Optimal Portfolio

OptimalPortfolio is an open source library for portfolio optimisation. This library implements classical portfolio optimisation techniques for equities, but is also extendable for non-equity products given the right adjustments in invariants. Furthermore, certain modern advances in portfolio optimisation, such as Hierarchical Risk Parity is also implemented.

Regardless of whether you are a fundamental investor, or an algorithmic trader, this library can aid you in allocating your capital in the most risk efficient way, allowing to optimise your utility. For more details on the project design and similar content, please check out Engineer Quant

Disclaimer: This is not trading or investment advice. Trading involves significant risk and do so at your risk.

Contents

• Contents
• Overview
• Full Sequence
• Functionality
  • Expected Returns
  • Risk Models
  • Objective Functions
  • Constraints
• Market Invariants
• Moment Estimation
  • Nonparametric Estimators
  • Maximum Likelihood Estimators
  • Shrinkage Estimators
• Optimal Allocations
  • Higher Moment Optimisation
  • Compared to Sharpe Ratio
• Roadmap

Overview

This library aims to make optimising portfolios accessible to every trader and investor. To install this library, download it and run

bash install.sh

Full Sequence

The pipeline for using this library is delibrately modular, so as to allow users to incorporate their own proprietary stacks and code into the optimisation. The full sequence from data to weights is in the examples/full_sequence.ipynb notebook, but for clarity, it is

• Acquire stock (or other asset class) data
• Calculate market invariants (for stocks is returns)
• Calculate moments of invariants (for simple mean-variance optimisation it is mean and covariance)
• Optimise weights according to a utility function, subject to constraints, with moments of invariants as inputs

import pandas as pd
import numpy as np

import PortOpt.invariants as invs
import PortOpt.moment_est as moments
from PortOpt.opt_allocations import Optimiser
import PortOpt.utility_functions as utils

tickers = ['AAPL', 'MSFT', 'CVX', 'GE', 'GOOGL']
stock_data = utils.get_data(tickers, '1d', '2015-01-01', '2021-01-01')

# Compute market invariants

stock_returns = invs.stock_invariants(stock_data)

riskmodel = moments.RiskModel(tickers)
stock_cov = riskmodel.avg_hist_cov(stock_data)

# Optimise weights according to mean-variance

optimiser = Optimiser(tickers, exp_returns, stock_cov)
weights = optimiser.mean_variance(threshold=0.1, type='variance')
print(weights)
weight_tearsheet(weights)

This should have the following output:

AAPL     0.182566
MSFT     0.143655
CVX      0.190582
GE       0.117543
GOOGL    0.365654

Annual Return: 15.026
Annual Volatility: 23.203
Annual Sharpe: 0.561

Functionality

Listed below is the current overall functionality of the library, split into the various parts of the pipeline.

Expected Returns

• Nonparametric
  • Mean Historical Returns
  • Mean Exponentially Weighted Returns
  • Capital Asset Pricing Model (CAPM)
• Shrinkage
  • James-Stein Mean Shrinkage

Risk Models

• Nonparametric
  • Historical Covariance
  • Exponentially Weighted Covariance
• Covariance Shrinkage
  • Identity Shrinkage
  • Scaled Variance Shrinkage
  • Ledoit-Wolf Single Index
  • Ledoit-Wolf Constant Correlation

Objective Functions

• Mean-Variance Optimisation
  • Maximise Returns with a risk limit
  • Minimise risk with a returns limit
• Minimum Volatility
• Maximum Sharpe Ratio

Constraints

• Long Short Neutral
• Maximum Position on an asset

Adding Custom Constraints/Objectives

Since constraints are very case specific, I have added a functionality to be able to add your own constraints to the optimisation problem. We add a long/short neutral constraint using the custom method to demonstrate

optimiser = Optimiser(tickers, exp_returns, stock_cov)

# Add LS neutral constraint
optimiser.add_constraints([lambda x: cp.sum(x) == 0])

# Optimise for mean-variance
weights = optimiser.mean_variance(threshold=0.1, type='variance')
print(weights)
weight_tearsheet(weights)

Market Invariants

The first step to optimising any portfolio is calculating market invariants. Market invariants are defined as aspects of market prices that have some determinable statistical behaviour over time. For stock prices, the compounded returns are the market invariants. So when we calculate these invariants, we can statistically model them and gain useful insight into their behaviour. So far, calculating market invariants of stock prices. The same can be implemented for options and bonds, but data acquisition is an issue.

Moment Estimation

Once the market invariants have been calculated, it is time to model the statistical properties of the invariants. This is an actively researched and studied field and due to the nature of the complexity involved in modelling the statistical properties of large market data, there are several limitations in estimating the moments of the distributions.

Nonparametric Estimators

The simplest method of estimating the mean and covariance of invariants are the sample mean and covariance. However, this can be extended by introducing weightage for the timestamps, i.e giving more weight to recent data than older data. One interesting approach I have taken is introducing exponentially weighted mean and covariance, which I read about here.

Shrinkage Estimators

Nonparametric estimators only converge to the population estimate as the number of independent, identically distribution (IID) data points tends to infinity. However, as anyone who has experimented with financial data will be aware, market data is everchaning, and in some cases the volume of data is minimal. In these circumstances, nonparametric estiamtors do not work as well as intended, and this will cause issues later on in the optimiser.

One method to circumvent this is to impose some known structure in the market data. This is the essence of shrinkage estimation. We shrink the sample moment towards a structure that we know a-priori.

An example of shrinkage will be Ledoit-Wolf Single-Index Model Covariance Shrinkage. This approach assumes that stock returns are significantly influenced by market returns. So we model the stock returns as a linear regression of market returns:

$$ \hat{r}{i,t} = \alpha{i} + \beta_{i} \hat{r}{m, t} + \epsilon{i,t} $$

We calculate (\beta_i) and (\alpha_i) from regression, and we assume the error (\epsilon) is independent and normally distributed, i.e., (\text{Cov}(\epsilon_i, \epsilon_j) = 0), (\text{Cov}(\hat{r}_m, \epsilon_i) = 0), and (\mathbb{E}[\epsilon] = 0), (\text{Var}(\epsilon) = \sigma_i^2). Given this, we let the shrinkage matrix be:

$$ F = \beta \beta^T \hat{\sigma}m^2 + \Sigma{\epsilon} $$

where (\Sigma_{\epsilon}) is the diagonal matrix of error variances. We can now calculate the shrunk covariance matrix as:

$$ \Sigma = \alpha F + (1 - \alpha) S $$

We can go a step further by choosing (\alpha) optimally, but for the sake of brevity, I will leave that to the interested reader to find. The source papers are in the references directory for those interested.

Optimal Allocations

Classical asset allocation is the efficient frontier allocation. This is also known as the mean-variance optimisation as it takes into account the estimators of the mean and variance. The procedure of optimisation involves choosing an utility function and optimising it for portfolio weights. So far, I have implmented Mean-Variance, Minimum Volatility and Maximum Sharpe Optimisation.

Mean-Variance

Mean-variance optimisation can take on two forms: maximising returns with a variance limit, or minimising variance with a return limit. Both forms call the same mean_variance() method in the Optimiser class.

Minimum Volatility

Minimum volatility optimisation minimises volatility whilst constraining the weights to fit a certain portfolio profile (Long only, Long/Short)

Maximum Sharpe

Maximum Sharpe optimisation requires a little more nuance as maximising sharpe ratio is not a convex optimisation problem. Hence, we have to do a variable transformation, with a few assumptions, to convert the problem into a convex optimisation one.

Roadmap

I have the following planned out and am working on implementing them:

• Nonparametric Estimators
  • Exponentially weighted skew and kurtosis

• Shrinkage Estimators

  • Shrinkage for higher moments
  • Optimal shrinkage coefficient for custom shrinkage matrices

• Optimisations

  • Extending Hierarchical Risk Parity
  • Higher Moment Optimisation
  • Backtesting optimisation