There is a branche of stochastic portfolio theory (see also this question). Fernholz and Karatzas have published research in this field (e.g. "Diversity and relative arbitrage in equity markets") and just recently I stumbled upon this new paper.

It seems that one of the main (theoretical) findings is that one can construct a portfolio that outperforms the market (relative arbitrage).

Is there applied work published about this? Does any (besides I think Fernholz) manager apply this theory?

EDIT: today a preprint was published by Philip Ernst, James Thompson, Yinsen Miao where they show that weights proportional to transformations of the markte cap weigth (e.g. $1/x^2$, where $x$ is the market cap weight) deliver portfolios that outperform the market. Is this an example? Isn't this was SPT is about?

EDIT: no input for this question with bounty?

  • 2
    $\begingroup$ I find SPT interesting, but I don't know enough to really provide any sort of comment. I looked over that paper you cited in the edit. It strikes me that the strategy is basically a way to overweight the smaller stocks. It doesn't mention any real-world difficulties with this strategy (increased turnover and liquidity issues with smaller cap stocks). It also doesn't reference Fama-French and any of the small cap literature. $\endgroup$
    – John
    Commented Mar 25, 2016 at 18:45
  • $\begingroup$ @John You are right about the paper mentioned above .. I think they lack a lot of important info (vola, draw down, Sharpe ratio, rolling performance, TE, ...) that I would be interested in ... and all the SC and turnover issues are missing too. $\endgroup$
    – Richi Wa
    Commented Mar 26, 2016 at 7:34
  • 1
    $\begingroup$ @Richard It is quite an exciting theory. I listened to Karatzas give a talk in April on the frontiers of this work, but it was entirely theoretical. In my opinion people are still trying to digest it all, but talking with others in industry it seems many are only vaguely aware of it. In my experience, as I'm sure yours, even if PMs try this technique you won't hear a peep about it without being in the loop, especially if it is working :) $\endgroup$
    – bcf
    Commented Aug 7, 2016 at 14:59
  • 1
    $\begingroup$ @Richard, I was drawn to the comment in the abstract of "Tukey’s Transformational Ladder for Portfolio Management": "we take care to differentiate it from the well-known "small-firm effect." My initial impression is extrapolating Tukey's ladder to find weights simply exploits the size effect. It sounds like optimization. Moreover, the paper assumes no transaction fees, slippage, or liquidity effects. I am fairly sure that if anyone were to model these effects, the expected arbitrage would all but disappear. I could be wrong, though. $\endgroup$ Commented Mar 8, 2017 at 0:36
  • $\begingroup$ @DavidAddison my impression about a lot of the literature is that few of them have these add-ons. So maybe a cross comparison on performance with other literature makes sense. Also the transaction fees highly depends on the firm, and the slippage depends on the execution strategy and order size, so does the liquidity effect ... $\endgroup$
    – Will Gu
    Commented Mar 10, 2017 at 19:42

2 Answers 2


SPT refines MPT by introducing the notion of stochastic variation into expected returns, whereby allocators can determine optimal bet sizes that maximize the long-run rate of return.

Previously, under MPT, allocators operated under the assumption that the mean rate of return would equivocate to the expected long run logarithmic rate. SPT refines this understanding by demonstrating that observed (arithmetic) rates of return overestimate the long-run rate. This ties back into the idea of discrete measurement error and anticipates the observed phenomenon of "volatility drag"(not coincidentally, SPT provides an explanation here as to why low beta and low volatility portfolios are likely out perform). Also, the stochastic drag is basically a restatement of Jensen's Inequality which states that the a secant line drawn on a convex curve overestimates its value.

For a real-world portfolio with continuous and stochastic pay-offs, the arithmetic expected return over-states the long-run expected return of a risky payoff. This result can be recovered from Ito's Lemma. For a single pay-off of $X$ with expected long-run growth, $\gamma$:

$$\gamma = \mu - \frac{\sigma^2}{2}$$

$$X_t = X_0e^{\gamma t + \sigma W_t}$$

SPT expands on the single-case by defining the expected long-run growth for a logarithmic portfolio of continuous semi-martingales under the risk-neutral measure as $\gamma^*$; securities weights are given by $\pi$:

(1) $$\gamma _{{\pi }}^{*}(t):={\frac {1}{2}}\sum _{{i=1}}^{n}\pi _{i}(t)\sigma _{{ii}}(t)-{\frac {1}{2}}\sum _{{i,j=1}}^{n}\pi _{i}(t)\pi _{j}(t)\sigma _{{ij}}(t)$$

Function (1) can be used to optimize bet-sizes within a portfolio in order to maximize the long-run expected return as a function of securities' individual variances. Under the special case that long-term expected returns are optimized with respect to the logarithmic utility function, function (1) leads to convergence with Kelly betting for a stochastic portfolio. For more on Kelly convergence, I recommend Kelly Capital Growth Criterion, for which Ed Thorp is an editor.

So, from here, it is easy to see that the goals of SPT are aligned with those MPT, except that SPT uses much milder assumptions regarding the optimal risk vs reward, asset comovement, etc. In SPT, the long run logarithmic rate of return discounts all other decision criteria (including gambler's ruin).

As far as practical applications, formula (1) suggests a variety of portfolio construction schemes which are likely to outperform the market portfolio. The intuition that security size is inversely proportional to variance leads to the case in which inverting market weights optimizes formula (1) which could be interpreted as a form of arbitrage.

After accounting for slippage, impact and trading costs, I do not believe that something as simple as inverting market weights will lead to an arbitrage with a unitary probability of outperforming the market.

Another, more practical approach is to optimize expected return under formula (1) given forward looking assumptions (e.g., regarding factor-based and/or fundamentally derived estimates for expected returns and variance).

  • $\begingroup$ Thank you for this interesting answer. The formula $E[X_t]$ is not correct as stated - is it? $\endgroup$
    – Richi Wa
    Commented Jan 29, 2018 at 8:26
  • $\begingroup$ No. It was not correct. It should've just been $X_t$. $E[X_t]$ would just be $X_0 e^{\gamma t}$. My bad. $\endgroup$ Commented Jan 29, 2018 at 9:11
  • 1
    $\begingroup$ As an aside, I've had some success integrating these ideas into my allocation framework by penalizing expected (discretely observed) returns by one-half their expected variance. In simulation, where weights are determined by the expected growth rate, this improves the long-term growth rate quite surprisingly. I don't pretend that my simplistic approach is anything fancy. $\endgroup$ Commented Jan 29, 2018 at 9:19
  • $\begingroup$ Thank you. I understand the long run growth rate and I am somehow fond of low vol strategies. I just think that one should rebalance (monthly, quaterly - something like this) and then I wonder whether we are still in a "long run" setting. Maybe the lesson is simply that variance (risk) matters for expected returns. It does so even more if you want to take money out every now and then (e.g. if you are retired). Some guy once said: "on the long run we are all dead" this is true as well :) $\endgroup$
    – Richi Wa
    Commented Jan 29, 2018 at 9:28

There is, but it is rarely used, for reasons that are beyond me. The Kelly criterion or the Kelly bet is in real world use. You can find the original article at: Kelly, J. L. (1956). "A New Interpretation of Information Rate". Bell System Technical Journal. 35 (4): 917–926. doi:10.1002/j.1538-7305.1956.tb03809.x.

There is a small body of literature built around this and this is the optimal stochastic calculus solution. The difficulty, that is generally ignored in option pricing, is determining the limiting distributions for assets, though this problem may have been solved recently. See the proceedings of the Southwestern Finance Association conference at https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=SWFA2017&paper_id=144

A Kelly bet is always the optimal intertemporal bet for both continuous and for discrete time-based portfolios. Note that due to Donsker's scale invariance, it doesn't actually matter which way you conceptualize the problem. Warren Buffet is one of the few advocates for it, and you can check to see how poorly that has worked out for Berkshire Hathaway, having taken it from $\$19$ per share to $\$262,491$ per share over 52 years. Of course, Berkshire has to adjust for liquidity costs and that results in a smaller bet than the costless method advocated by pure stochastic calculus.

  • $\begingroup$ Thanks for your answer, I will go through the article in the proceedings that you posted, I just wonder: is this related to stochstic porfoloio theory in the sense of Fenholz and Karatzas? $\endgroup$
    – Richi Wa
    Commented Mar 12, 2017 at 19:07
  • $\begingroup$ Probably not, but I have never heard of their stuff being used and your question was whether or not any of this was used. The Kelly bet is used. $\endgroup$ Commented Mar 12, 2017 at 21:43
  • $\begingroup$ Sorry, but this is really different to what I have asked. The right answer to another question. $\endgroup$
    – Richi Wa
    Commented Mar 13, 2017 at 12:43
  • $\begingroup$ I think there may be a stronger connection between Kelly and SPT than many of us realize. $\endgroup$ Commented Mar 21, 2017 at 17:03
  • $\begingroup$ There are articles in the framework of SPT in continuous time that follow directly from Kelly. There is, at least theoretically, a continuous Kelly portfolio as long as prices have changed. Nonetheless, because of Donsker's Scale Invariance theorem, it is pointless to talk about continuous time when there is no mathematical difference between continuous and discrete time and we live in a discrete world. Unless you have a proof that depends upon continuity, there is no actual difference. $\endgroup$ Commented Mar 22, 2017 at 3:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.