If you are reading this, you already know that the covariance matrix represents **unconditional** linear dependency between the variables. Far less mentioned is the bewitching fact that the elements of the *inverse* of the covariance matrix (i.e. the precision matrix) encode the **conditional** linear dependence between the variables. This post shows why that is the case. I start with the motivation to even discuss this, then the math, then some code.

# Tag: Portfolio consturction

## Statistical Shrinkage (4) – Covariance estimation

A common issue encountered in modern statistics involves the inversion of a matrix. For example, when your data is sick with multicollinearity your estimates for the regression coefficient can bounce all over the place.

In finance we use the covariance matrix as an input for portfolio construction. Analogous to the fact that variance must be positive, covariance matrix must be positive definite to be meaningful. The focus of this post is on understanding the underlying issues with an unstable covariance matrix, identifying a practical solution for such an instability, and connecting that solution to the all-important concept of statistical shrinkage. I present a strong link between the following three concepts: regularization of the covariance matrix, ridge regression, and measurement error bias, with some easy-to-follow math.

## Beta in the tails

Every form of strength is also a form of weakness^{*}. I love statistics, but I focus to much on methodology, which is not for everyone. Some people (right or wrong) question: “wonderful sir, but what can I do with it?”.

A new paper titled *“Beta in the tails”* is a showcase application for why we should focus on correlation structure rather than on average correlation. They discuss the question: *Do hedge funds hedge?* The reply: No, they don’t!

The paper *“Beta in the tails”* was published in the *Journal of Econometrics* but you can find a link to a working paper version below. We start with a figure replicated from the paper, go through the meaning and interpretation of it, and explain the methods used thereafter.

## Correlation and correlation structure (4) – asymmetric correlations of equity portfolios

Here I share a refreshing idea from the paper “Asymmetric correlations of equity portfolios” which was published in the *Journal of financial Economics*, a top tier journal in this field. The question is how much the observed conditional correlation on the downside (say) differs from the conditional correlation you would expect from a symmetrical distribution. You can find here an explanation for the H-statistic developed in the aforementioned paper and some code for illustration.

## Portfolio Construction Tilting towards Higher Moments

When you build your portfolio you must decide what is your risk profile. A pension fund’s risk profile is different than that of a hedge fund, which is different than that of a family office. Everyone’s goal is to maximize returns given the risk. Sinfully but commonly risk is defined as the variability in the portfolio, and so we feed our expected returns and expected risk to some optimization procedure in order to find the optimal portfolio weights. Risk serves as a decision variable. You choose the risk, and (hope to) get the returns.

A new paper from Kris Boudt, Dries Cornilly, Frederiek Van Hollee and Joeri Willems titled Algorithmic Portfolio Tilting to Harvest Higher Moment Gains makes good progress in terms of our definition of risk, and risk-return trade-off. They propose a quantified way in which you can adjust your portfolio to account not only for the variance, but also for higher moments, namely skewness and kurtosis. They do that in two steps. The first is to simply set your portfolio based on whichever approach you follow (e.g. minvol, equal risk contribution or other). In the second step you tilt the portfolio such that the higher moments are brought into focus and get the attention they deserve. This is done by deviating from the original optimization target so that higher moments are utility-improved: less variance, better skew and lower kurtosis.

## Curse of dimensionality part 3: Higher-Order Comoments

Higher moments such as Skewness and Kurtosis are not as explored as they should be.

These moments are crucial for managing portfolio risk. At least as important as volatility, if not more. Skewness relates to asymmetry risk and Kurtosis relates to tail risk.

Despite their great importance, those higher moments enjoy only a small portion of attention compared with their lower more friendly moments: the mean and the variance. In my opinion, one reason for this may be the impossibility of estimating those moments, estimating them accurately that is.

It is yet another situation where Curse of Dimensonality rears its enchanting head (and an idea for a post is born..).

## Portfolio Construction with R

## Preview

Constructing a portfolio means allocating your money between few chosen assets. The simplest thing you can do is evenly split your money between few chosen assets. Simple as it is, good research shows it is just fine, and even better than other more sophisticated methods (for example Optimal Versus Naive Diversification: How Inefficient is the 1/N). However, there is also good research that declares the opposite (for example Large Dynamic Covariance Matrices) so go figure.

Anyway, this post shows a few of the most common to build a portfolio. We will discuss portfolios which are optimized for:

- Equal Risk Contribution
- Global Minimum Variance
- Minimum Tail-Dependence
- Most Diversified
- Equal weights

We will optimize based on half the sample and see out-of-sample results in the second half. Simply speaking, how those portfolios have performed.

## On the 60/40 portfolio mix

Not sure why is that, but traditionally we consider 60% stocks and 40% bonds to be a good portfolio mix. One which strikes decent balance between risk and return. I don’t want to blubber here about the notion of risk. However, I do note that I feel uncomfortable interchanging risk with volatility as we most often do. I am not unhappy with volatility, I am unhappy with realized loss, that is decidedly not the same thing. Not to mention volatility does not have to be to the downside (though I just did).

Let’s take a look at this 60/40 mix more closely.

## Multivariate volatility forecasting, part 2 – equicorrelation

Last time we showed how to estimate a CCC and DCC volatility model. Here I describe an advancement labored by Engle and Kelly (2012) bearing the name: *Dynamic equicorrelation*. The idea is nice and the paper is well written.

Departing where the previous post ended, once we have (say) the DCC estimates, instead of letting the variance-covariance matrix be, we force some structure by way of averaging correlation **across** assets. Generally speaking, correlation estimates are greasy even without any breaks in dynamics, so I think forcing some structure is for the better.

## Linking backtesting with multiple testing

The other day, Harvey Campbell from Duke University gave a talk where I work. The talk- bearing the exciting name “Backtesting” was based on a paper by the same name.

The authors tackle the important problem of data-snooping; we need to account for the fact that we conducted many trials until we found a strategy (or a variable) that ‘works’. Accessible explanations can be found here and here. In this day and age, the ‘story’ behind what you are doing is more important than ever, given the things you can do using your desktop/laptop.

## Volatility forecast evaluation in R

In portfolio management, risk management and derivative pricing, volatility plays an important role. So important in fact that you can find more volatility models than you can handle (Wikipedia link). What follows is to check how well each model performs, in and out of sample. Here are three simple things you can do: