Homework 1
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Homework 1
MAFS6010R- Portfolio Optimization with R
MSc in Financial Mathematics
Fall 2018-19, HKUST, Hong Kong
Prof. Daniel P. Palomar
Hong Kong University of Science and Technology (HKUST)
Shrinkage estimator for \(\boldsymbol{\mu}\)
After Week 5 on shrinkage and the Black-Litterman model, you have learned how to improve the naive
estimation of \(\boldsymbol{\mu}\) and \(\boldsymbol{\Sigma}\) (i.e., the sample mean and sample covariance
matrix, respectively). In particular, we have seen that the estimation error in \(\boldsymbol{\mu}\) is much
more signicative
than that of \(\boldsymbol{\Sigma}\).
The purpose of this homework is to explore the possible improvements on the estimation of \
(\boldsymbol{\mu}\).
Outline
Step 1: Load market data (you can also start with synthetic data, but eventually you need to try with real
market data).
Step 2: Compute the sample mean estimator for \(\boldsymbol{\mu}\) and evaluate its performance by
computing the estimation error compared to the real parameter (in case of synthetic data) or the sample
estimation from the test data (in case of real data).
Step 3: Design the Markowitz mean-variance portfolio based on the sample mean estimator for \
(\boldsymbol{\mu}\) and the sample covariance matrix estimator for \(\boldsymbol{\Sigma}\). Evaluate it and
plot its performance.
Step 4: Consider some way to improve the estimation of \(\boldsymbol{\mu}\). For example, you could try the
James-Stein estimator (but have some imagination for the target) or the Black-Litterman model (but have
some imagination for the views).
Step 5: Evaluate its performance by computing the estimation error compared to the real parameter (in case
of synthetic data) or the sample estimation from the test data (in case of real data). Compare with the
estimation error in Step 2.
Step 6: Design the Markowitz mean-variance portfolio based on your estimator for\(\boldsymbol{\mu}\) and
the sample covariance matrix estimator for \(\boldsymbol{\Sigma}\). Evaluate it and plot its performance.
Compare with the performance in Step 3.
Step 7: Try more ideas. You will get additional points if you can clarify some different methods from class.
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Hint: There are tons of R packages improving estimation for mean and covariance matrix. Figure out what
they are doing by tracking their references. Some prior knowledge like sector information (partially available
via function getSectorInfo() in package covFactorModel ) could be helpful. Clarify your method even if it is
heuristic because we will count if you can make it reasonable.
Format for homeworks in R Markdown
Use the R Markdown (http://rmarkdown.rstudio.com/index.html) format (with le
extension .Rmd) to prepare
your homework. It is an extremely versatile format that allows the combination of formattable text,
mathematics based on Latex codes, R code (or any other language), and then automatic inclusion of the
results from the execution of the code (plots or just other type of output). This type of format also exists for
Python and they are generally referred to as Notebooks and have recently become key in the context of
reproducible research (because anybody can execute the source .Rmd le
and reproduce all the plots and
output). This document that you are now reading is an example of an R Markdown script.
R Markdown les
can be directly created or opened from within RStudio. To compile the source .Rmd le,
just
click the button called Knit and an html will be automatically generated after executing all the chunks of code
(other formats can also be generated like pdf).
The following is a simple template that can be used to prepare the homework and projects in this course:
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---
title: "Title"
subtitle: "Subtitle"
author: "Author"
date: '2018-10-09'
output: html_document
---
Summary of this document here.
# First header
## First subheader
# Second header
* bullet list
* bullet list
- more
- more
This is a link: [R MArkdown tutorial](http://rmarkdown.rstudio.com)
```r
# here some R code
```http://www.daixie0.com/contents/18/1885.html
For more information on the R Markdown formatting:
R Markdown tutorial (http://rmarkdown.rstudio.com)
R Markdown Cheat Sheet (https://www.rstudio.org/links/r_markdown_cheat_sheet)
R Markdown Reference Guide (https://www.rstudio.com/wp-content/uploads/2015/03/rmarkdownreference.pdf)
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