Stochastic Portfolio Theory, Long-Run Growth, and Portfolio Construction

# Real world application of stochastic portfolio theory


# Real world application of stochastic portfolio theory












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?

## Answer by David Addison (score 2, accepted)

https://quant.stackexchange.com/a/33143

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).

## Answer by Dave Harris (score -2)

https://quant.stackexchange.com/a/32953

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.