Asset return oments

Intermediate Portfolio Analysis in R

Ross Bennett

Instructor

Optimization Inputs

Portfolio optimization problem inputs:

Assets
Constraints
Objectives
Moments of asset returns

Asset return moments

First Moment: expected returns vector
Second Moment: variance-covariance matrix
Third Moment: coskewness matrix
Fourth Moment: cokurtosis matrix

Asset return moments

Moments to estimate are determined by objectives and constraints:

Mean - Variance
- Expected returns vector
- Covariance matrix
Minimum Variance
- Covariance matrix

Asset return moment estimates

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."

Methods:
- Sample
- Shrinkage estimators
- Factor model
- Expressing views
- Robust statistics

20 Asset Portfolio:

Method	Sample	k = 3 factors
# of parameters	210	86

Calculating moments in PortfolioAnalytics

set.portfolio.moments(R, 
                      portfolio,
                      method = c("sample", "boudt", "black_litterman", "meucci"),   
                      ...)

set.portfolio.moments() supports several methods:

Sample
Boudt
Black-Litterman
Meucci

Example: moments in PortfolioAnalytics

# Sample vs Boudt
sample_moments <- set.portfolio.moments(R = asset_returns,
                                        portfolio = port_spec)

boudt_moments <- set.portfolio.moments(R = asset_returns,
                                       portfolio = port_spec,
                                       method = "boudt",
                                       k = 1)

Example: moments in PortfolioAnalytics

round(sample_moments$sigma, 6)

          [,1]      [,2]      [,3]  ...
[1,]  0.000402 -0.000034  0.000262  ...
[2,] -0.000034  0.000632 -0.000037  ...
[3,]  0.000262 -0.000037  0.000337  ...
[4,]  0.000429 -0.000010  0.000568  ...

round(boudt_moments$sigma, 6)

          [,1]      [,2]      [,3]  ...
[1,]  0.000403 -0.000016  0.000224  ...
[2,] -0.000016  0.000636 -0.000019  ...
[3,]  0.000224 -0.000019  0.000337  ...
[4,]  0.000523 -0.000044  0.000614  ...

Let's practice!

Intermediate Portfolio Analysis in R