Calculation Expecting Credit Loss from a Portfolio

Question

I have the following question:

An investor holds a portfolio of 50 million dollars. This portfolio consists of 'A' rated bonds (30 million dollars) and 'BBB' rated bonds (20 million dollars). Assume that the one-year probabilities of default for 'A' rated and 'BBB' rated bonds are 3 and 5 percent, respectively, and that they are independent. If the recovery value for 'A' rated bonds in the event of default is 70% and the recovery value for 'BBB' rated bonds is 50%, what is the one-year expected credit loss from this portfolio?

How is this calculated with two differently rated bonds?

Answer 1

The expected loss (in dollars) is defined as

$$ \mathbb{E} (L)= \underbrace{PD}_{\text{default probability }} \times \underbrace{LGD}_{\text{loss given default }} \times \underbrace{EAD}_{\text{exposure at default}}$$

For your portfolio, the expected credit loss is \begin{aligned} \mathbb{E} (L_{portfolio}) & {} = \mathbb{E} (L_{A})+\mathbb{E} (L_{BBB}) \\ &{} = PD_A \times LGD_A \times EAD_A+ PD_{BBB} \times LGD_{BBB} \times EAD_{BBB} \\ & {} = 0.03 \times 0.30 \times $30m + 0.05 \times 0.50 \times \\\$ 20m \\ & {} = $0.77mn \\ \end{aligned}