Consider a portfolio of d=100 defaultable corporate bonds. We assume that the defaults of different bonds are independent. The default probability is identical for all bonds and is equal to 1.5%. The current price of the bond is $100. If there is no default, the bond payment after one year is $104; otherwise the payment is $0. Consider portfolio A as 100 units of the first bond while portfolio B as the collection of one unit of each bond. Choose the risk level α=0.05.

What is the Value-at-Risk (VaR) for portfolio B? How about portfolio A?

What is the expected-shortfall (ES) for portfolio B? How about portfolio A?

What if the defaults of different bonds are dependent? We can think about the total number of defaults follows a compounded binomial distribution or so-called beta-binomial distribution. Try different beta-binomial distributions for specific beta distribution parameter α and β so as to get the situation of correlated defaults; then figure out under such parameter setup, how the answers to (a) and (b) change.