Conditional quantile models for asset returns

• Thesis title in Czech: Podmíněné kvantilové modely pro výnosy aktiv
• Thesis title in English: Conditional quantile models for asset returns
• Academic year of topic announcement: 2018/2019
• Thesis type: Bachelor's thesis
• Thesis language: angličtina
• Department: Institute of Economic Studies (23-IES)
• Supervisor: doc. PhDr. Jozef Baruník, Ph.D.
• Author: hidden - assigned by the advisor
• Date of registration: 10.05.2019
• Date of assignment: 10.05.2019
• Date and time of defence: 08.09.2020 09:00
• Venue of defence: Opletalova - Opletalova 26, O206, Opletalova - místn. č. 206
• Date of electronic submission: 31.07.2020
• Date of proceeded defence: 08.09.2020
• Opponents: Mgr. Nicolas Fanta

References

• Christoffersen, P., Hahn, J. and Inoue A. 2001 Testing and comparing Value-at-Risk measures. Journal of Empirical Finance 8:325–342
• Engle, R.F. and S. Manganelli. 2004. Caviar: Conditional Autoregressive Value at Risk by Regression Quantiles. Journal of Business & Economic Statistics 22:367-381
• Engle, R.F. and S. Manganelli. 2001. Value at Risk Models in Finance. ECB Working Paper No. 75
• Christoffersen, P. F., Hahn, J., and Inoue, A. 2001. Testing and Comparing Value-at-Risk Measures. Journal of Empirical Finance 8:325-342.
• Koenker, R. and Bassett, G. 1978. Regression Quantiles. Econometrica 46:33-50.
• Koenker, R. and Xiao, Z. 2002. Quantile Autoregression. Journal of the American Statistical Association 101:980-990
• Tibshirani, R. 1996. Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society 58:267-288
• Žikeš F. and Baruník J. 2016 Semi-parametric Conditional Quantile Models for Financial Returns and Realized Volatility. Journal of Financial Econometrics 14:185–226

Preliminary scope of work in English

Research question and motivation

The research question I intend to answer is whether behavior of future stock returns at specific quantile can be written as a linear function of remaining stocks from a S&P 500 index. In other words, if there are some specific groups of stocks that tend to mode together or vice versa. We anticipate finding some connection because we expect quantiles of the asset to be influenced by other assets. Finding this linear dependence among equities in a portfolio can help us explain conditional behavior at the tails of return distribution.
The quantile regression (QR) was first proposed by Koenker and Basset (1978). Where the basic ordinary least square method provides an estimate of the conditional mean of the endogenous variable, the quantile regression estimates the various conditional quantiles of interest directly. Quantile regression provide us an appropriate methodology to estimate value at risk, since it can be viewed as a conditional function of a given return series.
Value at risk (VaR) is an estimate of how much a certain portfolio can lose within a given time period and at a given confidence level. Since its introduction in late 1980s it has become a standard measure of risk for financial institutions or regulators. Mainly due its simple interpretation and easy usage among wide class of assets. Although the concept of VaR is quite straightforward none of the methodologies developed so far yields satisfactory results (Engle and Manganelli 2004). Therefore, endeavor to find proper methodology behind VaR is significant for future risk management.


Contribution

Papers written regarding estimation of VaR via quantile regression tried to explain future returns by past variation and endogenous variables (e.g. Christoffersen at al. 2001). In my paper I would like to extrapolate price changes of remaining shares from endogenous variables and use them as regressors. With this approach we can investigate whether volatility of one stock can be attributable to the price changes of the remaining shares in the S&P 500 index as well as to the past volatility. This will help us to clarify behavior of shares during rare market conditions with increased volatility.
Methodology
I use dataset of daily closing prices of stock in the S&P 500 index for the past 10 years. After estimatimation of daily volatility we run regression as proposed above. We enhance prediction with regression shrinkage via the lasso. The nature of this constraint tends to produce coefficients that are exactly 0. That helps us increase accuracy of the modeland makes it more interpretable (Tibshirani 1996).

Outline
1. Introduction
2. Theoretical Framework
a. Literature review
b. Value at risk
i. Parametric approach
ii. Historical approach
iii. Quantile regression approach
c. Shrinkage methods
i. Ridge
ii. Lasso
3. Empirical research
a. Data description
b. Elaboration on model used
c. Regression results
d. Backtesting
4. Conclusion