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Blending convergence with divergence

Release date: 23 Dec 2013 | Eurex Group

Blending convergence with divergence

Thought leadership by Kathryn M. Kaminski, PhD, CAIA

Kathryn M. Kaminski, PhD
CAIA, KTH Royal Institute of Technology,
MIT Sloan and the Stockholm School of Economics

Risk is defined as the chance that things do not turn out as you expect. Uncertainty, perhaps even more ominous, is the chance the circumstances, the extent or magnitude of events are unknown to you. As individuals, in both finance and in our personal lives, we are constantly engaged in some form of risk taking. This varies from what we decide to eat for lunch to when to go bearish in equities. Given that risk taking is at the core of human experience, we can take a closer look at the two types of risk taking and how they impact our financial performance and expectations.

Just a few weeks ago, the Nobel Prize in Economic Sciences was awarded to three tremendous contributors in the world of finance. Their contributions represent two approaches which give a fundamentally different view of financial markets.1

First, the view that financial markets are efficient: a Eugene Fama world. Second, the view that financial markets are made up of people, people who may suffer from“irrational exuberance”: the Robert Shiller world of behavioral finance. Given this dichotomy of perspectives, it is only natural to revisit the basic tenets of risk taking and discuss how these impact our lives and our path towards success in financial markets.

In a seminal article on this topic, Mark Rzepczynski outlined how two very different world views impact the style of risk taking.2 The two types of risk taking are divided based on our underlying frame of reference: convergent and divergent. We are convergent risk takers when we believe that the world is well structured, stable and somewhat dependable. We are divergent risk takers when we profess our own ignorance to the true structure of potential risks/benefits with some level of skepticism for what is or is not dependable. To explain this point it is easier to first use an example.

We are convergent risk takers when we believe that the world is well structured, stable and somewhat dependable. We are divergent risk takers when we profess our own ignorance to the true structure of potential risks/benefits.


Examples of convergent and divergent risk taking

Imagine two simple strategies: strategy C for convergent and D for divergent. Each strategy is applied to a simple game of chance which is played consecutively over time.3 If we start with strategy C, each time you win you take your money and start a new subsequent game. When you lose, you keep playing the same game until you win again then start a new game. This game doubles up on losers and takes profit on winners. Strategy C believes in the profits and takes them but reconfirms its conviction by doubling up with losses. A strategy like this will have many small gains with the occasional catastrophic loss. A person who uses a strategy C believes in the system, trusting that in the long run they will win. When they are shown to be incorrect, they simply wait until things get better again to reconfirm their beliefs.

When we turn to strategy D, in this strategy each time you lose you quit, cut your losses and start a new game. When you start to win instead of quitting, you double your bet. This strategy has little faith in positions that are losing and tries to follow the prevailing run when they seem to have found a string of luck. In each particular game, strategy D has little faith in a losing position. Strategy D faces many small losses with the occasional huge win.

When we compare the distributions of convergent strategy C and divergent strategy D, we see that strategy C is comfortable with winning almost expecting things to go as they suppose. Strategy D is rather skeptical taking lots of risks but never investing too much in any particular game. These two approaches are mirror opposite in their outcomes. The convergent strategy has many small gains and catastrophic losses and the divergent strategy has many small losses and euphoric wins. When we make some simple assumptions about the underlying game, the distribution of an example of these two is shown in Figure 1. These two extreme yet simple examples show that convergent strategies have negative skewness and fat left tails while divergent strategies have positive skewness and fat right tails.

Reflection on convergent, divergent

If you turn to many activities in life, our behavior is based on convergent and divergent risk taking. For example, when you cross the street, you are taking a convergent risk. In most cases, you get to the other side earning a small gain. In the rare catastrophic event that you are run over, the consequences are disastrous. In this example, we tend to believe that crossing the street is generally safe.4 If we turn to social networking as another example, successful social networkers often use divergent risk taking strategies. They talk to as many people as possible quickly and often stealthily cutting their losses with those of less interest. A powerful social networker understands that they never know how many people it could take to hit the key contract persons which lead to new business deals. Social networkers which use convergent risk taking strategies only speak to the people they know and already consider interesting, developing those relationships but creating no new ones. There are many fields where ivergent risk taking is the name of the game. Entrepreneurs, venture capitalists, and researchers try lots of different ideas and approaches until they actually find one that is the big winner.5

The convergent strategy has small gains and catastrophic losses and the divergent strategy has many small losses and euphoric wins.


If we turn to financial risk taking and investment, taking a simple financial example, investment in equities is something that most investors believe in. They believe in both the existence of an equity risk premium over the long run driven by fundamental value and they believe in the efficiency of financial markets. In this framework, “investing” in equity markets is a convergent risk taking activity. In distribution, this is also true. Equity returns are positive in expectation yet negatively skewed with fat left tails. If we turn to divergent risk taking in financial markets, the obvious example is trend following. Trend followers do not believe in anything but opportunity. They profess their ignorance to the fundamental structure of market prices, guessing that on occasion markets may be driven by so called Keynesian “animal spirits” leading to periods of opportunity. When they see a trend they follow it, they give no consideration to fundamentals.6 In fact, the distribution of trend following is positive in expectation with positive skewness with fat right tail.

Trend followers do not believe in anything but opportunity. They profess their ignorance to the fundamental structure of market prices, guessing that on occasion markets may be driven by so called Keynesian “animal spirits” leading to periods of opportunity.


How should we invest?  Which risk taking strategies are prudent?

Risk taking and the actions we take in response to risks are directly linked to our belief structure and financial world view. As a result, this may lead us to invest in different things especially across time. This causes us to ask ourselves, what does it mean to invest? What is speculation and what is investment or does this even matter? If we turn to the definition of investment, “An investment is an asset or item that is purchased with the hope that it will appreciate in the future” (Investopedia). If we take a polarized view, thinking that markets are either efficient or irrational, we will only be convergent or divergent in our strategies. Yet during this cold December in Stockholm, the Nobel Prize selection committee sends us a reminder that we need to acknowledge the importance of two very different world views. If both of these views are important, this means that both styles of risk taking are important for success and performance in financial markets.

To make this point more concrete. If markets are efficient, convergent strategies are prudent in assets which maintain a core fundamental structure. For example in this context, long term investment in equities without cutting your losses makes sense. On the other hand, we must also profess our ignorance to the true structure of financial markets yielding to the “animal spirits” of the market. In this case, divergent risk taking strategies that respond well to the unknown and the uncertain. The optimal combination will depend of course on the state of the financial markets and the players who participate in them.7

The best overall strategy will be to combine some of both risk taking approaches. 8 To summarize, convergent risk taking allows us to compete and maintain value over time while exposing us to hidden risks (the so called black swans) whereas divergent risk taking allows us to adapt, innovate, and hopefully survive during periods of market distress.9


Chung, S., Rosenberg, M., and J. Tomeo, Hedge Fund of Fund Allocations Using a Convergent and Divergent Approach, The Journal of Alternative Investments, Summer 2004.
Lo, A. W., Adaptive Markets and the New World Order, Financial Analysts Journal, Vol. 68 (2012), pp. 18–29.
Lo,A.W., The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective, Journal of Portfolio Management, Vol. 30 (2004), pp. 15–29.
Rzepczynski, M., Market Vision and Investment Styles: Convergent versus Divergent Trading, The Journal of Alternative Investments, Winter 1999.
Taleb, N., Antifragility: Things that gain from disorder, Random House, New York, 2012.
Trendspotting in asset markets, Press Release, The Royal Swedish Academy of Sciences, October 2013.


1 The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel was given to Eugene Fama, Lars Peter Hansen, and Robert Shiller. To quote the Nobel Prize Selection Committee,“The Laureates have laid the foundation for the current understanding of asset prices. It relies in part on fluctuations in risk and risk attitudes, and in part on behavioral biases and market frictions.” Press Release, The Royal Swedish Academy of Science October 2013.
2 Reviews risk taking from the trading perspective: Rzepczynski (1999)
3 Here we make no assumptions about the distribution of the game, its stationarity in time or other details allowing for generality.
4 In the U.K., it is particularly important for right looking tourist who crosses traffic without looking taking substantial risks without even considering it.
5 This is a process which Nassim Taleb labels as tinkering. In his new book Antifragility, he discusses how tinkering leads to most important discoveries.
6 In practice most trend following strategies have a 2/3 failure rate of positions. This means that most trades may lose money but the ones that win seem to outweigh the smaller losses over time.
7 In the previous newsletter, we discussed Andrew Lo’s Adaptive Markets Hypothesis. This view of markets is consistent with both classic views suggesting instead that markets adapt and evolve over time. This framework lends its structure and principles from the field of evolutionary biology. The forces of competition, natural selection, adaptation, and reproduction can be seen to drive the market ecology. These factors combine to determine who will succeed or fail in financial markets. See Lo (2004, 2012).
8 As an example, Chung, Rosenberg, and Tomeo (2004) discuss how to allocate capital using a convergent and divergent approach in hedge fund portfolios.Their article shows, both conceptually and empirically, how this can be done in practice.
9 This article is the subject of a TEDx talk given by the author and available at

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