Question:

Assume we manage to characterize a portfolio as a linear combination of risk factors that are each normally distributed. For example, our portfolio return = aX1 + bX2 + cX3 ... where X1, X2, etc are normally distributed random variables.

(i) Do we typically describe parametric value at risk (VaR) in terms of a PDF, PMF, and/or CDF?
(ii) Can we say anything about the distribution of the portfolio? Under what condition(s)?

Answer:

(i) Value at risk is given by a CDF: P [loss < level ] = significance (which is equal to 1 - confidence). For example,

P [loss < -2.33 standard normal deviations] = 5%

That’s a CUMULATIVE probability distribution.

(ii) (Gujarati p 79) “A linear combination (function) of two (or more) normally distributed random variables is itself normally distributed” So, if our linear combination qualified (don’t hold your breath), it would allow us to treat the PORTFOLIO as normally distributed.

David

When I think of Var I think of the left side of a pdf and that area which is at -2.33 or more.  I have trouble thinking of it as a CDF

Frank, you are right about that. The area *under the curve* and to the left of, say, -2.33 in the PDF, that area becomes the line in the CDF. The pdf only gives you a point estimate, but losses anything less than, so technically “less than” makes it a CDF question but it is the same as you say, the area under the PDF curve. David

Great.  These questions do make you think