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Bayesian Portfolio Allocation

Thomas Wiecki, PhD

@twiecki

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Disclaimer

This presentation is for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation for any security; nor does it constitute an offer to provide investment advisory or other services by Quantopian, Inc. ("Quantopian"). Nothing contained herein constitutes investment advice or offers any opinion with respect to the suitability of any security, and any views expressed herein should not be taken as advice to buy, sell, or hold any security or as an endorsement of any security or company. In preparing the information contained herein, Quantopian, Inc. has not taken into account the investment needs, objectives, and financial circumstances of any particular investor. Any views expressed and data illustrated herein were prepared based upon information, believed to be reliable, available to Quantopian, Inc. at the time of publication. Quantopian makes no guarantees as to their accuracy or completeness. All information is subject to change and may quickly become unreliable for various reasons, including changes in market conditions or economic circumstances.

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Bayesian consultancy by the inventors of PyMC3
Solving advanced analytical problems
Team consists of PhDs, mathematicians, neuroscientists, and a former SpaceX rocket scientist
Clients in pharma (Roche), fintech, mortgage, agriculture, adtech, biotech...
www.pymc-labs.io

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Markowitz mean-variance optimization

Optimal portfolio - in theory - taking mean return, volatility and correlations into account
Highly sub-optimal in practice, why?

Estimates are very noisy, but we do not quantify that noise
Leads to lack of diversification

Many hacks… err solutions, exist:

Equal-weight (MVO but assuming means and vol are equal and no correlations exist)
Inverse-variance weighting (MVO but assuming means are equal and no correlations exist)
Hierarchical Risk Parity
...

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Bayesian statistics allows to build models flexibly

vs

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Bayesian statistics allows specification of prior information

+

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Not single most likely solution, but all probable solutions

Instead of point-estimates (scalar values) of e.g. the mean or variance, we use probability distributions that quantify uncertainty.

Point estimates

Probability distributions

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Given 16 strategies, how to weight them?

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Where we are

Data

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Bayesian Modeling: Coin flipping

Parameters

Prior p(θ)

Likelihood p(x | θ)

Model construction:

How parameters relate to data

Inference: Bayes Formula

most likely parameters given data

Data x

(Heads / Tails)

Parameters

Posterior p(θ | x)

p(heads)

Observe:

HTTHTTT

belief

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Probabilistic Programming

Parameters

Prior p(θ)

Likelihood p(x | θ)

Model construction:

How parameters relate to data

Inference: Bayes Formula

most likely parameters given data

Data x

(Heads / Tails)

Parameters

Posterior p(θ | x)

p(heads)

Observe:

HTTHTTT

belief

code

automatic

(MCMC)

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Modeling financial returns

T-Distribution

Inference: Bayes Formula

probability of parameters given data

Latent causes

(Parameters)

Distribution

of Data

Observed Data

mean returns
volatility
tails

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Where we are

Data

Model

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Probabilistic Programming framework for Python, FOSS
Specify arbitrary models in code by plugging probability distributions into each other
Intuitive model specification syntax

For example: x ~ N(0,1) translates to x = Normal('x', 0, 1)

Inference Button: Automatic and powerful inference for any model

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The model in

Parameters / Priors

Inference

Model specification

Lets look at how this model would look like in PyMC3 -- a new probabilistic programming framework written in Python and which I helped develop.
After importing and setting up the model context we have to define the unobservable parameters of our model.
As this is a Bayesian framework we have to specify priors which allow me to define my belief about possible parameter values before I have seen any data. Here I’m just assigning equal belief in a fixed interval but you have many choices here. Informative priors are very powerful to constrain the model and apply regularization.
Once the parameters are defined we have to relate them to the distribution of the data which is T as we said before.
Note how the first parameter is a name I can give to the distribution and how the three parameters of the T distribution connect up with the definition of the parameters above.
Using the observed keyword argument we pass in our observed data -- which is just an array of the daily returns of either the in or out-of-sample returns. We will run this model on the two data sets independently.
So far this code only specifies the model and under the hood builds up a compute graph in theano which we then just-in-time-compile to C for speed.
So that part is the model specification.
To actually run MCMC sampling to infer the parameters we instantiate a sampler object. Here we are using the NUTS sampler which is a very recent development that revolutionized probabilistic programming as it this sampler works on models that are way more complex than this one.
We then draw 5000 samples from the so-called posterior which will contain likely values of the parameters given the data.

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Where we are

Data

Model

Posterior

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Posterior probability that strategy is profitable (SR > 0)

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Where we are

Data

Model

Posterior

Predictions

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Bayesian Decision Making

So far we only have probability distributions for our strategies.
How to construct a portfolio from them?
Use model to generate all kinds of possible future scenarios (prediction)
Define loss function that rates how good a solution is given a scenario
Use optimizer to find best solution across all possible future scenarios

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Bayesian Decision Making

Data

Model

Posterior

Predictions

Optimizer

Loss function

Decision

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Predictions

Generate possible future scenarios by drawing parameter set from posterior & sampling from likelihood
Two sources of variability: Likelihood & uncertainty

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Bayesian Decision Making

Data

Model

Posterior

Predictions

Optimizer

Loss function

Decision

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Loss function for Mean-Variance

Utility theory tells us to minimize our expected losses (maximizing wins leads to overly risky behavior)
Black-Littermann: −exp(−λr(ω)), where r(ω) are the expected returns if we used portfolio weights ω, λ is how averse to losses we are

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Example

def loss_function(ω): # weight vector, e.g. [1/16, 1/16, …]

loss = 0

for r in sampled_returns:

# compute portfolio returns

port_rets = sum(r * ω)

loss += -exp(-port_rets)

return loss

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Bayesian Decision Making

Data

Model

Posterior

Predictions

Optimizer

Loss function

Decision

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Optimization → Output

Finds optimal portfolio weights ω which minimize expected loss
In our case: loss function is convex so we can use convex solvers (cvxpy) which are much faster, otherwise, use scipy.optimizer.fmin().

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The full model

Changes in volatility and mean over time using GPs
Hierarchical estimation to pool information from batch of algorithms
Correlations

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Benefits

Robust due to using posterior distributions rather than point-estimates
Different length track-records are automatically handled

Short but great track-record: high uncertainty -> many potentially bad outcomes -> low weight

Model can be improved to include all kinds of structure, like risk-factors, prior information we might have (e.g. knowing a certain manager well).

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Disclaimer

This presentation is for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation for any security; nor does it constitute an offer to provide investment advisory or other services by Quantopian, Inc. ("Quantopian"). Nothing contained herein constitutes investment advice or offers any opinion with respect to the suitability of any security, and any views expressed herein should not be taken as advice to buy, sell, or hold any security or as an endorsement of any security or company. In preparing the information contained herein, Quantopian, Inc. has not taken into account the investment needs, objectives, and financial circumstances of any particular investor. Any views expressed and data illustrated herein were prepared based upon information, believed to be reliable, available to Quantopian, Inc. at the time of publication. Quantopian makes no guarantees as to their accuracy or completeness. All information is subject to change and may quickly become unreliable for various reasons, including changes in market conditions or economic circumstances.