The yearly R in Finance conference is one of my favorites:
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..).
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.
Is bitcoin a bubble? I don’t know. What defines a bubble? The price should drastically overestimate the underlying fundamentals. I simply don’t know much about blockchain to have an opinion there. A related characteristic is a run-away price. Going up fast just because it is going up fast.
Bitcoin is a cryptocurrency created in 2008. I have never belonged with team “gets it” when it comes to Bitcoin investing, but perhaps time has come to reconsider.
Tail risk conventionally refers to the risk of a large and sharp draw down of the portfolio. How large is subjective and depends on how you define what is a tail.
A lot of research is directed towards having a good estimate of the tail risk. Some fairly new research also now indicates that investors perceive tail risk to be a stand-alone risk to be compensated for, rather than bundled together with the usual variability of the portfolio. So this risk now gets even more attention.
In trading and in trading-related research one could be quickly overwhelmed with the sea of ink devoted to trading strategies and the like. It is essential that you “pick your battles” so to speak. I recently finished reading Machine Trading, by Ernest Chan. Here is what I think about the book.
Density estimation belongs with the literature of non-parametric statistics. Using simple bootstrapping techniques we can obtain confidence intervals (CI) for the whole density curve. Here is a quick and easy way to obtain CI’s for different risk measures (VaR, expected shortfall) and using what follows, you can answer all kind of relevant questions.
The evaluation of volatility models is gracefully complicated by the fact that, unlike other time series, even the realization is not observable. Two researchers would never disagree about what was yesterday’s stock price, but they can easily disagree about what was yesterday’s stock volatility. Because we don’t observe volatility directly, each of us uses own proxy of choice. There are many ways to skin this cat (more on volatility proxy here).
In a previous post Univariate volatility forecast evaluation we considered common ways in which we can evaluate how good is our volatility model, dealing with one time-series at a time. But how do we evaluate, or compare two models in a multivariate settings, with two covariance matrices?
You probably know that even a trading strategy which is actually no different from a random walk (RW henceforth) can perform very well. Perhaps you chalk it up to short-run volatility. But in fact there is a deeper reason for this to happen, in force. If you insist on using and continuously testing a RW strategy, you will find, at some point with certainty, that it has significant outperformance.
This post explains why is that.
GARCH models are very responsive in the sense that they allow the fit of the model to adjust rather quickly with incoming observations. However, this adjustment depends on the parameters of the model, and those may not be constant. Parameters’ estimation of a GARCH process is not as quick as those of say, simple regression, especially for a multivariate case. Because of that, I think, the literature on time-varying GARCH is not yet at its full speed. This post makes the point that there is a need for such a class of models. I demonstrate this by looking at the parameters of Threshold-GARCH model (aka GJR GARCH), before and after the 2008 crisis. In addition, you can learn how to make inference on GARCH parameters without relying on asymptotic normality, i.e. using bootstrap.
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.
First things first.
What do we mean by sparse estimation?
Sparse – thinly scattered or distributed; not thick or dense.
The term ‘curse of dimensionality’ is now standard in advanced statistical courses, and refers to the disproportional increase in data which is needed to allow only slightly more complex models. This is true in high-dimensional settings. Here is an illustration of the ‘Curse of dimensionality’ in action.
In multivariate volatility forecasting (4), we saw how to create a covariance matrix which is driven by few principal components, rather than a complete set of tickers. The advantages of using such factor volatility models are plentiful.