Cornering Kelly
Thomas Schmelzer
Portfolio construction and technology @ ADIA | Commodities and LS Equities | Visiting Scholar at Stanford.
Univariate Trading Systems: Simple & Practical Position Sizing
In the world of trading, particularly within trend-following or signal-based strategies, many practitioners avoid using Markowitz optimization (or other complex optimization techniques) in favor of univariate systems. These systems scale positions based on a single signal, such as moving averages or momentum indicators, making them simpler to implement and more flexible compared to portfolio-level optimization.
How Univariate Trading Systems Work
In univariate trading systems, the decision-making process is driven by a single signal that dictates the trade. The position size typically scales in response to the strength of that signal, e.g. a stronger signal implies a larger position. This simplicity is one of the reasons these systems are so attractive to traders. While they are data-driven, they avoid the complexity of portfolio optimization. Position sizing is often determined using the Kelly Criterion, a powerful formula used to balance risk and reward.
The Kelly Criterion in Position Sizing
The Kelly Criterion is an optimal method for determining position size that maximizes long-term growth. The formula is:
w?=G / vola^2 × μ
Where
Interpretation and Scaling
At this stage, w? is dimensionless, meaning it doesn't directly tell you how much capital to allocate. To convert this to an actual cash position, you would typically:
Risk Implications: Isolation vs. Correlation
A key consideration is that univariate systems can lead to unhedged exposure. Since these systems act in isolation:
However, in a world where assets are uncorrelated, a group of such univariate systems could be very effective. In this scenario, the lack of correlation between positions allows the systems to scale in a way that balances risk across uncorrelated assets.
The Kelly Criterion effectively balances risk (volatility) and reward (signal strength) by adjusting position size. The scaling factor G serves as a way to control the trader's risk appetite, enabling dynamic position sizing based on signal strength.
Cornering Kelly
The approach is simple. We interpret the formula above as the (unscaled) solution of a convex program. Once done that we can move away in a continuous way and explore solutions respecting global risk limits or trading costs.
Introducing the risk-adjusted return
L[w] = μ w - 1 / (2*G) vola^2 *w^2
we observe that the gradient is zero exactly when
领英推荐
w?=G / vola^2 × μ
and we have reproduced the Kelly criterion. In a world with only one asset this would be still be perfect.
We can now interpret the vola^2 terms as the diagonal entries of a covariance matrix Q and ignore all off-diagonal entries, e.g.:
L[w] = μ w - 1 / (2*G) w’ Q *w
The gradient disappears for
w* = G’ inv(Q)*μ
Relaxing the Diagonal Covariance Matrix
So far, we’ve used a diagonal covariance matrix Q, meaning we’ve treated the assets as uncorrelated. This is a very aggressive shrinkage assumption—essentially the most extreme form of shrinkage estimator, where we assume no correlation between assets and only focus on their individual volatilities.
By relaxing the shrinkage assumption (i.e., allowing for correlations between assets), the covariance matrix Q can have off-diagonal entries that represent the covariances between the assets. The objective function remains:
L[w] = μ w - 1 / (2*G) w’ Q *w
where Q is full (not diagonal) and includes both variances (on the diagonal) and covariances (in the off-diagonal elements). This relaxed covariance structure allows the model to account for asset correlations and more complex risk interactions.
By exploring the space of feasible solutions in this way, it is possible to navigate a wide range of risk-return trade-offs, adjusting assumptions about asset correlations and risk tolerance. This is a key advantage of moving beyond the simple diagonal assumption.
Introducing Additional Constraints and Penalties
Once you have the base Kelly Criterion solution in this multidimensional form, you can further enhance the model by introducing additional constraints or penalties, such as:
Conclusions
By relaxing the assumption of a diagonal covariance matrix (i.e., assuming that assets can have correlations), we move from a simplified, univariate system to a multidimensional, more flexible portfolio optimization framework. This allows us to explore a much larger space of feasible solutions, accounting for the correlation structure between assets while still maintaining a convex, solvable problem.
This extension lets us dynamically adjust the portfolio's risk-return trade-offs, and by introducing constraints or penalties, we can tailor the solution to match real-world trading conditions and risk management objectives.
Univariate trading systems are very much a corner case of Markowitz. This little note is certainly not a surprise for people embracing optimization but I hope practitioners still doing the univariate route may find the interpretation given here enlightening.
Quantitative Researcher | PhD Astrophysics
3 周Thank you for the insightful post, Thomas. Allow me to ask a couple of questions please: 1. Global risk constraints: Is limiting (or targeting) overall portfolio variance same as choosing a smaller risk scaling G? By extension, if we were to constrain, say, the expected max drawdown (for a specified horizon), we should be able to do it with suitably choosing G? 2. If we are combining univariate signals across multiple assets, such that the diagonal entries of Q (individual variances of assets) are already accounted for in the respective signals, would it make sense to once again re-normalize capital allocations across assets with vec(1) Q^inv? Here, vec(1) is a row vector of ones. I ask this because a multi-variate approach at the outset for calculating signals can be very hard to solve, particularly if we are doing multi-period optimization to account for future trading costs. Thanks.
Portfolio construction and technology @ ADIA | Commodities and LS Equities | Visiting Scholar at Stanford.
3 周Some die-hard trend-followers will be confused about me dividing by the variance (the square of the volatility). It’s in general a good idea to estimate the expected volatility adjusted return. Volatility adjusted returns are homoscedastic! Hence one volatility is hiding in your $\mu$.
It is interesting but it reduces the Kelly criterion to an optimisation of position size concerning the first two moments, but a bet with two outcomes and one success probability has a mean, a variance AND skewness that can be tweaked by varying the bet's terms. In a way this is what makes the power of this criterion. Formulating the solution for multiple bets using the first 3 moments I am expecting is very challenging. My point here is to say that's why it mostly remains a univariate tool.