture of what type of risk levels are appropriate to these objectives; volatility levels significantly below these ranges will almost guarantee a failure to achieve associated targets, whereas figures vastly above these thresholds give rise to potential inconsistencies between return targets and risk tolerances.
Method 2: Managing Volatility as a Percentage of Trading Capital
As you may have surmised, the Inverted Sharpe Ratio method is best adapted to determining the lower bound of exposure that is consistent with reaching your targeted objectives. The main insight that it will pro- vide you in determining the appropriate associated upper bound is that if your volatility is too high, it will project out a return that far exceeds your objectives. While this information may be useful, it doesn’t address the binding constraint to risk assumption, which, as we all know by now, is the limited availability of risk capital. Clearly, in determining an upper bound of acceptable exposure, we would want to be motivated by a methodology that both minimizes the probability of our reaching our largest allowable loss and ensures that if, in fact, we do experience hard times, we don’t end up with a number that considerably exceeds this figure.
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 114 Sudhir 27A:GK040:Chapter:Chapter-05:
Setting Appropriate Exposure Levels (Rule 1) 115
The challenge, therefore, is to identify a threshold of exposure that expends, without exhausting, the full measure of risk capital allocated to the account. Reviewing this concept intuitively, the brain (well, at least my brain) moves fluidly to the following premise: In a rational portfolio risk management program, maximum exposure threshold should be tied to P/L performance over routine and identifiable cycles (e.g., one year). As a matter of good business judgment, I’m sure that you’ll agree that your risk- taking prerogatives are much broader when you have accumulated some bankable P/L to work with than they are when you’re down money and struggling to trade on.
The logical extension of this argument is that your maximum risk tolerance is fungible against your P/L, and the process of establishing an upper bound for your exposure begins with quantifying this interplay.
Our methodological approach begins by introducing the concept of trading capital, which is defined simply as the sum of your maximum loss toler- ance and your period-to-date P/L:
Trading Capital Risk Capital P/L
where Risk Capital largest allowable loss for the period
This simple equation defines the current amount of “losable” dollars asso- ciated with a given account until it hits the previously defined maximum loss threshold. For example, if an institutional trader has been given the right to trade until he or she has lost $5 million, then the trading capital on the first day of the year is $5 million. If the account subsequently makes $2 million, trading capital increases to $7 million. Similarly, if this account loses
$2 million, trading capital is diminished to $3 million.
With the concept of trading capital established, we can apply it as a measure of risk-taking capacity. As the figure rises, we have more room for exposure and can expand our risk taking accordingly: by taking on larger position sizes or by deploying larger amounts of capital into new ideas, or some combination thereof. By the same token, as trading capital is diminished, so too is our ability to bear the negative consequences of risk taking, and our upper bound for market risk assumption must contract accordingly.
The next step is to formalize this into an equation that expresses risk- taking capacity as a function of trading capital. While some may argue for complex formulas that may (or may not) bring some added precision, it is my experience that you can operate quite effectively by simply fixing upper exposure bands as a fixed percentage of trading capital. Once you have framed the issue in this manner, the exercise of establishing a maxi- mum is reduced to one of selecting the appropriate percentage threshold.
Here, again, there is room for divergent opinion; but based on years of
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 115 Sudhir 27A:GK040:Chapter:Chapter-05:
116 TRADING RISK
experience in monitoring traders in drawdown, I think that this number should not exceed 10%. In other words, as a rule, maximum volatility (expressed in one-day, one-standard-deviation terms) should not exceed 1/10 of trading current capital. In mathematical terms:
P/L Trading Capital 10%
where P/LVaR calculation at one-standard-deviation confidence level, or 20 standard deviations of P/L.
Again, if you have access to a VaR program and can convert the output to a one-standard-deviation confidence interval, it represents probably the best proxy for P/L, as it is tied directly to the risks that currently charac- terize your portfolio. However, the 20-day average volatility is a very good proxy of the kinds of risk that you are currently taking. In either case, here’s my rationale behind establishing 10% as an appropriate upper bound:
IfP/Lreaches 10% of the amount of capital you have left to lose at any given time, then we can expect with relative frequency (one observation out of six, or better, if we’ve figured it right) a loss that meets or exceeds 10% of your remaining risk-taking capacity. While a loss of this size would clearly be acceptable from time to time, we must be mindful of the fact that according to our standard deviation theory, market moves of two, three, and even four standard deviations occur routinely throughout the trading cycle. Indeed, most market mavens agree that these types of outlying observations happen with more frequency than is indicated by the proba- bilities typically associated with a normal distribution. As we have dis- cussed briefly, this phenomenon is commonly referred to as kurtosis, or
“fat tails” for the more pedestrian among us. Like it or not, fat tails are a reality in the markets. And I for one am glad for their presence, for the world would certainly lose some if its charm without them.
When a fat-tailed event, say, to the tune of three or four standard deviations does occur in the markets, if our volatility assumptions hold, a portfolio that has volatility of 10% of its trading capital may be exposed to losses that are between one-quarter and one-half of its remaining risk capacity. Needless to say, this is a potentially very serious erosion of risk capital. Moreover, as you have no doubt experienced, these types of extreme market movements tend to happen on multiple days in succession (or at least in very close proximity to one another), meaning that markets that have three- or four-standard-deviation moves in a single day can often experience much larger fluctuations before the dust settles and the market stabilizes. Thus, at volatility levels much above 10% of trading capital, traders can literally find themselves one adverse market event away from having blown through the lion’s share of their reserve. It is for this reason that I view 10% as an appropriate upper bound.
GK040-1502G-C05[109-132].qxd 7/23/04 4:53 PM Page 116 Sudhir 27A:GK040:Chapter:Chapter-05: