Alex Dannenberg, president of Pine Mountain Capital Management, outlines some of the characteristics of—and the logic behind—trend-following and relative-value trading strategies.

No strategy that depends only on historical prices can be consistently profitable if markets are efficient and the random-walk hypothesis holds true—that is, if future price movements are entirely unaffected by the past. So let me begin with the following claim: Price movements are not entirely random.

Price movements are instead characterized by extended periods of positive serial autocorrelation, during which markets “trend,” and extended periods of negative serial autocorrelation, during which markets are choppy and fluctuate around a slowly moving mean. The existence of these hypothesized periods of nonzero serial autocorrelation is necessary, but not sufficient, for systematic trading strategies to be consistently profitable. Trend-following strategies, for example, must be able to profit more from the periods of positive serial autocorrelation than they lose during choppy markets. It is far from obvious that this is possible, even if one accepts that market dynamics are not those of a random walk.

This hypothesis of nonzero serial autocorrelation is difficult to prove or disprove. Over many years (and a large sample of price changes), the hypothesis implies that markets will show an average serial autocorrelation close to zero, consistent with a random walk. Over shorter periods of a month or two (with much smaller samples of price changes), markets are assumed to evolve with nonzero serial autocorrelation. But random walks will also appear to have nonzero serial autocorrelation over short time periods because the sample error is large.

To illustrate this effect, consider a synthetic 14-year price series generated using a random walk with 7 percent lognormal price volatility. As shown in Figure 1, the serial autocorrelation can reach large positive and negative values when measured over time frames of several months—even when the process generating the data is defined as having zero serial autocorrelation.

Rather than detailing the statistical evidence for my hypothesis of nonzero serial autocorrelation, let me instead present a simple economic rationale for the hypothesis, which can be reduced to a sound bite: “Prices trend when information trends. Prices chop when information chops.”

Note that I am assuming a form of market inefficiency. If markets were perfectly efficient, trending information would not lead to trending prices. The fact that information sometimes trends would be priced in, and economic data releases would differ from investors’ expectations in a random, rather than biased, way—leading to random, rather than trending, price movement. My experience and intution is that markets are not efficient in this way.

Consider a simplified market with 10 bond traders, each of whom can be long or short one bond. Each trader has in mind a “fair value” for the bond based on his or her view of what inflation will be, what the Fed will do, what Microsoft’s earnings will be and so on. Now suppose that one trader thinks the fair value for the bond is 98, two traders think fair value is 99, four traders think fair value is 100, two are at 101, and one trader thinks fair value is 102. In this simple world, the market price for the bond will be 100. Why? Not because it’s the “average fair value,” but because it’s the price that clears the market. Only when the market price is 100 can all 10 traders express their views. The three traders who think that fair value is above 100 are long, the three traders who think that fair value is below 100 are short, and the four traders who think 100 is the fair price are flat.

So, in a nutshell, there’s a distribution of prices that traders regard as fair value. Supply comes into the market from traders who think that fair value is lower than the market price; demand comes from the traders who think that the market is undervalued (the market price is lower than fair value). Market prices move in order to balance supply and demand.

Nothing new so far. Now suppose that the underlying reality is that the U.S. economy is weak, but that this is not yet fully accepted or understood. Over time, evidence of the weak economy will build—low payroll numbers, high unemployment numbers, bad earnings reports and so on. As each new piece of evidence is delivered to the market, traders will revise their views of inflation, the Fed and so on, and raise their fair value price. As the whole distribution of fair value prices shifts upward, so will the market price. The market will have a long, drawn-out rally because the information demonstrating a weak economy will trickle in over a long, drawn-out period of time. All we have to assume in this simple analysis is that there is some genuine, underlying economic reality that isn’t widely understood and isn’t likely to change rapidly, and so can be gradually “uncovered” by news released into the market.

Now suppose that weak U.S. economic data have been rolling in for some months and that traders have received enough information to form accurate opinions about fair value and the economy. Under these circumstances, new information is as likely to raise traders’ ideas of fair value as it is to lower their ideas of fair value, and the market will trade in a range, exhibiting negative serial autocorrelation as it fluctuates around fair value. This will continue until another big gap arises between economic perception and reality, and ignites another trend.

Of course, the real world is much more complicated than our simple example. Markets are highly coupled, with shocks in one reverberating in others. Information released into the market is not always representative or accurate, giving price dynamics a genuine noise component as an overlay to the trending and mean-reverting components described earlier. But these complications don’t change the essence of the argument.

Trading Strategies

Assuming that market dynamics are not completely random, it now makes sense to look at systematic trading more closely. I won’t argue that systematic trading strategies will continue to be profitable—this is too open-ended a subject and has been dealt with extensively elsewhere. Instead, I will describe in plain terms what performance features an investor might expect from a fund that engages primarily in relative-value trading or trend-following.

In my parlance, the term “relative-value” strategy will include any scheme to lock in the difference between the market price and the fair value of a security by hedging with a related, but fairly priced security. Trading collateralized mortgage obligations against Treasuries and mortgage pass-throughs falls into this category, as does statisticcal arbitrage of the price spreads between elements of a basket of similar equities.

I argue that these and other relative-value strategies generate returns similar to those of a short position in an option straddle. Both the seller of a straddle the relative-value trader are betting that deviations from the status quo will be small and that there is a tendency for the status quo, or equilibrium, to be restored. A frequent seller of straddles always takes in a small premium and occasionally has to make a large payoff to the straddle buyer when the market makes a big move. Similarly, the relative-value trader usually makes small amounts of money when price spreads fluctuate around their equilibrium values, and occasionally loses a lot when the market makes a statistically rare big move or when fundamentals change the equilibrium levels. The straddle seller and relative-value trader have profit-and-return distributions that look remarkably alike, as Figure 2 indicates.

The correspondence between the relative-value trading strategy and the short straddle position depends on the assumption that the relative-value strategy does not include a stop-loss provision. If the relative-value fund were to trade with a stop-loss, then its profit-and-return distributions would not include unbounded losses. Unfortunately, relative-value traders rarely use stop-losses, because their positions seem to have greater profit potential and are therefore more painful to exit when the market goes further out of equilibrium and a stop-loss would normally be triggered.

Note, once again, that I am not making any assertions about the expected return from relative-value trading. I am only pointing out that the distribution of returns can be expected to have negative skew with larger, less-frequent losses balanced against smaller, more-frequent gains. In light of the tumult of 1998, it is also worth observing that the small and bounded upside potential often leads to the use of a lot of leverage—which in turn increases liquidity risk. The painful experiences of Long-Term Capital Management and other hedge funds highlight the importance of this liquidity risk as well as the necessity of a stop-loss strategy.

Let me now turn to trend-following strategies. This is well-trod territory and I will simply summarize the beautiful analysis of William Fung and David Hsieh, who showed that the performance of an investment in a trend-following fund closely resembles that of the purchase of a lookback straddle. Their analysis is both empirical and conceptual, and readers can turn to their paper for the data that support their claim. But Fung and Hsieh’s reasoning goes roughly as follows:

The lookback straddle buyer pays a premium to receive a payoff equal to the market high minus the market low during the lookback period. The straddle buyer’s downside is limited to the premium spent, and his upside is equal to the biggest market range experienced, whether the market ends up higher or lower. The straddle’s payoff only rarely covers the premium, but when it does, it typically pays out much more than the premium. In other words, the lookback straddle buyer usually loses a little but occasionally makes a lot.

By analogy, the investor in a trend-following fund also has a bounded downside equal to the sum of the management fee he or she pays and the loss level at which the fund will stop itself out. The investor’s upside is much greater than the downside and, typically, is expected to grow in proportion to the market’s range, whether the market goes up or down. In a given market, 55 percent to 60 percent of trend-following trades are losers and 40 percent to 45 percent are winners. Trend-following strategies, therefore, bear all the hallmarks of a typical long-gamma position: positive skew with larger, less-frequent gains balanced against smaller, more-frequent losses.

In both cases, the profit-and-return distributions are expected to—and do—resemble Figures 4 and 5.

Note that the upside potential of a trend-following position is much larger than that of a relative-value trade, and thus requires less leverage to earn a given return. This is why trend-following strategies have less exposure to liquidity risk than do relative-value strategies.

Markets are not entirely blind to history; instead, they go through eras of significantly positive and negative serial autocorrelation. These are the periods toward which trend-following and relative-value strategies, respectively, are aimed. I have mentioned that relative-value strategies behave like short-gamma positions and can be expected to generate negatively skewed return distributions. I have also mentioned that trend-following strategies behave like long-gamma positions and can be expected to generate positively skewed return distributions.

If a trend-following strategy and a relative-value strategy were to have the same expected return and Sharpe ratio, an investor should prefer the trend-following strategy, because of the positive skew in its return distribution. A proper comparison of the two strategies should use a skew-sensitive risk-adjusted return statistic that distinguishes loss-aversion from variance-aversion.

I will end with a conjecture: It seems likely that the risk-adjusted returns from trend-following and relative-value strategies are sufficiently large and uncorrelated that an optimal portfolio will contain investments in each. Furthermore, by combining both types of strategy an investor can construct a “gamma-neutral” investment that is easily combined with more traditional investment vehicles in any standard, mean-variance-based efficient portfolio construction.

William Margrabe, president of the William Margrabe Group, explains how options theory can be adopted to real-world business decisions.

Dear Dr. Risk,
I’m a general manager with a rudimentary academic background in finance. I’ve read recent articles in the Harvard Business Review and the McKinsey Quarterly suggesting that option-pricing models have applications in the real world. They say that managers can budget capital in the face of uncertainty much better with options than with tools like net present value (NPV) and decision trees.
Can you review the use of options models for managerial decision-making, highlighting their advantages and limitations, and let me know if any practical textbooks exist for a manager who wants to put these techniques to use?

Dear Alex,
A growing number of practitioners agree that many corporate investments and personal opportunities involve choices that are, in effect, “real options.”

What is a real option? A real option involves (a) tangible objects, such as bricks, mortar, equipment and people, and (b) physical actions, such as excavation, construction, demolition, hiring and firing. However, the boundary between real options and standard derivative products is fuzzy, and some business opportunities are hybrids.

For example, a pharmaceutical company’s research program is a portfolio of real, compound options. The cost of basic research is like an option premium. The cost of testing a promising drug is like a strike price. If the research turns up the next Viagra, then the option is deep-in-the-money. Most likely, it will expire worthless.

Two more examples: the owner of land next to Rockefeller Center has the real option either to erect a building today or continue operating a ground-level parking lot. When Con Ed buys a gas-fired electricity generator and puts it on a barge in the East River, it buys the real option to press it into service on hot summer days to meet peak demand.

Has anyone applied real options theory to business?

I asked one academic and one practitioner for real-option success stories. According to Lenos Trigeorgis, professor of finance at the University of Cyprus, “the earliest successful applications were recorded in the natural resources field, where the high price volatilities, long maturities, and the existence of futures and forward prices made it possible to obtail certainty-equivalent value estimates that could then be discounted at the riskless rate. More recent successes are now found in pharmaceuticals and other industries.”

Tom Copeland, chief corporate finance officer with Monitor Co. in Cambridge, Mass., says that real options usually come up “with clients with large investments, when they think flexibility is part of the story.” He gives an example: “There’s a major integrated oil company. In this case, it had a large natural gas field, 60 percent explored and 40 percent unexplored. It was trying to decide whether to develop the field now or explore more—a classic debate. Development now pushes cash flows nearer the present. The company knows it will have to put in pipelines, refineries and storage facilities. The question is how much of each. The company would normally look at geological reports and say that expected production is X. But the possible range around X is huge, so it is almost sure to be wrong. Either it will get more or less natural gas than expected. The problem is, it’s expensive to build an extra half a refinery. Or maybe production will be small and the firm will have excess capacity. Real-option theory tells us, If volatility goes up, defer. In this case, you can resolve the uncertainty by exploring.” At McKinsey and Monitor, Copeland applied real-option analysis to a dozen clients. Sometimes, he says, clients make decisions based on other models and then, after seeing the results of real-option analysis, “decisions get reversed.”

So how do you model a real option? The main business application for real options seems to be capital budgeting—investment—often related to strategic planning. One investment may open doors to other opportunities that may or may not pay off, and neither traditional NPV nor simple decision-tree methods are up to the task of evaluating such sequential investment decisions. Real-option analysis combines elements of NPV, decision trees and option pricing in an effort to help make better decisions.

The NPV approach consists of simplifying an investment opportunity into a sequence of (in and out) cash flows, “discounting” all future cash flows back to the present, and adding up all the present values to see whether the project is in the red or black. This is analogous to looking into a crystal ball to see how much money you would make if you paid a fee to walk through a garden, then picked every piece of fruit and vegetable, and sold the harvest at market prices.

The NPV approach allows us to apply standard techniques in the theory of choice to consumption at different times. The Capital Asset Pricing Model (CAPM) extends NPV to handle simple uncertainty. In their common forms, however, NPV and CAPM don’t easily allow for contingencies and options along the way.

A simple decision tree is like a map of a road system that starts at one point and then forks repeatedly, with foreseeable prizes and/or penalties along the way. At each fork, you deliberately choose which way to go. Where you end up and how much booty you collect depend entirely on your choices. A simple decision tree has no randomness to it, and the time value of money is not an issue. A sophisticated decision tree lets Mother Nature make the decision at some forks by “flipping a coin.” It reflects the time value of money, and uses something like dynamic programming to work back through the tree to make optimal decisions.

A sophisticated decision tree may be a practical tool for real-option analysis, because of its generality and flexibility, but creating the right tree may require a master modeler.

A binomial price tree looks like a decision tree, but at each fork Mother Nature essentially flips a coin to decide which way the underlying price goes. You may be able to make a decision at each node—not about which way to proceed, but perhaps about whether or not to exercise the option early. The binomial price tree in option-pricing fully embodies the idea of the time value of money. The binomial tree allows us to extend the theory of choice from consumption at different times to consumption in different states of the world, as well as at different times.

In the limit, as steps in time and space approach zero in the right way, the binomial price tree turns into a continuous-time diffusion process. Binomial option pricing approaches option pricing of the Black-Scholes-Merton sort.

If you’re lucky, your real option is an analogue of a standard financial option and you have all the data you need to plug into the standard pricing model. According to Trigeorgis, “almost all real options are special cases of compound exchange options.” A compound option is an option to buy an option, such as a call on a call option. An exchange option is an option to exchange one thing for another.

Soon, you’ll be thinking of your child’s education as a compound real option, with daily strike prices paid in hours of homework and the payoff in terms of admission to Ivy League schools.

Even if this is true, getting a good value out of the model isn’t a slam dunk. You have to know enough about options to recognize a classic situation when you’re staring right at it. If your key risk factor isn’t a traded commodity or asset, you’ll have to make some assumptions about inputs. Volatility numbers are typically hard to come by.

The question then becomes, “Which models work, in practice?” Except in the simplest cases, you’ll need to build your own model, typically a sophisticated decision tree. Here is what Trigeorgis says: “I view real-options valuation, especially in its binomial version, as a ‘market-adjusted’ version of decision-tree analysis. Mechanistically, it works in the same backward (averaging-and-folding-back) way as dynamic programming. Those firms that are already comfortable using decision-tree analysis, simulation or scenario analysis will find real options a natural and intuitive extension.”

“We use a lattice approach,” notes Copeland. “It allows us to deal with compound options and rainbow options with a small number of uncertainties. We find (a) we can get numbers, (b) because it’s a lattice approach, it’s algebraic and can be explained to management, whereas stochastic calculus conclusions are difficult to explain, and (c) we always have the net present value of a project as a base case, so the lattice approach without volatility reduces to the base case.”

You may also have to make some rough assumptions about the inputs to your model. While this may shake your confidence in the model, ask yourself where you’d be without even a rational conceptual framework.

With a little reading, thought and practice, you’ll see real options all around you, in business and your private life. Soon, you’ll be thinking of your child’s education as a compound real option, with daily strike prices paid in hours of homework and the payoff in terms of admission to Ivy League schools. The two main ways to analyze a real option are a standard option-pricing model and a sophisticated decision tree. The right model depends on the problem at hand. Finding or building it will require some expertise, which you might find in one of the three textbooks I reviewed on my web site. (www.margrabe.com/DrRisk.asp ).