Thomas Wiecki gave a presentation on Bayesian portfolio allocation. He discussed how Bayesian statistics allows flexible modeling that accounts for uncertainty, unlike traditional mean-variance optimization. His company PyMC Labs uses probabilistic programming to specify portfolio allocation models and infer posterior distributions over parameters. This allows generating predictions across many possible future scenarios and making decisions by optimizing over a loss function. The full Bayesian model can incorporate changes over time, pool information across strategies, and correlations to provide a robust allocation.

1. Bayesian Portfolio Allocation
Thomas Wiecki,PhD
PyMC Labs
QWAFAFEW
Boston, MA

2. ANNOUNCEMENTS
Hugh Crowther

3. QU Winter school 2021
● Theme: AI and ML Enablement
○ Data Science with Python
○ AI & Machine Learning for Financial Professionals
○ Model Risk & Governance
● https://quwinterschool.splashthat.com/

4. Slides and Code
● QuAcademy: www.qu.academy

5. Thomas Wiecki is the Chief Executive Officer at PyMC Labs, a
Bayesian consultancy (www.pymc-labs.io). Prior to that Thomas
was the VP of Data Science at Quantopian, where he used
probabilistic programming and machine learning to help build
the world’s first crowdsourced hedge fund. Among other open
source projects, he is involved in the development of PyMC3—a
probabilistic programming framework for Python. He holds a
PhD from Brown University.

6. Bayesian Portfolio Allocation
Thomas Wiecki, PhD
@twiecki

7. 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.

8. ● 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
8

9. 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
○ ...

10. Bayesian statistics allows to build models flexibly
vs

11. Bayesian statistics allows specification of prior information
+

12. 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

13. Given 16 strategies, how to weight them?

14. Where we are
Data

15. 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

16. 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
a
u
t
o
m
a
t
i
c
(
M
C
M
C
)

17. T-Distribution
Modeling financial returns
Inference:
Bayes
Formula
probability
of
parameters
given
data
Latent causes
(Parameters)
Distribution
of Data
Observed Data
● mean returns
● volatility
● tails

18. Where we are
Data
Model

19. ● 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

20. The model in
Parameters /
Priors
Inference
Model specification

21. Where we are
Data
Model
Posterior

22. Posterior probability that strategy is profitable (SR > 0)

23. Where we are
Data
Model
Posterior
Predictions

24. 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

25. Bayesian Decision Making
Data
Model
Posterior
Predictions
Optimizer
Loss
function
Decision

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

27. Bayesian Decision Making
Data
Model
Posterior
Predictions
Optimizer
Loss
function
Decision

28. 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

29. 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

30. Bayesian Decision Making
Data
Model
Posterior
Predictions
Optimizer
Loss
function
Decision

31. 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().

32. The full model
● Changes in volatility and mean over time using GPs
● Hierarchical estimation to pool information from batch of algorithms
● Correlations

33. 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).

34. Further reading
www.pymc-labs.io
@twiecki
https://docs.pymc.io
Bayesian Decision Making blog post: https://twiecki.io/blog/2019/01/14/supply_chain/

35. 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.

36. Thanks for joining!
Questions