Dreamtime Finance

In 1956, John Kelly introduced his “Kelly criteria” betting strategy: bet on each possible outcome in proportion to (your estimate of) that outcome’s chances of winning, regardless of the betting odds offered. More generally, a Kelly rule invests in each possible asset in proportion to its expected future payout, regardless of current asset prices. For example, if you estimate land will be worth 30% of world wealth in the distant future, you put 30% of your investments into land today, regardless of today’s land prices.

It turns out that the Kelly rule is close to the optimal long run investment plan, i.e., the one that would win an evolutionary competition. The exact best strategy would consider current prices and expected future price trajectories and carefully choose investments to max expected growth, i.e., the expected log of a distant future portfolio. But Kelly’s rule is far simpler, gets better than average growth regardless of state, time, or prices, and approaches the exact best strategy as good strategies come to dominate prices. In fact:

A stock market is evolutionary stable if and only if stocks are [price] evaluated by [Kelly rule] expected relative dividends. Any other market can be invaded in the sense that there is a portfolio rule that, when introduced on the market with arbitrarily small initial wealth, increases its market share at the incumbent’s expense. (more)

(More on evolutionary finance here, here, here, here; see especially this review.) We’ve had big financial markets for at least a century. Has that been long enough for near-optimal strategies to dominate? Not remotely. John Cochrane explains just how bad things are:

We thought returns were uncorrelated over time, so variation in price-dividend ratios was due to variation in expected cash flows. Now it seems all price-dividend variation corresponds to discount-rate variation. We thought that the cross-section of expected returns came from the CAPM. Now we have a zoo of new factors. … For stocks, bonds, credit spreads, foreign exchange, sovereign debt and houses, a yield or valuation ratio translates one-for-one to expected excess returns, and does not forecast the cash flow or price change we may have expected. In each case our view of the facts have changed 100% since the 1970s. …

All of these facts and theories are really about discount rates … and risk premiums. None are fundamentally about slow or imperfect diffusion of cash-flow information, i.e. informational “inefficiency.” Informational efficiency isn’t wrong or disproved. Efficiency basically won, and we moved on. When we see information, it is quickly incorporated in asset prices. … Informational efficiency is much easier for markets and models to obtain than wide risk sharing or desegmentation, which is perhaps why it holds more broadly. (more)

Got that? Finance prices today do a great job of aggregating info – relative prices between similar assets are great predictors of relative payouts. But when it comes to broad price aggregates, such as stocks in general or land in general, price changes basically reflect crazily-changing values. While in markets dominated by near-optimal traders, prices would only change when expected future payouts changed, in fact aggregate prices changes have almost no relation to matching future payouts changes. For example, land prices change plenty (as in the recent real estate bubble), but aggregate land price changes say almost nothing about future land rents.

I’ve talked before about how our era is a rare extreme “dreamtime,” with fast change and behavior quite out of equilibrium with evolutionary selection pressures. We not only have dreamtime fertility, i.e., far fewer kids per couple than selection would favor, we also have crazy-price dreamtime finance. This allows a relatively clear prediction of the future: finance will eventually “equilibrate.” Either the world will coordinate to block the creation of investment funds following near Kelly rules that reinvest most gains, or financial prices will eventually come to be dominated by such near-Kelly funds.

Once dominated by near-Kelly funds, finance prices will no longer suffer huge crazy booms and busts, like the recent dotcom boom or real-estate crash. Furthermore, interest rates should fall dramatically — future returns will no longer be discounted intrinsically, but only for opportunity cost reasons.

Apparently many funds today do now follow near Kelly rules:

The claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. (more)

So the main barrier seems to be fund ability and inclination to reinvest most gains. As I wrote a year ago:

Many folks would be willing to create trusts that accumulated funds long after their death and then paid distant descendants (perhaps indirectly) to do things like remember their ancestor’s name, pray to his gods, etc. Unless stolen, such funds would eventually come to dominate the world economy and dramatically lower interest rates. With lower interest rates … businesses and governments would have far stronger incentives to attend to the interests of distant future folks, such as via global warming policies. But we in fact refuse to enforce a great many such long term deals. (more)

In a large decentralized world, however, I doubt this barrier will stand. Nor can I see why it should. I for one welcome our new financial overlords. Seriously.

I wonder if anyone could estimate how long it should take Buffett/Gross size Kelly funds to dominate finance prices. More Kelly rule details from that review:

The most striking observation is that the investment strategy λ is given by the (conditional) expected value of the relative asset payoffs. This recipe is similar to the Kelly principle of “betting your beliefs” … Only the (objective) probabilities and the relative payoffs are needed in the calculation of λ. …

The [Kelly rule] locally evolutionary stable investment strategy … yields a superior growth rate at its own prices, and it is the only strategy with this property. The result holds in complete as well as in incomplete asset markets, which is remarkable given that a simple analysis using the overtaking criterium does not apply in the latter case. In general however this rule will not maximize the one-period logarithmic growth rate because away from a steady state the composition of the market matters. The wealth distribution and the particular strategies employed by all investors impact the price and thus the log-optimum investment. …

Kelly rule λ can be linked to utility maximization. Indeed there is a strong connection to logarithmic utility functions in a competitive equilibrium. Suppose prices are given by λ and an investor maximizes log utility given these prices (such as in a competitive equilibrium). Then his optimal strategy is λ. …

The only locally evolutionary stable investment strategy is the Kelly rule. A market in which a Kelly investor is the incumbent, relative asset prices are given by their fundamental value in terms of their relative payoffs. The ro-bustness of this market against any stationary mutant strategy implies that deviations from the fundamental relative valuation are corrected over time. …

If all investors are constrained by being required to choose constant investment strategies, there is exactly one strategy that will do best in the long term. It is the rule that divides an investor’s wealth in proportions given by the expected relative dividends. … This [Kelly rule] investment strategy does not match the growth optimal portfolio in general. The former is constant while the latter would depend on the price process and, thus, vary over time. The important exception is the case in which asset prices are constant and equal [Kelly fractions]. Then the [Kelly rule] investment strategy maximizes the expected logarithmic growth rate … the [Kelly rule] investor’s relative wealth will, on average, grow: the investor’s logarithmic growth rate is strictly positive if the current asset prices do not match [expected dividends]. A positive growth rate can be interpreted as experiencing faster growth than the ‘average investor.’ …

The price dynamics induced in a pool of constant investment strategies (and i.i.d. dividend payoffs) favors a λ investor for every distribution of wealth shares. The above-average expected growth of the λ investor’s wealth holds in every period in time and for every current price system. … Identifying assets that are underpriced resp. overpriced relative to the λ benchmark, one could construct a self-financing portfolio by going long resp. short in these assets. This should potentially boost the growth rate, but, on the other hand, increases the risk.

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  • Talosaga

    I think something’s missing from your argument. Just because the Kelly rule is an evolutionary stable strategy doesn’t mean the stockmarket will drift towards using it. It may be that in this dreamtime stock market environment, the kelly rule preforms badly. Thus we can’t expect the market to eventually reach its evolutionary stable strategy. Instead the market equalibrium may just travel around dreamtime forever.

    What’s your argument that the kelly rule is profitable in a dreatime environment?

    Or are you saying that prices will jump around like crazy and eventually this random process will land on the “kelly spot”, at which point the kelly rule will become most profitable and the stockmarket will be locked into its evolutionary stable strategy?

  • More generally, a Kelly rule invests in each possible asset in proportion to its expected future payout, regardless of current asset prices.

    That’s not true. Kelly most certainly *does* take prices into account. If a certain investment has negative expected value (i.e. price is higher than your estimate of its fair value) the optimal stake (according to both Kelly and common sense) is exactly $0. If, continuing your example, the land representing 30% of the world’s future wealth is selling for $100*X and the other 70% of assets combined sell for $700*X, the land is significantly undervalued and the optimal allocation (again according to both Kelly and common sense) will be heavily concentrated in land.

    Not sure how this relates to your larger evolutionary finance argument, but thought it needed to be clarified.

  • I would be curious to see any market data on the ratio of “smart” or institutional money vs individual investors over time. Also note that the kelly criterion isn’t a sufficient condition for accumulation of wealth.. you also need a sustainable edge in the markets, which are presumably getting ever more efficient and thus more difficult to beat.

  • The thing I don’t understand about the Kelly Rule argument, is how long the long term has to be and how good one’s estimate has to be for the argument to carry. It seems like there have to be some (“ignorant”) estimates for which holding a market neutral portfolio would be an improvement, and likewise markets that are out of balance for a long enough lifetime that following a Kelly Rule, while a theoretical win, in the short run of an actual lifetime would often lead to a gambler’s ruin.

    Even cryonicists, with a relatively long outlook, mostly don’t have a very high estimate of the actual chances of cryonics’ success, so we should be putting most of the weight of our expectation on a relatively normal lifetime. Does the Kelly Rule argument strongly show that every person’s best bet, even with a 40-, 50, or 60- year outlook, is to project out to their extremely long-run expectations of the actual values of alternative opportunities?

  • Chip Morningstar

    The Kelly criterion sounds a like a principle that is both true and useless.

    All this is predicated on investors’ estimates of the probabilities of winning on each bet bearing some resemblance to reality (at least on evolutionary time scales). But if you had any kind of effective strategy for reasonably making such estimates, you would have already solved the problem. I suspect the majority of people who made bad bets during the real estate bubble, for example, lost out not because they had a bad rule for aligning their investment strategy with their expectations, but because their expectations were invalid.

    Also, I believe it is very rare for most people to be able access their own expectations in quantifiable form, which, if I understand it correctly, the Kelly rule requires.

    This whole argument puts me in mind of the Austrian critique of socialism in the 1930s: the socialists said “we’ll just put all the data into this model and calculate what to do”, to which the Austrians said, “yes, but you can’t actually get the data”. It seems to me that the Kelly rule requires knowledge that is not generally available.

    • The book Fortune’s Formula by William Poundstone contains many examples of successful real-world applications of the Kelly principle. (btw, if you are *that* CM, I have my students read your “Lessons from Habitat” paper, nice to meet you.)

  • I deduced the the Kelly criterion before hearing about it, and I still have problems remembering or understanding it. It seems to be one of those concepts that is slippery to the mind, like minimax with a-b pruning, or Gödels incompleteness proof.

  • Robin, Wikipedia says that Kelly criterion is equivalent to maximizing expected log returns, which reminds me that we never resolved our debate about whether evolution “selects for” creatures that maximize expected number of descendants, or creatures that maximize expected log descendants. In particular I don’t think you responded to the point I made in this comment, beyond saying “that seems worth pondering”.

    To restate that point, creatures that maximize expected number of descendants will end up with a higher number of descendants averaged over all possible worlds, but their descendants are concentrated into a few possible worlds with very large total populations, whereas creatures that maximize log descendants end up dominating the population in most possible worlds.

    Similarly, when you say Kelly rule “would win an evolutionary competition” you mean that portfolios following the Kelly rule would dominate the market in most possible worlds. But portfolios that maximize expected return instead of expected log return would dominate in the few possible worlds that turn out to be extremely rich.

    Do you see any reason why we should be more concerned about “dominating in most possible worlds” instead of “taking over the few possible worlds that are really worth taking over”? It seems that you are more motivated by the former, but perhaps someone with a different psychological makeup could claim the latter as a legitimate preference?

    • Dániel Varga

      Wei Dai: At the other end of this spectrum there is quantum lottery. Would you say that quantum lottery is also something that someone with a different psychological makeup could claim as a legitimate strategy?

  • Talosaga, it “gets better than average growth regardless of state, time, or prices.”

    Matthew, I think you misunderstand the rule, and its implications.
    Chris, Kelly rule can’t go bankrupt, and Cochrane reviewed how prices now are way ignorant.

    Chip, Cohcrane reviewed how very simple stat track records give plenty good enough expectations for the Kelly rule to have better than average growth.

    Wei, Kelly is not *equivalent* to log returns, though it can be close. Yes creatures that consistently max expect distant future returns will have the most expected returns, but we have little reason to expect to see the selection of such creatures.

    • Robin, the self indication assumption (which you previously said you “embraced”) says that we should reason as if we are randomly selected from all creatures across all possible worlds (instead of first sampling a possible world, then sampling a creature within the chosen world). I pointed out in the linked comment that under such a sampling scheme you would expect the creature selected to be the descendent of an expected-descendants maximizer instead of an expected-log-descendants maximizer.

    • Wei this post is about financial prices, not the number of future descendants. Selection can reward the Kelly rule of investment even with zero effect of prices on the actual assets returns. So all these possible financial states could have exactly the same number of descendants.

      • I understand the point you’re making in this post, and was making a connection to the previous post. Someone who wants to maximize expected number of descendants would not want to apply the Kelly rule, despite it being “optimal” in some sense. It seems worth pointing this out.

  • Eric Falkenstein

    Empirically, fund inflows are convexly related to relative returns (mutual and hedge). Given this, I think there’s an incentive to greater volatility than the Kelly Criterion in asset markets.

  • Hul-Gil

    This article is extremely interesting. I’ve been a longtime lurker at LessWrong, but have not ventured over here; now I see what I’ve been missing.

    If, however, someone could simplify this for me… I’m about to invest in the stock market, as I have come into some money ($5,000 – I know, very little for most people). Should I be attempting to follow this Kelly Rule, or is my original strategy of investing in an index fund like the S&P 500 a better bet?

    • > Should I be attempting to follow this Kelly Rule, or is my original strategy of investing in an index fund like the S&P 500 a better bet?

      With all due respect, reading this post and the comments should have given you at least 3-5 arguments why you would not want to follow it for your investing. If you can’t think of any, then you probably shouldn’t do it in the first place because you don’t actually understand it.

      • Hul-Gil

        It seems to me that the post is advocating it, but some people in the comments disagree with this assessment. Wei Da appears to be talking about something else entirely; only Chip seems to offer an actual criticism. Can you explain one or two of these 3-5 arguments to me?

      • Sure.

        1. Kelly requires you to be committed to it as a strategy, you can’t really ‘time’ it or anything. (If it was optimal at the start, it’s optimal now even if you lost.) You may not have the emotional fortitude for it. (I learned this in practice betting Kelly on Intrade: http://www.gwern.net/Prediction%20markets#my-intrade-trading) This is a common criticism of Kelly in practice.
        2. Kelly requires you to be committed for a long time. In particular, as pointed out here already I think, you need to be able to think in centuries or millennia. With Kelly, as with many optimal strategies or results, ‘in the long run we are all dead’. How soon do you need that money…?
        3. You need an edge of some kind, otherwise Kelly has no way to maximize your growth rate. As decades of research goes to show, most people trade too much (I think Hanson has linked in the past one paper on how overtrading damages returns and a gender correlation with overtrading) and are overconfident and cannot beat the markets, which are pretty efficient. The smaller your edge, obviously, the less Kelly can grow your investment.
        4. You need to estimate your edge correctly! If you think your edge is better than it is, that’s as bad as thinking your edge is worse than it is.

        Hopefully that’s enough to explain why you probably don’t want to use this particular strategy. Leave it to the Buffets and Swiss Res of the world.

      • Hul-Gil

        Thanks – that all makes sense. #2 wouldn’t be a problem, apparently, if I did have such an edge… but since I’m not even entirely sure what an edge is in this context (inside information? some sort of predictive ability? gobs of cash?), I probably don’t!

        I’m going to go off on two tangents that always puzzled me, because you seem knowledgeable and your mention of Buffet reminded me. Feel free to ignore them if you don’t have the time or inclination, though; no worries.

        1.) Are economists rich? Shouldn’t they be?

        They’re not the highest-paid by salary, I know. But I always assumed that the knowledge required – about value, markets, investing, and the like; basically, “the science of money”(?) – would translate into turning small piles of money into big piles of money better than any other profession. I’m not quite certain this is so, though.

        I’ve been meaning to go through and look at what the profession ends up making people the richest on average (and of course discounting things like inheritance)… if you know any insight on this, too, please let me know!

        2.) Why is Buffet so good at investing? Does anyone know, beyond assumption of some sort of Kelly rule?

        That is – it seems his “edge” is abnormally large, even accounting for obvious stuff like time, dedication, and education. I’ve read that his performance in the stock market is wildly, improbably good. I’ve also read that attempting to “beat the market” with any sort of investing strategy will always fail (except for lucky breaks); yet it seems he is clearly doing it. Or did it at one time, anyway. This might just be a misinterpretation of his career, though.

      • > 1.) Are economists rich? Shouldn’t they be?

        I think they tend to be pretty well paid and pretty well off, more so than most other professions (all that is needed). And to some extent, being an economist is about learning how you *can’t* just become rich with a little extra knowledge (no free lunches, efficient markets, that sort of thing).

        As for Buffet, I have no idea.

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