Thursday, July 4, 2013

Thinking again about time

I have a column coming out in Bloomberg in the next few days. The topic is time -- literally -- and how to think about it. For people reading the column, I thought I'd offer some links to things I've written here before exploring the topic in more detail, and also to some original papers that explore the ideas in much greater depth.

Much of the recent research on this topic has been done by Ole Peters of the London Mathematical Laboratory and Santa Fe Institute. The gist of his overall argument is that the usual ensemble averages used to compute "expected" returns in finance and economics are, in many cases, simply wrong. This is not the correct way to make decisions in the real world. We're all used to the idea of averaging over possible outcomes, weighted by the appropriate probabilities, and asking: is the result positive or negative? If positive, then the gamble or investment is worth it, if we can accept the risk. But this is not a legitimate way of thinking, for it averages over parallel worlds, not through time -- where we actually live.

The trouble is that usual average over different outcomes mixes potential worlds in which we go broke with others in we get rich, and does this mixing all at once, immediately, so that good outcomes coming from one path cancel out bad outcomes coming from other paths. Importantly, this mixing takes the often irreversible consequences of bad outcomes (bankruptcy, for example) out of the picture. If you make hugely risky investments, this average gives you full credit for all the wonderful possible outcomes, weighted appropriately for their likelihood, which of course seems sensible. What it doesn't do is account for the very real fact that the bad outcomes may effectively wipe you out entirely and take you out of the game, making it impossible to play again -- in which case you will never get to experience those eventual big payoffs.

I've written three earlier blog posts -- here, here and here -- exploring his idea from different angles. These give links to Peters original papers and also to some other valuable discussions of this fundamental way of thinking. One further paper of interest is this recent work by Peters and William Klein which looks at how the assumption of ergodicity is systematically violated by the process of geometric Brownian motion, which is of course a workhorse model in finance and economics.

I'd like to just quote one paragraph from the latter paper to make the point of how counterintuitive this stuff is. We're all very used to thinking in terms of ensemble averages. Average over all the outcomes, weighted by the appropriate probabilities, and this should give you a good handle on whether the gamble is worth taking or not. But this view is totally mistaken. Indeed, as Peters and Klein point out,
In Geometric Brownian Motion, it is possible for the ensemble average to grow exponentially, while any individual trajectory decays exponentially on su fficiently long time scales [1]. Multiplicative growth is manifestly non-ergodic. But precisely the opposite is often assumed in economics, for instance in [2], p.98: "If a gamble is `favorable' from the point of view of the expectation value [ensemble average] and you have the choice of repeating it many times [time average], then it is wise to do so. For eventually, your amount of money [is] bound to increase."

That first sentence really sums up the problem. You can do the ensemble average and say, "Hey, I expect to get exponential growth! Let's go for it." Then you actually play and find you get wiped out. Try again and still get wiped out. See that everyone who plays the same game also gets wiped out. Strange. You may eventually accept that thinking in terms of parallel worlds isn't quite the right thing to do.

Of course, Peters analyses are also closely linked to the famous Kelly criterion introduced by IBM mathematician John Larry Kelly in 1956. The theory of practical betting based on this criterion has been extensively developed by Ed Thorpe and many others. See this summary paper, for example. There is a huge literature on this, and I don't in any way want to imply that people in finance have not been thinking about this. Many have. However, the failures of the expected value approach through ensembles is not appreciated widely enough.

It goes without saying, of course, that everything becomes more uncertain and complicated in the usual case from real life -- especially in finance -- where one does not know that actual probability distribution from which outcomes are being drawn. Perhaps there is no such distribution. In this cases, Thorpe and colleagues rightly counsel even greater caution:
The theory and practical application of the Kelly criterion is straightforward when the underlying probability distributions are fairly accurately known. However, in investment
applications this is usually not the case. Realized future equity returns may be very different from what one would expect using estimates based on historical returns. Consequentlypractitioners who wish to protect capital above all, sharply reduce risk as their drawdown increases


  1. Conscience of a ConservativeJuly 5, 2013 at 1:51 AM

    Felix Mandelbrot has also written about finance and Brownian motion. He disputes whether Brownian motion applies. Considering the multiple crises(100 year events)we seem to have every few years, I tend to agree.

    1. There will always be crises, and that's part of market correction and also human error. During crises, asset prices should inflate or deflate. I actually believe that Brownian motion is accurate in conceptualizing the behavioral component in asset pricing.

  2. "Of course, Peters analyses are also closely linked to the famous Kelly criterion introduced by IBM mathematician John Larry Kelly in 1956. "

    Confusing Bell Labs and IBM?,_Jr

    You have been told before why Peters's work does not make any sense and it is peculiar why you continue pushing his ideas. Nobody ever looks just at the expectation to place bets. Only Peters odes that. This is a straw man argument. People also consider the probability of the outcomes and many other parameters.

  3. you're exactly right... should have been Bell Labs. Sorry for that. But I do find it hard to accept your claim that no one uses expected utility theory as a basis to try to understand and guide decisions. It is the standard view. And it does not attempt a straightforward analysis of processes through time. Why does Peters work "not make any sense." ???

    1. "But I do find it hard to accept your claim that no one uses expected utility theory as a basis to try to understand and guide decisions."

      I said the opposite exactly. No one looks only at expectation. I think you may be a little confused about these matters and so is Peters or maybe Peters confused you?

      See ivansml's post below.

  4. From page 2 of Peters & Klein: "i.e. the logarithm is taken outside the average. This is crucial to leave the non-ergodic properties of the process, x(t), intact."

    In other words, authors have pretty much assumed their result from the beginning and rediscovered Jensen's inequality log(E[X]) > E[log(X)] in the process. At best, all this brings attention to the fact that with multiplicative growth, arithmetic average of gross growth rates can be greater than one, but their geometric average can be less than one. I fail to see how this trivial result relates to ergodicity as usually defined, much less how it could show "wide-spread conceptual inconsistency" in economics.

    The paper contains some further results which are perhaps worth publishing by themselves, but as a whole it's a confused mess, and I'm surprised it made its way into PRL, supposedly the top physics journal. More generally, the whole premise of Peters' research is that 1) standard economic theory is only about maximizing expected return - which is false, as anon above already pointed out, and 2) maximizing long-run geometric growth rate of a portfolio is the only correct criterion - but this is never proved, just simply assumed (which makes the whole argument rather circular). Waxing poetic about irreversibility of time may sound cool in a TED talk, but it's hard to see how it has anything to do with equations in his papers.

    1. Thanks Ivan for your comments as always. In this case I don't agree.

      First, I don't think it is fair to say that Peter's is claiming that "standard economic theory is only about maximizing expected return". He's made it pretty clear that others have considered these issues, and that the use of logarithmic utility gives the same result as the time-focussed perspective, only doesn't give a fully natural (in his view) justification for this. Here's a paragraph from a paper of his ( from a couple years ago (which may, I admit, make this point more clearly than the recent paper):
      "In conclusion, utility functions were introduced in the early 18th century to solve a problem that arose from using ensemble averages where time averages seem more appropriate. Much of the subsequently developed economic formalism is limited by a similar use of ensemble averages and often overlooks the general problem that time- and ensemble averages need not be identical. This issue was treated in detail only in the 20th century in the fi eld of ergodic theory. Making use of this work, a privileged portfolio uniquely speci fied by an optimal leverage and a maximized time-average growth rate is seen to exist along the e fficient frontier, the advantages of which have also been discussed elsewhere (Breiman 1961, Merton and Samuelson 1974, Cover and Thomas 1991).
      The concept of many universes is a useful tool to understand the limited signifi cance of ensemble averages. While modern portfolio theory does not preclude the use of, in its nomenclature,
      logarithmic utility, it seems to underemphasize its fundamental signi ficance. It was pointed out
      here that the default choice to optimize the time-average growth rate is physically motivated by the passage of time and the non-ergodic nature of the multiplicative process."

      On the other point, I don't at all see how Peter's has assumed his result at the outset. He simply sets up a calculation to estimate an ensemble average, which should be done by averaging over N different realizations of the process, and then taking the logarithm to get a growth rate for the ensemble average. He shows that this isn't the same as the time average growth rate (and notes that Kelly was first to point out this long ago). He's also very clear that pushing the logarithm inside the ensemble average leads to Bernoulli's result by effectively introducing logarithmic utility. None of this seems illegitimate to me -- and certainly does pertain to ergodicity as I understand it, which rests on the equivalence of time and ensemble averages.

    2. "It was pointed out here that the default choice to optimize the time-average growth rate is physically motivated by the passage of time and the non-ergodic nature of the multiplicative process."

      And this is what I have problems with. Why is maximizing log of wealth justified by "passage of time", but maximizing, say, square root of wealth is not? If there is some deeper argument, it should be made explicit, but I don't think there is.

      Regarding ergodicity, I've tried to understand Peters' point, but I'm convinced he's wrong. Yes, ergodicity is about time and ensemble average being equal - but those averages must be computed from the same expression. Taking log out of the expectation effectively means you're computing averages for two different things, mixing apples with oranges.

      Consider a counterexample - model where price in each period (time is discrete, t=1,2,...) has lognormal distribution, i.e. X(t) = exp(e(t)), and e(t) ~ N(mu, sigma^2), and e(t) is independent across time (so there is no growth, price just fluctuates completely randomly). Now if you computed or simulated ensemble and time average in the same way as defined in the paper (equations 4-5), you'd also get different values (if my math is correct, the first would be sigma^2, while the second zero). But X(t) is just a transformation of white noise, so it's stationary and ergodic according to any usual definition.

      By the way, you may find this criticism by Tim Johnson also relevant:

    3. "And this is what I have problems with. Why is maximizing log of wealth justified by "passage of time", but maximizing, say, square root of wealth is not? If there is some deeper argument, it should be made explicit, but I don't think there is."

      Because maximizing the expected log of wealth will, in the long run, after repeated "gambles" and assuming perfect knowledge of probabilities of potential outcomes, maximize actual wealth. This is true regardless of your utility function. You want to maximize the geometric mean, because you will make repeated gambles/investment decisions that compound over time.

      One property of using the expected log of wealth is that having $0 remaining in even the most unlikely outcome is infinitely bad, so it prevents going "all in" on any one gamble (or even risking 100% of savings on many diversified simultaneous gambles). Living to fight another day is built in.

      Maximizing the expected square root (or any other measure) of wealth does not do the same.

    4. Dave:

      It may maximize wealth in a sufficiently long horizon, but that's not the same as maximizing utility, as Samuelson showed. And if I'm for example investing in a pension plan, I care about distribution of my assets at specific date when I retire, not some asymptotic growth rate.

      Any utility function that goes to minus infinity at zero (of which there are infinitely many) will also prevent "all in" gambles.

  5. it's amazing. what you are talking about is the same idea that Fischer Black had in 1994 when he decided not to be a principal in LTCM because "time tends to cost more than it is worth in a way that can be exploited by markets"

  6. I have to think the results of these fancy formulas are akin to Russian roulette without a bullet. Test these formulas on Russian roulette with a live bullet if you want reality.

    Stephen P.

  7. I wonder. A portfolio performance presentation (following GIPS guidance) utilized time average method to explain the past. To get a general understanding of the portfolio future risk(s), most ex ante portfolio risk analysis utilize ensemble average method. Is there a disconnection here?


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