Equity Factor Portfolio Case Study

Harvey, Liu, and Zhu (2015) report 59 new factors discovered between 2010 and 2012.

• Key Criteria
  • Substantiated by academic and practicioner research.
  • Demonstrated historical return premium expected to persist.
  • Investable and defined with a systematic, rules based approach.
• Academic and practicioner research has identified a set of factors that have been shown to deliver a premium over the long-run.
  • Market, Size, Value, Momentum, Yield, Volatility, Quality, and Liquidity

spoiler

Qualitative Features

• Similar factor definitions, but unique index construction process across providers.
• Different universe of assets depending on provider and parent index.

Quantitative Features

• Risk and return characteristics
• Caveat of relying on historical backtest data, several of the indexes have less than 5 years of live performance.

SPOILER: No! Do your due dilligence.

Multi-Factor Index Construction

• Different factors targeted
• Top-down vs. Bottom-up Construction Process

Optimization

• Top-down approach with a set of single-factor indexes as building blocks
  • Portfolio optimization with objectives to minimize risk, minimize component contribution to risk, and maximize constant relative risk aversion
  • Heuristics: equal weight and inverse volatility
• Historical daily total return data from 2002-06-24 through 2016-09-30

Use the package PortfolioAnalytics to set up and run the optimizations. Define the portfolio specification and constraints.

library(PortfolioAnalytics)
R.4f <- R[,four.factor]
rp.seq <- generatesequence(min = 0, max = 1, by = 0.001)
port.spec <- portfolio.spec(assets = colnames(R.4f), 
                            weight_seq = rp.seq)
port.spec <- add.constraint(portfolio = port.spec, type = "weight_sum", 
                            min_sum = 0.99, max_sum = 1.01)
port.spec <- add.constraint(portfolio = port.spec, type = "box",
                            min = 0.15, max = 0.4)
# generate set of random portfolios
rp <- random_portfolios(port.spec, 5000)

# minimum standard deviation portfolio
port.min.sd <- add.objective(portfolio = port.spec, type = "risk", 
                             name = "StdDev")
# minimum expected shortfall
port.min.es <- add.objective(portfolio = port.spec, type = "risk", 
                             name = "ES", arguments = list(p = 0.95))
# component contribution to risk (standard deviation)
port.erc.sd <- add.objective(portfolio = port.min.sd, 
                             type = "risk_budget", name = "StdDev",
                             min_concentration = TRUE)
# component contribution to risk (expected shortfall)
port.erc.es <- add.objective(portfolio = port.min.es, 
                             type = "risk_budget", name = "ES", 
                             arguments = list(p = 0.95), 
                             min_concentration = TRUE)
# constant relative risk aversion
port.crra <- add.objective(portfolio = port.spec, type = "return", 
                           name = "CRRA", arguments = list(lambda = 5))

Fourth order expansion of the Constant Relative Risk Aversion (CRRA) Utility Function as in Martellini and Ziemann (2010) and Boudt, Lu, and Peeters (2015).

CRRA <- function(R, weights, lambda, sigma, m3, m4){
  weights <- matrix(weights, ncol=1)
  M2.w <- t(weights) %*% sigma %*% weights
  M3.w <- t(weights) %*% m3 %*% (weights %x% weights)
  M4.w <- t(weights) %*% m4 %*% (weights %x% weights %x% weights)
  term1 <- 0.5 * lambda * M2.w
  term2 <- (1 / 6) * lambda * (lambda + 1) * M3.w
  term3 <- (1 / 24) * lambda * (lambda + 1) * (lambda + 2) * M4.w
  out <- -term1 + term2 - term3
  out
}

training <- 504
rolling <- 504
rebal <- "quarters"
opt.min.sd <- optimize.portfolio.rebalancing(R.4f, 
                                             portfolio=port.min.sd, 
                                             optimize_method="random", 
                                             trace=TRUE, rp=rp, 
                                             rebalance_on=rebal, 
                                             training_period=training, 
                                             rolling_window=rolling)
w.min.sd <- normalizeWeights(extractWeights(opt.min.sd))
R.min.sd <- Return.portfolio(R.4f, w.min.sd)
colnames(R.min.sd) <- "min.sd"

chart.Weights(opt.min.sd, main = 'Optimal Weights: Min SD')

R.4f.opt <- cbind(R.min.sd, R.erc.sd, R.min.es, R.erc.es, R.crra, R.4f.ew)
charts.PerformanceSummary(R.4f.opt, main = '4-Factor Performance')

• 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
• Understand impact of constraints

port.lo <- add.constraint(portfolio = port.spec, type = "box",
                          min = 0, max = 1)
rp.lo <- random_portfolios(port.lo, 5000)
port.box <- add.constraint(portfolio = port.spec, type = "box",
                           min = 0.15, max = 0.4)
rp.box <- random_portfolios(port.box, 5000)

sim.lo.d <- simulate.portfolio(R.4f['2004-06-30/'], rp.lo, 
                               simulations = 1000, 
                               rebalance_on = rebal)
sim.lo.f <- simulate.portfolio(R.4f['2004-06-30/'], rp.lo, 
                               dynamic = FALSE, simulations = 1000, 
                               rebalance_on = rebal)
sim.lo <- cbind(sim.lo.d, sim.lo.f)

# repeat for box constraints

Simulation

Simulation Standard Deviation

Simulation Sharpe Ratio

Further work

• Bottom-up optimization
• Better estimates and forecasts
• Factor timing

Tools available to apply quantitative rigor

Do your due dilligence!