# 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.