148 The Basics of financial economeTrics
Applications of Quantile Regressions in Finance
One of the first applications of quantile regressions was in the area of risk
management, especially in the context of the popular risk measure value at
risk (VaR). VaR is a statistical measure that indicates under a certain set of
assumptions the most that a portfolio manager can lose within a reasonable
bound. For example, if the one-month VaR (0.05) of a portfolio is $55 million,
one can infer with 95% confidence that there is only a 5% chance that the
portfolio’s value will decrease more than $55 million in one month. Several
approaches have been proposed for measuring VaR. A quantile regression is
a natural tool to handle VaR problems because this type of regression can be
used to examine the relationship between portfolio returns and its determi-
nants over the entire return distribution. Engle and Manganelli were among
the first to consider the quantile regression for the VaR model.^5 Using daily
stock return data for General Motors, IBM, and the S&P 500, they show that
the tails follow a different behavior than the middle of the distribution. They
conclude that VaR calculations using a quantile regression outperform the
alternate approaches that have been proposed.
There are studies that suggest how quantile regressions can be used to
create portfolios that outperform traditional portfolio construction meth-
odologies. For instance, Ma and Pohlman employ a quantile regression to
forecast returns and construct portfolios.^6 Using time series data on 1,100
individual stock returns, they showed that portfolios constructed based on
return forecasts generated by a quantile regression outperformed the portfo-
lios created based on traditional approaches. Quantile regressions can also
be used to test the performance of a portfolio between higher and lower
return quantiles. For example, Gowland, Xiao, and Zeng, using a sample
of small cap stocks, showed that the performance of a 90th return quantile
differs from that of a 10th return quantile of a portfolio created on the basis
of book-to-market.^7
Below we provide two illustrations of applications to finance in more
detail. In the first, we look at how a quantile regression can be used to
(^5) Robert Engle and Simone Manganelli, “CAViaR: Conditional Autoregressive Value
at Risk by Regression Quantiles,” Journal of Business and Economic Statistics 22
(2004): 367−387.
(^6) Lingjie Ma and Larry Pohlman, “Return Forecasts and Optimal Portfolio Construc-
tion: A Quantile Regression Approach,” European Journal of Finance 14 (2008):
409 −425.
(^7) Chris Gowland, Zhijie Xiao, and Qi Zeng, “Beyond the Central Tendency: Quantile
Regression as a Tool in Quantitative Investing,” Journal of Portfolio Management
35, no. 3 (2009): 106−119.