The question:

A credit asset has a principal value of $6.0 with probability of default (PD) of 3.0% and a loss given default (LGD) characterized by the following probability density function (pdf): f(x) = x/18 such that 0 ≤ x ≤ $6. Let expected loss (EL) = E[PD*LGD]. If PD and LGD are independent, what is the asset's expected loss? (note: why does independence matter?)

Answer:

If PD and LGD are not independent, then E[PD*LGD] <> E(PD) * E(LGD); for example, if they are positively correlated, then E[PD*LGD] > E(PD) * E(LGD).

For the E[LGD], we integrate the pdf: if f(x) = x/18 s.t. 0 < x < $6, then F'(x) = (1/18)*(1/2)*x^2 = x^2/36

(note this satisfied the definition of a probability over the domain (0,6) as 6^2/36 = 1.0).

The mean of f(x) integrates xf(x) where xf(x) = x*x/18 = x^2/18, which integrates to 1/18*(x^3/3) = x^3/54, so E[LGD] = 6^3/54 = $4.0.

E[PD * LGD] = 3.0%*$4.0 = $0.120.

I'm confused as to why we need to integrate. Isn't f(x) the probability by itself?

Thanks!