Scenario Analysis of Risk Parity using...

Scenario Analysis of Risk Parity using RcppParallel

Jason FosterCRAN - Package ‘roll’

May 20, 2017

roll: rolling statistics

The roll package provides parallel functions for computing rolling statisticsof time-series data; it uses Rcpp, RcppArmadillo, and RcppParallel:

install.packages("roll")

Function Description

roll_sum Sumsroll_prod Productsroll_mean Meansroll_var Variancesroll_cov Covariancesroll_sd Standard deviations

Function Description

roll_cor Correlationsroll_scale Z-scoresroll_lm Linear modelsroll_eigen Eigendecompsroll_pcr PCA regressionsroll_vif Multicollinearity

Jason Foster CRAN - Package ’roll’ 2 / 15

mailto:[email protected]

https://CRAN.R-project.org/package=roll

Rolling beta for 2,000 portfolios!Speed gains from parallel computing help decrease the trade-off betweennumber of observations and window size:

0

3

6

9

12

500 1,000 1,500 2,000 2,500

Number of observations

Min

utes

FunctionfastLmPureroll_lm

Window size1,260882504126

Less than 1−minute!




RcppArmadillo versionDownload the S&P 500 index using quantmod and calculate returns:

library(quantmod)getSymbols("^GSPC")

Then use RcppArmadillo and estimate a portfolio’s sensitivity to the S&P500 index by rolling the fastLmPure function:

library(RcppArmadillo)fastLm_coef <- function(data) {

return(coef(fastLmPure(cbind(1, data$x), data$y)))}

arma_lm_coef <- rollapplyr(data = returns, width = 252,FUN = fastLm_coef,by.column = FALSE)




RcppParallel version

Repeat the exercise and take advantage of parallel processing in C++ byusing the roll_lm function in the roll package:

library(roll)roll_lm_coef <- roll_lm(x = returns$x,

y = returns$y,width = 252)$coefficients

And test that the coefficients from the fastLmPure and roll_lm functionsare equal:

all.equal(arma_lm_coef, roll_lm_coef)

## [1] TRUE




RcppParallel: parallel programming tools for ‘Rcpp’

RcppParallel, developed by the Rcpp Core team, makes it easy to createsafe, portable, high-performance parallel algorithms using C++ and Rcpp:

install.packages("RcppParallel")

In particular, RcppParallel provides two high-level parallel algorithms:

parallelFor(): convert the work of a standard serial “for” loop intoa parallel oneparallelReduce(): compute and aggregate multiple values in parallel

For more information, visit the RcppParallel website:http://rcppcore.github.io/RcppParallel/


http://rcppcore.github.io/RcppParallel/



Style analysis of multi-asset portfolios

Download daily returns for 150 portfolios in the Tactical Allocation andWorld Allocation categories and apply a 15 asset class factor model on adaily rolling window:

Stocks Bonds & other

US Large Cap Value US Gov 1-10YUS Large Cap Growth US Gov 10Y+US Mid Cap Non-US GovUS Small Cap IG CorpEurope HY CorpJapan MBSPacific ex Japan CommoditiesEmerging Markets

Sharpe, William F. 1988. “Determining a Fund’s Effective Asset Mix.” Investment Management Review.




Asset allocation

Daily returns are readily available and help make inferences about style:

71.4 secs

2.9 secs

0

20

40

60

80

0 250 500 750 1,000

Number of simulations

Sec

onds

fastLmPureroll_lm

25x over RcppArmadillo version!

Speed test uses univariate betas

0.0

0.2

0.4

0.6

0.8

2013 2014 2015 2016 2017

Coe

ffici

ents


Median portfolio




Concentrated in equity risk

A significant driver of return variability is the concentration of equity risk:

94.9 secs

2.2 secs

0

28

55

82

110

0 250 500 750 1,000


Sec

onds

cov.wtroll_cov

44x over R version!


0.0

2.5

5.0

7.5

10.0

2013 2014 2015 2016 2017

Ris

k co

ntrib

utio

n (%

)


Median portfolio




Decline in realized volatilityFrom mixed economic data to rising geopolitical risk, volatility acrossmarkets and asset classes is near an all-time. . . low?

78.8 secs

0.8 secs0

22

45

68

90

0 250 500 750 1,000


Sec

onds

sdroll_sd

93x over R version!


0

9

18

27

36

2013 2014 2015 2016 2017

Vol

atili

ty (

%)


Asset classes




Simple risk parity strategy

Consider a portfolio of N assets and let w = (w1, . . . ,wN) be a vector ofasset weights. Also define Σ to be the covariance matrix of returns. Thenthe volatility of the portfolio is defined as:

σ =√

wT Σw

and, by Euler’s theorem, the risk contribution of asset i is given by:

σi = wi∂σ

∂wi

Goal: solve for weights such that each asset contributes equal risk

However, in practice, it is possible for managers to have discretion over thetiming of investment decisions!




Scenario analysisCompute rolling statistics using the roll package and feed into a simplerisk parity strategy:

Instruments: 15 asset classesRisk settings: long-term; exponential decayApproach: passive; equal risk contributionsVolatility target: 10% volatility levelRebalancing: monthly; systematic

Now, against the backdrop of the current environment of low realizedvolatility, consider a few generic scenarios:

Scenario Volatility Correlation

Scenario 1 Average AverageScenario 2 Low LowScenario 3 High High




A toy example: return to average

Simulate scenarios and mechanics of the simple risk parity strategy:

83.4 secs

1 secs0

24

48

71

95

0 250 500 750 1,000


Sec

onds

RRcppParallel

81x over R version!


1.8

2.0

2.2

2.4

2.6

2014 2015 2016 2017 2018

Leve

rage

Volatility Correlation Total

Scenario 1




Explore the scenarios

Parallel computing via RcppParallel helps to dig deeper and faster:

2.1

2.3

2.4

2.6

2016−07 2017−01 2017−07 2018−01

Leve

rage


Scenario 2

2.1

2.3

2.4

2.6

2016−07 2017−01 2017−07 2018−01

Leve

rage


Scenario 3




roll: rolling statistics

Get the released version from CRAN:

install.packages("roll")

Or the development version from GitHub:

# install.packages("devtools")devtools::install_github("jjf234/roll")




Scenario Analysis of Risk Parity using...

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