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❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

P❤✐❧✐♣♣ ●s❝❤ö♣❢ ❲♦❧❢❣❛♥❣ ❑❛r❧ ❍är❞❧❡ ❆♥❞r✐❥❛ ▼✐❤♦❝✐ ▲❛❞✐s❧❛✉s ✈♦♥ ❇♦rt❦✐❡✇✐❝③ ❈❤❛✐r ♦❢ ❙t❛t✐st✐❝s ❈✳❆✳❙✳❊✳ ✕ ❈❡♥t❡r ❢♦r ❆♣♣❧✐❡❞ ❙t❛t✐st✐❝s ❛♥❞ ❊❝♦♥♦♠✐❝s ❍✉♠❜♦❧❞t✕❯♥✐✈❡rs✐tät ③✉ ❇❡r❧✐♥ ❤tt♣✿✴✴❧✈❜✳✇✐✇✐✳❤✉✲❜❡r❧✐♥✳❞❡ ❤tt♣✿✴✴❝❛s❡✳❤✉✲❜❡r❧✐♥✳❞❡ ❤tt♣✿✴✴✐rt❣✶✼✾✷✳❤✉✲❜❡r❧✐♥✳❞❡

▼♦t✐✈❛t✐♦♥ ✶✲✶

▼♦t✐✈❛t✐♦♥

❋✐❣✉r❡ ✶✿ ❚❤❡ t❡r❡s ♠❛❥♦r ♠✉s❝❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✷

❚❛✐❧ ❘✐s❦

❋✐❣✉r❡ ✷✿ ◆❡③❤❛ ✭❧✐♥❦✮

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✸

❱❛❘ ❛♥❞ ❊❙

⊡ ❱❛❧✉❡ ❛t ❘✐s❦ ✭❱❛❘✮

◮ ❇❛s❡❧ ■■■ ◮ ◆♦t ❝♦❤❡r❡♥t

❈♦❤❡r❡♥❝❡

⊡ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✭❊❙✮

❊❙ ❉❡✜♥✐t✐♦♥

◮ ❇❛s❡❧ ❈♦♠♠✐tt❡❡ ✭✷✵✶✹✮ ◮ ❈♦❤❡r❡♥t✱ ❢♦❝✉s ♦♥ t❛✐❧ str✉❝t✉r❡

❊①❛♠♣❧❡✿ ❉❡✉ts❝❤❡ ❇❛♥❦ ✲ r✐s❦ ❧❡✈❡❧s {✵.✵✵✵✷, ✵.✵✵✶, ✵.✵✶}✱ ❑❛❧❦❜r❡♥♥❡r ❡t ❛❧✳ ✭✷✵✶✹✮

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✹

◗✉❛♥t✐❧❡ ❱❛❘

❋✐❣✉r❡ ✸✿ ❉✐str✐❜✉t✐♦♥ ♦❢ r❡t✉r♥s✱ VaR✵.✵✺ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞ ✉♥❞❡r ❝❤❛♥❣✲ ✐♥❣ t❛✐❧ str✉❝t✉r❡✱ ❝❧♦✉❞✐♥❣ t❤❡ ✐♥✈❡st♦rs r✐s❦ ♣❡r❝❡♣t✐♦♥

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✺

❖❜❥❡❝t✐✈❡s

✭✐✮ ❯♥❞❡rst❛♥❞✐♥❣ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✭❊❙✮

◮ ❊①tr❡♠❡ ❡✈❡♥ts ❛♥❞ ❛ss♦❝✐❛t❡❞ r✐s❦ ◮ ❉✐str✐❜✉t✐♦♥❛❧ ❡♥✈✐r♦♥♠❡♥ts ✲ ✐♠♣❧✐❝❛t✐♦♥s

✭✐✐✮ ❚❊❘❊❙

◮ ❚❛✐❧ ❞r✐✈❡♥ r✐s❦ ❛ss❡ss♠❡♥t✱ r♦❜✉st♥❡ss ♦❢ ❊❙ ◮ ❆❞✈❛♥t❛❣❡s ♦❢ ❡①♣❡❝t✐❧❡s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✻

❚❛✐❧ ❘✐s❦

❊①❛♠♣❧❡✿ ✷✵✵✽ s✉❜♣r✐♠❡ ♠♦rt❣❛❣❡ ❝r✐s✐s

◮ ❙✫P ✺✵✵ ❧♦♥❣ ♣♦s✐t✐♦♥ ✐♥ ✷✵✵✽ ✭✷✻✶ ❞❛✐❧② r❡t✉r♥s✮ ◮ ◗✉❛♥t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ✶✪ ♣♦rt❢♦❧✐♦ r✐s❦

❙❝❡♥❛r✐♦ ❛♥❛❧②s✐s✱ µ = −✵.✵✵✷, σ = ✵.✵✷✺

◮ ❙❝❡♥❛r✐♦ ✭✐✮ ◆♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ VaR = −✺.✾%✱ ES = −✻.✽% ◮ ❙❝❡♥❛r✐♦ ✭✐✐✮ ▲❛♣❧❛❝❡ ❞✐str✐❜✉t✐♦♥ VaR = −✶✵.✵%✱ ES = −✶✷.✺%

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✼

❚❛✐❧ ❘✐s❦

✷✵✵✽ ✷✵✶✵ ✷✵✶✷ ✷✵✶✹ ◆♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ❱❛❘ ✲✺✳✾ ✲✷✳✻ ✲✶✳✽ ✲✶✳✻ ❊❙ ✲✻✳✽ ✲✸✳✵ ✲✷✳✶ ✲✶✳✾ ▲❛♣❧❛❝❡ ❞✐str✐❜✉t✐♦♥ ❱❛❘ ✲✶✵✳✵ ✲✹✳✹ ✲✸✳✶ ✲✷✳✹ ❊❙ ✲✶✷✳✺ ✲✺✳✻ ✲✸✳✾ ✲✸✳✺ ❊st✐♠❛t❡❞ ❱❛❘ ❛♥❞ ❊❙ ✐♥ ✪ ❢♦r ❙✫P ✺✵✵ ✐♥❞❡① r❡t✉r♥s ❛t ❧❡✈❡❧ α = ✵.✵✶

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

▼♦t✐✈❛t✐♦♥ ✶✲✽

❘❡s❡❛r❝❤ ◗✉❡st✐♦♥s

❲❤❛t ❛r❡ t❤❡ t❤r✐❧❧s ❢♦r ❊❙ ❡st✐♠❛t✐♦♥❄ ❲❤❛t ❛r❡ t❤❡ r♦❜✉st♥❡ss ♣r♦♣❡rt✐❡s ♦❢ ❊❙❄ ❲❤✐❝❤ r✐s❦ r❛♥❣❡ ✐s ❡①♣❡❝t❡❞ ✉♥❞❡r ❞✐✛❡r❡♥t t❛✐❧ s❝❡♥❛r✐♦s❄

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❖✉t❧✐♥❡

✶✳ ▼♦t✐✈❛t✐♦♥

• ✷✳ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

✸✳ ❚❊❘❊❙ ✹✳ ❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✺✳ ❈♦♥❝❧✉s✐♦♥s

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✶

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

⊡ ❋✐♥❛♥❝✐❛❧ r❡t✉r♥s Y

◮ ♣❞❢ f (y) ❛♥❞ ❝❞❢ F(y) ◮ ❍❡r❡✿ ❧♦✇❡r t❛✐❧ ✭❞♦✇♥s✐❞❡✮ r✐s❦

⊡ ❊①♣❡❝t❡❞ s❤♦rt❢❛❧❧ sη = ❊[Y |Y < η]

◮ ❇❛s❡❧✿ ❱❛❧✉❡ ❛t ❘✐s❦ t❤r❡s❤♦❧❞ η = qα = F −✶ (α)

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✷

❊①♣❡❝t✐❧❡s

⊡ ❊❙ ❡st✐♠❛t✐♦♥

◮ ❯s✐♥❣ ❡①♣❡❝t✐❧❡s✱ ❚❛②❧♦r ✭✷✵✵✽✮ ◮ ❊①♣❡❝t✐❧❡s r❡✢❡❝t t❤❡ t❛✐❧ str✉❝t✉r❡

⊡ ▲♦ss ❢✉♥❝t✐♦♥ ρα,γ (u) = |α − ■ {u < ✵}| |u|γ ✭✶✮

◮ ❊①♣❡❝t✐❧❡ eα = ❛r❣ ♠✐♥

θ ❊ ρα,✷ (Y − θ)

◮ ◗✉❛♥t✐❧❡ qα = ❛r❣ ♠✐♥

θ ❊ ρα,✶ (Y − θ)

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✸

▲♦ss ❋✉♥❝t✐♦♥

u 2 4 −6 −3 3 6 u 2 4 −6 −3 3 6 ❋✐❣✉r❡ ✹✿ ❊①♣❡❝t✐❧❡ ❛♥❞ q✉❛♥t✐❧❡ ❧♦ss ❢✉♥❝t✐♦♥s ❛t α = ✵.✵✶ ✭❧❡❢t✮ ❛♥❞ α = ✵.✺✵ ✭r✐❣❤t✮

▲◗❘❝❤❡❝❦

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✹

❚❛✐❧ ❙tr✉❝t✉r❡

⊡ ◗✉❛♥t✐❧❡s ❛♥❞ ❡①♣❡❝t✐❧❡s ✲ ♦♥❡✲t♦✲♦♥❡ ♠❛♣♣✐♥❣

❉❡t❛✐❧s

◮

• ♦❛❧✿ ew(α) = qα

◮ ❋✐♥❞ ❡①♣❡❝t✐❧❡ ❧❡✈❡❧ w (α)

⊡ ❊❙ ✉s✐♥❣ ❡①♣❡❝t✐❧❡s✱ ❚❛②❧♦r ✭✷✵✵✽✮

Pr♦♦❢

sqα = ew(α) + ew(α) − ❊[Y ] ✶ − ✷w(α) w(α) α

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✺

❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s

⊡ ❏♦♥❡s ✭✶✾✾✸✮✱ ●✉♦ ❛♥❞ ❍är❞❧❡ ✭✷✵✶✶✮

◮ ❆♥❛❧②t✐❝❛❧ ❢♦r♠✉❧❛ ❢♦r ❧❡✈❡❧ w(α)

❉❡t❛✐❧s

◮ ❆ss✉♠♣t✐♦♥✿ ❦♥♦✇♥ r❡t✉r♥ ❞✐str✐❜✉t✐♦♥ F(·)

❊①❛♠♣❧❡✱ ◆(✵, ✶) w(α) = −ϕ(qα) − qαα −✷ {ϕ(qα) + qαα} + qα − ❊[Y ]

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✷✲✻

❙❡♥s✐t✐✈✐t② ❆♥❛❧②s✐s

⊡ ❋♦r t❤❡ ♠♦r❡ ❣❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ❣✐✈❡♥ ✐♥ ✭✶✮

Pr♦♦❢

w(α, γ) = qα

−∞ |y − qα|γ−✶dF(y)

∞

−∞ |y − qα|γ−✶dF(y),

γ ≥ ✶ ⊡ ◗✉❛♥t✐❧❡ ❝❛s❡ w(α, ✶) = α✱ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ γ

◮ |y − qα| > ✶✿ ❡①♣♦♥❡♥t✐❛❧ ❝♦♥✈❡r❣❡♥❝❡ t♦✇❛r❞s α ❛s γ → ✶ ◮ |y − qα| < ✶✿ r♦♦t ❝♦♥✈❡r❣❡♥❝❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✶

❍❡❛✈② ❚❛✐❧❡❞ ❘❡t✉r♥s

−4 −2 2 4 −4 −2 2 4 Standard Normal Quantiles Std Return Quantiles ❋✐❣✉r❡ ✺✿ ❙✫P ✺✵✵ r❡t✉r♥ q✉❛♥t✐❧❡s ❢r♦♠ ✷✵✵✺✵✶✵✸✲✷✵✶✹✶✷✸✶✱ st❛♥❞❛r❞✐③❡❞ ✉s✐♥❣ ❛ ●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧

❚❊❘❊❙❴❙t❛♥❞❛r❞✐③❛t✐♦♥

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✷

◗✉❛♥t✐❧❡ ✲ ❊①♣❡❝t✐❧❡ ❘❡❧❛t✐♦♥

⊡ Pr♦♣❡rt✐❡s ♦❢ ❊❙

◮ ❊❙ ❞❡♣❡♥❞s ♦♥ t❤❡ eα t♦ qα ❞✐st❛♥❝❡ ◮ ❖t❤❡r ❝♦♠♣♦♥❡♥t✿ ❱❛❘

⊡ ■♠♣❧✐❝❛t✐♦♥s ♦❢ t❤✐❝❦❡♥✐♥❣ t❤❡ t❛✐❧

◮ ❊①♣❡❝t✐❧❡✲q✉❛♥t✐❧❡ r❡❧❛t✐♦♥ ❣✐✈❡♥ ❛ ❞✐str✐❜✉t✐♦♥ ◮ ❊①❛♠♣❧❡s✿ ◆♦r♠❛❧ ❛♥❞ ▲❛♣❧❛❝❡ ❝❛s❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✸

◗✉❛♥t✐❧❡ ✲ ❊①♣❡❝t✐❧❡ ✲ ◆♦r♠❛❧

0.5 1 −3 3

α qα, eα

Quantiles vs. Expectiles 0.5 1 −1 1

α

Difference

qα − eα

❋✐❣✉r❡ ✻✿ ❚♦♣✿ ◗✉❛♥t✐❧❡ ✭❜❧✉❡✮ ❛♥❞ ❊①♣❡❝t✐❧❡ ✭r❡❞✮✱ ❜♦tt♦♠✿ ❞✐✛❡r❡♥❝❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✹

◗✉❛♥t✐❧❡ ✲ ❊①♣❡❝t✐❧❡ ✲ ▲❛♣❧❛❝❡

0.5 1 −3 3

α qα, eα

Quantiles vs. Expectiles 0.5 1 −1 1

α

Difference

qα − eα

❋✐❣✉r❡ ✼✿ ❚♦♣✿ ◗✉❛♥t✐❧❡ ✭❜❧✉❡✮ ❛♥❞ ❊①♣❡❝t✐❧❡ ✭r❡❞✮✱ ❜♦tt♦♠✿ ❞✐✛❡r❡♥❝❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✺

◗✉❛♥t✐❧❡ ✲ ❊①♣❡❝t✐❧❡ ✲ ❊q✉❛❧✐t②

⊡ ■s t❤❡r❡ ❛ ❞✐str✐❜✉t✐♦♥ s✉❝❤ t❤❛t eα = qα ⊡ ❈♦♥s✐❞❡r ❛ ✭✈❡r② ❤❡❛✈② t❛✐❧❡❞✮ ❝❞❢✱ ❑♦❡♥❦❡r ✭✶✾✾✸✮ F(x) =    ✵.✺ − ✵.✺

• ✶ −

✹ ✹+x✷

✵.✺ , ✐❢ x < ✵ ✵.✺ + ✵.✺

• ✶ −

✹ ✹+x✷

✵.✺ , ❡❧s❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✻

◗✉❛♥t✐❧❡ ✲ ❊①♣❡❝t✐❧❡ ✲ ❊q✉❛❧✐t②

0.5 1 −3 3

α qα, eα

Quantiles vs. Expectiles 0.5 1 −1 1

α

Difference

qα − eα

❋✐❣✉r❡ ✽✿ ❚♦♣✿ ◗✉❛♥t✐❧❡ ✭❜❧✉❡✮ ❛♥❞ ❊①♣❡❝t✐❧❡ ✭r❡❞✮✱ ❜♦tt♦♠✿ ❞✐✛❡r❡♥❝❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✼

❚❊❘❊❙

⊡ ❋❧❡①✐❜❧❡ st❛t✐st✐❝❛❧ ❢r❛♠❡✇♦r❦ ✲ ❊❙ t❛✐❧ s❝❡♥❛r✐♦s

◮ Pr♦♣❡rt✐❡s ♦❢ ❊❙ ✐♥ ❛♥ ❡♥✈✐r♦♥♠❡♥t ◮ ❘✐s❦ ❝♦rr✐❞♦r✱ s❝❡♥❛r✐♦ ❛♥❛❧②s✐s

⊡ ❋❛♠✐❧② ♦❢ ❞✐str✐❜✉t✐♦♥s ✲ ❡♥✈✐r♦♥♠❡♥t

◮ ❉✐str✐❜✉t✐♦♥❛❧ ❢❛♠✐❧✐❡s✱ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ ◮ ▼✐①t✉r❡s✱ ❡✳❣✳ t✇♦✲❝♦♠♣♦♥❡♥t ❧✐♥❡❛r ♠✐①t✉r❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✽

❊①❛♠♣❧❡✿ ❈♦♥t❛♠✐♥❛t❡❞ ◆♦r♠❛❧ ❊♥✈✐r♦♥♠❡♥t

⊡ δ−❡♥✈✐r♦♥♠❡♥t✱ ❍✉❜❡r ✭✶✾✻✹✮ fδ (y) = (✶ − δ) ϕ (y) + δh (y) , δ ∈ [✵, ✶] ⊡ Pr❛❝t✐❝❡✿ ♥♦r♠❛❧✐t② ❛ss✉♠♣t✐♦♥✱ ✜♥❞✐♥❣s✿ ❤❡❛✈② t❛✐❧s

◮ ❋✐♥❛♥❝✐❛❧ ♠❛r❦❡ts✿ h(·) ✐s s②♠♠❡tr✐❝❛❧ ❛♥❞ ❤❡❛✈② t❛✐❧❡❞ ◮ ❈♦♥t❛♠✐♥❛t✐♦♥ ❞❡❣r❡❡ δ

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✾

❊①❛♠♣❧❡✿ ◆♦r♠❛❧✲▲❛♣❧❛❝❡ ▼✐①t✉r❡

10% 5% 1% 0.5 1 −6 −4 −2

α δ

ES ES, S&P500

❋✐❣✉r❡ ✾✿ ❚❤❡♦r❡t✐❝❛❧ ❊❙qα ❢♦r ❞✐✛❡r❡♥t ❝♦♥t❛♠✐♥❛t✐♦♥ δ ❛♥❞ r✐s❦ ❧❡✈❡❧ α

❚❊❘❊❙❴❊❙❴❆♥❛❧②t✐❝❛❧

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✶✵

❘❡❧❛t✐♦♥ t♦ ▼▲❊

⊡ ❆s②♠♠❡tr✐❝ ●❡♥❡r❛❧✐③❡❞ ❊rr♦r ❉✐str✐❜✉t✐♦♥ ✭❆●❊❉✮✱ ❆②❡❜♦ ❛♥❞ ❑♦③✉❜♦✇s❦✐ ✭✷✵✵✸✮ f (x) = γ σΓ( ✶

γ )

κ ✶ + κ✷ ❡①♣

• −κγ

σγ ■{x − µ ≥ ✵} − ✶ κγσγ ■{x − µ < ✵}

• |x − µ|γ
• Γ(x) =

∞

✵

xt−✶ ❡①♣(−x)dx ⊡ ❙❝❛❧❡ σ✱ s❦❡✇♥❡ss κ✱ ❧♦❝❛t✐♦♥ µ ❛♥❞ s❤❛♣❡ γ ✭❆s②♠♠❡tr✐❝ ▲❛♣❧❛❝❡ γ = ✶ ❛♥❞ ♥♦r♠❛❧ γ = ✷✮

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✸✲✶✶

▼✲◗✉❛♥t✐❧❡s ❛s ▲♦❝❛t✐♦♥ ❊st✐♠❛t❡

⊡ ❆●❊❉ ▲♦❣✲❧✐❦❡❧✐❤♦♦❞ − ❧♦❣{f (x|µ, σ, γ, κ)} =c(γ, σ, κ) + κγ σγ ■{x − µ ≥ ✵} + ✶ κγσγ ■{x − µ < ✵}

• |x − µ|γ

⊡ ❚❤❡ ❧♦❝❛t✐♦♥ ▼▲❊ µOPT ✐s ❡q✉❛❧ t♦ t❤❡ τ✲▼✲◗✉❛♥t✐❧❡ ✐❢ κ(δ) =

• τ

✶ − τ ✶

✷δ

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❙t♦❝❦ r❡t✉r♥s

❙t♦❝❦ ❊①❛♠♣❧❡

❉❆❳✱ ❋❚❙❊ ✶✵✵ ❛♥❞ ❙✫P ✺✵✵

■♥tr❛❞❛② ▼❛r❣✐♥

❉✐✛❡r❡♥t r✐s❦ ❧❡✈❡❧s ❋♦r❡✐❣♥ ❊①❝❤❛♥❣❡

❋♦r❡① ❊①❛♠♣❧❡

❊❯❘✴❯❆❍ ❡①❝❤❛♥❣❡ r❛t❡ ◆♦t r❡❧②✐♥❣ ♦♥ st❛♥❞❛r❞✐③❛t✐♦♥ P♦rt❢♦❧✐♦ ❙❡❧❡❝t✐♦♥

❚❊❉❆❙ ❊①❛♠♣❧❡

❚❊❉❆❙ ✭❚❛✐❧ ❊✈❡♥t ❉r✐✈❡♥ ❆❙s❡t ❛❧❧♦❝❛t✐♦♥✮ ❙♠❛❧❧ s❛♠♣❧❡ s✐③❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✷

❉❛t❛

⊡ ❉❆❳✱ ❋❚❙❊ ✶✵✵ ❛♥❞ ❙✫P ✺✵✵ ❞❛✐❧② r❡t✉r♥s

◮ ❘✐s❦ ❧❡✈❡❧ α✿ ✵✳✵✶✱ ✵✳✵✺ ❛♥❞ ✵✳✶✵ ◮ ❱❛r②✐♥❣ t❛✐❧ t❤✐❝❦♥❡ss δ

⊡ ❙♣❛♥✿ ✷✵✵✺✵✶✵✸✲✷✵✶✹✶✷✸✶ ✭✷✻✵✾ tr❛❞✐♥❣ ❞❛②s✮

◮ ❖♥❡✲②❡❛r t✐♠❡ ❤♦r✐③♦♥ ✭✷✺✵ tr❛❞✐♥❣ ❞❛②s✮ ✲ ♠♦✈✐♥❣ ✇✐♥❞♦✇ ◮ ❙t❛♥❞❛r❞✐③❡❞ r❡t✉r♥s

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✸

❘❡t✉r♥s

2005 2010 −5 5 DAX 2005 2010 −5 5 FTSE 100 2005 2010 −5 5 SP 500

❋✐❣✉r❡ ✶✵✿ ❙t❛♥❞❛r❞✐③❡❞ r❡t✉r♥s ♦❢ t❤❡ s❡❧❡❝t❡❞ ✐♥❞✐❝❡s ❢r♦♠ ✷✵✵✺✵✶✵✸✲ ✷✵✶✹✶✷✸✶

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❙t❛♥❞❛r❞✐③❛t✐♦♥

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✹

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

δ ❉❆❳ ❋❚❙❊ ✶✵✵ ❙✫P ✺✵✵ ✵✳✵ ✲✷✳✾✶ ✲✸✳✶✶ ✲✸✳✷✻ ✵✳✵✵✶ ✲✷✳✾✶ ✲✸✳✶✶ ✲✸✳✷✻ ✵✳✵✵✷ ✲✷✳✾✶ ✲✸✳✶✷ ✲✸✳✷✼ ✵✳✵✵✺ ✲✷✳✾✷ ✲✸✳✶✸ ✲✸✳✷✽ ✵✳✵✶ ✲✷✳✾✹ ✲✸✳✶✹ ✲✸✳✸✵ ✵✳✵✷ ✲✷✳✾✼ ✲✸✳✶✼ ✲✸✳✸✸

❚❛❜❧❡ ✶✿ ❊st✐♠❛t❡❞ ❊❙qα ❢♦r s❡❧❡❝t❡❞ ✐♥❞✐❝❡s ❛t α = ✵.✵✶✱ ❢r♦♠ ✷✵✶✹✵✶✶✻✲ ✷✵✶✹✶✷✸✶ ✭✷✺✵ tr❛❞✐♥❣ ❞❛②s✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✺

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

δ ❉❆❳ ❋❚❙❊ ✶✵✵ ❙✫P ✺✵✵ ✵✳✵✺ ✲✸✳✵✺ ✲✸✳✷✻ ✲✸✳✹✷ ✵✳✶ ✲✸✳✶✻ ✲✸✳✸✽ ✲✸✳✺✹ ✵✳✶✺ ✲✸✳✷✹ ✲✸✳✹✻ ✲✸✳✻✸ ✵✳✷✺ ✲✸✳✸✷ ✲✸✳✺✺ ✲✸✳✼✷ ✵✳✺ ✲✸✳✸✵ ✲✸✳✺✸ ✲✸✳✼✵ ✶✳✵ ✲✸✳✶✾ ✲✸✳✹✶ ✲✸✳✺✼

❚❛❜❧❡ ✷✿ ❊st✐♠❛t❡❞ ❊❙qα ❢♦r s❡❧❡❝t❡❞ ✐♥❞✐❝❡s ❛t α = ✵.✵✶✱ ❢r♦♠ ✷✵✶✹✵✶✶✻✲ ✷✵✶✹✶✷✸✶ ✭✷✺✵ tr❛❞✐♥❣ ❞❛②s✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✻

❊❙ ❉②♥❛♠✐❝s

❙❡t✉♣ ⊡ ❘✐s❦ ❧❡✈❡❧ α✿ ✵✳✶✵✱ ✵✳✵✺ ❛♥❞ ✵✳✵✶ ⊡ ❙❝❡♥❛r✐♦s✿ ▲❛♣❧❛❝❡ ❛♥❞ ♥♦r♠❛❧ ❊♠♣✐r✐❝❛❧ ❙t✉❞② ⊡ ❘♦❧❧✐♥❣ ✇✐♥❞♦✇ ❡①❡r❝✐s❡ ✲ ♦♥❡✲②❡❛r t✐♠❡ ❤♦r✐③♦♥ ✭✷✺✵ ❞❛②s✮ ⊡ ❙t♦❝❦ ♠❛r❦❡ts✿ ●❡r♠❛♥✱ ❯❑✱ ❯❙

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✼

❊❙ ❉②♥❛♠✐❝s

2005 2010 −4 −2 DAX 2005 2010 −4 −2 FTSE 100 2005 2010 −4 −2 SP 500 2005 2010 −4 −2 2005 2010 −4 −2 2005 2010 −4 −2

❋✐❣✉r❡ ✶✶✿ ❊❙qα ❛♥❞ ❱❛❘ ❛t α = ✵.✶✵❀ δ = ✵ ✭t♦♣✮ ❛♥❞ δ = ✶ ✭❜♦tt♦♠✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

2005 2010 −4 −2 DAX 2005 2010 −4 −2 FTSE 100 2005 2010 −4 −2 SP 500 2005 2010 −4 −2 2005 2010 −4 −2 2005 2010 −4 −2

❋✐❣✉r❡ ✶✷✿ ❊❙qα ❛♥❞ ❱❛❘ ❛t α = ✵.✵✺❀ δ = ✵ ✭t♦♣✮ ❛♥❞ δ = ✶ ✭❜♦tt♦♠✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✾

❊❙ ❉②♥❛♠✐❝s

2005 2010 −4 −2 DAX 2005 2010 −4 −2 FTSE 100 2005 2010 −4 −2 SP 500 2005 2010 −4 −2 2005 2010 −4 −2 2005 2010 −4 −2

❋✐❣✉r❡ ✶✸✿ ❊❙qα ❛♥❞ ❱❛❘ ❛t α = ✵.✵✶❀ δ = ✵ ✭t♦♣✮ ❛♥❞ δ = ✶ ✭❜♦tt♦♠✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✵

■♥tr❛❞❛② ▼❛r❣✐♥

❊①❛♠♣❧❡✿ ❋✐♥❛♥❝❡ ✲ ♣♦rt❢♦❧✐♦ ❡①♣♦s✉r❡ ❆♥ ✐♥✈❡st♦r ❡♥t❡rs ❛ ✶ ▼✐♦ ❯❙❉ ❧♦♥❣ ♣♦s✐t✐♦♥ ✭❡✳❣✳✱ ❙✫P ✺✵✵✮ ♦♥ ✷✵✶✹✶✷✸✶✳ ❯s✐♥❣ t❤❡ ❧❛st ✷✺✵ st❛♥❞❛r❞✐③❡❞ r❡t✉r♥s✱ t❤❡ r❡s❝❛❧❡❞ ❊❙ ✐s ♦❜t❛✐♥❡❞ ❛s ESqα ✐♥ ❯❙❉ α = ✵.✵✺ α = ✵.✵✶ α = ✵.✵✵✺ δ = ✵ ✲✶✵✱✸✺✽ ✲✶✻✱✻✾✹ ✲✶✾✱✶✽✽ δ = ✶ ✲✶✶✱✽✽✵ ✲✶✽✱✸✶✾ ✲✷✵✱✽✷✶

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✶

■♥tr❛❞❛② ▼❛r❣✐♥

❊①❛♠♣❧❡✿ ❋✐♥❛♥❝❡ ✲ ♣♦rt❢♦❧✐♦ ❡①♣♦s✉r❡ ❆♥ ✐♥✈❡st♦r ❡♥t❡rs ❛ ✶ ▼✐♦ ❯❙❉ ❧♦♥❣ ♣♦s✐t✐♦♥ ✭❡✳❣✳✱ ❙✫P ✺✵✵✮ ❛t t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ ✜♥❛♥❝✐❛❧ ❝r✐s✐s ✭✷✵✵✼✶✶✵✶✮✳ ❯s✐♥❣ t❤❡ ❧❛st ✷✺✵ st❛♥❞❛r❞✐③❡❞ r❡t✉r♥s✱ t❤❡ r❡s❝❛❧❡❞ ❊❙ ✐s ♦❜t❛✐♥❡❞ ❛s ESqα ✐♥ ❯❙❉ α = ✵.✵✺ α = ✵.✵✶ α = ✵.✵✵✺ δ = ✵ ✲✶✹✸✱✶✷✷ ✲✶✽✺✱✾✹✶ ✲✶✾✹✱✼✸✽ δ = ✶ ✲✶✻✷✱✺✷✾ ✲✷✵✸✱✵✵✽ ✲✷✶✵✱✹✸✷

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✷

❊①❝❤❛♥❣❡ ❘❛t❡ ❆♣♣❧✐❝❛t✐♦♥

2010 2015 −0.25 0.25 EUR/UAH Returns

❋✐❣✉r❡ ✶✹✿ ❘❡t✉r♥s ♦❢ t❤❡ ❊❯❘✴❯❆❍ ❡①❝❤❛♥❣❡ r❛t❡ ❢r♦♠ t❤❡ ✷✵✵✺✵✶✵✸ t♦ ✷✵✶✺✵✻✵✶

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✸

❊❙ ❉②♥❛♠✐❝s

2005 2010 2015 −0.1 −0.05 EUR−UAH ES corridor Corridor Ratios ❋✐❣✉r❡ ✶✺✿ ❊❙q✵.✵✶ ◆♦r♠❛❧✲▲❛♣❧❛❝❡ r✐s❦ ❡①tr❡♠❛ ✭❝♦rr✐❞♦r✮✳ ◆♦r♠❛❧ δ = ✵ ❛♥❞ ✧✇♦rst ❝❛s❡✧✱ ✐✳❡✳ δ = ✶

✸ s❝❡♥❛r✐♦s ✉s✐♥❣ ❛ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s✳

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✹

❊❙ ❉②♥❛♠✐❝s

2005 2010 2015 2005 2010 2015 1.15 1.2 Corridor Ratios ❋✐❣✉r❡ ✶✻✿ ❘❛t✐♦ ♦❢ t❤❡ ♠✐♥✐♠❛❧ ❛♥❞ ♠❛①✐♠❛❧ r✐s❦ ✐♥❞✐❝❛t✐♦♥s ✐♥ ❛ ◆♦r♠❛❧✲ ▲❛♣❧❛❝❡ ❡♥✈✐r♦♥♠❡♥t ✉s✐♥❣ r✐s❦ ❧❡✈❡❧s α = ✵.✵✶ ✭❜❧✉❡✮ ❛♥❞ α = ✵.✵✺ ✭❜❧❛❝❦✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✺

P♦rt❢♦❧✐♦ ✭❚❊❉❆❙✮ ❆♣♣❧✐❝❛t✐♦♥

01/31/2008 01/31/2013 −0.1 0.1 SP500

❋✐❣✉r❡ ✶✼✿ ✼✸ r❡t✉r♥ ♦❜s❡r✈❛t✐♦♥s ♦❢ ❛ ❣❧♦❜❛❧❧② s❡❧❡❝t❡❞ ❚❊❉❆❙ ♣♦rt❢♦❧✐♦ ✭❜❧✉❡✮ ✈❡rs✉s t❤❡ ❜❡♥❝❤♠❛r❦ ❙✫P✺✵✵ ✐♥❞❡① ✭r❡❞✮✱ ❍är❞❧❡ ❡t ❛❧✳ ✭✷✵✶✹✮

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✻

❙♠❛❧❧ ❙❛♠♣❧❡

0.001 0.05 0.1 0.15 0.2 0.25 −0.08 −0.04

Quantile based ESqα

α sqα

❋✐❣✉r❡ ✶✽✿ ❱❛❘α ✭❜❧❛❝❦✮ ❛♥❞ q✉❛♥t✐❧❡ ❜❛s❡❞ ❊❙qα ✉s✐♥❣ ❛ ♥♦r♠❛❧ s❝❡♥❛r✐♦ ✭❜❧✉❡✮ ❢♦r t❤❡ ❣❧♦❜❛❧❧② s❡❧❡❝t❡❞ ❚❊❉❆❙ ♣♦rt❢♦❧✐♦

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✼

❙♠❛❧❧ ❙❛♠♣❧❡

0.001 0.05 0.1 0.15 0.2 0.25 −0.08 −0.04

Expectile based ESqα

α sewα

❋✐❣✉r❡ ✶✾✿ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❊❙qα ✉s✐♥❣ ♥♦r♠❛❧ ✭❜❧✉❡✮ ❛♥❞ ✶

✸ ▲❛♣❧❛❝❡ ❝♦♥✲

t❛♠✐♥❛t✐♦♥ ✭r❡❞✮ s❝❡♥❛r✐♦ ❢♦r t❤❡ ❣❧♦❜❛❧❧② s❡❧❡❝t❡❞ ❚❊❉❆❙ ♣♦rt❢♦❧✐♦

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶✽

❘✐s❦ ❈♦rr✐❞♦r

0.001 0.05 0.1 1.1 1.5 1.9

Ratio of Maximal ES Variation

α

❋✐❣✉r❡ ✷✵✿ ❘❛t✐♦ ♦❢ t❤❡ ❧♦✇❡st ❛♥❞ ❤✐❣❤❡st r✐s❦ ✐♥❞✐❝❛t✐♦♥✱ ✐✳❡✳ ♠❛①✐♠❛❧ ✈❛r✐❛t✐♦♥ r❛t✐♦✱ ✐♥ ❛ ◆♦r♠❛❧✲▲❛♣❧❛❝❡ ❡♥✈✐r♦♥♠❡♥t ✉s✐♥❣ t❤❡ ❚❊❉❆❙ s❛♠♣❧❡

❋✐♥❛♥❝✐❛❧ ❆♣♣❧✐❝❛t✐♦♥s

❚❊❘❊❙❴❘♦❧❧✐♥❣❲✐♥❞♦✇

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❈♦♥❝❧✉s✐♦♥s ✺✲✶

❈♦♥❝❧✉s✐♦♥s

✭✐✮ ❯♥❞❡rst❛♥❞✐♥❣ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✭❊❙✮

◮ ❊①♣❡❝t✐❧❡s ❛r❡ s✉❝❝❡ss❢✉❧❧② ✉s❡❞ ❢♦r ❊❙ ❡st✐♠❛t✐♦♥ ◮ ❉✐str✐❜✉t✐♦♥❛❧ ❢❛♠✐❧✐❡s✱ ♠✐①t✉r❡s

✭✐✐✮ ❚❊❘❊❙

◮ ❊❙qα ❢♦r ❞✐✛❡r❡♥t r✐s❦ ❧❡✈❡❧s α ❛♥❞ s❝❡♥❛r✐♦s ◮ ❘♦❜✉st♥❡ss ♦❢ ❊❙ ✐♥ ❛ r❡❛❧✐st✐❝ ✜♥❛♥❝✐❛❧ s❡tt✐♥❣

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

P❤✐❧✐♣♣ ●s❝❤ö♣❢ ❲♦❧❢❣❛♥❣ ❑❛r❧ ❍är❞❧❡ ❆♥❞r✐❥❛ ▼✐❤♦❝✐ ▲❛❞✐s❧❛✉s ✈♦♥ ❇♦rt❦✐❡✇✐❝③ ❈❤❛✐r ♦❢ ❙t❛t✐st✐❝s ❈✳❆✳❙✳❊✳ ✕ ❈❡♥t❡r ❢♦r ❆♣♣❧✐❡❞ ❙t❛t✐st✐❝s ❛♥❞ ❊❝♦♥♦♠✐❝s ❍✉♠❜♦❧❞t✕❯♥✐✈❡rs✐tät ③✉ ❇❡r❧✐♥ ❤tt♣✿✴✴❧✈❜✳✇✐✇✐✳❤✉✲❜❡r❧✐♥✳❞❡ ❤tt♣✿✴✴❝❛s❡✳❤✉✲❜❡r❧✐♥✳❞❡ ❤tt♣✿✴✴✐rt❣✶✼✾✷✳❤✉✲❜❡r❧✐♥✳❞❡

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘❡❢❡r❡♥❝❡s ✻✲✶

❘❡❢❡r❡♥❝❡s

❆②❡❜♦✱ ❆✳ ❛♥❞ ❑♦③✉❜♦✇s❦✐✱ ❚✳ ❏✳ ❆♥ ❛s②♠♠❡tr✐❝ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ●❛✉ss✐❛♥ ❛♥❞ ▲❛♣❧❛❝❡ ❧❛✇s ❏♦✉r♥❛❧ ♦❢ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❙t❛t✐st✐❝❛❧ ❙❝✐❡♥❝❡ ✶ ✭✷✮✱ ✶✽✼✲✷✶✵✱ ✷✵✵✸ ❇❡❧❧✐♥✐✱ ❋✳ ✱ ❑❧❛r✱ ❇✳✱ ▼✉❧❧❡r✱ ❆✳ ❛♥❞ ●✐❛♥✐♥✱ ❊✳ ❘✳

• ❡♥❡r❛❧✐③❡❞ q✉❛♥t✐❧❡s ❛s r✐s❦ ♠❡❛s✉r❡s

■♥s✉r❛♥❝❡✿ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❊❝♦♥♦♠✐❝s ✺✹✱ ✹✶✲✹✽✱ ✷✵✶✹✱ ■❙❙◆✿ ✵✶✻✼✲✻✻✽✼

• ✉♦ ▼✳ ❛♥❞ ❍är❞❧❡✱ ❲✳ ❑✳

❙✐♠✉❧t❛♥❡♦✉s ❈♦♥✜❞❡♥❝❡ ❇❛♥❞ ❢♦r ❊①♣❡❝t✐❧❡ ❋✉♥❝t✐♦♥ ❆❞✈❛♥❝❡s ✐♥ ❙t❛t✐st✐❝❛❧ ❆♥❛❧②s✐s✱ ✷✵✶✶ ❉❖■✿ ✶✵✳✶✵✵✼✴s✶✵✶✽✷✲✵✶✶✲✵✶✽✷✲✶

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘❡❢❡r❡♥❝❡s ✻✲✷

❘❡❢❡r❡♥❝❡s

❇r❡❝❦❧✐♥❣✱ ❏✳ ❛♥❞ ❈❤❛♠❜❡rs✱ ❘✳ ▼✲q✉❛♥t✐❧❡s ❇✐♦♠❡tr✐❝❛ ✼✺✭✹✮✿ ✼✻✶✲✼✼✶✱ ✶✾✽✽ ❉❖■✿ ✶✵✳✶✵✾✸✴❜✐♦♠❡t✴✼✺✳✹✳✼✻✶ ❋✐s❤❜✉r♥✱ P✳ ❈✳ ▼❡❛♥✲❘✐s❦ ❆♥❛❧②s✐s ✇✐t❤ ❘✐s❦ ❆ss♦❝✐❛t❡❞ ✇✐t❤ ❇❡❧❧♦✇✲❚❛r❣❡t ❘❡t✉r♥s ❚❤❡ ❆♠❡r✐❝❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇ ✸✼✭✷✮✿ ✶✶✻✲✶✷✻✱ ✶✾✼✼ ❍är❞❧❡✱ ❲✳ ❑✳✱ ◆❛s❡❦✐♥✱ ❙✳✱ ▲❡❡✱ ❉✳ ❑✳ ❈✳ ❛♥❞ P❤♦♦♥✱ ❑✳ ❋✳ ❚❊❉❆❙ ✲ ❚❛✐❧ ❊✈❡♥t ❉r✐✈❡♥ ❆❙s❡t ❆❧❧♦❝❛t✐♦♥ ❙♦♥❞❡r❢♦rs❝❤✉♥❣s❜❡r❡✐❝❤ ✻✹✾✿ Ö❦♦♥♦♠✐s❝❤❡s ❘✐s✐❦♦ ✭❙❋❇ ✻✹✾ P❛♣❡rs✮ ✷✵✶✹✲✭✸✷✮

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘❡❢❡r❡♥❝❡s ✻✲✸

❘❡❢❡r❡♥❝❡s

❍✉❜❡r✱ P✳ ❏✳ ❘♦❜✉st ❊st✐♠❛t✐♦♥ ♦❢ ❛ ▲♦❝❛t✐♦♥ P❛r❛♠❡t❡r ❚❤❡ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙t❛t✐st✐❝s ✸✺✭✶✮✿ ✼✸✲✶✵✶✱ ✶✾✻✹ ❉❖■✿ ✶✵✳✶✷✶✹✴❛♦♠s✴✶✶✼✼✼✵✸✼✸✷ ❍✉❜❡r✱ P✳ ❏✳ ❛♥❞ ❘♦♥❝❤❡tt✐✱ ❊✳▼✳ ❘♦❜✉st ❙t❛t✐st✐❝s ❙❡❝♦♥❞ ❊❞✐t✐♦♥✱ ✷✵✵✾✱ ■❙❇◆✿ ✾✼✽✲✵✲✹✼✵✲✶✷✾✾✵✲✻ ❏♦♥❡s✱ ▼✳❈✳ ❊①♣❡❝t✐❧❡s ❛♥❞ ▼✲q✉❛♥t✐❧❡s ❛r❡ q✉❛♥t✐❧❡s ❙t❛t✐st✐❝s ✫ Pr♦❜❛❜✐❧✐t② ❧❡tt❡rs ✷✵✭✷✮✿ ✶✹✾✲✶✺✸✱ ✶✾✾✸✱ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✶✵✶✻✴✵✶✻✼✲✼✶✺✷✭✾✹✮✾✵✵✸✶✲✵

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘❡❢❡r❡♥❝❡s ✻✲✹

❘❡❢❡r❡♥❝❡s

❑❛❧❦❜r❡♥♥❡r✱ ▼✳✱ ▲♦tt❡r✱ ❍✳ ❛♥❞ ❖✈❡r❜❡❝❦✱ ▲✳ ❙❡♥s✐❜❧❡ ❛♥❞ ❡✣❝✐❡♥t ❝❛♣✐t❛❧ ❛❧❧♦❝❛t✐♦♥ ❢♦r ❝r❡❞✐t ♣♦rt❢♦❧✐♦s ❘■❙❑ ✶✾✲✷✹✱ ✷✵✶✹✱ ❏❛♥ ❑♦❡♥❦❡r✱ ❘✳ ❲❤❡♥ ❛r❡ ❡①♣❡❝t✐❧❡s ♣❡r❝❡♥t✐❧❡s❄ ❊❝♦♥♦♠✐❝ ❚❤❡♦r② ✾✭✸✮✿ ✺✷✻✲✺✷✼✱ ✶✾✾✸ ❉❖■✿❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✶✵✶✼✴❙✵✷✻✻✹✻✻✻✵✵✵✵✼✾✷✶ ◆❡✇❡②✱ ❲✳ ❑✳✱ P♦✇❡❧❧ ❏✳▲✳ ❆s②♠♠❡tr✐❝ ▲❡❛st ❙q✉❛r❡s ❊st✐♠❛t✐♦♥ ❛♥❞ ❚❡st✐♥❣✳ ❊❝♦♥♦♠❡tr✐❝❛ ✺✺✭✹✮✿ ✽✶✾✲✽✹✼✱ ✶✾✽✼ ❉❖■✿ ✶✵✳✷✸✵✼✴✶✾✶✶✵✸✶

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘❡❢❡r❡♥❝❡s ✻✲✺

❘❡❢❡r❡♥❝❡s

❚❛②❧♦r✱ ❏✳ ❲ ❊st✐♠❛t✐♥❣ ✈❛❧✉❡ ❛t r✐s❦ ❛♥❞ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✉s✐♥❣ ❡①♣❡❝t✐❧❡s ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊❝♦♥♦♠❡tr✐❝s ✭✻✮✱ ✷✱ ✷✵✵✽ ❨❛♦✱ ◗✳ ❛♥❞ ❚♦♥❣✱ ❍✳ ❆s②♠♠❡tr✐❝ ❧❡❛st sq✉❛r❡s r❡❣r❡ss✐♦♥ ❡st✐♠❛t✐♦♥✿ ❆ ♥♦♥♣❛r❛♠❡tr✐❝ ❛♣♣r♦❛❝❤ ❏♦✉r♥❛❧ ♦❢ ◆♦♥♣❛r❛♠❡tr✐❝ ❙t❛t✐st✐❝s ✭✻✮✱ ✷✲✸✱ ✶✾✾✻ ❨❡❡✱ ❚✳ ❲✳ ❚❤❡ ❱●❆▼ P❛❝❦❛❣❡ ❢♦r ❈❛t❡❣♦r✐❝❛❧ ❉❛t❛ ❆♥❛❧②s✐s ❘ r❡❢❡r❡♥❝❡ ♠❛♥✉❛❧ ❤tt♣✿✴✴✶✷✼✳✵✳✵✳✶✿✶✻✽✵✵✴❧✐❜r❛r②✴❱●❆▼✴❞♦❝✴❝❛t❡❣♦r✐❝❛❧❱●❆▼✳♣❞❢

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✶

❈♦❤❡r❡♥❝❡

⊡ ❈♦❤❡r❡♥t r✐s❦ ♠❡❛s✉r❡ ρ (·) ♦❢ r❡❛❧✲✈❛❧✉❡❞ r✳✈✳✬s ✇❤✐❝❤ ♠♦❞❡❧ t❤❡ r❡t✉r♥s

◮ ❙✉❜❛❞❞✐t✐✈✐t②✱ ρ(x + y) ≥ ρ(x) + ρ(y)

❉❡t❛✐❧s

◮ ❚r❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥❝❡✱ ρ(x + c) = ρ(x) ❢♦r ❛ ❝♦♥st❛♥t c ◮ ▼♦♥♦t♦♥✐❝✐t②✱ ρ(x) < ρ(y), x < y ◮ P♦s✐t✐✈❡ ❤♦♠♦❣❡♥❡✐t②✱ ρ(kx) = kρ (x) , k > ✵

❱❛❘ ❛♥❞ ❊❙

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✷

❙✉❜❛❞❞✐t✐✈✐t②

❈♦❤❡r❡♥❝❡

⊡ ρ (x + y) ≤ ρ (x) + ρ (y) ⊡ ❉✐✈❡rs✐✜❝❛t✐♦♥ ♥❡✈❡r ✐♥❝r❡❛s❡s r✐s❦ ⊡ ◗✉❛♥t✐❧❡s ❛r❡ ♥♦t s✉❜❛❞❞✐t✐✈❡ ⊡ ❊①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✐s s✉❜❛❞❞✐t✐✈❡✱ ❉❡❧❜❛❡♥ ✭✶✾✾✽✮

❱❛❘ ❛♥❞ ❊❙

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✸

❊❙ ✉s✐♥❣ ❡①♣❡❝t✐❧❡s

⊡ ❊①♣❡❝t✐❧❡ ✭s❡❡ ❛❧s♦ ✼✲✹ ❛♥❞ ✼✲✺✮ eτ = ❛r❣ ♠✐♥

θ ❊ ρτ,✷ (Y − θ)

ρτ,✷ (u) = |τ − ■ {u < ✵}| |u|✷ ⊡ ❋✐rst ♦r❞❡r ❝♦♥❞✐t✐♦♥ (✶ − τ) s

−∞

(y − s)f (y)dy − τ ∞

s

(y − s)f (y)dy =✵

❚❛✐❧ ❙tr✉❝t✉r❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✹

❊❙ ✉s✐♥❣ ❡①♣❡❝t✐❧❡s

⊡ ❊①t❡♥s✐♦♥ ❛♥❞ r❡❢♦r♠✉❧❛t✐♦♥ (✶ − τ) s

−∞

(y − s)f (y)dy − τ s

−∞

(y − s)f (y)dy = τ ∞

−∞

(y − s)f (y)dy ⊡ ❘❡❛rr❛♥❣✐♥❣ eτ − ❊(Y ) = ✶ − ✷τ τ eτ

−∞

(y − eτ)f (y)dy

❚❛✐❧ ❙tr✉❝t✉r❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✺

❊❙ ✉s✐♥❣ ❡①♣❡❝t✐❧❡s

⊡ ❊①♣❡❝t❡❞ s❤♦rt❢❛❧❧ eτ − ❊[Y ] = ✶ − ✷τ τ ❊ [(Y − eτ) ■{Y < eτ}] ❊[Y |Y < eτ] = eτ + τ(eτ − ❊[Y ]) (✷τ − ✶)F(eτ) ⊡ ❯s❡ ew(α) = qα ❊[Y |Y < qα] = ew(α) + (ew(α) − ❊[Y ])w(α) (✷w(α) − ✶)α

❚❛✐❧ ❙tr✉❝t✉r❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✻

❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s✱ w (α)

⊡ ❘❡❧❛t✐♦♥ ♦❢ ❡①♣❡❝t✐❧❡s ❛♥❞ q✉❛♥t✐❧❡s ✭♣r♦♦❢ ✼✲✼ ❛♥❞ ✼✲✽✮ w(α) = ▲P▼ew(α)(y) − ew(α)α ✷

• ▲P▼ew(α)(y) − ew(α)α
• + ew(α) − ❊[Y ]

⊡ ❲✐t❤ t❤❡ ❧♦✇❡r ♣❛rt✐❛❧ ♠♦♠❡♥t ▲P▼u(y) = u

−∞

yf (y)dy

❚❛✐❧ ❙tr✉❝t✉r❡ ❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✼

❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s✱ w (α)

⊡ ❊①♣❡❝t✐❧❡ ✭❆◆❉ ❧♦❝❛t✐♦♥ ❡st✐♠❛t❡✮ s♦❧✈❡s {α − ✶} eα

−∞

(y − eα)f (y)dy = α ∞

eα

(y − eα)f (y)dy

• +

−α

eα

−∞(y−eα)f (y)dy

⊡ ❘❡❛rr❛♥❣❡ α

• eα − ✷

eα

−∞

eαf (y)dy

• +

eα

−∞

eαf (y)dy =α ∞

−∞

yf (y)dy − ✷ eα

−∞

yf (y)dy

• +

eα

−∞

yf (y)dy

❚❛✐❧ ❙tr✉❝t✉r❡ ❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✽

❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s✱ w (α)

⊡ ❖r❞❡r✐♥❣ t❡r♠s α

• ✷

eα

−∞

yf (y)dy − eα eα

−∞

f (y)dy

• + eα − ❊[Y ]
• =

eα

−∞

yf (y)dy − eα

−∞

eαf (y)dy ⊡ ❙♦❧✈✐♥❣ ❢♦r t❤❡ r✐s❦ ❧❡✈❡❧✱ F(ew(α)) = α w(α) = ▲P▼ew(α)(y) − ew(α)α ✷

• ▲P▼ew(α)(y) − ew(α)α
• + ew(α) − ❊[Y ]

❚❛✐❧ ❙tr✉❝t✉r❡ ❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✾

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ❛♥❞ ❱❛❧✉❡ ❛t ❘✐s❦

⊡ ❱❛❧✉❡ ❛t ❘✐s❦ ✭❱❛❘✮

◮ ❋♦r ❛ ❝❞❢ F(·) ♦❢ ❛♥ r✳✈✳ Y

❱❛❘α = qα = F(α)−✶ ⊡ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✭❊❙✮

◮ ❇❛s❡❧✿ ❱❛❘ t❤r❡s❤♦❧❞ η = qα

ESη = ❊[Y |Y < η]

❱❛❘ ❛♥❞ ❊❙

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❆♣♣❡♥❞✐① ✼✲✶✵

• ❡♥❡r❛❧ ◗✉❛♥t✐❧❡ ❘❡❧❛t✐♦♥ ▲❡✈❡❧

⊡ ❙♦❧✉t✐♦♥ zα t♦ ✭✶✮ α ✶ − α = zα

−∞ |y − zα|γ−✶ dF(y)

∞

zα |y − zα|γ−✶ dF(y)

⊡ ❖r❞❡r✐♥❣ t❡r♠s α = zα

−∞ |y − zα|γ−✶ dF(y)

∞

∞ |y − zα|γ−✶ dF(y)

❙❡♥s✐t✐✈✐t② ❆♥❛❧②s✐s

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t✐❧❡ ❜❛s❡❞ ❙❤♦rt❢❛❧❧

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✽

❊❙ ❉②♥❛♠✐❝s

❋✐❣✉r❡ ✾✿ ❊❙q ❛♥❞ ❱❛❘ ❛t ✵ ✵✶❀ ✵ ✭t♦♣✮ ❛♥❞ ✶ ✭❜♦tt♦♠✮❀ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ♦❢ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s ❢♦r t❤❡ q✉❛♥t✐❧❡

❚❊❘❊❙ ✲ ❚❛✐❧ ❊✈❡♥t ❘✐s❦ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧