1. Asset return oments
Recall the diagram from the chapter 1 slide with the representation of a portfolio optimization framework.
2. Optimization Inputs
The inputs of the portfolio optimization problem are the assets, constraints, objectives, and asset returns. Generally, the universe of assets, constraints, and objectives are defined by the portfolio manager or the fund mandate. There is no uncertainty in these inputs. However, the moments of asset returns must be estimated. In theory, better estimates lead to better results. It is important to understand that estimating values introduces estimation error and model risk.
3. Asset return moments
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.
4. Asset return moments
The objectives and constraints defined in the portfolio problem determine which moments must be estimated. For example, a portfolio optimization problem in the mean-variance framework to maximize mean return and minimize portfolio standard deviation requires estimates for the both the expected returns vector and covariance matrix. However, a problem to minimize portfolio variance only requires an estimate of the covariance matrix.
5. Asset return moment estimates
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. The other methods shown here attempt to address the issues of sample estimates by reducing estimation error and imposing structure to reduce the number of values to estimate. For example, a 20 asset portfolio 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. In this course, you will work through portfolio optimization problems using sample, factor model, and robust statistics to estimate portfolio moments.
6. Calculating moments in PortfolioAnalytics
The PortfolioAnalytics package supports several methods for calculating moments of the asset returns. You will use the set-dot-portfolio-dot-moments function which calculates moments and supports the following methods: sample, Boudt, Black-Litterman, and Meucci. You can specify which to use with the method argument. The default behavior is to use sample estimates for the moments. Sample estimates are the most basic method of estimating moments. More advanced methods include factor models or models for expressing views such as the Black-Litterman or Meucci models. You will learn how to set the portfolio moments using sample estimates as well as with a statistical factor model using what we refer to as the Boudt method.
7. Example: moments in PortfolioAnalytics
This slide demonstrates using set-dot-portfolio-dot-moments with the default behavior as well as a more advanced technique by specifying method = "boudt" to calculate the asset return moments. For this example, you are just interested in calculating the moments to learn about the set-dot-portfolio-dot-moments function. In later exercises and videos, you will learn how to use these in an optimization. This example uses the first 4 columns of the edhec data as the asset returns and a portfolio specification with an objective to minimize portfolio standard deviation. The set-dot-portfolio-dot-moments function needs the asset returns, portfolio specification, and optional arguments depending on the method to calculate the moments. The default behavior is to calculate moments using the sample method so you only need to pass in the asset returns and portfolio specification. The Boudt method calculates moments by fitting a statistical factor model with k factors. Using a factor model is a way of imposing structure on the variance-covariance matrix. To use the Boudt method with 1 factor, you call set-dot-portfolio-dot-moments and specify method="boudt" and k=1.
8. Example: moments in PortfolioAnalytics
This slide shows the output of the variance-covariance matrix estimates using the two methods from the previous slide. The set.portfolio.moments function returns a named list so you can extract the second moment with dollar sign sigma as shown in the slide. Note that for the purpose of display on the slide, the values are rounded to 6 digits. Now you know how to estimate moments using the sample method as well as more advanced methods. You also know how to extract those estimates for further analysis.
9. Let's practice!
Let's move on to the exercises.