Introduction to R for Quantitative Finance: R is a statistical computing language that's ideal for answering quantitative finance questions. This book gives you both theory and practice, all in clear language with stacks of real-world examples. Ideal for R beginners or expert alike.

The Mean-Variance model by Markowitz (Markowitz, H.M. (March 1952)) is practically the ice-cream/umbrella business in higher dimensions. For the mathematical formulation, we need some definitions.

They are explained as follows:

Let be the random return variables with a finite variance, be their covariance matrix, and .

We will focus on the following optimization problems:

It is clear that is the variance of the portfolio and is the expected...

In the last 50 years, many great algorithms have been developed for numerical optimization and these algorithms work well, especially in case of quadratic functions. As we have seen in the previous section, we only have quadratic functions and constraints; so these methods (that are implemented in R as well) can be used in the worst case scenarios (if there is nothing better).

However, a detailed discussion of numerical optimization is out of the scope of this book. Fortunately, in the special case of linear and quadratic functions and constraints, these methods are unnecessary; we can use the Lagrange theorem from the 18th century.

It is useful to know that portfolio optimization is totally integrated in various R packages that we will discuss later. However, it's better to walk before we run; so let's start with a simple self-made R function that we would also itemize line by line as follows:

minvariance <- function(assets, mu = 0.005) {
    return  <- log(tail(assets, -1) / head(assets, -1))
    Q       <- rbind(cov(return), rep(1, ncol(assets)),
               colMeans(return))
    Q       <- cbind(Q, rbind(t(tail(Q, 2)), matrix(0, 2, 2)))
    b       <- c(rep(0, ncol(assets)), 1, mu)
    solve(Q, b)
}

This is a direct implementation of the algorithm that we discussed in the Theorem (Lagrange) section.

For demonstration purposes, we have fetched some IT stock prices from a Quandl superset (http://www.quandl.com/USER_1KR/1KT), which is a public service providing an easy access to a large amount of quant data. Although the URL points to a downloadable comma-separated values (CSV...

What happens when a riskless asset is added to the model? If and X is any risky portfolio, then and obviously, . This means that those portfolios form a straight line on the mean-standard deviation plane. Any portfolio on this line is available by investing into R and X. It is clear that the best choice for X is the point where this line is tangent to Efficient Frontier. This tangency point is called the market portfolio or tangency portfolio, and the tangent of Efficient Frontier of risky assets at this point is called Capital Market Line (CML), which consists of the efficient portfolios of all the assets in this case. The last question that we address regarding the mean-variance model is how the market portfolio (or equivalently, the CML) can be determined.

We can easily modify the variance minimization code to accomplish this. First of all, if we add a riskless asset, a full-zero row and column is added to the covariance matrix (where n is...

When we optimize a portfolio, we don't have the real covariance matrix and the expected return vector (that are the inputs of the mean-variance model); we use observations to estimate them, so Q, r, and the output of the model are also random variables.

Without going into the details, we can say that this leads to surprisingly great uncertainty in the model. In spite of the strong law of large numbers, optimal portfolio weights sometimes vary between . Fortunately, if we have a few years' data (daily returns), the relative error of the measured risk is only 20-25 %.

Variance as a risk measure is convenient, but has some drawbacks. For instance, when using variance, positive changes in the return can be considered as the increase of risk. Therefore, more sophisticated risk measures have been developed.

For example, see the following short demo about various methods applied against the previously described IT_return assets for a quick overview about the options provided by the fPortfolio package:

> Spec <- portfolioSpec()
> setSolver(Spec) <- "solveRshortExact"
> setTargetReturn(Spec) <- mean(colMeans(IT_return))
> efficientPortfolio(IT_return, Spec, 'Short')
> minvariancePortfolio(IT_return, Spec, 'Short')
> minriskPortfolio(IT_return, Spec)
> maxreturnPortfolio(IT_return, Spec)

These R expressions return different portfolio weights computed by various methods not discussed in this introductory chapter. Please refer to the package bundled documentation, such as ?portfolio, and the relevant articles...

This chapter covered portfolio optimization. After presenting the main idea, we introduced the Markowitz model and its mathematical formulation. We discussed the methods for possible solutions and implemented a simple algorithm to demonstrate how these methods work on real data. We have also used pre-written R packages to solve the same problem. We broadly discussed other important subjects like the market portfolio, the uncertainty in the estimation of the covariance matrix, and the risk measures beyond variance. We hope that this was a useful first run on the topic and you are encouraged to study it further or check out the next chapter, which is about a related subject—asset pricing models.