Introduction to Hedge Ratio Estimation Methods

by Joohwan Ko

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Introduction

The hedge ratio estimation problem is one of the most important issues for portfolio managers. The key concept of the hedging problem can be posed as the following equation:

$S_{t}=P_{1, t}+\sum_{n=2}^{N} \omega_{n} P_{n, t}$

where $P_1$ represents the market value at observation $t$ of a portfolio we wish to hedge and $P_n$ represents a set of variables(instruments or portfolios) available for building a hedge. The hedging problem is in computing the vector $w_n$ (holdings of each variable).

Hedging error is denoted as below, which implies the error after $h$ observation. The hedging error is also important in hedge ratio estimation as to whether it is stationary or not results in different results of the methods.

$e(h)=S_{T+h}-S_{T}$

The hedge ratio estimation methods can be divided into two:

Single Period Method
Multi-Period Method

In this blog post, we’ll simply go through the main concepts of each method and closely follow a paper by Lopez de Prado, M.M. and Leinweber, D. (2012). Advances in Cointegration and Subset Correlation Hedging Methods. Therefore, for further details and implementation, we would highly recommend you to read individual papers for each of the methods provided.

Single Period Method

The single period method is also known as the static method as it assumes that the returns are IID(independent and identically distributed random variables). Here, in this blog post, we’ll deal with the three main methods which are widely used in the field.

OLS in Differences (OLSD)
Minimum Variance Portfolio (MVP)
Principal Components Analysis (PCA)

OLS in Differences (OLSD)

Because of its simplicity, this is one of the most widely used methods. As the name implies, it utilized OLS, Ordinary Least Squared, in the differences of market values of portfolios. As the equation shows where $\Delta P$ represents the change in market value between observations, the solution is $w$ equals $-\beta$ .

$\Delta P_{1, t}=\alpha+\sum_{n=2}^{N} \beta_{n} \Delta P_{n, t}+\varepsilon_{t}$

A necessary condition for the method is that the $\alpha$ needs to be statistically insignificant. Despite its simplicity, it still has a lot of limitations. As it has extremely restrictive conditions, it assumes that any change in the target portfolio must be offset by the hedging portfolio.

Minimum Variance Portfolio (MVP)

The method is introduced by Markowitz (1952). The settings for this method are: $\Delta P$ observations are IID Normal, $V$ is the covariance matrix of $\Delta P$ , where its first column represents the covariances against the portfolio we wish to hedge( $P_1$ ). The main objective function is the equation below.

$\begin{array}{l}\operatorname{Min}_{\beta} \quad \beta^{\prime} V \beta\\\text { s.t. } \quad \beta^{\prime} a=1\end{array}$

This optimization problem can be solved by using Lagrangian and the optimal beta and weights are as follows:

$L(\beta, \lambda)=\frac{1}{2} \beta^{\prime} V \beta-\lambda\left(\beta^{\prime} a-1\right)$

$\beta=\frac{V^{-1} a}{a^{\prime} V^{-1} a} \quad \omega_{j}=\frac{\beta_{j}}{\beta_{1}}$

It is worth noting that the MVP method delivers the minimum risk solution under the assumption of Normality, but beyond that assumption, a number of alternative objective functions could be chosen as well.

Principal Components Analysis (PCA)

The last method in a single period is Principal Components Analysis. Here, the basic settings are similar to the MVP method. The $\Delta P$ is assumed to be IID Normal. The target is to compute the vector of weightings beta such that $\Delta P$ times $\beta$ is hedged against moves of the $m$ largest principal components where $m$ is typically $N-1$ .

The goal of this method is to find $\beta$ which follows the equation below where $(W^{*})^{'}$ is the transposed eigenvector matrix after removing the columns associated with the unhedged eigenvectors. This approach shows the advantage of searching for a solution that hedges against the principal sources of risk.
Like the two methods we have discussed above, it doesn’t guarantee that the source of risk is stationary. It could be argued however that, having the smallest variance (in differences), the stationarity of the eigenvectors with the smallest eigenvalues is not a major issue as the components we remain exposed to have the smallest variance in differences. This makes PCA a valid method of hedging estimation.

Multi-Period Method

The multi period method is also known as the dynamic method as it does not assume IID random perturbations for the returns. Here are the two most popular methods in the multi period.

OLS in Levels (OLSL)
Error Correction Model (ECM)

OLS in Levels (OLSL)

First, let’s take a look at OLSL. The goal itself is similar to OLSD method but the condition that the hedge is effective when $S$ is stationary in mean and variance gives a different approach to OLSD. However, as the error correction component is not separated from the observed levels in the equation, the calculated weight, $\beta$ , may not be the optimal solution to the problem in the OLSD.

$P_{1, t}=\sum_{n=2}^{N} \beta_{n} P_{n, t}+S_{t}$

In other words, in the OLSD, the hedging errors are not corrected over time. This is because of the specification in differences, which does not contain any memory of the process as it goes. Although OLSL corrects the error as time goes as it does not use differences, it still does not separate the terms of error correction in the estimation method. This is the main reason why ECM has become popular after this.

Error Correction Model (ECM)

Next is the error correction model, ECM. This model improves some limitations of the previous methods as it corrects errors as time goes. Although this model only shows a hedge ratio between two portfolios, the extension of this method will be more discussed in the advanced methods later. In the equation, $p$ is the natural log of market value $P$ and $\gamma$ has to be tested positive in order to be effectively hedge.

$\Delta p_{1, t}=\beta_{0}+\beta_{1} \Delta p_{2, t}+\gamma\left(p_{2, t-1}-p_{1, t-1}\right)+\varepsilon_{t}$

The optimal holdings will be as follows where $K$ equals e to the $\beta_0$ divided by $\gamma$ .

$\left(\omega_{1}, \omega_{2}\right)=(1,-K)$

Advanced Methods

We have discussed some of the basic methods in hedge ratio estimation. However, as there are hundreds of researches done regarding hedge ratio estimations, we’ll see two more methods which are classified as “advanced” methods.

Box-Tiao Canonical Decomposition (BTCD)
Dickey-Fuller Optimal (DFO)

Box-Tiao Canonical Decomposition (BTCD)

First, Box and Tiao introduced a canonical transformation of an N-dimensional stationary autoregressive process. The components of the transformed process can then be ordered from least to most predictable according to the research by Box and Tiao.

The estimation of this method goes as follows. For the VAR(L) equation, which is called as forecasting equation, this method fits $\beta$ and estimates $\hat{P_t}$ from the $beta$ . With the estimated $P_t$ , it undergoes decomposition process and solves for optimal weight. In short, the objective is to come up with the matrix of coefficients that deliver a vector of forecasts with the most predictive power over the next observation.

$\sum_{l=1}^{L} \sum_{i=1}^{N} \beta_{i, l, n} P_{t-l, i}+\beta_{n, 0} X_{t-1, n}+\varepsilon_{t, n}$

The whole process of BTCD is very complicated as it can be another whole blog post, please take a look at the original paper if you are interested!

Dickey-Fuller Optimal (DFO)

In the multi period method, the ECM is a dynamic model limited to two dimensions and we have mentioned that we’ll see the expansion of the method. This limitation could be solved through a canonical transformation of a multivariate, multi-equation specification as we did for BTCD and it is called Dickey-Fuller Optimal(DFO).

$S_{t}=P_{1, t}+\sum_{n=2}^{N} \omega_{n} P_{n, t}$

This approach will give a stronger structure than ECM through a system of equations with multiple variables, each of them with an individual autoregressive equilibrium condition. The target of this method is to find an optimal $w$ where the probability of having a unit root in the spread( $S$ ) is minimized. In short, this can be understood as multi-dimensional version of ECM.

This is the end of the brief introduction to hedge ratio estimation! We know this is just a simple overview of the few concepts but more details for each of the methods will be discussed in the later blog posts.