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Stochastic Process

A Gentle Introduction to Stochastic Portfolio Theory (and its Inverse)

I made a brief comment about Stochastic Portfolio Theory that sparked some interest and a post or two by Alberto Bueno-Guerrero recently. Few, it would seem, are aware of this body of work so this article comprises:

1. A short teaser for stochastic portfolio theory

And for fun:

2. An illustration of what we might call inverse stochastic portfolio theory, which is where we take a heuristic portfolio construction technique and show that it would arise as the result of an optimization.

Stochastic Portfolio Theory (e.g. AbeBooks). Markowitz is the first name to appear in the acknowledgements.

A Characterisation of Stochastic Portfolio Theory

Stochastic Portfolio Theory comprises a collection of observations (theorems) about growth rates of portfolios that can be rebalanced in continuous time. Creator Robert Fernholz describes it thus:

A novel mathematical framework for analyzing portfolio behaviour and equity market structure … providing insight into questions of market equilibrium and arbitrage [that] can be used to construct portfolios with controlled behaviour … and has been the basis for successful investment strategies employed for over a decade by the institutional equity manager INTECH, where I have served as chief investment officer.

The crucial approximation of continuous time rebalancing is one that in theory has become more relevant over the decades since Fernholz’ original work, not less, due to plummeting trading costs. And treated only as a series of results in stochastic calculus Ferholz’ observations are incontrovertible.

For instance, the theory offers a decomposition of the return of a continuously rebalanced portfolio, and analysis of the same for portfolios that are specified in various ways, subject to assumptions about the generative model for asset prices. Stochastic Portfolio Theory provides an analytic perspective on the return of diversified portfolios relative to a cap-weighted index, for example, but many things besides (see the references).

How much one chooses to take literally some candidate portfolio methods and how much one morphs mathematical results into empirical opinion is largely up to the reader. As with Markowitz’ characterisation of the problem, care should be taken with the use of these observations and in particular the estimation of parameters such as covariance matrices. (I’ll simply pass the reader my own reading list on robust portfolio and estimation methods.)

Also as with Markowitz’s work, it is relatively easy to poke holes in models (or be cavalier with their use) and a little harder to build workable, useful extensions that don’t discard the insight. I pushed back on the more facile rejection of mathematical tooling in this note and the discussion therein, and I’ll continue that theme in the second half of this article.

Fernholz writes:

Stochastic Portfolio Theory is a descriptive theory that is applicable under the wide range of assumptions and conditions that may hold in actual equity markets. Unlike dynamic asset pricing theory, stochastic portfolio theory is consistent with either equilibrium or disequilibrium, with either arbitrage or no-arbitrage, and it remains valid regardless of the existence of an equivalent martingale measure.

Stochastic Portfolio Theory intends a relatively non-normative set of tools for building models of the stock market and exploring their implications for portfolio management. As with any set of mathematical insights it can be augmented, perturbed, extended and used in ways that are not necessarily obvious (and proprietary, as it happens).

Stochastic Portfolio Theory by Example

This introductory post draws on some notes I made for myself more than ten years ago, when my interest was first piqued and I read Fernholz’ book. I believe it is sufficient to introduce the reader to an analysis of long-term returns of continuously rebalanced portfolios using stochastic calculus, and if a few readers come away thinking “oh, Ito’s Lemma might be useful” for the first time, then I’ll call it a success.

We shall consider the instantaneous return of a continuously rebalanced portfolio.

Here Z(t) is the value of the portfolio and X(t) the price process vector, assumed lognormal.

Notation for lognormal stock price processes

I’m aware there is a segment of the community that takes any assumption such as this as an excuse to leave the auditorium, just moments before an opportunity to learn something. I think that’s foolish in the grand scheme of things if for no other reason that models like this can often be used as a first order term (an example is given here, in a slightly different setting).

In what follows we will consider, using Ito’s Lemma, both sides of the instantaneous portfolio return equation. The goal is to test whether our intuition for continuous return is sufficient or — to the contrary —determine if there are terms that arise that reveal something we did not expect.

To that end, first consider the right hand side — the individual assets:

After an application of Ito’s Lemma ot the stock price process

This part is unsurprising since the right hand side merely expresses the benefit of volatility if your baseline is the logarithmic return. You benefit from variance comes from the exponential (upward moves help more than downward hurt). Let’s proceed to multiply this by the portfolio weights, however, and we define the portfolio drift:

Logarithmic portfolio drift

Notice that we know the stochastic term whereas we just label the drift, for now. The reason for the former is that dZ/Z(t) and d(log Z(t)) have the same Brownian terms and the stochastic part of the former is obvious from the previous equation.

The next trick is applying Ito’s Lemma once more, this time to log(Z(t)) exponentiated to get back to Z(t). This will allow us to relate the portfolio drift to the stocks. Take log Z(t) from above and apply the map exp:

A neat little trick to remember … applying Ito’s Lemma to an inverse

This time Ito’s Lemma yields, after division:

After using Ito’s Lemma to turn log Z(t) back into Z(t)…

and behold we are on the left hand side of the instantaneous portfolio return equation we started with. All that remains is to equate both expressions for dZ(t)/Z(t):

After equating two expressions for instantaneous portfolio return

We have pulled the logarithmic portfolio return to one side and can now interpret this fairly easily. The right hand side breaks down as:

1. The combination of the logarithmic returns of the individual stocks.
2. A weighted combination of the covariance diagonal terms.
3. A portfolio variance term that provides a “volatility drag”.

Only two out of three were to be expected, no?

That’s pretty much all I have for you at this juncture as far as Stochastic Portfolio Theory is concerned. Is it enough to justify further exploration? Perhaps it sheds some light on Markowitz’ and minimum variance portfolios too, whose empirical properties and relationship to heuristic alternatives is something I grapple with.

I suppose that if you already had a financial intuition involving the weighted diagonals of the covariance matrix and the excess return process shown, then perhaps this exercise has gained you nothing. However most of us mortals might seek to improve our intuition from observations of this sort.

Inverse Stochastic Portfolio Theory

But I promised something novel so next, I provide an observation of my own that is vaguely motivated by the excess return equation above and in a different way bolsters, I hope, the assertion that stochastic calculus can yield insight.

Perhaps surprisingly, stochastic calculus can be useful in helping us understand portfolio construction (and critique it) even if the portfolio method is heuristic and, on the surface, devoid of any mathematics at all.

Should we label this Inverse Stochastic Portfolio Theory because it resembles inverse reinforcement learning? Here we are trying to back into the long term objective that is being optimized by an agent … even if the agent is blissfully unaware they are behaving the same way as another optimization-using agent. Inverse Stochastic Portfolio Theory is fun because you can use it to tie up faux philosophers of finance in knots of their own making and then casually walk away.

Allow me to illustrate with reference to a barbell portfolio — this time a portfolio of bonds, not stocks. It is called barbell because the prescription calls for investing half the portfolio in the longest dated bond and half in the shortest dated bond, and nothing in-between. (There is also barbell portfolio advice applying to asset allocation that should not be confused with the fixed income variant — we shall shred it another day).

Should a sensible portfolio construction ever eschew the middle-ground? A barbell bond portfolio splits the investment equally between the shortest dated bond and the longest dated bond. Ostensibly it is a heuristic approach intended to avoid the perils of optimization, but in this note we show it is equivalent to an optimization whose objective is rather hard to justify financially.

I have to keep this example as simple as possible so it works, and I assume a lattice of zero coupon bonds with prices denoted:

where t is the current time and tau the time to maturity in years. We shall assume that all bonds are priced off the same piecewise constant forward curve with knot points also at integer years:

and assume further that the changes in forward rates f(t,s) at time t for different years are independent. We presume the forward rates are driven by standard Brownian motion with the same standard deviation η. They may also have non-trivial drift but here it suffices to observe that the vector of bonds has dynamics given by

or more succinctly:

Log price dynamics for zero coupon bonds

We consider a portfolio of these bonds with weights π summing to unity. By analogy with Stochastic Portfolio theory we consider a modified excess return given by

A strange modified excess return definition mostly defying financial sense

where, following Stochastic Portfolio Theory notation, σij is the log-asset covariance, here equal to η2 multiplied by the i,jth element of JJ’. We make no statement as to what modified excess return represents, except to compare it to excess return noted in the first part of this post, namely:

Excess return

which is most certainly meaningful, as noted. Indeed, the log-optimal investor may seek to maximize excess return. In contrast, the modified excess return makes the covariance term more important so one might reason that, all else being equal, choosing modified excess return represents a sacrifice of long term growth in exchange for reduced portfolio variance (though the difference picks up the between-asset terms only, not the variances).

Trying to understand the difference between excess return as derived in Stochastic Portfolio Theory and a modifed excess return, which we will show to be optimized by a barbell heuristic.

Whatever the modification contemplated may imply, we proceed towards its surprising implication by mentally multiplying J above and noting that (JJ′)i,j=min(i,j) because:

Proof that JJ’ = min(i,j)

Thus in this slightly contrived forward rate model we have modified excess return proportional to the following:

Modified excess return — special case of zero coupon bond portfolio

This leaves us with a cute little optimization. We claim that ψ(π) and hence the modified excess return is maximized, subject to ∑πi=1, by setting the portfolio equal to a “barbell”. In a barbell bond portfolio half the portfolio is invested in the first (shortest maturity) bond and the other half on the last (longest maturity) bond.

The barbell portfolio, which surprisingly, arises as the solution to optimizing modified excess return

And by the way this portfolio corresponds to a modified excess return of ψ(π∗)=(n−1)/4. To prove this observe that ψ(π) can be re-written as follows (count the number of times each πi and πi πj occurs)

Simplified expression for modified excess return of a barbell zero coupon bond portfolio, that makes is obvious how it should be optimized and what the optimal modified excess return is.

I have introduced:

as the sum of portfolio weights leaving out the first i. The expression is clearly maximized by setting all u1…un equal to 1/2. By back substitution beginning with πn this implies π=π∗ as claimed.

Forgive those bearing alternatives to optimization, for they know not what they do

To recap:

1. We have shown that a barbell zero coupon bond portfolio would be chosen by someone who believes in a very simple short rate model and wishes to maximise “modified excess return”. We are lucky in this example that the optimization is so clean, and that the objective can be revealed.

2. Moreover, we know from the exhibit of the use of Ito’s Lemma in the first half of the note that in contrast, excess return can be interpreted with respect to long term portfolio logarithmic growth rate and we can therefore nail down exactly the difference between that objective and the unwitting objective that would give rise to a barball portfolio. To repeat:

The right hand side of this equation betrays the accidental emphasis. And who knows, perhaps there is a sound interpretation of modified excess return therein. However, this certainly complicates the task for “meta-critics” of quantitative finance!

Further reading

See:

• Stochastic Portfolio Theory, an Overview. Fernholz and Karatzas. (pdf).
• The wikipedia page provides a somewhat terse introduction to functionally-generated portfolios (scroll down) which is a focus of Stochastic Portfolio Theory.
• The definitive reference is still the book. (e.g. AbeBooks, Amazon).

As noted, stochastic calculus is hardly the subject of data science hype these days … but hopefully a few more people like Quasar Chunawala follow Alberto’s lead in posting about their journey though it. I always liked Oksendal’s book due to the emphasis on calculation but that’s just taste.

Nothing in this post should be construed as investment advice.