….but inside he was terrified. “I can’t believe this is happening.” he said – Today’s NY Times reporting on a Katie Couric interview with Captain C.B. Sullenberger.

Not surprisingly, the hero pilot of flight 1549 unconsciously used a technique that works – his feelings were put into words – even if only in his head. This kind of acknowledgement of anxiety actually helps the brain to focus because it isn’t wasting energy on trying to suppress a feeling that is a very real reflection of the situation.

This works in much less dire circumstances where the only risk is a loss of capital. Giving voice – verbally or internally – to one’s acutal feelings – without edit or judgment – allows one to much more easily see (and execute on) what needs to be done. I have said it before and will say it again, FDR was wrong. It is not that the only thing we have to fear is fear itself, the only thing we have to fear is no fear at all.

With apologies for being out of touch, I would like to make a couple of notes on psych cap. First, and I have said this before in different way, it really does appear that our brains have special “curcuits” for detecting imprecision or uncertainty. In other words, that feeling that someone or something is there – even if you can’t see or hear anyone? (which would have come in handy while searching for food in the woods) most likely has its own chip.

And this chip works.

Ever wonder why you review your plan and within minutes do exactly the opposite? It is because your brain detects that the plan doesn’t perfectly fit the circumstances and it goes into “ambiguity” mode. In ambiguity mode it relies on gut-feel more than anything. …. yet who among us has really been taught to systematically use gut-feel?

In that mode most of us are lost. We can’t tell the difference between a tickle that the market is going to turn on us and residual fear left over the last time the market burned us.

Now this game ain’t easy – and to make it harder is this conflict between planning for markets in estimated probabilities while fighting our brains which are using the uncertainty circuit. There is only one solution to this – get better at differentiating amongst our feelings. Don’t blame the messenger but the data lies there.

Someone named Robert Skidlesky wrote a book called John Maynard Keynes: 1883-1946: Economist, Philosopher, Statesman. He says that Keynes said “not all future events could be reduced to measurable risk. There was a residue of genuine uncertainty and this made disaster an ever-present possibility, not a once-in-a-lifetime ’shock’. Investment was more an act of faith than a scientific calculation of probabilities.” (The New York Times Sunday Magazine 12.14.08)

We couldn’t agree more – regardless of timeframe. And neuroeconomist research (Check out Hsu or our French PhD chick’s posts) is even providing pictures of our brains acting on this faith versus “scientific calculation of probabilities”.

Not that us traders shouldn’t attempt to create market strategies and tactics that divine the probabilities but when we do so, we will be better served to remember that in effect we are using a crude tool – no matter how precise it may appear. Any effort to predict where the market will be is indeed a prediction on the future – be it in 10 minutes or 10 years.

Furthermore, our brains seem to be wired to rely on feelings when they detect inherent uncertainty – or imprecise probabilities. So… all the more reason we need to understand that we are working with only an approximation no matter how complex our algorithm, chart or trading tactics are.

In short, anything can (and clearly does) happen at any time. We need our BRAINS – which happen to be really good at unconscious pattern recognition and dealing with imprecision to kick in – not just the sophisticated overlay we have concocted.

After all we are betting on other brains to pay more – or sell for less. It can never hurt to know what game we are really playing.

Can it?

Portfolio selection: Let’s exhume the buried man!

In his milestone paper “Portfolio Selection” published in the Journal of Finance in 1952, Harry Markowitz, the pioneer of “modern finance,” recommends to use the Expected return-Variance (E-V) rule, both as a working hypothesis to explain investment behavior and as a guide to “investment” – as distinguished from speculative behavior. This rule implies that an actor who considers yield to be good, risk to be bad, and speculation to be banned, should diversify in such a way that his portfolio lies in the “efficient frontier.” The idea is very simple. When building your portfolio, combine the securities in such a way that for a given expected return (E) of your portfolio, its variance (V) is minimum. This defines all the efficient combinations (E,V). Then, according to your degree of risk aversion, pick one such combination – a risk averse person will prefer a low E – low V portfolio, whereas a risk lover will choose a high E-high V combination. The beauty of this rule lies in its apparent readiness. Given a probability distribution of yields of the various securities, computing the set of efficient (E,V) combinations is straightforward. But this is misleading, because it dodges the main issue: Where do the expected returns and variances estimates come from? In other terms, how do we set our probability beliefs?

At the end of the paper, Markowitz himself recognizes that he has been silent about the origin of these beliefs throughout:

“To use the E-V rule in the selection of securities, we must have procedures for finding reasonable expected returns and variances [...] I will not pursue the subject here, for this is “another story.” It is a story of which I have read only the first page of the first chapter. In this paper we have considered the second stage in the process of selecting a portfolio. This stage starts with the relevant beliefs about the securities involved. We have not considered the first stage: the formation of the relevant beliefs on the basis of information.”

Dodging the question of belief formation is murderous

It seems that fifty years from this paper, we are still stuck in “the first page of the first chapter.” Quite ironically, after Markowitz’s paper, such E-V rule has become ubiquitous in real-world finance, despite this inherent indeterminacy. Why does such belief indeterminacy matter? After all, is it so complicated to use a mix of statistical analysis and practical judgment, so that we derive sensible probability beliefs? In the first instance, one can use observed yields and volatilities from the past, to get statistical estimates of the true yields and variances. The problem is elsewhere. By emphasizing belief formation related to the expected returns and variances, we are missing the key point. Belief formation about world uncertainty is the issue, which is totally buried by Markowitz here. Markowitz implicitly assumes we live in a Gaussian world of “mild uncertainty,” where price changes are characterized by stability around the average. In such a world, Markowitz’s “E” and “V” are relevant objects. But what if randomness is not mild at all in our world? What if there is not such a thing as “value,” and returns have infinite variance? In such a world, sensible people do not average but rather arbitrage between times and places. If so, by putting forward “E” and “V,” maybe Markowitz leads us to wrongly interpret the world we live in.

Returns uncertainty, viewed from the empirical side

And there is good evidence that it is the case. If returns were Gaussian, we would have 68% of small price changes within one standard deviation of the mean, 95% within two standard deviations of the mean; and outliers (large changes) would be extremely rare: according to the Gaussian model, index swings of more than 7% should come once every 300,000 years… Rather, empirically, we observe too many large and too many small changes in the prices (this is what we call “fat tails”). Moreover, we observe irregular trends of large changes followed by clusters of small changes. That is, trouble runs in streaks (a wild day might be followed by a wilder day). This means that “persistence” is far larger than expected, would the world be Gaussian. As such – at least for investing horizons from two hours to six months, the Gaussian hypothesis is the wrong interpretation.

Uncertainty in the finance world: Where do we stand?

What is the correct model of uncertainty in finance then? There is no definite answer to that paramount question. Actually, two distinct routes are possible.

Nonstationarity

The first one assumes nonstationarity. For instance, the widespread use of GARCH models is to capture the foregoing phenomenon of persistence, while staying within the Gaussian boundaries. The idea is to introduce changes in volatility – that is, instead of considering one single Gaussian distribution for the returns, consider multiple ones, each characterized by its own level of volatility. When real world volatility soars (resp settles), make the Gaussian curve grow (resp shrink). This indeed enables to fit the data. Poissonian uncertainty has also been suggested as a model to replace the Gaussian (Brownian) model. Gaussian risk involves a high probability of a small change, while Poissonian risk involves a small probability of a large change (jump).

The multifractal model

The second route is the one taken by Beno?t Mandelbrot, the father of fractal geometry. It avoids to assume nonstationarity. Rather, he proposes real world randomness to be best described as “slow.” The Gaussian model entails a randomness that is too mild. Conversely, “wild randomness” is characterized by an extreme degree of unpredictability: tails are huge, “everything can happen,” whereby there is no way to forecast the returns (both expectation and variance of price changes are infinite).

Slow randomness is in between: there is no stability around the average, and tails are fatter than in the Gaussian world. Mandelbrot’s model of real world uncertainty is remarkably elegant – and I’m not (only) saying that because Mandelbrot is French. Not only his framework generates the fat tails and persistence phenomena observed in real data, it also suggests using quantitative tools to rigorously measure (1) how fat the tails are, and (2) the degree of persistence in the returns.

Alpha and H

Two parameters summarize these two dimensions, alpha for the size of the returns, and H for their sequences (dependence). The first parameter comes from modeling the tails with a Power Law. For x large (we are at the tails), It sets the ratio of probability of a return larger than n x over the probability of a return larger than x to be n^{-alpha}. Intuitively, the smaller alpha, the fatter the tails (i.e, the larger the instability: the realization of an outlier moves the average).

With a Gaussian distribution, alpha equals 2; under wild randomness, it is 1. Alpha in between points to a world of slow randomness, the one we presumably live in many instances. The second parameter is very intuitive too. It says something about the sequences of the price changes (runs) rather than their size. The question is how much the past shapes. In a Gaussian world, over a given period (say, 10 years), the range between the highest return and the lowest one is sqrt(5) times the empirical standard deviation of the returns from one year to the next. However, when randomness is characterized by persistence, the high-to-low range widens not by a square-root law but as a H power, with H larger than 1/2. This means long term dependence, and captures the fact that turbulence clusters.

Blueprint: Assessing real uncertainty is like constructing “dikes”

No model is universal. Plainly, interpreting properly the nature of the randomness we face in our investment decisions is context-specific. So, how do we choose between the models? We don’t have to be blind about uncertainty. Once we no longer take for granted the Gaussian assumption, all we have to do is appraising properly the world we invest in. That is, we can track — and perhaps forecast – how turbulent the market is becoming, using fractal geometry and the alpha and H measures.

In “The Prince,” N Machiavelli compares “Fortune” to a violent river, and suggests ours constructing dikes to protect ourselves. What does it means for finance to construct such dikes?

Just that we need to assess alpha and H.

And in practice?

If alpha is smaller than 2, then we should definitely think again about the E-V rule. (Same thing if H is different from 1/2.) If so, we should ignore “E” and “V” and merely focus on both alpha and H, with which to evaluate true risk.

Sadly, things are not that simple, because measuring? alpha and H is in practice very difficult.
In 1991 Andrew Lo reported that Mandelbrot’s tests for H can confound long-term memory with the effects of short-term memory. And the measures of H are not robust: there is no consensus as for the S&P 500 index for example: the estimates for H vary from 0.53 to 0.74. Furthermore, the degree of dependence varies a lot from one type of financial asset to another (gold prices, oil markets, foreign exchanges, might have long memory, whereas cotton, British government bonds, do not). So, overall, it is unrealistic to hail alpha and H as new yardsticks for finance.

We’d better acknowledge our ignorance: Doubt is good

To be honest, it is impossible to claim with certainty that one model is the correct one at a specific moment in time. Accounting for model uncertainty is the hallmark of modern econometrics and (truly) modern finance. Is it a retreat into – at least lucid – blindness? I don’t think so.

The only relevant question is referred to by Nassim Taleb as “tinkering:” How to make sound investment choices in a world we don’t understand? Modern finance tackles this problem, by putting forward rules of behavior under “ambiguity.” By “ambiguity,” we academics precisely refer to these situations of missing information about “the rules” of financial investment, the one we model as a big game of betting under unknown odds (more on this here).

F.Nietzsche wrote in his “Ecce Homo:”

“No doubt, certainty is what drives one insane.”

I think “modern finance” drove us insane because it has forgotten to doubt.