Thread by @HangukQuant

THREAD: Stochastic Optimization in Dynamic Environments: Portfolio Allocation by a Quant ▼

1\ Combining alpha signals are an essential part of portfolio management, with extensive literature on integrating alpha. Famous examples include the (Half) Kelly, Markowitz portfolios.

2\ We provide a review of these methods and offer glimpses at our unique, proprietary robust signal-weighting scheme. Let us consider the problem statement and inherent characteristics of dynamic optimisation.

3\ The obvious, and most problematic behaviour is the presence of stochasticity in a dynamic environment.

For an academic treatment of stochastic optimization, a lesson from the Department of Statistics at Columbia.

stat.columbia.edu/~liam/teaching…

4\ The ideal set of weights for integrated modelling is itself changing; introducing an objective conflict in modelling approach; increasing look-back durations for empirical modelling increase statistical significance but reduce sensitivity to regimes in market structure.

5\ A quick review (including static, Fama-MacBeth, Markowitz, Grinold ++) on existing methods developed for signal weighting by Nomura. The resulting outcome is that:

As asset/alpha universes become more complex, non-static schemes and complexity have additional benefits.

6\ Conversely, when the problem setting is simpler (asset counts, non-linearity et cetera), complexity is punished.

nomura.com/events/global-…

An analogous study shows that dynamic signal weighting approach adds significant value compared to the static approach under 2 conditions:

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(1) The discrimination of the performance of signals in different contexts.
(2) The correlation of the signals performance in each context.

papers.ssrn.com/sol3/papers.cf…

8\ These conditions reiterates Nomura’s view on dynamic modelling being rewarded as opposed to static allocations under complexity of the contextual environment.

9\ The contextual modelling not only refers to techniques such as identification of regimes, but also identifying seasonality effects etc in switching the tactical allocation of your strategies.

10\ It is not difficult to understand why this is so, with some mathematics. The (1) condition is trivial to understand, and let’s provide mathematical proofs for (2).

11\ For multivariate portfolios, you can calculated their risk-adjusted returns (Sharpe) using linear algebra. You may also plot their expected returns against the risk, and obtain the efficient frontier. The tangency portfolio gives us the highest Sharpe ratio.

12\ In practical trading conditions, where the weight vector does not have to sum to 1, then regardless of the desired return exposure, one can just target the tangency and lever the risk for highest Sharpe. How do you obtain tangency weight vector?

13\ This constrained optimisation problem can be solved with a bit of calculus and algebra.

14\ However, a few objections come to mind.
(ONE) Assumption is that alphas have positive expected returns in the first place. There is no reason to think that an alpha is exhibiting consistent negative returns. Weight sums should be one, but each entry should also be > 0.

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Our constraints should change to something like
l1-norm(w) = 1, where w_i >= 0 \forall w_i \in w.
However, this does not operate on the frontier and we do not have closed form solution for this.

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(TWO) The tangency vectors that allowed for short positioning allows for weird entries like (-10, 10, 1), which sum up to 1. However, in reality, margin requirements do not “net out”, so our tangency solutions may not be tradable.

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(THREE) We do did not solve our issue with stochasticity. Our E(R) vector is approximated with something like rolling means, but we still have the objective tradeoff between statistical significance and regime sensitivity of our walk-forward analysis.

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Let us first solve ONE, TWO. Solving ONE solves TWO, since there are the sum(w) = l1-norm(w). Solving one is a constrained optimisation and we can use optimisation algorithms in AI.

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Techniques like Gradient Descents, Simulated Annealing (modelling Ising spin glasses), Evolutionary (modelling Mendelian genetics), Neural Nets (modelling brain synapses) each have their places in identifying candidate solutions in large search spaces.

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So, how difficult is it to identify global optima for static problems? How do you find “best solutions” for problems without closed form solution? Using swarm intelligence (modelling flight path of birds), less that 20 iterations are required to find the best choice.

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Using some “tricks”, the birds (green) signal to their leader (blue) who quickly find the optimal weights, represented by vibrating red particles. The static problems are not too difficult to solve, particularly in lower dimensions. Now let’s consider dynamic conditions.

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We pick 10 alphas from the our weekly publications, of varying strength. We test 4 variants (vol as control)

1) 1/N (Equal)
2) Technical Momentum (TA)
3) Tangencies (Markowitz)
4) HangukQuant

In complex settings,
i) HangukQuant, Markowitz performs better, static/TA lagged

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ii) HangukQuant consistently beat other variants in drawdown.

In simpler settings (3 alphas),
i) HangukQuant performed best, TA/equal weighting both performed closely.
ii) Markowitz suffered in performance and drawdown, reinforcing Nomura’s complexity claim.

24\ We find that our proprietary modelling method (analysing return surface likelihoods) were robust to both simple and complex settings, with benefits increasing per complexity. In both cases, risk-adjusted returns were highest.

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In 24 hours, we publish the full report to our proprietary modelling theory. Get access with a paid subscription today! You are NOT going to want to miss this.

hangukquant.substack.com