Hi Sipani,
Kudos for your careful looks.
Re 4.2: It is to make his key points about recovery rate:
* Goes in cycles
* Average (through the cycle) of about 40% is a decent assumption
* But, most important, highly variable (high standard deviation). Hard to pin down recovery/LGD!
Re 4.6, he doesn’t show he derived (probably a variation on the one factor copula in Hull), but if you start with constant recovery, it is just a plot of (credit) VaR increasing with confidence. Then he compares to same plot but now LGD has a distribution. As EL and UL are function of [PD,LGD], increasing the uncertaintly in LGD is going to increase the unexpected loss; importantly, unexpected loss is CVaR [at some confidence]- EL. This is the point: going from constant LGD (e.g., 40% or 50%) to distributed LGD/recovery (e.g., beta) increases CVaR.
Re the Beta distribution (4.7): it is a bit tricky to interpret the plot of continuous PDF (e.g., normal, beta). Really, the point here is that beta can be shaped in many ways (the “utility” distribution). The x runs from 0 to 100% by definition in the Beta: it’s support is [0,1].
The y-axis, IMO, is not so critical. For a given interval; e.g., [0.4,0.5], then the probability of falling between the interval is here f(x)(0.5-0.4) or y*0.1. Roughly, I am rounding. The area under the interval is the probability, and the total area under the PDF sums to 1.0.
But for exam purposes, I’d offer that the graphs illustrates the key point: the graph is versatile (many shapes) and is a good fit for a parameter (recovery/LGD) that is notoriously difficult to parameterize (plug into a model with an assumption). So, you’ll see the AIMS are appropriately *qualitative* not quantitative in regard to LGD/recovery
David
