With the bond investment example, show that the Expected Shortfall risk metric is coherent for both = 0.05 and = 0.03.

- Bond Face value: $1,000 - Default probability: 0.04, loss = N(1000,1) Not default probability: 0.96, loss = N(-50,1) Two Portfolios Pi: Buy two bonds from the same company P2: Buy one bond each from two different companies assuming independent between them - Expected loss is identical E(L) = 0.96(-502) +0.04(1,000 2) --16 E(L) = 0.96*(-502)+2x0.96 x 0.04 x(950) +0.04 (2,000) = -16 Note: S(2x)/(x) dx = S(x)/(x)dx +S(y)f(y)dy = S(x)f(x)dx [ f(y)dy +S(y)f(y)dy f(x)dx = SS (x+y)f(x)f(y)dxdy - Bond Face value: $1,000 - Default probability: 0.04, loss = N(1000,1) Not default probability: 0.96, loss = N(-50,1) Two Portfolios Pi: Buy two bonds from the same company P2: Buy one bond each from two different companies assuming independent between them - Expected loss is identical E(L) = 0.96(-502) +0.04(1,000 2) --16 E(L) = 0.96*(-502)+2x0.96 x 0.04 x(950) +0.04 (2,000) = -16 Note: S(2x)/(x) dx = S(x)/(x)dx +S(y)f(y)dy = S(x)f(x)dx [ f(y)dy +S(y)f(y)dy f(x)dx = SS (x+y)f(x)f(y)dxdy