PortfolioAnalytics Tutorial

May 19, 2017

Overview

Discuss Portfolio Optimization
Introduce PortfolioAnalytics
Demonstrate PortfolioAnalytics with Examples

Modern Portfolio Theory

"Modern" Portfolio Theory (MPT) was introduced by Harry Markowitz in 1952.

In general, MPT states that an investor's objective is to maximize portfolio expected return for a given amount of risk.

Common Objectives

Maximize a measure of gain per unit measure of risk
Minimize a measure of risk

How do we define risk? What about more complex objectives and constraints?

Portfolio Optimization Objectives

Minimize Risk
- Volatility, Tail Loss (VaR, ES), Max Drawdown, etc.
Maximize Risk Adjusted Return
- Sharpe Ratio, Information Ratio, etc.
Risk Budgets
- Equal Component Contribution to Risk (i.e. Risk Parity)
- Limits on Component Contribution
Maximize a Utility Function
- Quadratic, Constant Relative Risk Aversion (CRRA), etc.
Minimize Tracking Error
- Replicating portfolio to track an index or basket of assets

Challenges of Portfolio Optimization in R

Many solvers are not specific to portfolio optimization
Understanding the capabilities and limits of solvers to select the appropriate solver for the problem or formulate the problem to fit the solver.
Lack of a common interface to a wide range of solvers in R, difficult to switch between solvers.
Closed-Form vs. Global Solvers

Portfolio optimization is a hard problem.
One challenge is knowing what solver to use and the capabilities and limits of the chosen solver.
You can chose a solver based on the objective or formulate the objective to fit the chosen solver. Ideally, you want the flexibility of effortlessly switching between solvers depending on the problem.
In many cases, there is more than one solver that can solve the problem and you should evaluate both.
The main advantage of closed-form solvers is that they solve a given problem very fast and efficiently and return an exact solution. The main drawback is that the problem must be formulated in a very specific way that is typically unique to the solver.
Global solvers have the advantage of being able to solve more general types of problems and find the approximate solution of the global minimum or maximum of the objective function with local minima or maxima. However, the algorithms used in global solvers are relatively more complex and more compute intensive.

PortfolioAnalytics Overview

PortfolioAnalytics is an R package designed to provide numerical solutions and visualizations for portfolio optimization problems with complex constraints and objectives.

Supports:

multiple constraint and objective types
modular constraints and objectives
an objective function can be any valid R function
user defined moment functions (e.g. covariance matrix, return projections)
visualizations
solver agnostic
parallel computing

PortfolioAnalytics was designed specifically to address the problems discussed in the previous slide.
Key components of the architecture of PortfolioAnalytics are modularity and flexibility. The multiple types and modularity of constraints and objectives allow you to add, remove, and combine multiple constraint and objective types very easily.
An objective function can be defined as any valid R function. This means that you are not limited to a specific type or set of problems.
Like objective functions, you can define custom moment functions as any valid R function.
Visualization helps to build intuition about the problem and understand the feasible space of portfolios.
Another key benefit is that PortfolioAnalytics supports several solvers and allows you to specify different solvers with minimal or no changes to the portfolio specification.

PortfolioAnalytics Framework

Workflow: Specify Portfolio

p <- portfolio.spec(assets = c('AAPL', 'MSFT', 'GOOG'))
print(p)

## **************************************************
## PortfolioAnalytics Portfolio Specification 
## **************************************************
## 
## Call:
## portfolio.spec(assets = c("AAPL", "MSFT", "GOOG"))
## 
## Number of assets: 3 
## Asset Names
## [1] "AAPL" "MSFT" "GOOG"

Initializes the portfolio object that holds portfolio level data, constraints, and objectives

Workflow: Add Constraints

# weights sum to 1
p <- add.constraint(portfolio = p, type = 'weight_sum',
                    min_sum = 1, max_sum = 1)
# special type for full investment constraint
# add.constraint(portfolio = p, type = 'full_investment')
# box constraints
p <- add.constraint(portfolio = p, type = 'box',
                    min = 0.2, max = 0.6)

Supported Constraint Types

Sum of Weights
Box, Group
Factor Exposure
Position Limit
and many more

Workflow: Add Objectives

p <- add.objective(portfolio = p, type = 'risk', name = 'StdDev')

Supported Objective types

Return
Risk
Risk Budget
Weight Concentration

Workflow: Run Optimization

opt <- optimize.portfolio(R, portfolio=p, optimize_method='random',
                          search_size=2000)

opt.rebal <- optimize.portfolio.rebalancing(R, portfolio=p,
                                            optimize_method='random',
                                            search_size=2000,
                                            rebalance_on='quarters',
                                            training_period=60,
                                            rolling_window=60)

Workflow: Analyze Results

Visualization	Data Extraction
plot	extractObjectiveMeasures
chart.Concentration	extractStats
chart.EfficientFrontier	extractWeights
chart.RiskReward	print
chart.RiskBudget	summary
chart.Weights

Support Multiple Solvers

Linear and Quadratic Programming Solvers

R Optimization Infrastructure (ROI)
- GLPK (Rglpk)
- Symphony (Rsymphony)
- Quadprog (quadprog)

Global Solvers

Random Portfolios
Differential Evolution (DEoptim)
Particle Swarm Optimization (pso)
Generalized Simulated Annealing (GenSA)

Random Portfolios

PortfolioAnalytics has three methods to generate random portfolios.

The sample method to generate random portfolios is based on an idea by Pat Burns.
The simplex method to generate random portfolios is based on a paper by W. T. Shaw.
The grid method to generate random portfolios is based on the gridSearch function in the NMOF package.

Random portfolios allow one to generate an arbitray number of portfolios based on given constraints. Will cover the edges as well as evenly cover the interior of the feasible space. Allows for massively parallel execution.
The sample method to generate random portfolios is based on an idea by Patrick Burns. This is the most flexible method, but also the slowest, and can generate portfolios to satisfy leverage, box, group, and position limit constraints.
The simplex method to generate random portfolios is based on a paper by W. T. Shaw. The simplex method is useful to generate random portfolios with the full investment constraint, where the sum of the weights is equal to 1, and min box constraints. Values for min_sum and max_sum of the leverage constraint will be ignored, the sum of weights will equal 1. All other constraints such as the box constraint max, group and position limit constraints will be handled by elimination. If the constraints are very restrictive, this may result in very few feasible portfolios remaining. Another key point to note is that the solution may not be along the vertexes depending on the objective. For example, a risk budget objective will likely place the portfolio somewhere on the interior.
The grid method to generate random portfolios is based on the gridSearch function in NMOF package. The grid search method only satisfies the min and max box constraints. The min_sum and max_sum leverage constraint will likely be violated and the weights in the random portfolios should be normalized. Normalization may cause the box constraints to be violated and will be penalized in constrained_objective.

Comparison of Random Portfolio Methods (Interactive!)

Random Portfolios: Simplex Method

Hierarchical (i.e. Multilayer) Optimization

see PortfolioAnalytics Tutorial R/Finance 2016

Regime Switching Model

Define a model to characterize n regimes
Define n portfolio specifications, 1 for each regime

see PortfolioAnalytics Tutorial R/Finance 2016

Estimating Moments

Ledoit and Wolf (2003):

"The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization."

Sample
Shrinkage Estimators
Factor Model
Expressing Views
- PortfolioAnalytics Tutorial R/Finance 2015
See Custom Moment and Objective Functions vignette

What I am referring to as the asset return moments are properties of the distribution of asset returns. The first moment is the expected returns vector, the second moment is the variance-covariance matrix, the third moment is the coskewness matrix, and the fourth moment is the cokurtosis matrix.
There is a lot of research in the area of different models and methods for estimating asset return moments for portfolio optimization and risk models. In their 2003 paper, Ledoit and Wolf claim that the sample covariance matrix should not be used in portfolio optimization.
Methods for estimating moments include, but are not limited to sample estimates, shrinkage estimators, factor models, expressing views, and techniques from robust statistics.
The main drawbacks of sample estimates are estimation error and the curse of dimensionality. As an example for a portfolio of 20 assets, 210 elements must be estimated for the sample covariance matrix. Using a factor model with 3 factors requires 86 elements to be estimated for the covariance matrix. Risks of estimation error increase as the dimension of assets and parameters to estimate increases.
Estimating moments using shrinkage estimators, factor models, views are methods to address the disadvantages of using sample estimates. I am not making a claim that one method is better than another. The method chosen depends on your own situation and information/data available.

Example: U.S. Equity Sector ETF Portfolio

Consider a long only U.S. Equity portfolio allocation problem. Goal is to outperform the S&P 500 (SPY as a proxy).

Data

Monthly returns of 9 U.S. Equity Sector ETFs from 1999 to May 2016 (Source: Yahoo)

Approach

Baseline portfolio
Box constraints
Custom objective function
Analysis

Baseline Portfolio Specification

Baseline portfolio specification to minimize portfolio standard deviation, subject to full investment and long only constraints.

# base portfolio specification
rp.seq <- generatesequence(min = 0, max = 1, by = 0.002)
p <- portfolio.spec(assets = colnames(R), 
                    weight_seq = rp.seq)
p <- add.constraint(portfolio = p, type = 'weight_sum', 
                    min_sum = 0.99, max_sum = 1.01)
p <- add.constraint(portfolio = p, type = 'box', min = 0, max = 1)
p <- add.objective(portfolio = p, type = 'return', name = 'mean', 
                   multiplier = 0)
p <- add.objective(portfolio = p, type = 'risk', name = 'StdDev')

Box Constraints

Add box constraints such that for each asset the minimum weight is 5% and the maximum weight is 20%.

copy the baseline portfolio specification
replace the second index of the constraint object

# portfolio specification with box constraints
p.box <- p
p.box <- add.constraint(portfolio = p.box, type = 'box', 
                        min = 0.05, max = 0.20, indexnum = 2)

Single Period Optimization

Run a single period optimization for the portfolio specifications

# generate the random portfolios
rp <- random_portfolios(portfolio = p, permutations = rp.n, 
                        method = 'sample')
rp <- normalize.weights(rp)
rp.box <- random_portfolios(portfolio = p.box, permutations = rp.n, 
                            method = 'sample')
rp.box <- normalize.weights(rp.box)

# run the optimizations
opt.base <- optimize.portfolio(R = tail(R, 36), portfolio = p, 
                               rp = rp, optimize_method = 'random',
                               trace = TRUE)
opt.box <- optimize.portfolio(R = tail(R, 36), portfolio = p.box, 
                              rp = rp.box, optimize_method = 'random', 
                              trace = TRUE)

Feasible Space Comparison

Impact of box constraints on the feasible space of the in sample period.

Optimization with Periodic Rebalancing

opt.base.rebal <- optimize.portfolio.rebalancing(R = R, portfolio = p,
                                                 optimize_method = 'random',
                                                 rp = rp, trace = TRUE,
                                                 rebalance_on = 'quarters',
                                                 training_period = 36,
                                                 rolling_window = 36)
opt.base.rebal.r <- Return.portfolio(R, weights = extractWeights(opt.base.rebal))
colnames(opt.base.rebal.r) <- 'base'

opt.box.rebal <- optimize.portfolio.rebalancing(R = R, portfolio = p.box,
                                                optimize_method = 'random',
                                                rp = rp.box, trace = TRUE,
                                                rebalance_on = 'quarters',
                                                training_period = 36,
                                                rolling_window = 36)
opt.box.rebal.r <- Return.portfolio(R, weights = extractWeights(opt.box.rebal))
colnames(opt.box.rebal.r) <- 'box'

Baseline Optimal Weights

chart.Weights(opt.base.rebal, main = 'Baseline Portfolio Optimal Weights')

Box Constrained Optimal Weights

chart.Weights(opt.box.rebal, main = 'Box Constrained Portfolio Optimal Weights')

Baseline and Constrained

opt.r <- cbind(opt.base.rebal.r, opt.box.rebal.r)
charts.PerformanceSummary(opt.r, main = "Performance Summary")

Tracking Error Custom Objective Function

Define a custom objective function for penalized tracking error

te.target <- function(R, weights, Rb, min.te = 0.02, 
                      max.te = 0.05, scale = 12){
    r <- Return.portfolio(R = R, weights = weights)
    Rb <- Rb[index(r)]
    te <- sd(r - Rb) * sqrt(scale)
    # penalize tracking error outside of [min.te, max.te] range
    out <- 0
    if(te > max.te)
        out <- (te - max.te) * 10000
    if(te < min.te)
        out <- (min.te - te) * 10000
    out
}

Tracking Error Target Optimization

Add tracking error as an objective to the baseline portfolio specification
Specify the arguments to te.target as a named list
Target an Annualized Tracking Error in the range of 3% to 5%

p.te <- p
p.te <- add.objective(portfolio = p, type = 'risk', name = 'te.target',
                      arguments = list(Rb = R.mkt, scale = 12,
                                       min.te = 0.03, max.te = 0.05))

opt.te.rebal <- optimize.portfolio.rebalancing(R = R, portfolio = p.te,
                                                optimize_method = 'random',
                                                rp = rp, trace = TRUE,
                                                rebalance_on = rebal.period,
                                                training_period = n.train,
                                                rolling_window = n.roll) 
opt.te.rebal.r <- Return.portfolio(R, weights = extractWeights(opt.te.rebal))
colnames(opt.te.rebal.r) <- 'te.target'

Tracking Error Target

opt.r <- na.omit(cbind(opt.base.rebal.r, opt.box.rebal.r, 
                       opt.te.rebal.r, R.mkt))
charts.PerformanceSummary(opt.r, main = "Performance Summary")

Portfolio Simulation and Analysis

Random portfolios to simulate portfolios of zero skill managers
Feasible portfolios by construction
Monte Carlo requires a model specification
Time series dependency with bootstrap
No guarantee of a feasible portfolio with bootstrap or Monte Carlo Simulation

Simulate Portfolio Returns

Simulate portfolio returns using the random portfolios generated from the baseline portfolio specification and box constrained portfolio specification.

sim.base <- simulate.portfolio(R, rp = rp, simulations = 5000, 
                               rebalance_on = 'quarters')
sim.box <- simulate.portfolio(R, rp = rp.box, simulations = 5000, 
                              rebalance_on = 'quarters')
sim <- cbind(sim.box, sim.base)
charts.PerformanceSummary(sim, colorset = c(rep("salmon", ncol(sim.box)), 
                                            rep('gray', ncol(sim.base))), 
                          legend.loc = NULL, main = "Simulated Performance Summary")
legend('topleft', legend = c('constrained', 'unconstrained'), 
       fill = c('salmon', 'gray'), bty = 'n')

Simulated Portfolio

Simulated vs. Optimal Portfolios: Sharpe Ratio Distribution

Simulated vs. Optimal Portfolios: Sharpe Ratio Distribution

Simulated vs. Optimal Portfolios: Tracking Error Distribution

Improvements

Risk Measures
- Expected Shortfall
- Component contribution to risk (i.e. Risk Budgeting)
Better Estimates
- Factor Model
- GARCH
- Return Forecasts
More Data
- Weekly or daily frequency
  - downsampling to monthly frequency "throws away" a lot of data
- Estimates based on higher frequency data

Conclusion

Introduced the goals and summary of PortfolioAnalytics
Demonstrated the flexibility through examples
Plans for continued development
- Interface to \(parma\)
- Additional solvers
- Gallery of examples

Acknowledgements

Google: funding Google Summer of Code (GSoC) for 2013 and 2014
GSoC Mentors: Brian Peterson, Peter Carl, Doug Martin, and Guy Yollin
R/Finance Committee

PortfolioAnalytics Links

Source code for the slides

https://github.com/rossb34/PortfolioAnalyticsPresentation2017

and view it here

http://rossb34.github.io/PortfolioAnalyticsPresentation2017/

DataCamp Course

https://www.datacamp.com/courses/intermediate-portfolio-analysis-in-r

Overview

Modern Portfolio Theory

Portfolio Optimization Objectives

Challenges of Portfolio Optimization in R

PortfolioAnalytics Overview

PortfolioAnalytics Framework

Workflow: Specify Portfolio

Workflow: Add Constraints

Workflow: Add Objectives

Workflow: Run Optimization

Workflow: Analyze Results

Support Multiple Solvers

Random Portfolios

Comparison of Random Portfolio Methods (Interactive!)

Random Portfolios: Simplex Method

Hierarchical (i.e. Multilayer) Optimization

Regime Switching Model

Estimating Moments

Example: U.S. Equity Sector ETF Portfolio

Baseline Portfolio Specification

Box Constraints

Single Period Optimization

Feasible Space Comparison

Optimization with Periodic Rebalancing

Baseline Optimal Weights

Box Constrained Optimal Weights

Baseline and Constrained

Tracking Error Custom Objective Function

Tracking Error Target Optimization

Tracking Error Target

Portfolio Simulation and Analysis

Simulate Portfolio Returns

Simulated Portfolio

Simulated vs. Optimal Portfolios: Sharpe Ratio Distribution

Simulated vs. Optimal Portfolios: Sharpe Ratio Distribution

Simulated vs. Optimal Portfolios: Tracking Error Distribution

Improvements

Conclusion

Acknowledgements

PortfolioAnalytics Links

Any Questions?

References and Useful Links