# Market VaR and Future Returns, from Gabe Carbone

August 14, 2006 |

The idea that higher returns are preceded by periods of higher risk is pretty well known around these parts. This idea was successfully put to the test by the Chair and the Collab in PracSpec with the VIX swing system (p. 109).Value-at-Risk is a popular risk management (or possibly risk description) measure. It is widely used (and abused) in finance. Basically, the (1-day) VaR of a portfolio is the critical value in the left tail of the normal distribution, below which (usually) 5% of daily portfolio returns would fall. The VaR says nothing about the magnitude of the returns that fall below the critical value.

Instead of just using the normal distribution, VaR can be calculated using a Monte Carlo simulation of returns (the generating distribution can be one that is fatter-tailed than a normal one). Also, VaR can be calculated by re-sampling from the empirical distribution of returns, and ranking them according to magnitude. The VaR number then becomes the xth-percentile of the re-sampled returns. This can be repeated many times, and an average VaR number calculated from the many simulations.

I wondered whether the market’s VaR was in any way predictive of its future returns. To answer the question, I calculated the 1-day VaR for the QQQQs (NASDAQ 100 etf), and compared this to the subsequent QQQQ returns. The way I calculated the VaR number for each day was to resample from the empirical distribution of the previous 30-days of one-day returns. I did this 1000 times, and took the average VaR number from each simulation. This was done daily from 26 April 1999 through to June 23. This VaR number was compared to the subsequent QQQQ returns. Note that VaR calculation periods are overlapping.

There seems to be little correlation between VaR and subsequent returns. I looked at both subsequent 1-day returns and subsequent 30-day (overlapping) returns.

Found here is the (heavily commented) Matlab code I used to calculate the VaR numbers, for those who would like to try the same (or try to find errors).