What is a conditional quantile function? What is 

conditional quantile of $Y(t)$ given $Y(t-1)$ where $Y(t)$ is a univariate time series 

(they call it conditional value at risk in risk management).
Can anybody explain this?
Thanks
 A: Let's take the median, which is the 0.5 quantile, and is a kind of "typical middle value". The median is the value $Q_{.5}(Y)$ such that an equal number of observation are less than and greater than the value (for an odd sample size), or the average of the two central values (for an even sample size). For example, the median of $\{99,100,101\}$ is $100$. For a symmetric distribution, the mean and the median will be the same, but generally they are not equal. For financial data, I would wager that the median is usually lower since it is less influenced by extreme values. The unconditional median is the answer to the question what is the typical value of of my time series $Y(t)$. For example, what is the typical middle weekend revenue of a movie theatre. The average may be quite different.
The conditional median answers the question what is typical value of $Y(t)$ if I know the value of $Y(t-1)$. For instance, if $Y(t-1)$ was high and the time series had some persistence, you might guess that the conditional median would be higher than the unconditional one. One example might be the typical weekend earnings of a movie theatre if a popular blockbuster opened the week before. You can also condition on other factors, such as $t$ falling on holiday weekend or rain, and not just previous values.
You can perform similar calculations at other parts of the distribution. For example, what about the revenue of a movie theatre that is better than typical, say in the $.8$ quantile?  
In risk management, you are presumably interested in what your losses might be in the tails of the distribution and not just the typical middle, so conditional quantiles offer a way to look at the worst case scenarios. 
