Question 1. Suppose that the default rate of a portfolio has the triangular distribution: πππππ[ππ] = 2 − 2ππ. Suppose that in this portfolio πππ is a function of ππ: πππ[ππ] = ππ1/2. Derive and state the function ππππππ[πππ]. Create a single diagram containing plots of (πππππ[ππ] and ππππππ[πππ]) for variables in the range between 0 and 1.
Question 2. Making the same assumptions as in Question 1, derive and state ππππππ π [πππ π ]. Create a diagram containing the two plots from Question 1 along with the plot of ππππππ π [πππ π ] for variables in the range between 0 and 1; limit the vertical axis to the range from zero to 3. State the values of
• Expected loss, EL
• Expected LGD, ELGD
• “Time-weighted” LGD
3. Express the standard deviation of a Vasicek distribution as an integral that involves the Vasicek
PDF. For distributions with PD = 0.10, numerically integrate and plot the standard deviation for 0.05 < ο² < 0.95. On a separate diagram, plot two Vasicek distributions: PD = 0.10, ο² = 0.05 and PD =
0.10, ο² = 0.95, limiting the vertical axis to {0, 0.12}.
4. Suppose two loans have Vasicek distributions. One loan has PD = 0.06, ο² = 0.06, the second loan has PD = 0.03, ο² = 0.20, and both loans respond to the same systematic risk factor. Plot on a single diagram the two inverse CDFs. At the lower quantiles, the first loan has greater cPD than the second. The situation is reversed at very high quantiles. Estimate the quantile at which both loans have the same value of cPD.
