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st s t ts rt s Prts t r t sr

  1. ❍♦✇ ❘♦❜✉st ✐s t❤❡ ❱❛❧✉❡✲❛t✲❘✐s❦ ♦❢ ❈r❡❞✐t ❘✐s❦ P♦rt❢♦❧✐♦s❄ ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ▲✳ ❘üs❝❤❡♥❞♦r❢✱ ❙✳ ❱❛♥❞✉✛❡❧✱ ❏✳ ❨❛♦✮ ❈❛r♦❧❡ ❇❡r♥❛r❞✱ ●r❡♥♦❜❧❡ ❊❝♦❧❡ ❞❡ ▼❛♥❛❣❡♠❡♥t ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛②✱ ✸✵✳✵✹✳✷✵✶✺ ✶

  2. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❆❣❡♥❞❛ ❇❛❝❦❣r♦✉♥❞ ✶ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s ✷ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ ❉❡♣❡♥❞❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ✸ ❆♣♣r♦①✐♠❛t❡ ❱❛❘ ❇♦✉♥❞s ✹ ❈❛s❡ ❙t✉❞②✿ ❈r❡❞✐t ❘✐s❦ P♦rt❢♦❧✐♦ ✺ ✷

  3. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞ ❆❣❡♥❞❛ ❇❛❝❦❣r♦✉♥❞ ✶ ▲✐t❡r❛t✉r❡ ❖❜s❡r✈❛t✐♦♥s ✸

  4. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ❈r❡❞✐t r✐s❦ ♠❛♥❛❣❡♠❡♥t ✶ ▼❛♥❛❣❡♠❡♥t ♦❢ ❝r❡❞✐t r✐s❦ ✐s ♦❢ ✉t♠♦st ✐♠♣♦rt❛♥❝❡ ✭❈r✐s✐s ✷✵✵✽✮✳ ✷ P♦rt❢♦❧✐♦ ♠♦❞❡❧s ❛r❡ s✉❜❥❡❝t t♦ s✐❣♥✐✜❝❛♥t ♠♦❞❡❧ ✉♥❝❡rt❛✐♥t② ✭ ❞❡❢❛✉❧ts ❛r❡ r❛r❡ ❛♥❞ ❝♦rr❡❧❛t❡❞ ❡✈❡♥ts ✮✳ ✸ ❘❡❝❡♥t st✉❞✐❡s ✭❊♠❜r❡❝❤ts ❡t ❛❧✳ ✭✷✵✶✸✱✷✵✶✹✮✮ s❤♦✇ t❤❛t t❤❡ ✐♠♣❛❝t ♦❢ ♠♦❞❡❧ ✉♥❝❡rt❛✐♥t② ♦♥ ❱❛❧✉❡✲❛t✲❘✐s❦ ✭❱❛❘✮ ❡st✐♠❛t❡s ✐s ❤✉❣❡✳ ✹

  5. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ❈r❡❞✐t r✐s❦ ♠❛♥❛❣❡♠❡♥t✿ ◆♦t❛t✐♦♥ • n ✐♥❞✐✈✐❞✉❛❧ r✐s❦s ( L ✶ , L ✷ , ..., L n ) ✭r✐s❦② ❧♦❛♥s✮ • ❆ ♣♦rt❢♦❧✐♦ S := L ✶ + ... + L n • ❱❛❧✉❡✲❛t✲❘✐s❦ ♦❢ S ❛t ❧❡✈❡❧ q ∈ ( ✵ , ✶ ) ❱❛❘ q ( S ) = F − ✶ S ( q ) = ✐♥❢ { x ∈ R | F S ( x ) ≥ q } ✺

  6. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ▼♦t✐✈❛t✐♦♥ ♦♥ ❱❛❘ ❛❣❣r❡❣❛t✐♦♥ ❋✉❧❧ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ✿ L j ∼ F j ❛♥❞ r❡♣r❡s❡♥t r✐s❦s ❛s L j ❂ F − ✶ ( U j ) j ✇❤❡r❡ U j ✐s U ( ✵ , ✶ ) . + ❋✉❧❧ ■♥❢♦r♠❛t✐♦♥ ♦♥ ❞❡♣❡♥❞❡♥❝❡ ✿ ( U ✶ , U ✷ , ..., U n ) ∼ C ✭❈ ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♣✉❧❛✮ ⇒ ❱❛❘ q ( L ✶ + L ✷ + ... + L n ) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞✦ ✻

  7. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ▼♦t✐✈❛t✐♦♥ ♦♥ ❱❛❘ ❛❣❣r❡❣❛t✐♦♥ ❋✉❧❧ ✐♥❢♦r♠❛t✐♦♥ ♦♥ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ✿ L j ∼ F j ❛♥❞ r❡♣r❡s❡♥t r✐s❦s ❛s L j ❂ F − ✶ ( U j ) j ✇❤❡r❡ U j ✐s U ( ✵ , ✶ ) . + P❛rt✐❛❧ ♦r ♥♦ ■♥❢♦r♠❛t✐♦♥ ♦♥ ❞❡♣❡♥❞❡♥❝❡ ✿ ( U ✶ , U ✷ , ..., U n ) ∼ ??? ⇒ ❱❛❘ q ( L ✶ + L ✷ + ... + L n ) ❝❛♥♥♦t ❜❡ ❝♦♠♣✉t❡❞✦ ❖♥❧② ❛ r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❢♦r ❱❛❘ q ( L ✶ + L ✷ + ... + L n ) ✳ ✼

  8. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ▲✐t❡r❛t✉r❡ ▼❛①✐♠✉♠ ❱❛❘ ✉♥❞❡r ❉❡♣❡♥❞❡♥❝❡ ❯♥❝❡rt❛✐♥t② ❇♦✉♥❞s ♦♥ ❱❛❧✉❡✲❛t✲❘✐s❦ M := s✉♣ VaR q [ L ✶ + L ✷ + ... + L n ] , s✉❜❥❡❝t t♦ L j ∼ F j , ❝♦♣✉❧❛ ❈ ❂ ✉♥❦♥♦✇♥ • ❊①♣❧✐❝✐t s❤❛r♣ ❜♦✉♥❞s · n = ✷ ▼❛❦❛r♦✈ ✭✶✾✽✶✮✱ ❘üs❝❤❡♥❞♦r❢ ✭✶✾✽✷✮ · ❤♦♠♦❣❡♥❡♦✉s ♣♦rt❢♦❧✐♦s✿ ❘üs❝❤❡♥❞♦r❢ ✫ ❯❝❦❡❧♠❛♥♥ ✭✶✾✾✶✮✱ ❉❡♥✉✐t✱ ●❡♥❡st ✫ ▼❛r❝❡❛✉ ✭✶✾✾✾✮✱ ❊♠❜r❡❝❤ts ✫ P✉❝❝❡tt✐ ✭✷✵✵✻✮✱ ❲❛♥❣ ✫ ❲❛♥❣ ✭✷✵✶✶✮✱ ❇❡r♥❛r❞✱ ❏✐❛♥❣ ❛♥❞ ❲❛♥❣ ✭✷✵✶✹✮ · ❤❡t❡r♦❣❡♥❡♦✉s ♣♦rt❢♦❧✐♦s✿ ❲❛♥❣ ✫ ❲❛♥❣ ✭✷✵✶✺✮ • ❆♣♣r♦①✐♠❛t❡ s❤❛r♣ ❜♦✉♥❞s · ❚❤❡ ❘❡❛rr❛♥❣❡♠❡♥t ❆❧❣♦r✐t❤♠ ✭P✉❝❝❡tt✐ ✫ ❘üs❝❤❡♥❞♦r❢ ✭✷✵✶✷✮✱ ❊♠❜r❡❝❤ts✱ P✉❝❝❡tt✐ ✫ ❘üs❝❤❡♥❞♦r❢ ✭✷✵✶✸✮✮ ✽

  9. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ❖❜s❡r✈❛t✐♦♥s ❖❜s❡r✈❛t✐♦♥s • ❚❤❡ ❜♦✉♥❞ ▼ ♠❛② ❜❡ t♦♦ ✇✐❞❡ t♦ ❜❡ ♣r❛❝t✐❝❛❧❧② ✉s❡❢✉❧✿ ❛ ❢❡❛t✉r❡ t❤❛t ❝❛♥ ♦♥❧② ❜❡ ❡①♣❧❛✐♥❡❞ ❜② t❤❡ ❛❜s❡♥❝❡ ♦❢ ❞❡♣❡♥❞❡♥❝❡ ✐♥❢♦r♠❛t✐♦♥✳ • ❖✉r ♦❜❥❡❝t✐✈❡✿ ✐♥❝♦r♣♦r❛t❡ ❞❡♣❡♥❞❡♥❝❡ ✐♥❢♦r♠❛t✐♦♥ ✾

  10. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❇❛❝❦❣r♦✉♥❞✱ ❖❜s❡r✈❛t✐♦♥s ❇♦✉♥❞s ♦♥ ❱❛❧✉❡✲❛t✲❘✐s❦ ◮ ❱❛❘ q ✐s ♥♦t ♠❛①✐♠✐③❡❞ ❢♦r t❤❡ ❝♦♠♦♥♦t♦♥✐❝ s❝❡♥❛r✐♦✿ S c = L c ✶ + L c ✷ + ... + L c n ✇❤❡r❡ ❛❧❧ L c i ❛r❡ ❝♦♠♦♥♦t♦♥✐❝✳ M ≥ VaR q [ L c ✶ + L c ✷ + ... + L c n ] = VaR q [ L ✶ ] + VaR q [ L ✷ ] + ... + VaR q [ L n ] ✇❤❡r❡ ✭ L c ✶ , L c ✷ , ... L c n ) ✐s ❛ ❝♦♠♦♥♦t♦♥✐❝ ❝♦♣② ♦❢ ✭ L ✶ , L ✷ , ... L n ) , ✐✳❡✳ ( L c ✶ , L c ✷ , ... L c n ) = ( F − ✶ L ✶ ( U ) , F − ✶ L ✷ ( U ) , ..., F − ✶ L n ( U )) . ✶✵

  11. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s ❆❣❡♥❞❛ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s ✷ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ ✷ r✐s❦s ❱❛❘ ❇♦✉♥❞s ✇✐t❤ n r✐s❦s ❊①❛♠♣❧❡ ✶✶

  12. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s✱ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ ✷ r✐s❦s ✏❘✐s❦✐❡st✑ ❉❡♣❡♥❞❡♥❝❡✿ ♠❛①✐♠✉♠ ❱❛❘ q ✐♥ ✷ ❞✐♠s ■❢ L ✶ ❛♥❞ L ✷ ❛r❡ ❯✭✵✱✶✮ ❝♦♠♦♥♦t♦♥✐❝✱ t❤❡♥ VaR q ( S c ) = VaR q ( X ✶ ) + VaR q ( X ✷ ) = ✷ q . q q ✶✷

  13. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s✱ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ ✷ r✐s❦s ✏❘✐s❦✐❡st✑ ❉❡♣❡♥❞❡♥❝❡✿ ♠❛①✐♠✉♠ ❱❛❘ q ✐♥ ✷ ❞✐♠s ■❢ L ✶ ❛♥❞ L ✷ ❛r❡ ❯✭✵✱✶✮ ❛♥❞ ❛♥t✐♠♦♥♦t♦♥✐❝ ✐♥ t❤❡ t❛✐❧✱ t❤❡♥ VaR q ( S ∗ ) = ✶ + q ✳ q q VaR q ( S ∗ ) = ✶ + q > VaR q ( S c ) = ✷ q ✶✸ ⇒ t♦ ♠❛①✐♠✐③❡ ❱❛❘ q ✱ t❤❡ ✐❞❡❛ ✐s t♦ ❝❤❛♥❣❡ t❤❡ ❝♦♠♦♥♦t♦♥✐❝ ❞❡♣❡♥❞❡♥❝❡ s✉❝❤ t❤❛t t❤❡ s✉♠ ✐s ❝♦♥st❛♥t ✐♥ t❤❡ t❛✐❧

  14. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s✱ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ n r✐s❦s ❱❛❘ ❛t ❧❡✈❡❧ q ♦❢ t❤❡ ❝♦♠♦♥♦t♦♥✐❝ s✉♠ ✇✳r✳t✳ q VaR q (S c ) p q 1 ✶✹

  15. ✶✹t❤ ❙❝✐❡♥t✐✜❝ ❉❛② ✸✵✳✵✹✳✷✵✶✺ ❯♥❝♦♥str❛✐♥❡❞ ❱❛❘ ❇♦✉♥❞s✱ ❱❛❘ ❇♦✉♥❞s ✇✐t❤ n r✐s❦s ❱❛❘ ❛t ❧❡✈❡❧ q ♦❢ t❤❡ ❝♦♠♦♥♦t♦♥✐❝ s✉♠ ✇✳r✳t✳ q TVaR q (S c ) VaR q (S c ) p q 1 � ✶ ✶ ✇❤❡r❡ ❚❱❛❘ ✭❊①♣❡❝t❡❞ s❤♦rt❢❛❧❧✮✿❚❱❛❘ q ( X ) = ❱❛❘ u ( X ) ❞ u q ∈ ✵ ✶ ✶ − q q ✶✺

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