# Shedding Light Upon Forecasting Correlations Between Different Hedge Fund Strategies

*A review of “Hedge Fund Portfolio Construction: A Comparison of Static & Dynamic Approaches” by Daniel Giamouridis and Ioannis D. Vrontos - Journal of Banking & Finance 31 (2007) 199–217.*

### Introduction

Imagine going to sleep on the 19th of February 2020 and waking up on the 19th of March 2020. The “bad news buzz” from the tv program you watched before bedtime sits in a remote unapproachable part of your brain. You get up, one month later, and while sipping your morning coffee an unexpected notification pops up on your phone. “S&P 500 at $2,409”. The red text impresses itself into your mind. That is, roughly, a 33% decline from what you had gone to sleep with. Before having any more coffee, you decide to put your pajama back on and take another nap.

The article that we are going to review today is very related to the recent, stock market events, but let’s get back to this later.

### About the Article

In 2007, D. Giamouridis and I. D. Vrontos published a very interesting article. Briefly, they wanted to see how the performance of a portfolio could change by using different values of expected volatilities to rebalance in the correct direction.

Now here come a few trickier assumptions:

- The portfolio was rebalanced on a monthly basis.
- The portfolio consisted of a mix of hedge fund indexes.
- The forecasted volatilities in their portfolio were estimated using 5 different models.
- These volatilities were used to analyse the changes, on a monthly basis, of the correlations between different hedge fund indexes.

Why does doing this make sense though?

The answer is that, whenever you add or remove a data point from the data set used to analyse the correlation between two securities, the correlation itself changes. You can see this graphically below:

Here is the Beta of KKR’s stock compared to the S&P 500 on a daily basis from 15/04/2019 to 31/12/2019. Under is the same metric but for a longer time period: from 15/04/2019 to 15/04/2020.

There is a big difference, we can see the beta has changed from 1.34 to 0.84. Now, this data was taken from a very short time period (less than a year, and a year) so it is not the ideal for a beta analysis. What should be observed though, is how big the impact of the Covid-19 stock market fall was on the change in this metric. In the first regression, it’s excluded, while it’s included in the second regression.

Noticing this change is very important in understanding one of the concepts that Giamouridis and Vrontos worked on, and raises a few questions about portfolio rebalancing. Should the covariance between assets be assumed to stay constant? What is the best way to estimate it?

The authors of the paper had more good reasons to find a model that would suit a portfolio of hedge fund indexes. Firstly, they assumed that the spirit of hedge fund investing is quite an active one. Using derivatives, leverage and sophisticated strategies, hedge fund managers invest in a dynamic way. Secondly, the correlations of the financial time series of the fund’s underlying assets are themselves time-varying. Finally, they noticed there was evidence of high “kurtosis^{1}” levels and of “volatility clustering^{2}” in hedge fund returns.

The last concept, that of volatility clustering is particularly true in the current market. Take a look at the changes in the VIX, and see it yourself.

*Chart Showing the evolution of the Vix from 2017 to 2020. Clear clusters can be noticed, for example; in Q1 2018 and Q2020*

The current volatility situation shows us the importance of adjusting volatility expectations when we create our portfolios, even when creating hedge fund portfolios.

### Two Frameworks for Optimal Hedge Fund Portfolios

**I) The Mean-Variance Framework**

Here the authors attempt to create a portfolio that, given a specific expected return, rebalances the hedge fund indexes to minimise the expected variance. Practically, they are saying “let’s find the combination of proportions of the different indexes that creates the lowest variance (or risk), for a certain expected return.”

**II) The Mean-CVaR Framework**

This one is slightly more complex. CVaR is a metric used to measure the tail-risk of the fund. It makes an average of the losses in the tail of the security’s return distribution; we will not dig deeper into this though. The size of the current stock market crash though, shows us how important it is to measure tail-risk of investments.

### Predicting Covariances of Hedge Fund Returns

This is the trickier part. Since the models used are complex statistical machines, we will limit ourselves to an overview of their functionalities.

The models used include: a sample-covariance model (SAM), an implicit factor model (IFAC), an implicit factor GARCH model (IFAC-G), a full factor multivariate GARCH model (FFMG), and a regime-switching dynamic correlations model (RSDC). While we do not have to understand these fully, there are a number of important differences between them that we should know:

- The SAM model is the simplest one and is based on nothing more than the historical covariance to date.
- The IFAC adds a layer of complexity: it tests how much certain variables (from the fund’s return data) impacted its return and uses these to estimate the future return.
- The GARCH based models become more dynamic: they are more flexible and capture several kinds of heteroskedastic
^{3}behaviours. - The most powerful model though resulted in being the RSDC. This model contains multiple structures that can be switched to when at a certain moment in time this structure is most appropriate to fit the data. Given the “extreme” changes in volatility and its clustering property that we saw earlier, it’s easy to understand how we could improve our predictive ability with a more flexible model.
- Understanding which model to use becomes essential when we acknowledge the fact that market’s returns, and often hedge fund returns, do not represent a perfect normal distribution. The recent bear market is an example of this, as well as the clustering property of volatility.

### The Authors’ Datasets

The authors based their portfolio choices off of 8 hedge fund indexes:

- Equity hedge
- Event-driven
- Equity market neutral
- Macro
- Relative Value Arbitrage
- Convertible Arbitrage
- Distressed Securities
- Merger Arbitrage

*Summary statistics of the hedge fund indexes returns used in the study (from 1990 to 2005).*

They also used monthly time series data for the returns and variances of these hedge fund indexes. The dataset is split in two:

- An in-sample dataset (from 1990 to 2001) used to calculate historical trends in order to make first estimates.
- An out-of-sample dataset (from 2002 to 2005) used to assess the performance of the portfolio rebalancing methods they used.

This timeline is very interesting, as many crises happened and impacted the stock market in those years. This creates a good basis to test the more flexible covariance estimators. A few examples are the Mexican crisis, the Asian crisis, the Russian crisis, LTCM’s bailout, and the .com bubble (which can be clearly seen popping under).

*The Dow Jones’ Performance from 1995 to 2005.*

### Hedge Fund Portfolio Performance

To complete the study, the authors of the paper created two portfolios. One more conservative with zero expected return, but simply aiming to reduce volatility to its minimums, and a more aggressive one, aiming for a 15.5% return with minimum volatilities.

They then modelled the covariances between the hedge fund strategies and back-tested these two portfolios, using the various variance forecasts and rebalanced each month in order to achieve minimal volatility levels per unit of return.

The following step was assessing the performance of the portfolios. They did this by constructing a Sharpe Ratio (CSR in the image below) assessing what the best return-to-volatility ratios were. As they had expected, the RSDC model was the best performing one, and the results were significant to a statistical level. This model was also the best in reducing tail risk (through the CVaR metric from earlier).

*We can see how, while the rebalancing strategy that uses the RSDC model to predict covariances between the hedge fund indexes is not the one providing the greatest return in the aggressive portfolio, it is the one that offers, by far, the best Sharpe Ratio in both portfolios.*

### Conclusion

This study proves that certain assumptions made in the world of finance, particularly those used to predict covariance between assets, often are not the best ways of looking at markets. Things change, the world changes, and as they do so do expectations. This results in not only different returns, but also in changes in the quality of these returns. It is therefore very important when one is creating her/his own portfolio, to consider not only current expectations, but also future expectations. This is particularly true for hedge fund investing, where portfolio managers change their approaches actively and with fast dynamics.

It is true that the above-mentioned models might be too complex for the majority of us to use, yet they create a good picture of what it is like to think in market “phases”, to understand changes in financial markets, and to learn to apply greater flexibility when rebalancing portfolios.

^{1}Kurtosis: a measure of how likely it is for the security to have an “extreme” change in price.

^{2}Volatility Clustering: the concept that large changes tend to be followed by large changes, while small changes tend to be followed by small changes.

^{3}Heteroskedasticity: in this case, we’re talking about the fact that returns’ data of a security can have greater/lower variances for different return data of another security.

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