Consider a 5-year bond with a face value of $100 that pays an annual coupon of 6% and is currently rated BBB. The expected recovery rate in case of default is 51.13%. In the table below, you are given the probabilities that in one year (i) the bond issuer maintains its BBB rating, (ii) is upgraded to AAA, AA or A or downgraded to BB, B or CCC and (iii) defaults.

Consider also the one-year forward yield curve for zero-coupon bonds with

different maturities and credit ratings (rates are discretely compounded):

a) Compute the volatility of the above bond in one year.

b) Is the volatility a coherent risk measure? Explain.

c) Compute the 1-year Value-at-Risk and Expected Shortfall of the above

bond at a 99% confidence level.

Consider a 5-year bond with a face value of $100 that pays an annual coupon of 6% and is currently rated BBB. The expected recovery rate in case of default is 51.13%. In the table below, you are given the probabilities that in one year (i) the bond issuer maintains its BBB rating, (ii) is upgraded to AAA, AA or A or downgraded to BB, B or CCC and (iii) defaults. Rating Probability (%) AAA 0.02 AA 0.33 A 5.95 BBB 86.93 BB 5.3 B 1.17 CCC 0.12 Default 0.18 Rating AAA AA A BB B CCC 1 year 3.60% 3.65% 3.72% 4.10% 5.55% 6.05% 15.05% Years to Maturity 2 years 4.17% 4.22% 4.32% 4.67% 6.02% 7.02% 15.02% 3 years 4.73% 4.78% 4.93% 5.25% 6.78% 8.03% 14.03% 4 years 5.12% 5.17% 5.32% 5.63% 7.27% 8.52% 13.52% Consider a 5-year bond with a face value of $100 that pays an annual coupon of 6% and is currently rated BBB. The expected recovery rate in case of default is 51.13%. In the table below, you are given the probabilities that in one year (i) the bond issuer maintains its BBB rating, (ii) is upgraded to AAA, AA or A or downgraded to BB, B or CCC and (iii) defaults. Rating Probability (%) AAA 0.02 AA 0.33 A 5.95 BBB 86.93 BB 5.3 B 1.17 CCC 0.12 Default 0.18 Rating AAA AA A BB B CCC 1 year 3.60% 3.65% 3.72% 4.10% 5.55% 6.05% 15.05% Years to Maturity 2 years 4.17% 4.22% 4.32% 4.67% 6.02% 7.02% 15.02% 3 years 4.73% 4.78% 4.93% 5.25% 6.78% 8.03% 14.03% 4 years 5.12% 5.17% 5.32% 5.63% 7.27% 8.52% 13.52%