Liquidity management, fire sales and liquidity crises in banking

▲✐q✉✐❞✐t② ▼❛♥❛❣❡♠❡♥t✱ ❋✐r❡ ❙❛❧❡s ❛♥❞ ▲✐q✉✐❞✐t② ❈r✐s❡s ∗ ✐♥ ❇❛♥❦✐♥❣✿ t❤❡ ❘♦❧❡ ♦❢ ▲❡✈❡r❛❣❡ † ❋❛❜✐❛♥❛ ●♦♠❡③ ‡ ◗✉②♥❤✲❆♥❤ ❱♦ ▼❛r❝❤ ✷✾✱ ✷✵✶✾ ❆❜str❛❝t ❚❤✐s ♣❛♣❡r ♣r♦♣♦s❡s ❛ ♣♦s✐t✐✈❡ t❤❡♦r② ♦❢ t❤❡ ❧✐♥❦s ❜❡t✇❡❡♥ ❜❛♥❦s✬ ❝❛♣✐t❛❧✐s❛t✐♦♥ ❛♥❞ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ t❛❦✐♥❣✱ t❤❡ ❡①t❡♥t ♦❢ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠s✱ ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ■♥ ❛ ❜❛s✐❝ ❢r❛♠❡✇♦r❦ ✇✐t❤ ❛ s✐♥❣❧❡ ❜❛♥❦✱ ✇❡ ✜♥❞ t❤❛t ❜❛♥❦s✬ ✐♥❝❡♥✲ t✐✈❡s t♦ ❤♦❧❞ ❧✐q✉✐❞✐t② ❢♦r ♣r❡❝❛✉t✐♦♥❛r② r❡❛s♦♥s ❛r❡ ✐♥❝r❡❛s✐♥❣ ✇✐t❤ t❤❡✐r ❝❛♣✐t❛❧✳ ■♥ ❛ ❝♦♥t✐♥✉✉♠✲♦❢✲❜❛♥❦s s❡tt✐♥❣ ✐♥ ✇❤✐❝❤ ❜♦t❤ ♣r❡❝❛✉t✐♦♥❛r② ❛♥❞ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡s ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✱ ✇❡ ✜♥❞ t❤❛t ✇❤✐❧❡ t❤❡ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✱ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ t❤❡ ❧❛tt❡r ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s ✐s ♥♦t ♠♦♥♦t♦♥✐❝✳ ❏❊▲ ❈❧❛ss✐✜❝❛t✐♦♥✿ ❉✽✷✱ ●✷✶✳ ❑❡②✇♦r❞s✿ ▲❡✈❡r❛❣❡✱ ❲❤♦❧❡s❛❧❡ ❉❡❜ts✱ Pr❡❝❛✉t✐♦♥❛r② ▲✐q✉✐❞✐t② ❍♦❧❞✐♥❣s✱ ❙♣❡❝✉✲ ❧❛t✐✈❡ ▲✐q✉✐❞✐t② ❍♦❧❞✐♥❣s✱ ❛♥❞ ❈❛s❤✲■♥✲❚❤❡✲▼❛r❦❡t Pr✐❝✐♥❣✳ ❈♦rr❡s♣♦♥❞✐♥❣ ❛✉t❤♦r ◗✉②♥❤✲❆♥❤ ❱♦ ❛t t❤❡ ❇❛♥❦ ♦❢ ❊♥❣❧❛♥❞✳ ❆ ♣r❡✈✐♦✉s ✈❡rs✐♦♥ ✇❛s ❝✐r❝✉❧❛t❡❞ ✉♥❞❡r t❤❡ t✐t❧❡✿ ✧▲✐q✉✐❞✐t② ▼❛♥❛❣❡♠❡♥t ✐♥ ❇❛♥❦✐♥❣✿ ❲❤❛t ✐s t❤❡ ❘♦❧❡ ♦❢ ▲❡✈❡r❛❣❡❄✧ ❲❡ ❛r❡ ❣r❛t❡❢✉❧ t♦ ❚♦♥✐ ❆❤♥❡rt ✭❞✐s❝✉ss❛♥t✮✱ ❯rs ❇✐r❝❤❧❡r✱ ●✐❛♥♥✐ ❉❡ ◆✐❝♦❧♦✱ ❍❛♥s ●❡rs❜❛❝❤✱ ▼✐❝❤❡❧ ❍❛❜✐❜✱ ❋r❡❞❡r✐❝ ▼❛❧❤❡r❜❡✱ ❊tt♦r❡ P❛♥❡tt✐ ✭❞✐s❝✉ss❛♥t✮✱ ❇r✉♥♦ P❛r✐❣✐✱ ❙❡❜❛st✐❛♥ P❢❡✐❧ ❛♥❞ ❏❡❛♥✲❈❤❛r❧❡s ❘♦❝❤❡t ❢♦r t❤❡✐r ❤❡❧♣❢✉❧ ❝♦♠♠❡♥ts ❛♥❞ s✉❣❣❡st✐♦♥s✳ ❲❡ ❛❧s♦ t❤❛♥❦ s❡♠✐♥❛r ❛♥❞ ❝♦♥❢❡r❡♥❝❡ ♣❛rt✐❝✐♣❛♥ts ❛t t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❩✉r✐❝❤✱ ✷✵✶✺ P❛r✐s ❋✐♥❛♥❝✐❛❧ ▼❛♥❛❣❡♠❡♥t ❈♦♥❢❡r❡♥❝❡✱ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❇❛s❡❧✱ t❤❡ ❇❛♥❦ ♦❢ ❊♥❣❧❛♥❞✱ ■❋❆❇❙ ✷✵✶✻✱ ❘❊❙ ✷✵✶✼✱ ❊❋▼❆ ✷✵✶✼✱ ❊❊❆ ✷✵✶✼✱ ❋▼❆ ✷✵✶✼✱ t❤❡ ✜rst ❛♥♥✉❛❧ ✇♦r❦s❤♦♣ ♦❢ t❤❡ ❊❙❈❇ ❘❡s❡❛r❝❤ ❈❧✉st❡r ✸ ♦♥ ✜♥❛♥❝✐❛❧ st❛❜✐❧✐t②✱ ♠❛❝r♦♣r✉❞❡♥t✐❛❧ r❡❣✉❧❛t✐♦♥ ❛♥❞ ♠✐❝r♦♣r✉❞❡♥t✐❛❧ s✉♣❡r✈✐s✐♦♥✱ ✷✵✶✽ ■❇❊❋❆ ❙✉♠♠❡r ❝♦♥❢❡r❡♥❝❡✱ ❊❙❊▼ ✷✵✶✽ ❛♥❞ ■❲❋❙❆❙ ✷✵✶✽ ❢♦r t❤❡✐r ✉s❡❢✉❧ ❢❡❡❞❜❛❝❦✳ ❆ s✐❣♥✐✜❝❛♥t ♣❛rt ♦❢ t❤✐s r❡s❡❛r❝❤ ✇❛s ❞♦♥❡ ✇❤❡♥ t❤❡ s❡❝♦♥❞ ❛✉t❤♦r ✇❛s ❛✣❧✐❛t❡❞ ✇✐t❤ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❩✉r✐❝❤ ❛♥❞ r❡❝❡✐✈❡❞ ❢✉♥❞✐♥❣ ❢r♦♠ t❤❡ ❊❘❈ ✭t❤❡ ❣r❛♥t ❛❣r❡❡♠❡♥t ✷✹✾✹✶✺✲❘▼❆❈✮✱ ❢r♦♠ ◆❈❈❘ ❋✐♥❘✐s❦ ✭♣r♦❥❡❝t ✧❇❛♥❦✐♥❣ ❛♥❞ ❘❡❣✉❧❛t✐♦♥✧✮ ❛♥❞ ❢r♦♠ ❙✇✐ss ❋✐♥❛♥❝❡ ■♥st✐t✉t❡ ✭♣r♦❥❡❝t ✧❙②st❡♠✐❝ ❘✐s❦ ❛♥❞ ❉②♥❛♠✐❝ ❈♦♥tr❛❝t ❚❤❡♦r②✧✮✳ ❚❤❡ ✈✐❡✇s ❡①♣r❡ss❡❞ ✐♥ t❤✐s ♣❛♣❡r ❛r❡ t❤♦s❡ ♦❢ t❤❡ ❛✉t❤♦rs✱ ❛♥❞ ♥♦t ♥❡❝❡ss❛r✐❧② t❤♦s❡ ♦❢ t❤❡ ❇❛♥❦ ♦❢ ❊♥❣❧❛♥❞ ♦r ✐ts ❝♦♠♠✐tt❡❡s✳ † ❯♥✐✈❡rs✐t② ♦❢ ❇r✐st♦❧✳ ❊✲♠❛✐❧✿ ❢❛❜✐❛♥❛✳❣♦♠❡③❅❜r✐st♦❧✳❛❝✳✉❦✳ ‡ ❇❛♥❦ ♦❢ ❊♥❣❧❛♥❞✱ ❚❤r❡❛❞♥❡❡❞❧❡ ❙tr❡❡t✱ ▲♦♥❞♦♥✱ ❊❈✷❘ ✽❆❍✳ ❊✲♠❛✐❧✿ q✉②♥❤✲ ❛♥❤✳✈♦❅❜❛♥❦♦❢❡♥❣❧❛♥❞✳❝♦✳✉❦✳ ∗ Electronic copy available at: https://ssrn.com/abstract=2696126 ✶ ■♥tr♦❞✉❝t✐♦♥ ■♥ t❤❡ ❛❢t❡r♠❛t❤ ♦❢ t❤❡ ❣❧♦❜❛❧ ✜♥❛♥❝✐❛❧ ❝r✐s✐s✱ t✇♦ ♥❡✇ ❧✐q✉✐❞✐t② st❛♥❞❛r❞s✱ ♥❛♠❡❧② t❤❡ ▲✐q✉✐❞✐t② ❈♦✈❡r❛❣❡ ❘❛t✐♦ ✭▲❈❘✮ ❛♥❞ t❤❡ ◆❡t ❙t❛❜❧❡ ❋✉♥❞✐♥❣ ❘❛t✐♦ ✭◆❙❋❘✮✱ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② t❤❡ ❇❛s❡❧ ❈♦♠♠✐tt❡❡✳ ❚❤❡✐r ♠❛✐♥ ♦❜❥❡❝t✐✈❡s ❡♥❝♦♠♣❛ss ❝r❡❛t✐♥❣ ✐♥❝❡♥✲ t✐✈❡s ❢♦r ❜❛♥❦s t♦ ❜❡tt❡r ♠❛♥❛❣❡ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ ❛♥❞ ✐♠♣r♦✈✐♥❣ t❤❡ ❜❛♥❦✐♥❣ s❡❝t♦r✬s ❛❜✐❧✐t② t♦ ❛❜s♦r❜ ❧✐q✉✐❞✐t② s❤♦❝❦s✳ ❚❤❡s❡ ♥❡✇ ♣♦❧✐❝② ♠❡❛s✉r❡s r❛✐s❡ ❛ ♥✉♠❜❡r ♦❢ q✉❡s✲ t✐♦♥s✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ♣♦t❡♥t✐❛❧ s✉❜st✐t✉t❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡♠❡♥t❛r✐t② ❜❡t✇❡❡♥ t❤❡♠ ❛♥❞ t❤❡ ❛❧r❡❛❞②✲❡①✐st✐♥❣ ❝❛♣✐t❛❧ r❡q✉✐r❡♠❡♥ts✳ ❯s✐♥❣ ❛ ♣♦s✐t✐✈❡ ❛♣♣r♦❛❝❤✱ t❤❡ ❝✉rr❡♥t ♣❛♣❡r ❛✐♠s t♦ s❤❡❞ ❧✐❣❤t ♦♥ t❤✐s q✉❡st✐♦♥ ❜② ❡①❛♠✐♥✐♥❣ t❤❡ ❡✛❡❝ts ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤❡✐r ✐♥❝❡♥t✐✈❡s ✇✐t❤ r❡s♣❡❝t t♦ ❧✐q✉✐❞✐t② r✐s❦ ♠❛♥❛❣❡♠❡♥t✳ Pr❡❝✐s❡❧②✱ t❤✐s ♣❛♣❡r ❡①♣❧♦r❡s t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥s✿ ❉♦ ❜❡tt❡r ❝❛♣✐t❛❧✐s❡❞ ❜❛♥❦s ❤❛✈❡ ❜❡tt❡r ✐♥❝❡♥t✐✈❡s t♦ ♠❛♥❛❣❡ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦❄ ❆r❡ ❜❡tt❡r ❝❛♣✐t❛❧✐s❡❞ ❜❛♥❦✐♥❣ s②st❡♠s ♠♦r❡ ♦r ❧❡ss ✈✉❧♥❡r❛❜❧❡ t♦ ❧✐q✉✐❞✲ ✐t② ❝r✐s❡s❄ ❍♦✇ ❞♦❡s t❤❡ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❛✛❡❝t t❤❡ ❡①t❡♥t ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠❄ ■t t❤✉s ♣r♦♣♦s❡s ❛ ♣♦s✐t✐✈❡ t❤❡♦r② ♦❢ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❜❛♥❦s✬ ❝❛♣✐t❛❧✐s❛t✐♦♥ ❛♥❞ ❧✐q✉✐❞✐t② r✐s❦ t❛❦✐♥❣✱ ❛s ✇❡❧❧ ❛s t❤❡ s❡✈❡r✐t② ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ❲❡ ❞❡✈❡❧♦♣ t❤✐s t❤❡♦r② ✐♥ ❛ ♠♦❞❡❧ ✇❤❡r❡ ❜❛♥❦s ❡♥❣❛❣❡ ✐♥ ♠❛t✉r✐t② tr❛♥s❢♦r♠❛t✐♦♥✱ ❜♦rr♦✇✐♥❣ s❤♦rt ❛♥❞ ❧❡♥❞✐♥❣ ❧♦♥❣✳ ❚❤✐s ❧❡❛✈❡s t❤❡♠ ✈✉❧♥❡r❛❜❧❡ t♦ ❧✐q✉✐❞✐t② s❤♦❝❦s✳ ❙♦ ❢❛r✱ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ❜❛♥❦ ❧✐q✉✐❞✐t②✱ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦s ❛r❡ ✉s✉❛❧❧② ♠♦❞❡❧❧❡❞ ❛ ❧❛ ❉✐❛♠♦♥❞ ✲ ❉②❜✈✐❣✳ ■♥ t❤✐s ✇❛②✱ ❜❛♥❦s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ✜♥❛♥❝❡❞ ❜② r❡t❛✐❧ ❞❡♠❛♥❞❛❜❧❡ ❞❡♣♦s✐ts ❛♥❞ t❤✉s✱ ♠❛② ❜❡ ❤✐t ❜② ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐❢ ❛ ❤✐❣❤ ♥✉♠❜❡r ♦❢ t❤❡✐r ❞❡♣♦s✐t♦rs ❝♦♠❡ t♦ ✇✐t❤❞r❛✇✳ ❚❤✐s ✇✐t❤❞r❛✇❛❧ ✐s✱ ✐♥ t✉r♥✱ ♠♦❞❡❧❧❡❞ ❛s ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ❡✐t❤❡r ❜② t❤❡ t✐♠❡ ♣r❡❢❡r❡♥❝❡s ♦❢ r✐s❦✲❛✈❡rs❡ ❞❡♣♦s✐t♦rs ♦r ❜② t❤❡ ❝♦♦r❞✐♥❛t✐♦♥ ♣r♦❜❧❡♠ ❜❡t✇❡❡♥ t❤❡♠ ❡❛❝❤ r❡❝❡✐✈✐♥❣ s♦♠❡ ♣r✐✈❛t❡ s✐❣♥❛❧ ❛❜♦✉t t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❜❛♥❦s✬ ❛ss❡ts✳ ■♥ t❤✐s ♣❛♣❡r✱ ✐♥s♣✐r❡❞ ❜② s❡✈❡r❛❧ ♦❜s❡r✈❛t✐♦♥s ❢r♦♠ t❤❡ ✷✵✵✼ ✲ ✷✵✵✾ ❝r✐s✐s✱ ✇❡ ❛❞♦♣t ❛ ♠♦❞❡❧❧✐♥❣ ❛♣♣r♦❛❝❤ t❤❛t ❞✐✛❡rs ✐♥ t✇♦ ♠❛✐♥ ❛s♣❡❝ts✳ ❋✐rst✱ ✐♥ ♦✉r s❡t✉♣✱ ❜❛♥❦s✬ s❤♦rt✲t❡r♠ ❞❡❜ts t❛❦❡ t❤❡ ❢♦r♠ ♦❢ ✉♥s❡❝✉r❡❞ ✇❤♦❧❡s❛❧❡ ❞❡❜ts ✭✐♥st❡❛❞ ♦❢ ❜❡✐♥❣ r❡t❛✐❧ ❞❡♣♦s✐ts✮✱ s✉❝❤ ❛s ✉♥s❡❝✉r❡❞ ❝♦♠♠❡r❝✐❛❧ ♣❛♣❡rs✳ ❚❤❡s❡ t②♣❡s ♦❢ ❞❡❜t ✇❡r❡ ❛t t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ r❡❝❡♥t ❣❧♦❜❛❧ ✜♥❛♥❝✐❛❧ ❝r✐s✐s✳ ❙✐♥❝❡ ✉♥s❡❝✉r❡❞ ✇❤♦❧❡s❛❧❡ ❞❡❜ts ❛r❡ ✉s✉❛❧❧② ❤❡❧❞ ❜② s♦♣❤✐st✐❝❛t❡❞ ✐♥✈❡st♦rs✱ s✉❝❤ ❛s ✜♥❛♥❝✐❛❧ ✐♥st✐t✉t✐♦♥s ♦r ♠♦♥❡② ✷ Electronic copy available at: https://ssrn.com/abstract=2696126 ♠❛r❦❡t ❢✉♥❞s✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❜❛♥❦s✬ s❤♦rt✲t❡r♠ ❞❡❜t❤♦❧❞❡rs ❛r❡ r✐s❦✲♥❡✉tr❛❧✱ ❛♥❞ t❤❛t t❤❡ ❞❡❜t r❡♣❛②♠❡♥t ✐s ❡♥❞♦❣❡♥♦✉s❧② ❞❡t❡r♠✐♥❡❞✱ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❜❛♥❦s✬ ❝❤♦✐❝❡ ♦❢ ❛ss❡ts✳ ❚❤✐s ❡♥❞♦❣❡♥❡✐t② ✐s ❛ ❦❡② ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ♦✉r ♣❛♣❡r ❛♥❞ ♦t❤❡r ♣❛♣❡rs t❤❛t ♠♦❞❡❧ r❡t❛✐❧ ❞❡♣♦s✐ts ❛♥❞ ❛ss✉♠❡ ❛♥ ❡①♦❣❡♥♦✉s ❞❡♣♦s✐t r❛t❡✳ ❙❡❝♦♥❞✱ t❤❡ ❧✐q✉✐❞✐t② ❡✈❡♥t ✐s ♠♦❞❡❧❧❡❞ ❜② t❤❡ ❛rr✐✈❛❧ ♦❢ ✉♥❢❛✈♦r❛❜❧❡ ❛♥❞ ♣✉❜❧✐❝ ✐♥❢♦r✲ ♠❛t✐♦♥ ♦♥ t❤❡ q✉❛❧✐t② ♦❢ ❜❛♥❦s✬ ❛ss❡ts✳ ❚❤✐s ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ♥❡❣❛t✐✈❡❧② ❛✛❡❝ts ❜❛♥❦s✬ ❢✉♥❞✐♥❣ ❧✐q✉✐❞✐t②✱ ❛♥❞ t❤✉s✱ ♠❛❦❡s ✐t ❞✐✣❝✉❧t ❢♦r ❜❛♥❦s t♦ ♠❡❡t t❤❡✐r r❡♣❛②♠❡♥t ♦❜❧✐❣❛✲ t✐♦♥s✳ ❚❤✐s ✐s ❛♥❛❧♦❣♦✉s t♦ t❤❡ ✉♥❢♦❧❞✐♥❣ ❡✈❡♥ts ♦❢ t❤❡ r❡❝❡♥t ❝r✐s✐s✱ ❛❝❝♦r❞✐♥❣ t♦ ✇❤✐❝❤ t❤❡ tr✐❣❣❡r✐♥❣ ♦❢ t❤❡ ❧✐q✉✐❞✐t② ♣r♦❜❧❡♠ ✐♥✈♦❧✈❡❞ s♦♠❡ ♣✉❜❧✐❝ ✐♥❢♦r♠❛t✐♦♥ r❡❣❛r❞✐♥❣ ❛♥ ✐♥❝r❡❛s❡ ✐♥ s✉❜♣r✐♠❡ ♠♦rt❣❛❣❡ ❞❡❢❛✉❧ts✳ ❲❤❛t ✇❛s ❢♦❧❧♦✇❡❞ ✇❛s ❛ ❞❡t❡r✐♦r❛t✐♦♥ ♦❢ t❤❡ s❤♦rt✲t❡r♠ ❢✉♥❞✐♥❣ ♠❛r❦❡t✱ s✉❝❤ ❛s t❤❡ ❝♦♠♠❡r❝✐❛❧ ♣❛♣❡r ♠❛r❦❡t✳ ❍❡♥❝❡✱ t❤❡ ❝♦♥t❡①t ✇❡ ❤❛✈❡ ✐♥ ♠✐♥❞ ✐s ♦♥❡ ♦❢ ❜❛♥❦s t❤❛t ❛r❡ ✜♥❛♥❝❡❞ ❜② ❡q✉✐t② ❛♥❞ ✇❤♦❧❡s❛❧❡ ✉♥s❡❝✉r❡❞ s❤♦rt✲t❡r♠ ❞❡❜t t❤❛t ♠❛t✉r❡s ❛❢t❡r ♦♥❡ ♣❡r✐♦❞✳ ❇❛♥❦s ❝♦✉❧❞ ✐♥✈❡st ✐♥ t✇♦ t②♣❡s ♦❢ ❛ss❡ts✳ ❖♥❡ ✐s s❤♦rt✲t❡r♠ ❛ss❡ts✱ r❡❢❡rr❡❞ t♦ ❛s ❧✐q✉✐❞ ❛ss❡ts✱ ❛♥❞ t❤❡ ♦t❤❡r ✐s ❧♦♥❣✲t❡r♠ ❛ss❡ts✳ ❚❤❡ ❧❛tt❡r ✐s ♠♦r❡ ♣r♦✜t❛❜❧❡ t❤❛♥ t❤❡ ❢♦r♠❡r✱ ❜✉t t❛❦❡s t✇♦ ♣❡r✐♦❞s t♦ ②✐❡❧❞ ❛ ❝❛s❤ ✢♦✇✳ ❚❤✐s ♠❛t✉r✐t② ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ t❤❡ ♣❛②♦✛ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛♥❞ ❞❡❜t r❡♣❛②♠❡♥ts ❣✐✈❡s r✐s❡ t♦ ❛ ♥❡❡❞ ❢♦r ❜❛♥❦s t♦ ❛rr❛♥❣❡ ❢♦r s♦♠❡ ❧✐q✉✐❞✐t② ❛t t❤❡ ✐♥t❡r✐♠ ❞❛t❡ ✇❤❡♥ t❤❡✐r s❤♦rt✲t❡r♠ ❞❡❜t r❡♣❛②♠❡♥ts ❛r❡ ❞✉❡✳ ❲❡ ❛ss✉♠❡ t❤❛t ❜❛♥❦s ❝♦✉❧❞ r❛✐s❡ ❧✐q✉✐❞✐t② ❛t t❤❡ ✐♥t❡r✐♠ ❞❛t❡ ❜② ♣❧❡❞❣✐♥❣ ❢✉t✉r❡ ❝❛s❤ ✢♦✇s ♦❢ t❤❡✐r ❧♦♥❣✲t❡r♠ ❛ss❡ts ✲ ❢✉♥❞✐♥❣ ❧✐q✉✐❞✐t②✳ ❍♦✇❡✈❡r✱ t❤❡ ❜❛♥❦s✬ ❝❛♣❛❝✐t② t♦ ❣❡♥❡r❛t❡ ❧✐q✉✐❞✐t② ✐♥ t❤✐s ✇❛② ♠❛② ❜❡ r❡str✐❝t❡❞ ✐❢ ❜❛❞ ♥❡✇s ♦♥ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐s r❡✈❡❛❧❡❞ ❛t t❤❡ r❡♣❛②♠❡♥t ❞❛t❡✳ ❚❤✐s ❧✐♠✐t❡❞ ♣❧❡❞❣❡❛❜✐❧✐t② ♣r♦✈✐❞❡s✱ ✐♥ ♦✉r ♠♦❞❡❧✱ ❛ r❡❛s♦♥ ❢♦r ❜❛♥❦s t♦ ✐♥✈❡st ✐♥ t❤❡ ❧❡ss ♣r♦✜t❛❜❧❡✱ ❜✉t ♠♦r❡ ❧✐q✉✐❞ s❤♦rt✲t❡r♠ ❛ss❡ts ✐❢ t❤❡② ✇❛♥t t♦ ✐♥s✉r❡ t❤❡♠s❡❧✈❡s ❛❣❛✐♥st ❧✐q✉✐❞✐t② s❤♦❝❦s✳ ❲❡ st❛rt ✇✐t❤ ❛♥ ❡①❛♠✐♥❛t✐♦♥ ♦❢ ❜❛♥❦s✬ ✐♥❝❡♥t✐✈❡s t♦ ❤♦❧❞ ❧✐q✉✐❞ ❛ss❡ts t♦ ♣r♦t❡❝t t❤❡♠s❡❧✈❡s ❛❣❛✐♥st ❢✉t✉r❡ ❧✐q✉✐❞✐t② s❤♦❝❦s✱ ✐✳❡✳✱ ✐♥❝❡♥t✐✈❡s ❢♦r ♣r❡❝❛✉t✐♦♥❛r② ❧✐q✉✐❞✐t② ❤♦❧❞✲ ✐♥❣s✱ ❛♥❞ ❤♦✇ t❤❡s❡ ✐♥❝❡♥t✐✈❡s ❛r❡ ❛✛❡❝t❡❞ ❜② ❜❛♥❦s✬ ❧❡✈❡r❛❣❡✳ ❚❤✐s ✐s ❞♦♥❡ ✇✐t❤✐♥ ❛ s✐♠♣❧❡ ❢r❛♠❡✇♦r❦ ♦❢ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❜❛♥❦s✬ ❞❡❝✐s✐♦♥s✳ ❚❤❡♥✱ ✐♥ ♦r❞❡r t♦ ❛♥❛❧②s❡ t❤❡ ❧✐♥❦s ❜❡t✇❡❡♥ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❛♥❞ ❜♦t❤ t❤❡ ❡①t❡♥t ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠✱ ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✱ ✇❡ ❝❛st t❤✐s ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ♦❢ ✐♥❞✐✈✐❞✉❛❧ ❜❛♥❦s✬ ❞❡❝✐s✐♦♥s ✐♥t♦ ❛ ❝♦♥t✐♥✉✉♠✲♦❢✲❜❛♥❦s s❡tt✐♥❣ ✐♥ ✇❤✐❝❤ ✇❡ ❛❝❝♦✉♥t ❢♦r t❤❡ ❡①✐st❡♥❝❡✱ ❛t t❤❡ ✐♥t❡r✐♠ ✸ Electronic copy available at: https://ssrn.com/abstract=2696126 ❞❛t❡✱ ♦❢ ❛ s❡❝♦♥❞❛r② ♠❛r❦❡t ❢♦r ❧♦♥❣✲t❡r♠ ❛ss❡ts✳ ❚❤❡r❡❢♦r❡✱ ❜❛♥❦s ✇✐t❤ ❛ ❧✐q✉✐❞✐t② s❤♦rt✲ ❛❣❡ ❝♦✉❧❞ s❡❧❧ t❤❡✐r ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐♥ ♦r❞❡r t♦ r❛✐s❡ ❧✐q✉✐❞✐t②✳ ❚❤✐s ❛❞❞✐t✐♦♥❛❧ ❡❧❡♠❡♥t s❡r✈❡s t✇♦ ♣✉r♣♦s❡s✳ ■t ✜rst ❛❧❧♦✇s ✉s t♦ ❝❛♣t✉r❡ ❛♥♦t❤❡r s♦✉r❝❡ ♦❢ ❧✐q✉✐❞✐t② t❤❛t ❜❛♥❦s ❝❛♥ r❡❧② ♦♥✱ ♥❛♠❡❧② ♠❛r❦❡t ❧✐q✉✐❞✐t②✳ ■t ❛❧s♦ ❡♥❛❜❧❡s ✉s t♦ t❛❦❡ ✐♥t♦ ❝♦♥s✐❞❡r❛t✐♦♥ ❛♥♦t❤❡r ♠♦t✐✈❡ ❞r✐✈✐♥❣ ❜❛♥❦s✬ ❝❤♦✐❝❡ ♦❢ ❡①✲❛♥t❡ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♣r❡❝❛✉t✐♦♥✲ ❛r② ♠♦t✐✈❡✳ ❚❤❛t ✐s t❤❡ ✧str❛t❡❣✐❝✧ ♠♦t✐✈❡ ♦❢ ❜❡✐♥❣ ❛❜❧❡ t♦ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ✜r❡ s❛❧❡s ✲ t❤❡ s♦✲❝❛❧❧❡❞ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡ ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ■♥ ♦✉r ❝♦♥t✐♥✉✉♠✲♦❢✲❜❛♥❦s s❡tt✐♥❣✱ ❛ss❡t s❛❧❡s ❛r❡ ♠♦❞❡❧❧❡❞ ❛s ✐♥ ❆❝❤❛r②❛ ❛♥❞ ❱✐s✇❛♥❛t❤❛♥ ✭✷✵✶✶✮✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛r❡ s♣❡❝✐✜❝ ❛♥❞ ❝❛♥ ♦♥❧② ❜❡ ❛❝q✉✐r❡❞ ❜② ❜❛♥❦s t❤❛t s✉r✈✐✈❡ ❧✐q✉✐❞✐t② s❤♦❝❦s ❛♥❞ ❤❛✈❡ s♣❛r❡ ❧✐q✉✐❞✐t②✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r✐❝❡ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts✱ ✇❤✐❝❤ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♠❛r❦❡t✲❝❧❡❛r✐♥❣ ❝♦♥❞✐t✐♦♥✱ ✐s ♦❢ t❤❡ ✧❝❛s❤✲ ✐♥✲t❤❡✲♠❛r❦❡t✧ t②♣❡ ♣r♦♣♦s❡❞ ❜② ❆❧❧❡♥ ❛♥❞ ●❛❧❡ ✭✶✾✾✹✮✳ ▼♦r❡♦✈❡r✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ r❡t✉r♥s ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛r❡ ♣❡r❢❡❝t❧② ❝♦rr❡❧❛t❡❞ ❛❝r♦ss ❜❛♥❦s✳ ❍❡♥❝❡✱ t❤❡ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ✇✐❧❧ t♦✉❝❤ ♦♥ t❤❡ ❛ss❡ts ♦❢ ❛❧❧ ❜❛♥❦s s✐♠✉❧t❛♥❡♦✉s❧②✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐♥ ♦✉r s❡t✉♣ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ s②st❡♠✐❝ s❤♦❝❦✳ ❖✉r ❝♦♥tr✐❜✉t✐♦♥ ✐s t✇♦❢♦❧❞✳ ❋✐rst✱ ✇❡ ❤✐❣❤❧✐❣❤t ❛ ♥❡✇ ❝❤❛♥♥❡❧ t❤❛t ❧✐♥❦s ❜❛♥❦s✬ ❝❛♣✐t❛❧✐s❛t✐♦♥ ❛♥❞ t❤❡✐r ✐♥❝❡♥t✐✈❡s ❢♦r ♣r❡❝❛✉t✐♦♥❛r② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ❖✉r ♠❛✐♥ ✜♥❞✐♥❣ ✐s t❤❛t ❛ ❜❛♥❦ ✇✐❧❧ ❤♦❧❞ ❛♥ ❛❞❡q✉❛t❡ ❧❡✈❡❧ ♦❢ ❧✐q✉✐❞ ❛ss❡ts t♦ s❤✐❡❧❞ ✐ts❡❧❢ ❢r♦♠ ❧✐q✉✐❞✐t② s❤♦❝❦s ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✇❡❧❧ ❝❛♣✐t❛❧✐s❡❞✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❧✐❡s ✐♥ t❤❡ ❢❛❝t t❤❛t ✇❤❡♥ ❧❡✈❡r❛❣❡ ✐s ❤✐❣❤✱ t❤❡ ❜❛♥❦s✬ ❡①♣♦s✉r❡ t♦ ❧✐q✉✐❞✐t② s❤♦❝❦s ✐s ❧❛r❣❡✳ ❇✉②✐♥❣ ✐♥s✉r❛♥❝❡ ❜② s❡❝✉r✐♥❣ s♦♠❡ ❡①✲❛♥t❡ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ✐s t❤❡♥ t♦♦ ❝♦st❧②✱ ✇❤✐❝❤ ✐♥❞✉❝❡s ❜❛♥❦s t♦ ❢♦r❣♦ t❤❡ ✐♥s✉r❛♥❝❡ ♦♣t✐♦♥ ❛♥❞ ❣❛♠❜❧❡✳ ❚✇♦ ✐♥t❡r❡st✐♥❣ ✐♠♣❧✐❝❛t✐♦♥s r❡s✉❧t ❢r♦♠ t❤✐s ✜♥❞✐♥❣✳ ❋r♦♠ t❤❡ ♣❡rs♣❡❝t✐✈❡ ♦❢ ♣♦❧✐❝② ✐♠♣❧✐❝❛t✐♦♥s✱ ✐♥ ♦✉r s✐♠♣❧❡ s❡tt✐♥❣✱ t❤❡r❡ ❡①✐sts ❛ ❝✉t♦✛ ❝❛♣✐t❛❧ r❛t✐♦ ❧❡✈❡❧ ❛❜♦✈❡ ✇❤✐❝❤ ❜❛♥❦s ✇✐❧❧ ❝❤♦♦s❡ t♦ ♠❛♥❛❣❡ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ ♣r✉❞❡♥t❧②✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t ❛ ♣r♦♣❡r❧② ❞❡s✐❣♥❡❞ ❝❛♣✐t❛❧ r❡q✉✐r❡♠❡♥t ❝♦✉❧❞ ♣r♦✈✐❞❡ ❜❛♥❦s ✇✐t❤ t❤❡ r✐❣❤t ✐♥❝❡♥t✐✈❡s t♦ ❤❛✈❡ ❛❞❡q✉❛t❡ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ❲❡ ❛r❡ ♥♦t ❝❧❛✐♠✐♥❣ t❤❛t t❤✐s r❡s✉❧t ✐♠♣❧✐❡s t❤❛t✱ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝❛♣✐t❛❧ r❡q✉✐r❡♠❡♥ts✱ ❧✐q✉✐❞✐t② r❡q✉✐r❡♠❡♥ts s✉❝❤ ❛s ▲❈❘ ❛r❡ r❡❞✉♥❞❛♥t✳✶ ❆❧❧ ✇❡ ❛r❡ ❝❧❛✐♠✐♥❣ ✐s t❤❛t ❛ r❡str✐❝t✐♦♥ ♦♥ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❝❛♥ ❤❛✈❡ ♣♦s✐t✐✈❡ ✐♠♣❛❝t ♦♥ t❤❡✐r ✐♥❝❡♥t✐✈❡s t♦ ♠❛♥❛❣❡ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦✱ ❛♥❞ t❤✉s✱ ❝❛♣✐t❛❧ r❡q✉✐r❡♠❡♥ts ❛♥❞ ❧✐q✉✐❞✐t② r❡q✉✐r❡♠❡♥ts ❛r❡✱ t♦ s♦♠❡ ❡①t❡♥t✱ s✉❜st✐t✉t❛❜❧❡✳ ✶ ■♥ ♦✉r ✈✐❡✇✱ t❤❡ r❡❞✉♥❞❛♥❝② q✉❡st✐♦♥ ♥❡❡❞s t♦ ❜❡ ❛❞❞r❡ss❡❞ ✐♥ ❛ ❣❡♥❡r❛❧ ❡q✉✐❧✐❜r✐✉♠ s❡tt✐♥❣✳ ✹ Electronic copy available at: https://ssrn.com/abstract=2696126 ❋r♦♠ t❤❡ ❝♦♥❝❡♣t✉❛❧ ♣❡rs♣❡❝t✐✈❡✱ ♦✉r ✜♥❞✐♥❣ ❤✐❣❤❧✐❣❤ts t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ✐s ♥❡❝❡ss❛r② ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❧✐♥❦❛❣❡ ❜❡t✇❡❡♥ ❜❛♥❦s✬ ❝❛♣✐t❛❧ ❛♥❞ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ t❛❦✐♥❣✱ ❛s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❧✐♥❦❛❣❡ ❜❡t✇❡❡♥ ❜❛♥❦s✬ ❝❛♣✐t❛❧ ❛♥❞ t❤❡✐r ❝r❡❞✐t r✐s❦ t❛❦✐♥❣✳ ◆♦t❡ t❤❛t ✐♥ ♦✉r ♠♦❞❡❧✱ t❤❡ ❞❡❜t r❡♣❛②♠❡♥t ✐s ❡♥❞♦❣❡♥♦✉s❧② ❞❡t❡r✲ ♠✐♥❡❞ t♦ ♠❛❦❡ ❜❛♥❦s✬ ❞❡❜t❤♦❧❞❡rs ❜r❡❛❦✲❡✈❡♥ ✐♥ ❡①♣❡❝t❡❞ t❡r♠s✳ ❚❤❡r❡❢♦r❡✱ ❞✐✛❡r❡♥t❧② ❢r♦♠ t❤❡ ❝❛s❡ ♦❢ ❝r❡❞✐t r✐s❦✱ t❤❡ ❡✛❡❝t ♦❢ ❜❛♥❦s✬ ❝❛♣✐t❛❧ ♦♥ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ t❛❦✐♥❣ ❞♦❡s ♥♦t ❛r✐s❡ ❞✉❡ t♦ t❤❡ ❢❛✐❧✉r❡ ♦❢ ❜❛♥❦s✬ ❝r❡❞✐t♦rs t♦ ♣r♦♣❡r❧② ♣r✐❝❡ t❤❡ ❧❡✈❡❧ ♦❢ ❧✐q✉✐❞✐t② r✐s❦ t❛❦❡♥ ❜② ❜❛♥❦s✳ ■♥ r❡❧❛t✐♦♥ t♦ ♦✉r s❡❝♦♥❞ ❝♦♥tr✐❜✉t✐♦♥✱ t♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡✱ t❤✐s ♣❛♣❡r ✐s t❤❡ ✜rst ♦♥❡ t❤❛t t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t✱ ✐♥ ❛ ✉♥✐✜❡❞ s❡tt✐♥❣✱ ❛❧❧ t❤r❡❡ ♣♦ss✐❜❧❡ s♦✉r❝❡s ♦❢ ❧✐q✉✐❞✐t②✱ ❛s ✇❡❧❧ ❛s t✇♦ ♠♦t✐✈❡s ❢♦r ❜❛♥❦s t♦ ❤♦❧❞ ❧✐q✉✐❞ ❛ss❡ts ❛♥❞ ❡①❛♠✐♥❡s t❤❡ ✐♠♣❧✐❝❛t✐♦♥s ♦❢ t❤❡✐r ✐♥t❡r❛❝t✐♦♥ ❢♦r ❛ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ❲❡ ✜♥❞ t❤❛t ✇❤❡♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❜❡❝♦♠❡s ♠♦r❡ ❤✐❣❤❧② ❧❡✈❡r❛❣❡❞✱ t❤❡ ♣r❡❝❛✉t✐♦♥❛r② ♠♦t✐✈❡ ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❧❡❛❞s t♦ ❛ ✇❡❛❦ ❞❡❝r❡❛s❡ ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ♦❢ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛♥❞ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❜❛♥❦s t❤❛t ❢❛✐❧ s✉❜s❡q✉❡♥t t♦ t❤❡ ♠❛t❡r✐❛❧✐s❛t✐♦♥ ♦❢ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ ✲ ♦✉r ♠❡❛s✉r❡ ♦❢ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ❍♦✇❡✈❡r✱ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡ ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❤❛s t❤❡ ♦♣♣♦s✐t❡ ❡✛❡❝t✳ ❚❤❡ ♦✈❡r❛❧❧ ✐♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❞✐✛❡rs ❜❡t✇❡❡♥ t❤❡ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ❛♥❞ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❢❛✐❧❡❞ ❜❛♥❦s✳ ❲❡ s❤♦✇ ❛♥❛❧②t✐❝❛❧❧② t❤❛t t❤❡ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ✐s ✇❡❛❦❧② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ t❤❡ ❞❡❣r❡❡ ♦❢ ❧❡✈❡r❛❣❡ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ❍♦✇❡✈❡r✱ ❛♥❞ ✐♥t❡r❡st✐♥❣❧②✱ ♦✉r ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s ❤✐❣❤❧✐❣❤ts t❤❛t t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❜❛♥❦s t❤❛t ✇✐❧❧ ❢❛✐❧ ✇❤❡♥ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐s ♠❛t❡r✐❛❧✐s❡❞ ✐s ♥♦t ♠♦♥♦t♦♥✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ■♠♣r♦✈✐♥❣ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✬s ❝❛♣✐t❛❧✐s❛t✐♦♥ ✐s ❜❡♥❡✜❝✐❛❧✱ ❡①❝❡♣t ✇❤❡♥ t❤❡ s②st❡♠ ✐s ♣♦♦r❧② ❝❛♣✐t❛❧✐s❡❞✳ ▼♦r❡♦✈❡r✱ t❤❡ ❞✐✛❡r❡♥❝❡ ✐♥ t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s❡❝t♦r ♦♥ t❤❡ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ❛♥❞ ♦♥ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❢❛✐❧❡❞ ❜❛♥❦s s✉❣❣❡sts t❤❛t ❛ s❡✈❡r❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ ❛ ❤✐❣❤ ♣r♦♣♦rt✐♦♥ ♦❢ ❜❛♥❦ ❢❛✐❧✉r❡s ✐♥ t❤❡ s②st❡♠ ❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛♣♣❡♥ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❚❤❡ ♦r❣❛♥✐③❛t✐♦♥ ♦❢ t❤❡ ♣❛♣❡r ✐s ❛s ❢♦❧❧♦✇s✳ ❆❢t❡r ❞✐s❝✉ss✐♥❣ t❤❡ r❡❧❛t❡❞ ❧✐t❡r❛t✉r❡ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❡ ❛♥❛❧②s❡ ✐♥ ❙❡❝t✐♦♥ ✸ t❤❡ ❜❛♥❦s✬ ❝❤♦✐❝❡ ♦❢ ♣r❡❝❛✉t✐♦♥❛r② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛♥❞ t❤❡ ❡✛❡❝t ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤✐s ❞❡❝✐s✐♦♥✳ ❚❤❡♥ ✇❡ ♠♦✈❡ ♦♥✱ ❡①❛♠✐♥✐♥❣ ✐♥ ❙❡❝t✐♦♥ ✹ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ t❤❡ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❛♥❞ t❤❡ ✺ Electronic copy available at: https://ssrn.com/abstract=2696126 ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠✱ ❛s ✇❡❧❧ ❛s t❤❡ s❡✈❡r✐t② ♦❢ t❤❡ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ❋✐♥❛❧❧② ✇❡ ❝♦♥❝❧✉❞❡ ✐♥ ❙❡❝t✐♦♥ ✺✳ ❆❧❧ ♣r♦♦❢s ❛r❡ ♣r♦✈✐❞❡❞ ✐♥ t❤❡ ❆♣♣❡♥❞✐①✳ ✷ ❘❡❧❛t❡❞ ▲✐t❡r❛t✉r❡ ❚❤✐s ♣❛♣❡r ❝♦♥tr✐❜✉t❡s t♦ t❤❡ ❧✐t❡r❛t✉r❡ ♦♥ ❜❛♥❦ ❧✐q✉✐❞✐t② ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❡✛❡❝t ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤❡✐r ✐♥❝❡♥t✐✈❡s t♦ ♠❛♥❛❣❡ ❧✐q✉✐❞✐t② r✐s❦✳ ❆s s✉❝❤✱ ✐t ✐s ❞✐r❡❝t❧② r❡❧❛t❡❞ t♦ s❡✈❡r❛❧ ♣❛♣❡rs t❤❛t st✉❞② ❜❛♥❦s✬ ❝❤♦✐❝❡ ♦❢ ✐♥✈❡st♠❡♥t ❜❡t✇❡❡♥ ❧✐q✉✐❞ ❛♥❞ ✐❧❧✐q✉✐❞ ❛ss❡ts ✭s❡❡ ❡✳❣✳✱ ❆❝❤❛r②❛ ❡t ❛❧✳ ✭✷✵✶✶✮✱ ▼❛❧❤❡r❜❡ ✭✷✵✶✹✮✱ ❍❡✐❞❡r ❡t ❛❧✳ ✭✷✵✶✺✮✱ ❛♥❞ ❆❝❤❛r②❛ ❡t ❛❧✳ ✭✷✵✶✺✮✮✳ ❚❤❡s❡ ♣❛♣❡rs ❞✐✛❡r ✐♥ t❤❡ ❞❡t❡r♠✐♥❛♥ts ✉♣♦♥ ✇❤✐❝❤ t❤❡② ❢♦❝✉s✳ ❆❝❤❛r②❛ ❡t ❛❧✳ ✭✷✵✶✶✮ ❡①❛♠✐♥❡ t❤❡ ❡✛❡❝t ♦❢ ♣♦❧✐❝② ✐♥t❡r✈❡♥t✐♦♥s t♦ r❡s♦❧✈❡ ❜❛♥❦ ❢❛✐❧✉r❡ ♦♥ ❡①✲❛♥t❡ ❜❛♥❦ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ▼❛❧❤❡r❜❡ ✭✷✵✶✹✮ ♣r♦✈✐❞❡s ❛ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ t❤❡ ❢❡❛r ♦❢ ❢✉t✉r❡ ♠❛r❦❡t ✐❧❧✐q✉✐❞t② ❞✉❡ t♦ ❛❞✈❡rs❡ s❡❧❡❝t✐♦♥ ♠❛② tr✐❣❣❡r ❤♦❛r❞✐♥❣ ❜❡❤❛✈✐♦r t♦❞❛②✳ ❍❡✐❞❡r ❡t ❛❧✳ ✭✷✵✶✺✮ ❛♥❛❧②s❡ ❜❛♥❦s✬ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s t♦ s❤❡❞ ❧✐❣❤t ♦♥ ❤♦✇ ❜❛♥❦s✬ ♣r✐✈❛t❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ r✐s❦ ♦❢ t❤❡✐r ❛ss❡ts ❛✛❡❝ts t❤❡ tr❛❞✐♥❣ ❛♥❞ ♣r✐❝✐♥❣ ♦❢ ❧✐q✉✐❞✐t② ✐♥ t❤❡ ✐♥t❡r❜❛♥❦ ♠❛r❦❡t✳ ❆❝❤❛r②❛ ❡t ❛❧✳ ✭✷✵✶✺✮ st✉❞② ❤♦✇ t❤❡ ❧✐q✉✐❞✐t② ❝❤♦✐❝❡s ♦❢ ✜r♠s ❛r❡ s❤❛♣❡❞ ❜② r✐s❦✲ s❤❛r✐♥❣ ♦♣♣♦rt✉♥✐t✐❡s ✐♥ t❤❡ ❡❝♦♥♦♠②✳ ❖✉r ♣❛♣❡r ❝♦♥s✐❞❡rs t❤❡ ❡✛❡❝t ♦❢ ❜❛♥❦s✬ ❧✐❛❜✐❧✐t② str✉❝t✉r❡ ♦♥ t❤❡✐r ❝❤♦✐❝❡s ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ❚❤❡ ❝✉rr❡♥t ♣❛♣❡r ✐s ❛❧s♦ ❝♦♥♥❡❝t❡❞ t♦ s❡✈❡r❛❧ ❝♦♥tr✐❜✉t✐♦♥s t❤❛t ✉s❡ t❤❡ ✧❝❛s❤✲✐♥✲t❤❡✲ ♠❛r❦❡t✲♣r✐❝✐♥❣✧ ♠❡❝❤❛♥✐s♠ ♣r♦♣♦s❡❞ ❜② ❆❧❧❡♥ ❛♥❞ ●❛❧❡ ✭✶✾✾✹✱ ✷✵✵✹✱ ✷✵✵✺✮ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ✜♥❛♥❝✐❛❧ ❢r❛❣✐❧✐t② ✭s❡❡ ❡✳❣✳✱ ❇♦❧t♦♥ ❡t ❛❧✳ ✭✷✵✶✶✮✱ ❆❝❤❛r②❛ ❛♥❞ ❱✐s✇❛♥❛t❤❛♥ ✭✷✵✶✶✮✱ ❋r❡✐①❛s ❡t ❛❧✳ ✭✷✵✶✶✮✱ ❛♥❞ ●❛❧❡ ❛♥❞ ❨♦r✉❧♠❛③❡r ✭✷✵✶✸✮✮✳ ❚❤❡ ♠♦st ❝❧♦s❡❧② r❡❧❛t❡❞ ♣❛♣❡r t♦ ♦✉r ✇♦r❦ ✐s t❤❛t ♦❢ ❆❝❤❛r②❛ ❛♥❞ ❱✐s✇❛♥❛t❤❛♥ ✭✷✵✶✶✮✱ ✇❤✐❝❤ ❜✉✐❧❞s ❛ ♠♦❞❡❧ t♦ ✉♥❞❡r✲ st❛♥❞ t❤❡ ❞❡✲❧❡✈❡r❛❣✐♥❣ ♦❢ t❤❡ ✜♥❛♥❝✐❛❧ s❡❝t♦r ❞✉r✐♥❣ ❝r✐s❡s✳ ❚❤❡② ❡①❛♠✐♥❡ ❤♦✇ ❛❞✈❡rs❡ s❤♦❝❦s t❤❛t ♠❛t❡r✐❛❧✐③❡ ❞✉r✐♥❣ ❣♦♦❞ ❡❝♦♥♦♠✐❝ t✐♠❡s✱ r❡♣r❡s❡♥t❡❞ ❜② ❤✐❣❤ ❡①♣❡❝t❛t✐♦♥s ❛❜♦✉t ❡❝♦♥♦♠✐❝ ❢✉♥❞❛♠❡♥t❛❧s✱ ❧❡❛❞ t♦ ❣r❡❛t❡r ❞❡✲❧❡✈❡r❛❣✐♥❣ ❛♥❞ ❛ss❡t ♣r✐❝❡ ❞❡t❡r✐♦r❛✲ t✐♦♥✳ ❖✉r ❝♦♥t✐♥✉✉♠✲♦❢✲❜❛♥❦s s❡t✉♣ ✇✐t❤ ❛ss❡t s❛❧❡s ✐s✱ ✐♥ ❢❛❝t✱ ✐♥s♣✐r❡❞ ❜② ❆❝❤❛r②❛ ❛♥❞ ❱✐s✇❛♥❛t❤❛♥ ✭✷✵✶✶✮✬s s❡tt✐♥❣✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ✐s t❤❛t ✇❡ ❛❧❧♦✇ ❜❛♥❦s t♦ ❤♦❧❞ ❧✐q✉✐❞✲ ✐t② ❡①✲❛♥t❡ t♦ s❡❧❢✲✐♥s✉r❡ ❛❣❛✐♥st ❧✐q✉✐❞✐t② s❤♦❝❦s✱ ✇❤✐❝❤ ❡♥❛❜❧❡s ✉s t♦ s❤❡❞ ❧✐❣❤t ♦♥ ❤♦✇ t❤❡ ❜❛♥❦s✬ ✐♥❝❡♥t✐✈❡s t♦ ♠❛♥❛❣❡ ❧✐q✉✐❞✐t② r✐s❦ ❛r❡ ❛✛❡❝t❡❞ ❜② t❤❡✐r ❧✐❛❜✐❧✐t② str✉❝t✉r❡✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ✐♥s✐❣❤ts ♦♥ ✇❤✐❝❤ ♦✉r ♠♦❞❡❧ ✐s ❜✉✐❧t ❛r❡ ❧✐♥❦❡❞ t♦ s❡✈❡r❛❧ ♦t❤❡r ✻ Electronic copy available at: https://ssrn.com/abstract=2696126 ❧✐t❡r❛t✉r❡s✳ ■♥❞❡❡❞✱ t❤❡ ✐❞❡❛ t❤❛t t❤❡ ❧✐❛❜✐❧✐t② str✉❝t✉r❡ ♦❢ ❛ ❜❛♥❦ ♠❛② ✐♠♣❛❝t ✐ts ❝❤♦✐❝❡s ♦❢ ❛ss❡t ❝♦♠♣♦s✐t✐♦♥ ✐s ❧✐♥❦❡❞ t♦ t❤❡ ❡①t❡♥s✐✈❡ ❧✐t❡r❛t✉r❡ t❤❛t ❡①❛♠✐♥❡s t❤❡ r❛t✐♦♥❛❧❡ ❢♦r ✐♥tr♦❞✉❝✐♥❣ ❝❛♣✐t❛❧ r❡❣✉❧❛t✐♦♥✳ ❙❡❡✱ ❛♠♦♥❣ ♦t❤❡rs✱ ❘♦❝❤❡t ✭✶✾✾✷✮✱ ❇❡s❛♥❦♦ ❛♥❞ ❑❛♥❛t❛s ✭✶✾✾✻✮✱ ❇❧✉♠ ✭✶✾✾✾✮ ❛♥❞ ❘❡♣✉❧❧♦ ✭✷✵✵✹✮✳✷ ❚❤✐s ❧✐t❡r❛t✉r❡ ❡①❛♠✐♥❡s ❤♦✇ ❜❛♥❦s✬ ✐♥❝❡♥t✐✈❡s t♦ t❛❦❡ ❡①❝❡ss✐✈❡ r✐s❦ ❝❛♥ ❜❡ ❝✉r❜❡❞ ❜② r❡q✉✐r✐♥❣ ❜❛♥❦s t♦ ♠❛✐♥t❛✐♥ ❛♥ ❛❞❡q✉❛t❡ ❝❛♣✐t❛❧ r❛t✐♦✳ ❲❤✐❧❡ t❤❡ ❢♦❝✉s ♦❢ t❤✐s ❧✐t❡r❛t✉r❡ ✐s t❤❡ ✐♠♣❛❝t ♦♥ t❤❡ ❜❛♥❦s✬ ✐♥❝❡♥t✐✈❡s t♦ t❛❦❡ ❝r❡❞✐t r✐s❦✱ t❤✐s ♣❛♣❡r st✉❞✐❡s t❤❡ ❡✛❡❝t ♦♥ t❤❡✐r ✐♥❝❡♥t✐✈❡s t♦ ♠❛♥❛❣❡ ❧✐q✉✐❞✐t② r✐s❦✳ ■♥ ♦✉r ♣❛♣❡r✱ t❤❡ r❡❛s♦♥ ❢♦r ❜❛♥❦s t♦ ❤♦❧❞ ❧✐q✉✐❞✐t② ✐s ❜❛s❡❞ ♦♥ t✇♦ ❛ss✉♠♣t✐♦♥s✿ ❡①✲❛♥t❡ ✉♥❝❡rt❛✐♥t② ❛❜♦✉t ❧✐q✉✐❞✐t② ♥❡❡❞s ❛♥❞ ❧✐♠✐t❡❞ ♣❧❡❞❣❡❛❜✐❧✐t② ❞✉❡ t♦ ❛s②♠♠❡tr✐❝ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡s❡ t✇♦ ❛ss✉♠♣t✐♦♥s ❛r❡ s✐♠✐❧❛r t♦ t❤♦s❡ ✉s❡❞ ❜② ❍ö❧♠str♦♠ ❛♥❞ ❚✐r♦❧❡ ✭✶✾✾✽✮ t♦ ❛♥❛❧②s❡ t❤❡ ❧✐q✉✐❞✐t② ❞❡♠❛♥❞ ♦❢ t❤❡ ❝♦r♣♦r❛t❡ s❡❝t♦r ❛♥❞ t❤❡ r♦❧❡ ♦❢ ❣♦✈❡r♥✲ ♠❡♥t ✐♥ s✉♣♣❧②✐♥❣ ❧✐q✉✐❞✐t②✳ ❚❤❡ ♠❛✐♥ ❞✐✛❡r❡♥❝❡ ❧✐❡s ✐♥ t❤❡ ❢❛❝t t❤❛t ✐♥ ❍ö❧♠str♦♠ ❛♥❞ ❚✐r♦❧❡ ✭✶✾✾✽✮✱ ❧✐q✉✐❞✐t② s❤♦❝❦s ❛r✐s❡ ❛s ♣r♦❞✉❝t✐♦♥ s❤♦❝❦s t♦ t❤❡ ✜r♠s✬ t❡❝❤♥♦❧♦❣✐❡s✳ ❚❤❡ s✐③❡ ♦❢ t❤❡ s❤♦❝❦s ✐s ❡①♦❣❡♥♦✉s ❛♥❞ ❡s♣❡❝✐❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✜r♠s✬ ❜❛❧❛♥❝❡ s❤❡❡t ❝❤❛r❛❝t❡r✐st✐❝✳ ❲❡ r❛t❤❡r ❞❡r✐✈❡ ❧✐q✉✐❞✐t② ♥❡❡❞s ❛s ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ✐♥ ❡q✉✐❧✐❜r✐✉♠ ❜② ❛♥ ❛ss❡t✲❧✐❛❜✐❧✐t② ♠✐s♠❛t❝❤✳ ❙✉❝❤ ❛ ❞✐✛❡r❡♥❝❡ ❡①♣❧❛✐♥s ✇❤② ✐♥ ❍ö❧♠str♦♠ ❛♥❞ ❚✐r♦❧❡ ✭✶✾✾✽✮✱ t❤❡ ✜r♠s✬ ❧✐q✉✐❞✐t② ❞❡♠❛♥❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡✐r ❧✐❛❜✐❧✐t② str✉❝t✉r❡✱ ✇❤❡r❡❛s ✐♥ ♦✉r ❢r❛♠❡✇♦r❦✱ ✐t ❞♦❡s✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t ❧✐q✉✐❞✐t② s❤♦❝❦s ❛r✐s✐♥❣ ❢r♦♠ t❡❝❤♥♦❧♦❣② s❤♦❝❦s✱ ❛s ✐♥ ❍ö❧♠str♦♠ ❛♥❞ ❚✐r♦❧❡ ✭✶✾✾✽✮✱ ❛r❡ s✉✐t❛❜❧❡ ❢♦r ♥♦♥✲✜♥❛♥❝✐❛❧ ❡♥t❡r♣r✐s❡✱ ✇❤✐❧❡ ♦✉r ❢♦r♠✉❧❛t✐♦♥ ✐s ♠♦r❡ r❡❛s♦♥❛❜❧❡ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ✜♥❛♥❝✐❛❧ ✐♥st✐t✉t✐♦♥s✳ ❋✐♥❛❧❧②✱ ❛ ♥✉♠❜❡r ♦❢ r❡❝❡♥t ♣❛♣❡rs ❤❛✈❡ ❢♦❝✉s❡❞ ♦♥ t❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♦❢ ❜❛♥❦ ❧✐q✉✐❞✐t② r❡❣✉❧❛t✐♦♥ ✭s❡❡ ❡✳❣✳✱ ❈❛❧♦♠✐r✐s ❡t ❛❧✳ ✭✷✵✶✹✮✱ ❲❛❧t❤❡r ✭✷✵✶✻✮✱ ❙❛♥t♦s ❛♥❞ ❙✉❛r❡③ ✭✷✵✶✻✮✱ ❛♥❞ ❑❛s❤②❛♣ ❡t ❛❧✳ ✭✷✵✶✼✮✮✳ ❆❧❧ ♦❢ t❤❡s❡ ♣❛♣❡rs ♣r❡s❡♥t ❞✐✛❡r❡♥t r❛t✐♦♥❛❧❡s ❢♦r ✐♥tr♦❞✉❝✐♥❣ ❧✐q✉✐❞✐t② r❡q✉✐r❡♠❡♥ts✳ ❆♠♦♥❣ t❤❡s❡ ♣❛♣❡rs✱ t❤❡ ❝❧♦s❡st ♦♥❡ t♦ ♦✉rs ✐s ❑❛s❤②❛♣ ❡t ❛❧✳ ✭✷✵✶✼✮✱ ❛s t❤❡② ❝♦♥s✐❞❡r t❤❡ ✐♥t❡r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✬s ❧✐q✉✐❞✐t② ❛♥❞ ❧❡✈❡r❛❣❡ ❞❡❝✐s✐♦♥s✳ ■♥ t❤❡✐r s❡tt✐♥❣✱ t❤❡r❡ ✐s ♥♦ r♦♦♠ ❢♦r s♣❡❝✉❧❛t✐✈❡ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ ❛s t❤❡r❡ ✐s ♥♦ ✐♥t❡r❜❛♥❦ ♠❛r❦❡t ❢♦r t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳ ■♥ ♦✉r ♣❛♣❡r✱ ✇❡ t❛❦❡ t❤❡ ❧✐❛❜✐❧✐t② str✉❝t✉r❡ ❛s ❣✐✈❡♥✱ ❜✉t ❛♥❛❧②③❡ ❤♦✇ ✐t s❤❛♣❡s ❜❛♥❦s✬ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ ❝♦♥s✐❞❡r✐♥❣ ❜♦t❤ t❤❡ ♣r❡❝❛✉t✐♦♥❛r② ❛♥❞ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡s ❢♦r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✳ ❚❤✐s ❛❧❧♦✇s ✉s t♦ ❞r❛✇ ❝♦♥❝❧✉s✐♦♥s ♦♥ ❤♦✇ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❤❛s ❛♥ ✐♠♣❛❝t ♦♥ ✜r❡✲s❛❧❡s ♣r✐❝❡s✳ ✷ ❋♦r ❛♥ ❡①❝❡❧❧❡♥t r❡✈✐❡✇ ♦❢ t❤✐s ❧✐t❡r❛t✉r❡✱ s❡❡ ❋r❡✐①❛s ❛♥❞ ❘♦❝❤❡t ✭✷✵✵✽✮ ❛♥❞ ❱❛♥❍♦♦s❡ ✭✷✵✵✼✮✳ ✼ Electronic copy available at: https://ssrn.com/abstract=2696126 ✸ Pr❡❝❛✉t✐♦♥❛r② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛♥❞ ❧❡✈❡r❛❣❡ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤❡✐r ♣r❡❝❛✉t✐♦♥❛r② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ✐♥ ❛ s✐♠♣❧❡ s❡tt✐♥❣ ♦❢ ❛ s✐♥❣❧❡ ❜❛♥❦ t❤❛t s❡❡❦s t♦ ♠❛♥❛❣❡ ✐ts ❧✐q✉✐❞✐t② r✐s❦✳ ✸✳✶ ❙❡t✉♣ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❞❛t❡s t = 0, 1, ❛♥❞ ❛ ❜❛♥❦ ✇✐t❤ ❜❛❧❛♥❝❡ s❤❡❡t ♦❢ s✐③❡ ♥♦r♠❛❧✐s❡❞ t♦ 1✳ ❲❡ ❛ss✉♠❡ t❤❛t ❛t ❞❛t❡ 0✱ t❤❡ ❜❛♥❦ ❤❛s ❛ ♣r♦♣♦rt✐♦♥ E ♦❢ ❢✉♥❞s ❛s ❡q✉✐t② ❛♥❞ r❛✐s❡s t❤❡ r❡♠❛✐♥✐♥❣ ❢r❛❝t✐♦♥ − E ❜② ✐ss✉✐♥❣ ✉♥s❡❝✉r❡❞ s❤♦rt✲t❡r♠ ❞❡❜t t♦ r✐s❦✲♥❡✉tr❛❧ ✐♥✈❡st♦rs✳✸ ❚❤❡ ❢❛❝❡ ✈❛❧✉❡ ♦❢ t❤❡ s❤♦rt✲t❡r♠ ❞❡❜t t❤❛t ♥❡❡❞s t♦ ❜❡ r❡♣❛✐❞ ❛t ❞❛t❡ ✐s ❞❡♥♦t❡❞ ❜② D1 ❛♥❞ ✇✐❧❧ ❜❡ ❡♥❞♦❣❡♥♦✉s❧② ❞❡t❡r♠✐♥❡❞✳ ■♥✈❡st♠❡♥t ♦♣♣♦rt✉♥✐t✐❡s✳ ❚❤❡ ❜❛♥❦ ❤❛s ❛❝❝❡ss t♦ t✇♦ ✐♥✈❡st♠❡♥t ♦♣♣♦rt✉♥✐t✐❡s✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❛ s❤♦rt✲t❡r♠ ❛ss❡t✱ r❡❢❡rr❡❞ t♦ ❛s ❛ ❧✐q✉✐❞ ❛ss❡t✱ ✇❤✐❝❤ ♣r♦❞✉❝❡s ❛ ❣r♦ss ❞❡t❡r♠✐♥✐st✐❝ r❡t✉r♥ ♦❢ ♣❡r ♣❡r✐♦❞✳ ❚❤❡ s❡❝♦♥❞ ✐♥✈❡st♠❡♥t ♦♣♣♦rt✉♥✐t② ✐s ❛ ❝♦♥st❛♥t✲ r❡t✉r♥✲t♦✲s❝❛❧❡ ♣r♦❥❡❝t✱ r❡❢❡rr❡❞ t♦ ❛s ❛ ❧♦♥❣✲t❡r♠ ❛ss❡t✱ ✇✐t❤ t✇♦ ♠❛✐♥ ❢❡❛t✉r❡s✳ ❋✐rst✱ ✐t ❣❡♥❡r❛t❡s ❛ r❛♥❞♦♠ ❝❛s❤ ✢♦✇ y˜ ≥ ♦♥❧② ❛t ❞❛t❡ t = 2✳ ❙❡❝♦♥❞✱ ✐ts r❡t✉r♥s ❛r❡ ❡①♣♦s❡❞ t♦ ❛ s❤♦❝❦ t❤❛t ✐s r❡❛❧✐s❡❞ ❛t ❞❛t❡ t = 1✱ ❛s ❞❡s❝r✐❜❡❞ ❜❡❧♦✇✳ ▲✐q✉✐❞✐t② s❤♦❝❦✳ ❆t ❞❛t❡ 1✱ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ r❡❣❛r❞✐♥❣ t❤❡ r❡t✉r♥s ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t ❜❡❝♦♠❡s ♣✉❜❧✐❝❧② ❛✈❛✐❧❛❜❧❡✳ ❇❛❞ ♥❡✇s ✐s r❡✈❡❛❧❡❞ ✇✐t❤ t❤❡ ♣r♦❜❛❜✐❧✐t② − α ❛♥❞ ❣♦♦❞ ♥❡✇s ❤❛♣♣❡♥s ✇✐t❤ t❤❡ ❝♦♠♣❧❡♠❡♥t❛r② ♣r♦❜❛❜✐❧✐t②✳ ◆♦t❡ t❤❛t ❛❧t❤♦✉❣❤ t❤❡ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ✐s ♣✉❜❧✐❝❧② ♦❜s❡r✈❛❜❧❡✱ ✐t ✐s ♥♦t ✈❡r✐✜❛❜❧❡✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t t❤❡ s❤♦rt✲t❡r♠ ❞❡❜t r❡♣❛②♠❡♥t ❝❛♥♥♦t ❜❡ ❝♦♥t✐♥❣❡♥t ✉♣♦♥ ✐t✳ ❲❡ ✇✐❧❧✱ ❤❡r❡❛❢t❡r✱ r❡❢❡r t♦ t❤❡ r❡✈❡❧❛t✐♦♥ ♦❢ ❜❛❞ ♥❡✇s ❛s t❤❡ ♠❛t❡r✐❛❧✐s❛t✐♦♥ ♦❢ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ s✐♥❝❡ ✐t ✇✐❧❧ ❧✐♠✐t t❤❡ ❡①t❡♥t t♦ ✇❤✐❝❤ t❤❡ ❜❛♥❦ ❝❛♥ ♣❧❡❞❣❡ t❤❡ ❢✉t✉r❡ ❝❛s❤ ✢♦✇ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳ ■♥❞❡❡❞✱ ✇❡ ❛ss✉♠❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦s✐t✐✈❡ ✐♥❢♦r♠❛t✐♦♥✱ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t ②✐❡❧❞s ❛t ❞❛t❡ ❛ ♣❛②♦✛ ❡q✉❛❧ t♦ yH > ♣❡r ✉♥✐t ♦❢ ✐♥✈❡st♠❡♥t ✇❤❡♥ ✐t s✉❝❝❡❡❞s ✲ ✇❤✐❝❤ ✸ ❚❤❡ ❛❝❛❞❡♠✐❝ ❧✐t❡r❛t✉r❡ ❤❛s ♦✛❡r❡❞ t✇♦ ❡①♣❧❛♥❛t✐♦♥s ❛❜♦✉t ✇❤② ❜❛♥❦s ✉s❡ s❤♦rt✲t❡r♠ ❞❡❜t✳ ❚❤❡ ✜rst ♦♥❡ ❝♦♠❡s ❢r♦♠ t❤❡ ❜❡♥❡✜❝✐❛❧ ❡✛❡❝ts ♦❢ s❤♦rt✲t❡r♠ ❞❡❜t ✐♥ ❞✐s❝✐♣❧✐♥✐♥❣ ❜❛♥❦s✬ ♠❛♥❛❣❡rs✳ ❚❤❡ s❡❝♦♥❞ ❡①♣❧❛♥❛t✐♦♥ ❢♦❝✉s❡s ♦♥ t❤❡ r♦❧❡ ♦❢ ❜❛♥❦s ❛s ❧✐q✉✐❞✐t② ♣r♦✈✐❞❡rs✿ ❜❛♥❦s ✐ss✉❡ s❤♦rt✲t❡r♠ ❞❡❜t t♦ ♣r♦✈✐❞❡ ✢❡①✐❜✐❧✐t② t♦ ❝r❡❞✐t♦rs ✇❤♦ ♠❛② ❜❡ ❤✐t ❜② ❛ ❧✐q✉✐❞✐t② s❤♦❝❦✳ ■♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r✱ ✇❡ ❞♦ ♥♦t ❡①♣❧✐❝✐t❧② ♠♦❞❡❧ t❤❡ r❡❛s♦♥ ❢♦r ✇❤✐❝❤ t❤❡ ❜❛♥❦ ✉s❡s s❤♦rt✲t❡r♠ ❢✉♥❞✐♥❣✱ ❜✉t ❛ss✉♠❡ ✐t ❡①♦❣❡♥♦✉s❧②✳ ■♥ ❧✐♥❡ ✇✐t❤ t❤❡ s❡❝♦♥❞ ❡①♣❧❛♥❛t✐♦♥✱ ✇❡ ❥✉st✐❢② s✉❝❤ ✉s❡ ❛s ❛ ❜❛♥❦✬s r❡s♣♦♥s❡ t♦ t❤❡ ✐♥✈❡st♦rs✬ ❞❡♠❛♥❞ ❢♦r ❧✐q✉✐❞ ✐♥✈❡st♠❡♥t✳ ✽ Electronic copy available at: https://ssrn.com/abstract=2696126 ♦❝❝✉rs ✇✐t❤ t❤❡ ♣r♦❜❛❜✐❧✐t② θ ✲ ❛♥❞ ③❡r♦ ✇❤❡♥ ✐t ❢❛✐❧s✳ ◆❡❣❛t✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ❤❛s t✇♦ ✐♠♣❧✐❝❛t✐♦♥s ❢♦r ❛ ❧♦♥❣✲t❡r♠ ❛ss❡t✬s r❡t✉r♥✳ ❋✐rst✱ t❤❡ ✉♥✐t ❝❛s❤ ✢♦✇ yL ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❛ss❡t ✐♥ t❤❡ ❝❛s❡ ♦❢ s✉❝❝❡ss ✐s ❧♦✇❡r ✭✐✳❡✳✱ yL < yH ✮✳ ▼♦r❡♦✈❡r✱ t❤❡ s✉❝❝❡ss ♣r♦❜❛❜✐❧✐t② ✐♥ t❤✐s s✐t✉❛t✐♦♥ ❞❡♣❡♥❞s ♦♥ ❜❛♥❦✬s ♠♦♥✐t♦r✐♥❣✱ ❞❡♥♦t❡❞ ❜② m✱ ✇❤✐❝❤ ✐s ♥♦t ♦❜s❡r✈❛❜❧❡ ❜② ♦✉ts✐❞❡rs✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ ❜❛♥❦ ❝❛♥ ❝❤♦♦s❡ ❡✐t❤❡r t♦ ❡①❡rt ♠♦♥✐t♦r✐♥❣ ❡✛♦rt✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ m = 1✱ ♦r t♦ s❤✐r❦✱ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ m = 0✳ ■❢ ✐t ❞♦❡s ❡①❡rt ❡✛♦rt✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ✐s ❡q✉❛❧ t♦ θ✱ ❛s ✐♥ t❤❡ ❝❛s❡ ♦❢ ♣♦s✐t✐✈❡ ✐♥❢♦r♠❛t✐♦♥✳ ❍♦✇❡✈❡r✱ ✐❢ ✐t s❤✐r❦s✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ✐s r❡❞✉❝❡❞ t♦ θ − ∆✳ ▼♦♥✐t♦r✐♥❣ ✐s ❝♦st❧② ❢♦r t❤❡ ❜❛♥❦✱ ❛♥❞ ✇❡ ❝❛♣t✉r❡ ✐t ❜② ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❜❛♥❦ ♦❜t❛✐♥s s♦♠❡ ♣r✐✈❛t❡ ❜❡♥❡✜t B ♣❡r ✉♥✐t ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t ✐❢ ✐t s❤✐r❦s✳ ❋✐❣✉r❡ ✶ s✉♠♠❛r✐③❡s t❤❡ ♣❛②♦✛ str✉❝t✉r❡ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳ ❋✐❣✉r❡ ✶✿ P❛②♦✛ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t ❚✐♠✐♥❣✳ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ ❡✈❡♥ts✱ ✇❤✐❝❤ ✐s s✉♠♠❛r✐s❡❞ ✐♥ ❋✐❣✉r❡ ✷✱ ✐s ❛s ❢♦❧❧♦✇s✳ ❆t t = 0✱ t❤❡ ❜❛♥❦ ❞❡❝✐❞❡s ❤♦✇ ♠✉❝❤ t♦ ✐♥✈❡st ✐♥ ❡❛❝❤ ♦❢ t❤❡ ❛ss❡ts✳ ❉❡♥♦t❡ ❜② c t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❧✐q✉✐❞ ❛ss❡ts ❤❡❧❞ ❜② t❤❡ ❜❛♥❦❀ t❤❡ r❡♠❛✐♥✐♥❣ ♣r♦♣♦rt✐♦♥ − c ✐s ✐♥✈❡st❡❞ ✐♥ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳✹ ❆t t = 1✱ t❤❡ ✐♥❢♦r♠❛t✐♦♥ r❡❣❛r❞✐♥❣ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ✹ ■♥ t❤❡ ❝✉rr❡♥t s❡t✉♣✱ ✇❡ ❛ss✉♠❡ t❤❛t t❤❡ t♦t❛❧ s✐③❡ ♦❢ t❤❡ ❜❛♥❦✬s ❜❛❧❛♥❝❡ s❤❡❡t ✐s ✜①❡❞ ❛♥❞ ♥♦r♠❛❧✐s❡❞ t♦ 1✳ ❍❡♥❝❡✱ ✐❢ t❤❡ ❜❛♥❦ ❝❤♦♦s❡s t♦ ✐♥✈❡st ❛ ❢r❛❝t✐♦♥ c ♦❢ ✐ts ❜❛❧❛♥❝❡ s❤❡❡t ✐♥ ❧✐q✉✐❞ ❛ss❡ts✱ t❤❡ r❡♠❛✐♥✐♥❣ ❢r❛❝t✐♦♥ − c ✇✐❧❧ ❜❡ ✐♥ t❤❡ ❢♦r♠ ♦❢ ❧♦♥❣✲t❡r♠ ❛ss❡ts✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ♦♥❡ ❝♦✉❧❞ ❛ss✉♠❡ ❛ ✜①❡❞ s✐③❡ I ♦❢ ✐♥✈❡st♠❡♥t ✐♥ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛♥❞ t❤❡ ❜❛♥❦ ❝❛♥ ❝❤♦♦s❡ t❤❡ ✈♦❧✉♠❡ ♦❢ ❧✐q✉✐❞ ❛ss❡ts✱ ❡①♣r❡ss❡❞ ❛s ❛ ✾ Electronic copy available at: https://ssrn.com/abstract=2696126 ❛ss❡t ✐s r❡✈❡❛❧❡❞✱ ❛♥❞ t❤❡ s❤♦rt✲t❡r♠ ❞❡❜t ❝♦♥tr❛❝ts ♠❛t✉r❡✳ ■❢ t❤❡ ❜❛♥❦✬s ❤♦❧❞✐♥❣s ♦❢ ❧✐q✉✐❞ ❛ss❡ts ❛r❡ ♥♦t ❡♥♦✉❣❤ t♦ r❡♣❛② ❞❡❜t❤♦❧❞❡rs✱ t❤❡ ❜❛♥❦ ✇✐❧❧ ✐ss✉❡ ♥❡✇ ❞❡❜t ♣❧❡❞❣✐♥❣ t❤❡ ❢✉t✉r❡ ♣❛②♦✛ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳ ■♥ t❤❡ ❝❛s❡ t❤❡ ❜❛♥❦ ❝❛♥♥♦t ❜♦rr♦✇ ❡♥♦✉❣❤ t♦ r❡♣❛② ❞❡❜t❤♦❧❞❡rs✱ ✐t ✐s ❧✐q✉✐❞❛t❡❞✳ ❆t ❞❛t❡ 32 ✱ ❜❡t✇❡❡♥ t = ❛♥❞ t = 2✱ ✐❢ ♥❡❝❡ss❛r②✱ t❤❡ ❜❛♥❦ ❞❡❝✐❞❡s ✇❤❡t❤❡r t♦ ❡①❡rt ❡✛♦rt t♦ ♠♦♥✐t♦r t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t✳ ❆t t = 2✱ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡t r❡t✉r♥s ❛r❡ r❡❛❧✐s❡❞✱ ❛♥❞ ❛❧❧ ♣❛②♠❡♥ts ❛r❡ s❡tt❧❡❞✳ ❋✐❣✉r❡ ✷✿ ❚❤❡ t✐♠❡❧✐♥❡ ❲❡ ❛ss✉♠❡ t❤❛t ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❜❛♥❦ ✐s ❧✐q✉✐❞❛t❡❞✱ t❤❡ ❧✐q✉✐❞❛t✐♦♥ ✈❛❧✉❡ ♣❡r ✉♥✐t ♦❢ ❧♦♥❣✲t❡r♠ ❛ss❡t ✐s ❡q✉❛❧ t♦ ✱ r❡❣❛r❞❧❡ss ♦❢ t❤❡ ✐♥❢♦r♠❛t✐♦♥ r❡✈❡❛❧❡❞ ❛t ❞❛t❡ ✶✳ ◆♦t❡ t❤❛t ✐♥ ♦✉r ❢r❛♠❡✇♦r❦✱ ✐t ❞♦❡s ♥♦t ♠❛tt❡r ✇❤❡t❤❡r t❤❡ ❤♦❧❞❡rs ♦❢ t❤❡ ♥❡✇ s❤♦rt✲ t❡r♠ ❞❡❜t ✐ss✉❡❞ ❛t ❞❛t❡ ❛r❡ t❤❡ ❝✉rr❡♥t ❜❛♥❦✬s ❞❡❜t❤♦❧❞❡rs ♦r ♥❡✇ ✐♥✈❡st♦rs✳ ❲❤❛t ♠❛tt❡rs ✐s t❤❛t t❤❡ ♣r✐❝❡ ♦❢ t❤✐s ♥❡✇ s❤♦rt✲t❡r♠ ❞❡❜t ❞❡♣❡♥❞s ♦♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ r❡✈❡❛❧❡❞✳ ■❢ ♥❡✇ ❞❡❜t ✐s ✐ss✉❡❞ t♦ ♥❡✇ ✐♥✈❡st♦rs✱ t❤❡♥ ♦✉r ❛ss✉♠♣t✐♦♥ ✐s t❤❛t t❤❡ ♥❡✇ ✐♥✈❡st♦rs ✇✐❧❧ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ✇❤❡♥ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❞❡❜t r❡♣❛②♠❡♥ts✳ ■❢ ♥❡✇ ❞❡❜t ✐s ✐ss✉❡❞ t♦ t❤❡ ❝✉rr❡♥t ❜❛♥❦✬s ❞❡❜t❤♦❧❞❡rs✱ t❤❡♥ ✇❡ ❝❛♥ s❡❡ t❤❡ ✐ss✉❛♥❝❡ ♦❢ ♥❡✇ ❞❡❜t ❛s t❤❡ ❝✉rr❡♥t ❞❡❜t❤♦❧❞❡rs ❛❣r❡❡✐♥❣ t♦ r♦❧❧ ♦✈❡r t❤❡ ❞❡❜t ❛♥❞✱ ❝r✉❝✐❛❧❧②✱ t♦ r❡♣r✐❝❡ t❤❡ ❞❡❜t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ r❡✈❡❛❧❡❞✳✺ ♠✉❧t✐♣❧❡ n ♦❢ I ✱ ❤❡❧❞ ♦♥ t♦♣ ♦❢ t❤❛t✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❧✐q✉✐❞ ❛ss❡ts ♦✈❡r t♦t❛❧ ❛ss❡ts ✐s n/1 + n✳ ❆❧❧ ♦✉r r❡s✉❧ts ✇✐❧❧ ❝❛rr② ♦✈❡r ✇✐t❤ c r❡♣❧❛❝❡❞ ❜② n/1 + n✳ ✺ ❚❤✐s r❡♣r✐❝✐♥❣ ♣♦ss✐❜✐❧✐t② ✐s ❞✐✛❡r❡♥t t❤❛♥ ♦t❤❡r ❝♦♥tr✐❜✉t✐♦♥s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✇❤✐❝❤ ❛ss✉♠❡ t❤❛t ✶✵ Electronic copy available at: https://ssrn.com/abstract=2696126 ❞❡♣❡♥❞s ♦♥ ✇❤✐❝❤ ♦❢ t❤❡ t✇♦ ♣♦ss✐❜❧❡ ❡q✉✐❧✐❜r✐❛ ❞❡s❝r✐❜❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✸ ✇✐❧❧ ♣r❡✈❛✐❧✳ ■t ✐s ✇♦rt❤ ❡❧❛❜♦r❛t✐♥❣ ♠♦r❡ ♦♥ t❤❡ t✇♦ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮ t❤❛t ❞❡t❡r♠✐♥❡ t♦❣❡t❤❡r t❤❡ ❝✉t♦✛ ❧❡✈❡❧ ❛♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ✐♥ t❤❡ s❡❝♦♥❞ ♣♦ss✐❜❧❡ ❡q✉✐❧✐❜r✐✉♠✳ ❊q✉❛t✐♦♥ ✭✷✵✮ ✐s t❤❡ ❡q✉❛❧✐s❛t✐♦♥ ♦❢✱ ♦♥ t❤❡ ▲❍❙✱ t❤❡ t♦t❛❧ ❧✐q✉✐❞✐t② ❛✈❛✐❧❛❜❧❡ t♦ ❛❜s♦r❜ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛t ❞❛t❡ ❛♥❞✱ ♦♥ t❤❡ ❘❍❙✱ t❤❡ t♦t❛❧ ♠❛r❦❡t ✈❛❧✉❡ ♦❢ ❛❧❧ ❛ss❡ts ♦✛❡r❡❞ ❢♦r s❛❧❡✳ ❊q✉❛t✐♦♥ ✭✶✾✮ ❞❡t❡r♠✐♥❡s t❤❡ ♠❛r❣✐♥❛❧ ❜❛♥❦s t❤❛t ✇♦✉❧❞ ❢❛✐❧ ❛t ❞❛t❡ 1✳ ■t ✐s ❞❡r✐✈❡❞ ❜② ❡q✉❛❧✐s✐♥❣ t❤❡ ❡①♣❡❝t❡❞ ♣r♦✜t t❤❡② ✇♦✉❧❞ ❣❡t ✐❢ t❤❡✐r ❝❤♦✐❝❡ ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛❧❧♦✇s t❤❡♠ t♦ s✉r✈✐✈❡ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ ❛♥❞ ✐❢ ✐t ❞♦❡s ♥♦t✳ ◆♦t✐❝❡ t❤❛t t❤❡ s♦❧✉t✐♦♥s t♦ t❤❡ s②st❡♠ ♦❢ t❤❡ t✇♦ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮ ❞❡♣❡♥❞ ♦♥ h✳ ❉❡♥♦t❡ t❤❡♠ ❜② E(h) ❛♥❞ p(E(h), h)✳ ❚❤❡ ♣r❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ✇✐t❤ ❛ ♣r✐❝❡ str✐❝t❧② ❧♦✇❡r t❤❛♥ ρ∗ + δ ✐❢ p(E(h), h) s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ✭✷✶✮✳ ❖t❤❡r✇✐s❡✱ t❤❡ ♣r❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ✇✐t❤ ❛ ♣r✐❝❡ ❡q✉❛❧ t♦ ρ∗ + δ ✳ ■t ✐s t❤✉s ♥❡❝❡ss❛r②✱ ✐♥ ♦r❞❡r t♦ ❡st❛❜❧✐s❤ t❤❡ ♣r❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠✱ t♦ ❡①❛♠✐♥❡ ❤♦✇ p(E(h), h) ✈❛r✐❡s ✇✐t❤ h✳ ❲❡ ♣r❡s❡♥t t❤❡ ♠❛✐♥ r❡s✉❧ts ✐♥ t❤❡ ❜❡❧♦✇ ♣r♦♣♦s✐t✐♦♥✳ Pr♦♣♦s✐t✐♦♥ ✹✳ Pr❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❝♦✉♣❧❡ E(h), p(E(h), h) t❤❛t s♦❧✈❡s t❤❡ s②st❡♠ ♦❢ t❤❡ t✇♦ ✭✐✮ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮✳ ❲❡ ❤❛✈❡✿ • E(h) ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ h✳ • p(E(h), h) ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❜♦t❤ E ❛♥❞ h✳ ✭✐✐✮ ˆ h) ˆ = ρ∗ + δ s✉❝❤ t❤❛t✿ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ t❤r❡s❤♦❧❞ hˆ ❞❡✜♥❡❞ ❜② p(E(h), ˆ ✱ t❤❡ ♣r❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ✐♥ ✇❤✐❝❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ pe • ❋♦r h ≤ h ❡q✉❛❧s ρ∗ + δ ✳ ˆ ✱ t❤❡ ♣r❡✈❛✐❧✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ✇❤❡r❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ pe • ❋♦r h > h ❡q✉❛❧s p(E(h), h)✱ ✇❤✐❝❤ ✐s str✐❝t❧② ❧♦✇❡r t❤❛♥ ρ∗ + δ ✳ ❚❤❡ ❝✉t♦✛ ❝❛♣✐t❛❧ r❛t✐♦ ❛♥❞ t❤❡ ✉♥✐t ♣r✐❝❡ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝❛♥ t❤❡r❡❢♦r❡ ❜❡ s✉♠♠❛r✐s❡❞ r❡s♣❡❝t✐✈❡❧② ❛s ❢♦❧❧♦✇s✿ ✷✼ Electronic copy available at: https://ssrn.com/abstract=2696126   − ρ∗ − δ ✐❢ h ≤ h ˆ e ˆ E (h) =  E(h) ✐❢ h > hˆ ✭✷✷✮ ❛♥❞ pe Eˆ e (h), h =   ✐❢ h ≤ hˆ ρ∗ + δ ✭✷✸✮  p(E(h), h) ✐❢ h > h ˆ ❙✐♥❝❡ t❤❡ ❝❛♣✐t❛❧ r❛t✐♦ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❞✐str✐❜✉t✐♦♥ F (E, h)✱ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❦s t❤❛t ❢❛✐❧ ❛t ❞❛t❡ ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛s F (Eˆ e (h), h)✳ ✹✳✸ ❋✐r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s ❲❡ ♥♦✇ st✉❞② t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ✭✐✳❡✳✱ t❤❡ ✐♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ h✮ ♦♥ t❤❡ ❡①t❡♥t ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s ♠❡❛s✉r❡❞ ❜② t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❜❛♥❦s t❤❛t ✇✐❧❧ ❜❡ ❝❧♦s❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❝r②st❛❧❧✐s❛t✐♦♥ ♦❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦✳ ❲❡ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ pe (Eˆ e (h), h) ❛♥❞ F Eˆ e (h), h ✇✐t❤ r❡s♣❡❝t t♦ h ❛s ❢♦❧❧♦✇s✿ ≤0 dpe (Eˆ e (h), h) = dh hˆ ✭❜✮ P❛♥❡❧ ❇✿ ■♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ h ✇❤❡♥ h < hˆ ❋✐❣✉r❡ ✺✿ ■♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ h ❚❤❡ ✜❣✉r❡ s❤♦✇s t❤❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ t❤❡ ❝❛♣✐t❛❧ r❛t✐♦ ♦❢ t❤❡ ❜❛♥❦s ✐♥ t❤❡ s②st❡♠ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❜❡t❛ ❞✐str✐❜✉t✐♦♥✱ ♣❛r❛♠❡tr✐③❡❞ ❜② t✇♦ ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡rs✱ a ❛♥❞ b✳ ❲❡ ✜① a ❛t ❛♥❞ ✈❛r② t❤❡ ✈❛❧✉❡ ♦❢ h = b − a✳ ❚❤❡ ♥✉♠❡r✐❝❛❧ ✈❛❧✉❡s ❢♦r ♦t❤❡r ♣❛r❛♠❡t❡rs ❛r❡ ❛s ❢♦❧❧♦✇s✿ α = 0.6, θ = 0.7, yH = 1.8, yL = 1.5, ∆ = 0.4 ❛♥❞ B = 0.3 ✲ ■♥ P❛♥❡❧ ❆✱ ✇❤❡♥ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ s❤✐❢ts ❢r♦♠ t❤❡ s♦❧✐❞ ❧✐♥❡ t♦ t❤❡ ❞♦tt❡❞ ❧✐♥❡✱ ❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✱ t❤❡r❡ ❛r❡ t✇♦ ❝♦✉♥t❡r✈❛✐❧✐♥❣ ❡✛❡❝ts ❛t ♣❧❛②✳ ❋✐rst✱ ✐❢ ✇❡ ❦❡❡♣ Eˆ e ✜①❡❞✱ t❤✐s s❤✐❢t ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ✐♥❝r❡❛s❡s t❤❡ ♥✉♠❜❡r ♦❢ ❜❛♥❦s t❤❛t ✇♦✉❧❞ ❜❡ ✐❧❧✐q✉✐❞✱ ✇❤✐❝❤ ✐s ♥♦✇ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✹ ❛r❡❛s✱ ♥❛♠❡❧② ❜❧✉❡✱ ✶✸ ❋♦r♠❛❧❧②✱ ❧♦♦❦✐♥❣ ❛t t❤❡ t✇♦ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮✱ ✇❡ s❡❡ t❤❛t ❛ ❝❤❛♥❣❡ ✐♥ h ✜rst ✐♥❞✉❝❡s ❛ ❝❤❛♥❣❡ ✐♥ pe ✱ ✇❤✐❝❤ ✐♥ t✉r♥ ❧❡❛❞s t♦ ❛ ❝❤❛♥❣❡ ✐♥ E ✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ E ✇✐t❤ r❡s♣❡❝t t♦ pe ❛r❡ ♥♦♥ ♥❡❣❛t✐✈❡✳ ✷✾ Electronic copy available at: https://ssrn.com/abstract=2696126 ❣r❡❡♥✱ ②❡❧❧♦✇✱ ❛♥❞ r❡❞✳ ❚❤✐s ❞r✐✈❡s ❞♦✇♥ t❤❡ ❡①♣❡❝t❡❞ ♣r✐❝❡ ❛♥❞ ♠♦✈❡s t❤❡ t❤r❡s❤♦❧❞ Eˆ e t♦ t❤❡ ❧❡❢t✱ s✐♥❝❡ t❤❡ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❡①♣❡❝t❡❞ ❣❛✐♥s ❢r♦♠ ❜✉②✐♥❣ ❛ss❡ts ❝❤❡❛♣❧② ✐♥❞✉❝❡s ♠♦r❡ ❜❛♥❦s t♦ ❝❤♦♦s❡ t♦ ❤♦❧❞ ❧✐q✉✐❞✐t②✳ ❍❡♥❝❡✱ ✐♥ t❤❡ ♥❡✇ ❡q✉✐❧✐❜r✐✉♠✱ t❤❡ ❢r❛❝t✐♦♥ ♦❢ ❜❛♥❦s t❤❛t ✇✐❧❧ ❜❡ ❝❧♦s❡❞ ❛t ❞❛t❡ ✐s ❝♦♠♣♦s❡❞ ♦❢ t❤❡ ❣r❡❡♥ ❛♥❞ r❡❞ ❛r❡❛s✳ ✲ ■♥ P❛♥❡❧ ❇✱ t❤❡ s❡❝♦♥❞ ❡✛❡❝t ✐s ♠✉t❡✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ✇❤❡♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ✐s ✇❡❧❧ ❝❛♣✐t❛❧✐s❡❞ ✭✐✳❡✳✱ h ✐s s♠❛❧❧✮✱ t❤❡ s♣❛r❡ ❝❛♣❛❝✐t② ♦❢ t❤❡ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛♠♦♥❣ ✇❡❧❧✲❝❛♣✐t❛❧✐s❡❞ ❜❛♥❦s ✐s st✐❧❧ ❤✐❣❤✳ ❚❤❡r❡❢♦r❡✱ ❛ s♠❛❧❧ s❤✐❢t ✐♥ t❤❡ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❧❡❛❞ t♦ ❛ ❞❡❝r❡❛s❡ ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡✳ ❋♦r♠❛❧❧②✱ ❛s st❛t❡❞ ✐♥ Pr♦♣♦s✐t✐♦♥ ✸✱ ✇❤❡♥ h ✐s st✐❧❧ ❧♦✇ ✭✐✳❡✳✱ h < hˆ ✮✱ ❜❛♥❦s ✇✐t❤ ❛ ❝❛♣✐t❛❧ r❛t✐♦ ❣r❡❛t❡r t❤❛♥ − ρ∗ − δ ❛r❡ ✐♥❞✐✛❡r❡♥t t♦ ❛♥② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❜❡t✇❡❡♥ max 1−ρ∗ −Ei ,0 1−ρ∗ ❛♥❞ 1✳ ❋♦❧❧♦✇✐♥❣ ❛ s♠❛❧❧ s❤✐❢t t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡s❡ ❜❛♥❦s ❝❛♥ ✐♥❝r❡❛s❡ t❤❡✐r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ ❛♥❞ t❤✉s✱ t❤❡ ♣r❡✲s❤✐❢t ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❝❛♥ st✐❧❧ ❜❡ s✉♣♣♦rt❛❜❧❡✳ ❲❡ r❡❢❡r t♦ t❤❡ ✜rst ❡✛❡❝t ❛s t❤❡ ❡✛❡❝t ♦❢ t❤❡ ♣r❡❝❛✉t✐♦♥❛r② ♠♦t✐✈❡ ❢♦r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ s✐♥❝❡ ✐t r❡✢❡❝ts t❤❡ ♣✉r❡ ✐♠♣❛❝t ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤❡✐r ✐♥❝❡♥t✐✈❡s t♦ ❤♦❧❞ ❧✐q✉✐❞✐t②✱ ❛s ❛♥❛❧②s❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳✷✳ ❚❤❡ s❡❝♦♥❞ ❡✛❡❝t ✐s r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❡✛❡❝t ♦❢ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡ ❢♦r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ s✐♥❝❡ ✐t ❡①♣r❡ss❡s t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ❝❤❛♥❣❡s ✐♥ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❣❛✐♥s ❢r♦♠ ❜✉②✐♥❣ ❛ss❡ts✳ ❚❤❡r❡❢♦r❡✱ ✇❤❡♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ❜❡❝♦♠❡s ♠♦r❡ ❤✐❣❤❧② ❧❡✈❡r❛❣❡❞✱ t❤❡ ♣r❡❝❛✉t✐♦♥❛r② ♠♦t✐✈❡ ❢♦r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s r❡s✉❧ts ✐♥ ❛ ✇❡❛❦ ❞❡❝r❡❛s❡ ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❛♥❞ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❦s t❤❛t ❢❛✐❧ ❢♦❧❧♦✇✐♥❣ t❤❡ ♠❛t❡r✐❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦✱ ✇❤✐❧❡ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡ ❢♦r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❤❛s t❤❡ ♦♣♣♦s✐t❡ ❡✛❡❝t✳ ❚❤❡ ♦✈❡r❛❧❧ ✐♠♣❛❝t t❤✉s ❞❡♣❡♥❞s ♦♥ ✇❤✐❝❤ ♦♥❡ ♦❢ t❤❡ t✇♦ ❡✛❡❝ts ✐s str♦♥❣❡r✳ ❋r♦♠ ❊①♣r❡ss✐♦♥ ✭✷✸✮ ❛♥❞ P❛rt ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✱ ✇❡ s❡❡ t❤❛t t❤❡ ❡✛❡❝t ♦❢ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♠♦t✐✈❡ ♦♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ✐s ♥❡✈❡r str♦♥❣❡r t❤❛♥ t❤❡ ✐♠♣❛❝t ♦❢ t❤❡ ♣r❡❝❛✉t✐♦♥❛r② ♠♦t✐✈❡✳ ❆s ❛ r❡s✉❧t✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ♦❢ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐s ✇❡❛❦❧② ❞❡❝r❡❛s✐♥❣ ✇✐t❤ h✳ ❚❤✐s ✐s ✐♥t✉✐t✐✈❡✱ s✐♥❝❡ ❜❛♥❦s ✇✐❧❧ ♥❡✈❡r ✐♥❝r❡❛s❡ t❤❡✐r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s t♦ t❤❡ ❡①t❡♥t t❤❛t ✐t s❛t✉r❛t❡s t❤❡ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ❣❛✐♥s ❢r♦♠ ❜✉②✐♥❣ ❛ss❡ts✳ ❖t❤❡r✇✐s❡ t❤❡② ✇✐❧❧ ♥♦t ❜❡ ❛❜❧❡ t♦ ❛❝t✉❛❧❧② ❡♥❝❛s❤ ❛♥② ❛❞❞✐t✐♦♥❛❧ ♣r♦✜ts✳ ✸✵ Electronic copy available at: https://ssrn.com/abstract=2696126 ❈♦r♦❧❧❛r② ✷✳ ❚❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ♦❢ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐s ✇❡❛❦❧② ❞❡❝r❡❛s✐♥❣ ✇✐t❤ t❤❡ ❞❡❣r❡❡ ♦❢ ❧❡✈❡r❛❣❡ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ◆✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s✳ ■♥ r❡❧❛t✐♦♥ t♦ t❤❡ ♦✈❡r❛❧❧ ✐♠♣❛❝t ♦❢ ❛ ❝❤❛♥❣❡ ✐♥ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❦s t❤❛t ✇✐❧❧ ❢❛✐❧ ❛t ❞❛t❡ 1✱ ✉♥❢♦rt✉♥❛t❡❧② ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❛♥❛❧②t✐❝❛❧❧② ✈❡r✐❢② t❤❡ s✐❣♥ ♦❢ t❤❡ t♦t❛❧ ❡✛❡❝t✳ ❲❡ t❤✉s ❝♦♠♣❧❡♠❡♥t ♦✉r ❛♥❛❧②s✐s ✇✐t❤ s♦♠❡ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts✳ ❖✉r ❜❛s❡❧✐♥❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛r❡ ❛s ❢♦❧❧♦✇s✿ ❣♦♦❞ ♥❡✇s ✐s r❡✈❡❛❧❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ❡q✉❛❧ t♦ 0.6✳ ■♥ t❤❛t ❝❛s❡✱ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ②✐❡❧❞ ❛ ♣❛②♦✛ ❡q✉❛❧ t♦ 1.8 ✇✐t❤ ♣r♦❜❛❜✐❧✐t② θ ❡q✉❛❧ t♦ 0.7✳ ■❢ ❜❛❞ ♥❡✇s ❛rr✐✈❡s✱ t❤❡ s✉❝❝❡ss❢✉❧ ❝❛s❤ ✢♦✇ ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐s r❡❞✉❝❡❞ t♦ 1.5 ❛♥❞✱ ✇✐t❤♦✉t ❜❛♥❦s✬ ♠♦♥✐t♦r✐♥❣ ❡✛♦rt✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ s✉❝❝❡ss ✐s r❡❞✉❝❡❞ ❜② ∆ = 0.4✳ ❇❛♥❦s✬ ♣r✐✈❛t❡ ❜❡♥❡✜ts ✐♥ t❤❡ ❝❛s❡ ♦❢ s❤✐r❦✐♥❣ ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡q✉❛❧ t♦ 0.3✳ ❚❤❡ ❝❛♣✐t❛❧ r❛t✐♦ ♦❢ ❜❛♥❦s ✐♥ t❤❡ s②st❡♠ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❜❡t❛ ❞✐str✐❜✉t✐♦♥✱ ♣❛r❛♠❡tr✐③❡❞ ❜② t✇♦ ♣♦s✐t✐✈❡ ♣❛r❛♠❡t❡rs✱ a ❛♥❞ b✳ ■♥ ♦r❞❡r t♦ ✐❧❧✉str❛t❡ t❤❡ ❡✛❡❝t ♦❢ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❡①t❡♥t ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✱ ✇❡ ♥✉♠❡r✐❝❛❧❧② s♦❧✈❡ t❤❡ s②st❡♠ ♦❢ t❤❡ t✇♦ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮ ✇❤✐❧❡ ✈❛r②✐♥❣ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ s❤❛♣❡ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❜❡t❛ ❞✐str✐❜✉t✐♦♥✳ Pr❡❝✐s❡❧②✱ ✇❡ ✜① a ❛t ❛♥❞ ✈❛r② t❤❡ ✈❛❧✉❡ ♦❢ h = b − a✳ ◆♦t✐❝❡ t❤❛t ✇❤❡♥ a ✐s ✜①❡❞✱ t❤❡ ❤✐❣❤❡r t❤❡ ✈❛❧✉❡ ♦❢ h✱ t❤❡ ❜✐❣❣❡r t❤❡ ♠❛ss ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❧❡❢t ✐s✳ ❚❤✐s t❤❡r❡❢♦r❡ ❛❧❧♦✇s ✉s t♦ ✐♥t❡r♣r❡t ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ✈❛❧✉❡ ♦❢ h ❛s ❛ ♠♦r❡ ❤✐❣❤❧② ❧❡✈❡r❛❣❡❞ ❜❛♥❦✐♥❣ s②st❡♠✳ ❲❡ ♣✐♥ ❞♦✇♥ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ✈❛❧✉❡ ♦❢ h ❜② ❝❤❡❝❦✐♥❣ ✇❤❡t❤❡r t❤❡ r❡s✉❧t✐♥❣ ♣r✐❝❡ ❢r♦♠ s♦❧✈✐♥❣ t❤❡ t✇♦ ❊q✉❛t✐♦♥ ✭✶✾✮ ✲ ✭✷✵✮ s❛t✐s✜❡s ❈♦♥❞✐t✐♦♥ ✭✷✶✮✳ ■❢ s♦✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ❝❤❛r❛❝t❡r✐s❡❞ ✐♥ P❛rt ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉✐❧✐❜r✐✉♠ ✐s t❤❡ ♦♥❡ ✐♥ ✇❤✐❝❤ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❡q✉❛❧s ρ∗ + δ ✳ ❲❡ s❤♦✇ ✐♥ ❋✐❣✉r❡ ✻✭❛✮ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡✱ ❛♥❞ ✐♥ ❋✐❣✉r❡ ✻✭❜✮✱ t❤❡ ❢r❛❝t✐♦♥ ♦❢ t❤❡ ❜❛♥❦s t❤❛t ✇✐❧❧ ❜❡ ❝❧♦s❡❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❝r②st❛❧❧✐s❛t✐♦♥ ♦❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦✳ ❈♦♥s✐st❡♥t ✇✐t❤ ❈♦r♦❧❧❛r② ✷✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ✐s ♥♦♥ ✐♥❝r❡❛s✐♥❣ ✇✐t❤ h ❛♥❞ ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✇❤❡♥ h ✐s ❤✐❣❤ ❡♥♦✉❣❤✳ ■♥t❡r❡st✐♥❣❧②✱ ✇❡ s❡❡ t❤❛t t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ❜❛♥❦s t❤❛t ✇✐❧❧ ❢❛✐❧ ✇❤❡♥ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐s ♠❛t❡r✐❛❧✐s❡❞ ✐s ♥♦t ♠♦♥♦t♦♥✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞❡❣r❡❡ ♦❢ ✸✶ Electronic copy available at: https://ssrn.com/abstract=2696126 ❧❡✈❡r❛❣❡ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ✭❛✮ ❊q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ h ✭❜✮ ❋r❛❝t✐♦♥ ♦❢ ❝❧♦s❡❞ ❜❛♥❦s ❢♦r ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦❢ h ❋✐❣✉r❡ ✻✿ ❈❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ❝♦♠♣❡t✐t✐✈❡ ❡q✉✐❧✐❜r✐✉♠ ❚✇♦ ♠❛✐♥ ✐♥s✐❣❤ts ❡♠❡r❣❡ ❢r♦♠ ❋✐❣✉r❡ ✻✳ ❋✐rst✱ t❤❡ ❡✛❡❝ts ♦❢ t❤❡ ❝❤❛♥❣❡s ✐♥ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠ ♦♥ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥✐t✐❛❧ ❧❡✈❡❧ ♦❢ t❤❡ s②st❡♠✬s ❝❛♣✐t❛❧✐s❛t✐♦♥✳ ■♠♣r♦✈✐♥❣ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✬s ❝❛♣✐t❛❧✐s❛t✐♦♥ ✐s ❜❡♥❡✜❝✐❛❧✱ ❡①❝❡♣t ✇❤❡♥ t❤❡ s②st❡♠ ✐s ❤❛r❞❧② ❝❛♣✐t❛❧✐s❡❞✳ ❙❡❝♦♥❞✱ ❛ s❡✈❡r❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠ ❛♥❞ ❛ ❤✐❣❤ ♣r♦♣♦rt✐♦♥ ♦❢ ❜❛♥❦ ❢❛✐❧✉r❡s ✐♥ t❤❡ s②st❡♠ ❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② ❤❛♣♣❡♥ t♦❣❡t❤❡r✳ ✺ ❈♦♥❝❧✉s✐♦♥ ❚❤✐s ♣❛♣❡r ❞❡✈❡❧♦♣s ❛ ♠♦❞❡❧ ♦❢ ❜❛♥❦s✬ ❧✐q✉✐❞✐t② ♠❛♥❛❣❡♠❡♥t t♦ ❡①❛♠✐♥❡ t❤❡ ✐♠♣❛❝t ♦❢ ❜❛♥❦s✬ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦♥ t❤❡✐r ❧✐q✉✐❞✐t② r✐s❦ t❛❦✐♥❣✱ t❤❡ ❡①t❡♥t ♦❢ t❤❡ ✜r❡✲s❛❧❡ ♣r♦❜❧❡♠s ❛♥❞ t❤❡ s❡✈❡r✐t② ♦❢ ❧✐q✉✐❞✐t② ❝r✐s❡s✳ ❲❡ ✜♥❞ t❤❛t ❜❛♥❦s ❤❛✈❡ ✐♥❝❡♥t✐✈❡s t♦ ❤♦❧❞ ❛♥ ❛❞❡q✉❛t❡ ❛♠♦✉♥t ♦❢ ❧✐q✉✐❞✐t② t♦ ♣r♦t❡❝t t❤❡♠s❡❧✈❡s ❛❣❛✐♥st ❢✉t✉r❡ ❧✐q✉✐❞✐t② s❤♦❝❦s ♦♥❧② ✐❢ t❤❡② ❛r❡ ✇❡❧❧ ❝❛♣✐t❛❧✐③❡❞✳ ❲❡ ❛❧s♦ ✜♥❞ t❤❛t ✇❤❡♥ t❤❡ s♣❡❝✉❧❛t✐✈❡ ♦❢ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ✐s t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t✱ t❤❡ ✜r❡✲s❛❧❡ ❞✐s❝♦✉♥t ✐s ✇❡❛❦❧② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ t❤❡ ❞❡❣r❡❡ ♦❢ ❧❡✈❡r❛❣❡ ✐♥ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦♣♦rt✐♦♥ ♦❢ ❜❛♥❦s t❤❛t ✇✐❧❧ ❢❛✐❧ ✇❤❡♥ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐s ♠❛t❡r✐❛❧✐s❡❞ ✐s ♥♦t ♠♦♥♦t♦♥✐❝ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❛♣✐t❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✳ ■♠♣r♦✈✐♥❣ t❤❡ ❜❛♥❦✐♥❣ s②st❡♠✬s ❝❛♣✐t❛❧✐s❛t✐♦♥ ✐s ❜❡♥❡✜❝✐❛❧✱ ❡①❝❡♣t ✇❤❡♥ t❤❡ s②st❡♠ ✐s ♣♦♦r❧② ❝❛♣✐t❛❧✐s❡❞✳ ❲❡ ❜❡❧✐❡✈❡ t❤❛t t❤❡ ♣r❡s❡♥t ❢r❛♠❡✇♦r❦ ♣r♦✈✐❞❡s ❛ ✉s❡❢✉❧ s♣r✐♥❣❜♦❛r❞ ❢♦r ❢✉t✉r❡ r❡✲ s❡❛r❝❤ t❤❛t ❤❡❧♣s ❞❡❡♣❡♥ ♦✉r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ✐♠♣❛❝t ♦❢ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ♦♥ t❤❡✐r ✸✷ Electronic copy available at: https://ssrn.com/abstract=2696126 ✐♥❝❡♥t✐✈❡s ❢♦r ❧✐q✉✐❞✐t② ♠❛♥❛❣❡♠❡♥t✳ ❖♥❡ ♣r♦♠✐s✐♥❣ ❡①t❡♥s✐♦♥ ✇♦✉❧❞ ❜❡ t♦ ❡♥❞♦❣❡♥✐s❡ t❤❡ ❜❛♥❦s✬ ❝❤♦✐❝❡ ♦❢ ❧❡✈❡r❛❣❡✳ ❚❤✐s ✇♦✉❧❞ ❛❧❧♦✇ ✉s t♦ ❛♥❛❧②s❡✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❡✛❡❝ts ♦❢ ❜❛♥❦s✬ ❝❛♣✐t❛❧ ♦♥ t❤❡✐r ❡✛♦rt t♦ r❡❞✉❝❡ t❤❡ ❧✐❦❡❧✐❤♦♦❞ ♦❢ ❛ ❧✐q✉✐❞✐t② s❤♦❝❦✳ ❆♥♦t❤❡r ✐♥t❡r❡st✐♥❣ ❡①t❡♥s✐♦♥ ✇♦✉❧❞ ❜❡ t♦ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡ r♦❧❡ ♦❢ ❧♦♥❣✲t❡r♠ ❞❡❜t ❛♥❞ ❛s❦ ✇❤❡t❤❡r ❤♦❧❞✐♥❣ ❧✐q✉✐❞ ❛ss❡ts ❛♥❞ ❢✉♥❞✐♥❣ ❜② ❧♦♥❣✲t❡r♠ ❞❡❜t ❛r❡ ♣❡r❢❡❝t s✉❜st✐t✉t❡s ❢r♦♠ ❛ ❧✐q✉✐❞✐t② r✐s❦ ♣❡rs♣❡❝t✐✈❡✳ ❆♣♣❡♥❞✐① ❉❡r✐✈❛t✐♦♥ ♦❢ Pr♦❣r❛♠ ℘✳ ◆♦t❡ t❤❛t ❛t ❞❛t❡ 1✱ ❛❢t❡r t❤❡ r❡❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦✱ ❛ ❜❛♥❦ i ❡✐t❤❡r ❤❛s t♦ s❡❧❧ ❛❧❧ ♦❢ ✐ts ❧♦♥❣✲t❡r♠ ❛ss❡ts ❛♥❞ ❜❡ ❝❧♦s❡❞✱ ♦r ✐t s✉r✈✐✈❡s t❤❡ s❤♦❝❦ ❛❢t❡r s❡❧❧✐♥❣ ❛ ❢r❛❝t✐♦♥ ♦❢ ✐ts ❧♦♥❣✲t❡r♠ ❛ss❡ts✱ ♦r ✐t ✐s ❧✐q✉✐❞ ❛♥❞ ❝❛♥ ❜✉② t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts s♦❧❞ ❜② ✐❧❧✐q✉✐❞ ❜❛♥❦s✳ ❚❤❡r❡❢♦r❡✱ ✐ts ❡①♣❡❝t❡❞ ♣r♦✜t ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✿ Πi = N P V + Ei − ci N P V + (1 − α)(1 − ci )γi (θyL − p)1ρi ≤ρ∗ − (1 − α)(1 − ci )βi (θyL − p)1ρi >ρ∗ ✭❆✶✮ ❚❤❡ ♣r♦❜❧❡♠ t❤❛t ❞❡t❡r♠✐♥❡s t❤❡ ♦♣t✐♠❛❧ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ♦❢ ❜❛♥❦ i ✐s✿ ▼❛① Πi ci ∈[0,1] ✭❆✷✮ s✉❜❥❡❝t t♦ αD1i + (1 − α)D1i 1ρi ≤ρ∗ + (1 − α)min D1i , (1 − ci )βi p + (1 − ci ) (1 − βi )ρ∗ + ci ) 1ρi >ρ∗ = − Ei ✭❆✸✮ D1i − ci = ρi − ci ✸✸ Electronic copy available at: https://ssrn.com/abstract=2696126 ✭❆✹✮ βi = 1, γi = ρ∗ − ρi p − ρ∗ ρi − ρ∗ p − ρ∗ ✭❆✺✮ ✭❆✻✮ for ρ∗ < p < θyL ❚♦ ❡①♣r❡ss t❤❡ ❛❜♦✈❡ ♣r♦❣r❛♠ ✐♥ t❡r♠s ♦❢ εi ✱ ❧❡t ✉s ✜rst r❡✇r✐t❡ t❤❡ ❢♦✉rt❤ t❡r♠ ✐♥ t❤❡ ❘❍❙ ♦❢ ❊①♣r❡ss✐♦♥ ✭❆✶✮✳ ❲❡ ❤❛✈❡✿ (1 − ci )γi = (1 − ci )ρ∗ − + Ei + ci (1 − ci )ρ∗ − D1i + ci = p − ρ∗ p − ρ∗ ✭❆✼✮ ✇❤❡r❡ t❤❡ s❡❝♦♥❞ ❡q✉❛❧✐t② ❝♦♠❡s ❢r♦♠ ❈♦♥❞✐t✐♦♥ ✭❆✸✮ ✉♥❞❡r t❤❡ ❝❛s❡ ρi ≤ ρ∗ ✳ ❘❡♣❧❛❝✐♥❣ ci ❛s ❞❡✜♥❡❞ ✐♥ ❊①♣r❡ss✐♦♥ ✭✶✹✮ ✐♥t♦ ❊①♣r❡ss✐♦♥ ✭❆✼✮✱ ✇❡ ♦❜t❛✐♥✿ (1 − ci )γi = εi (1 − ρ∗ ) − p − ρ∗ − ρ∗ − Ei ,0 p − ρ∗ for ρ∗ < p < θyL ✭❆✽✮ ❲❡ ♣r♦❝❡❡❞ s✐♠✐❧❛r❧② ✇✐t❤ t❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ ❘❍❙ ♦❢ ❊①♣r❡ss✐♦♥ ✭❆✶✮ ❛♥❞ ♦❜t❛✐♥✿ (1 − ci )βi =   − εi (1−ρ∗∗ ) + p−ρ  − ε − max i 1−ρ∗ −Ei ,0 p−ρ∗ 1−ρ∗ −Ei ,0 1−ρ∗ ✐❢ ρ∗ < ρi < p ✐❢ ρi ≥ p ✭❆✾✮ ❚❤❡♥ ❜② ♣❧✉❣❣✐♥❣ ❊①♣r❡ss✐♦♥s ✭❆✽✮ ✲ ✭❆✾✮ ✐♥t♦ ❊①♣r❡ss✐♦♥ ✭❆✶✮✱ ✇❡ ❣❡t t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ Pr♦❣r❛♠ ℘✳ Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✷✳ ❚♦ ❞❡t❡r♠✐♥❡ t❤❡ ♦♣t✐♠❛❧ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s✱ ❧❡t ✉s ❝♦♠♣✉t❡ t❤❡ ✜rst ❞❡r✐✈❛t✐✈❡ ♦❢ ❊①♣r❡ss✐♦♥ ✭✶✼✮ ❛s ❢♦❧❧♦✇s✿ −N P V + (1 − α)(1 − ρ∗ )(θyL − p) 1ρi

ρ∗ + δ ✱ ❊①♣r❡ss✐♦♥ ✭✶✼✮ ✐s str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✇✐t❤ εi ❢♦r ❛❧❧ i✱ ✇❤✐❝❤ ♠❡❛♥s ✸✹ Electronic copy available at: https://ssrn.com/abstract=2696126 −Ei t❤❛t ❛t t❤❡ ♦♣t✐♠✉♠ εi = − 1−ρ ,0 ✳ 1−ρ∗ ∗ • ■❢ p < ρ∗ + δ ✱ ❊①♣r❡ss✐♦♥ ✭✶✼✮ ✐s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ✇✐t❤ εi ❢♦r ❜❛♥❦s ✇✐t❤ ρi < p✱ ❛♥❞ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ ✇✐t❤ εi ❢♦r ❜❛♥❦s ✇✐t❤ ρi ≥ p✳ ❚❤❡r❡❢♦r❡✱ ❛t t❤❡ ♦♣t✐♠✉♠✿ εi = Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✷✳ Ei − ρ∗ − Ei , 1 + − , 1ρi ≥p ρi

ρ∗ +δ ✱ t❤❡② ✇✐❧❧ ❛❧❧ ❤♦❧❞ ③❡r♦ ❧✐q✉✐❞✐t②✳ ❍❡♥❝❡✱ t❤❡r❡ ✐s ♥♦ ❧✐q✉✐❞✐t② ❛✈❛✐❧❛❜❧❡ ❡①✲♣♦st t♦ s✉♣♣♦rt t❤✐s ♣r✐❝❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛♥ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡r❡ p > ρ∗ + δ ❝❛♥♥♦t ❡①✐st✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✸✳ • ❲❡ st❛rt ✜rst ✇✐t❤ t❤❡ ❝❤❛r❛❝t❡r✐s❛t✐♦♥ ♦❢ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐♥ ✇❤✐❝❤ t❤❡ ♣r✐❝❡ ✐s str✐❝t❧② ❧♦✇❡r t❤❛♥ ρ∗ + δ ✳ ❲❡ ❦♥♦✇ ❢r♦♠ ▲❡♠♠❛ ✷ t❤❛t ✐❢ ❜❛♥❦s ❡①♣❡❝t t❤❡ ♣r✐❝❡ t♦ ❜❡ str✐❝t❧② ❧♦✇❡r t❤❛♥ ρ∗ + δ ✱ t❤❡ ❜❛♥❦s✬ ♦♣t✐♠❛❧ ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❛r❡ ❛s ❢♦❧❧♦✇s✿ εi = − ρ∗ − Ei Ei , 1 + − , 1ρi >p ρi ≤p − ρ∗ − ρ∗ ❚❤❡r❡❢♦r❡✱ ✇❡ ♥❡❡❞ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡♥ ❛ ❜❛♥❦ i ✇✐❧❧ ❝❤♦♦s❡ t❤❡✐r ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣ s✉❝❤ t❤❛t ρi ≤ p ♦r ρi > p ❜② ❝♦♠♣❛r✐♥❣ ✐ts ❡①♣❡❝t❡❞ ♣r♦✜ts ✐♥ t❤❡ ❝❛s❡ ✐t ✐s ❧✐q✉✐❞ ❛♥❞ ✐♥ t❤❡ ❝❛s❡ ✐t ✐s ♥♦t✳ ■❢ ❜❛♥❦ i ❝❤♦♦s❡s t♦ ❜❡ ✐❧❧✐q✉✐❞✱ ✐ts ❡①♣❡❝t❡❞ ♣r♦✜t ✐s ❛s ❢♦❧❧♦✇s✿ Πilli i = N P V + Ei − (1 − α)(θyL − p) ✭❆✶✸✮ ■❢ ❜❛♥❦ i ❝❤♦♦s❡s t♦ ❜❡ ❧✐q✉✐❞✱ ✐ts ❡①♣❡❝t❡❞ ♣r♦✜t ✐s ❛s ❢♦❧❧♦✇s✿ Πilli i = Ei + (1 − α)(θyL − p) Ei p − ρ∗ ✭❆✶✹✮ ❍❡♥❝❡✱ t❤❡ ❝✉t♦✛ ❝❛♣✐t❛❧ r❛t✐♦ E ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ E + (1 − α)(θyL − p) E = N P V + E − (1 − α)(θyL − p) p − ρ∗ ✸✺ Electronic copy available at: https://ssrn.com/abstract=2696126 ✭❆✶✺✮ ✇❤✐❝❤ ✐s ❡q✉✐✈❛❧❡♥t t♦✿ E NP V + = p − ρ∗ (1 − α)(θyL − p) ✭❆✶✻✮ ◆♦t❡ t❤❛t ❛❧❧ ❜❛♥❦s ✇✐t❤ Ei ≥ E ✇✐❧❧ ❝❤♦♦s❡ t♦ ✐♥✈❡st ❛❧❧ ♦❢ t❤❡✐r ❢✉♥❞s ✐♥ t❤❡ ❧✐q✉✐❞ ❛ss❡ts✱ ✇❤✐❝❤ ✐♠♣❧✐❡s t❤❛t t❤❡ s♣❛r❡ ❧✐q✉✐❞✐t② ♦❢ ❡❛❝❤ ❜❛♥❦ i ✐s Ei ✳ ❚❤❡r❡❢♦r❡✱ t❤❡ t♦t❛❧ s♣❛r❡ ❧✐q✉✐❞✐t② ✐s ❡q✉❛❧ t♦ E Ef (E, h)dE ✳ ❙✐♥❝❡ ❛❧❧ ❜❛♥❦s ✇✐t❤ Ei < E ✇✐❧❧ ❤♦❧❞ ③❡r♦ ❧✐q✉✐❞✐t② ❛♥❞ ✇✐❧❧ s❡❧❧ ❛❧❧ ♦❢ t❤❡✐r ❧♦♥❣✲t❡r♠ ❛ss❡ts✱ ✐❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦ ✐s r❡❛❧✐s❡❞✱ t❤❡ t♦t❛❧ s✉♣♣❧② ♦❢ t❤❡ ❧♦♥❣✲t❡r♠ ❛ss❡ts ✐♥ t❤❡ s❡❝♦♥❞❛r② ♠❛r❦❡t ✐s E f (E, h)dE ✳ ❍❡♥❝❡✱ t❤❡ ♠❛r❦❡t ❝❧❡❛r✐♥❣ ❝♦♥❞✐t✐♦♥ ✐♠♣❧✐❡s t❤❛t t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥✿ E Ef (E, h)dE = pe E f (E, h)dE ✭❆✶✼✮ ❚♦ s✉♠♠❛r✐s❡✱ ✐♥ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✇❤❡r❡ p < ρ∗ + δ ✱ t❤❡ ❝✉t♦✛ ❝❛♣✐t❛❧ ❧❡✈❡❧ ❛♥❞ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ♣r✐❝❡ ❛r❡ ❥♦✐♥t❧② ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s✿ NP V E +1= ∗ p−ρ (1 − α)(θyL − p) E Ef (E, h)dE = pe f (E, h)dE ✭❆✶✽✮ ✭❆✶✾✮ E • ❲❡ ♥♦✇ ❝❤❛r❛❝t❡r✐s❡ t❤❡ ❡q✉✐❧✐❜r✐✉♠ ✐♥ ✇❤✐❝❤ t❤❡ ♣r✐❝❡ ✐s ❡q✉❛❧ t♦ ρ∗ + δ ✳ ❋r♦♠ Pr♦❣r❛♠ ℘✱ ✇❡ s❡❡ t❤❛t ❜❛♥❦s ✇✐t❤ ρi > p ✇✐❧❧ ❤♦❧❞ ③❡r♦ ❧✐q✉✐❞✐t②✱ ❛♥❞✱ ❝♦♥s❡q✉❡♥t❧②✱ ✇✐❧❧ ❤❛✈❡ t♦ s❡❧❧ ❛❧❧ ♦❢ t❤❡✐r ❧♦♥❣✲t❡r♠ ❛ss❡ts✳ ❚❤❡✐r ♣r♦✜ts ✇✐❧❧ t❤✉s ❜❡ ❡q✉❛❧✿ ∗ Πilli i = N P V + Ei − (1 − α)(θyL − ρ − δ) ✭❆✷✵✮ ❋♦r ❜❛♥❦s t❤❛t ❤❛✈❡ ρi < p✱ ✇❡ ❛❧s♦ s❡❡ t❤❛t t❤❡② ✇✐❧❧ ❜❡ ✐♥❞✐✛❡r❡♥t t♦ ❛♥② ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ❜❡t✇❡❡♥ max 1−ρ∗ −Ei ,0 1−ρ∗ ❛♥❞ 1✳ ❚❤❡✐r ❡①♣❡❝t❡❞ ♣r♦✜t ✐s ❛s ❢♦❧❧♦✇s✿ Πlii = Ei + Ei NP V − ρ∗ ✸✻ Electronic copy available at: https://ssrn.com/abstract=2696126 ✭❆✷✶✮ ◆♦t❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ρi > p = ρ∗ + δ ✐♠♣❧✐❡s t❤❛t D1i −ci 1−ci > ρ∗ + δ ✳ ❙✐♥❝❡ ❛ ❜❛♥❦ i t❤❛t ❤❛s ρi > p ✇✐❧❧ ❜❡ ❝❧♦s❡❞ ❛t ❞❛t❡ 1✱ D1i ✐s ❞❡t❡r♠✐♥❡❞ ❛s ❢♦❧❧♦✇s ✭❆✷✷✮ αD1i + (1 − α)p = − Ei ❚❤❡r❡❢♦r❡✱ ❝♦♥❞✐t✐♦♥ ρi > p = ρ∗ + δ ✐s ❡q✉✐✈❛❧❡♥t t♦ Ei < − ρ∗ − δ ✳ ◆♦t✐❝❡ ❛❧s♦ t❤❛t ✐❢ Ei < − ρ∗ − δ ✱ t❤❡♥ ✇❡ ❤❛✈❡ ✭❆✷✸✮ Πlii < Πilli i ✇❤✐❝❤ ♠❡❛♥s t❤❛t ❜❛♥❦s ✇✐t❤ ❝❛♣✐t❛❧ r❛t✐♦ ❧♦✇❡r t❤❛♥ − ρ∗ − δ ✇✐❧❧ ✐♥❞❡❡❞ ♣r❡❢❡r t♦ ❤♦❧❞ ③❡r♦ ❧✐q✉✐❞✐t②✳ ❚❤❡② ✇✐❧❧ t❤✉s ❜❡ ❝❧♦s❡❞ ❛t ❞❛t❡ ❢♦❧❧♦✇✐♥❣ t❤❡ r❡❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❧✐q✉✐❞✐t② s❤♦❝❦✳ Pr♦♦❢ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹✳ • ❋✐rst✱ ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t t❤❡ s②st❡♠ ♦❢ t❤❡ t✇♦ ❊q✉❛t✐♦♥s ✭✶✾✮ ✲ ✭✷✵✮ ❤❛s ✉♥✐q✉❡ s♦❧✉t✐♦♥s✳ ■♥❞❡❡❞✱ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✷✵✮✱ ✇❡ ❝❛♥ ❞❡r✐✈❡ pe ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ E ❛♥❞ h ❛s ❢♦❧❧♦✇s✿ e p = Ef (E, h)dE E E f (E, h)dE ✭❆✷✹✮ ❈♦♠♣✉t✐♥❣ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ pe ✇✐t❤ r❡s♣❡❝t t♦ E ✱ ✇❡ ❤❛✈❡✿ f (E, h) E ∂pe =− ∂E E f (E, h)dE + E E Ef (E, h)dE ρ∗ ❢♦r E ≤ E ≤ E ✱ ✇❡ ❤❛✈❡✿ ∂g(E) > ∀E ≤ E ≤ E ∂E and lim g(E) = +∞ E−→E ✭❆✸✶✮ ❍❡♥❝❡✱ lim − G(E) = +∞ and E−→E lim + G(E) = −∞ ✭❆✸✷✮ E−→E ▼♦r❡♦✈❡r✱ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t G(E) ✐s ❛ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ E ❢♦r E ≤ E ≤ E ✳ ❚❤✐s✱ t♦❣❡t❤❡r ✇✐t❤ ❘❡s✉❧t ✭❆✸✷✮✱ ✐♠♣❧✐❡s t❤❛t ❊q✉❛t✐♦♥ ✭❆✷✾✮ ❤❛s ✉♥✐q✉❡ s♦❧✉t✐♦♥ E(h)✱ s❛t✐s❢②✐♥❣ E ≤ E ≤ E ✳ • ◆♦✇✱ ✇❡ ✇✐❧❧ s❤♦✇ t❤❛t E(h) ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ h✳ ✸✽ Electronic copy available at: https://ssrn.com/abstract=2696126 ❋r♦♠ ❊q✉❛t✐♦♥ ✭✶✾✮✱ ✉s✐♥❣ ✐♠♣❧✐❝✐t ❞✐✛❡r❡♥t✐❛t✐♦♥✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ❞❡r✐✈❛✲ t✐✈❡ ♦❢ E(h) ✇✐t❤ r❡s♣❡❝t t♦ h ❛s ❢♦❧❧♦✇s✿ e e − ∂p E(1 − α) + ∂p (1 − α)(θyL − pe ) dE ∂h ∂h =− e e p dh (1 − α)(θyL − pe ) − ∂h E(1 − α) + ∂p (1 − α)(θyL − p) ✭❆✸✸✮ ∂E ❆❢t❡r s♦♠❡ ❛rr❛♥❣❡♠❡♥ts✱ ✇❡ ♦❜t❛✐♥✿ ∂pe N P V − (1 − α)θyL + (1 − α)(E + 2pe − ρ∗ ) dE = e dh ∂h (1 − α)(θyL − pe ) − ∂p (N P V − (1 − α)θyL + (1 − α)(E + 2pe − ρ∗ )) ∂E ❙✐♥❝❡ pe ≥ ρ∗ ❛♥❞ s✐❣♥ ❛s ∂pe ∂h ∂pe ∂E ✭❆✸✹✮ ≤ 0✱ ❢r♦♠ ❊①♣r❡ss✐♦♥ ✭❆✸✹✮✱ ✇❡ s❡❡ t❤❛t dE dh ❤❛s t❤❡ s❛♠❡ ✳ ❙✐♥❝❡ h ♠❡❛s✉r❡s t❤❡ ♠❛ss ♦♥ t❤❡ ❧❡❢t s✐❞❡ ♦❢ t❤❡ ❜❛♥❦s✬ ❧❡✈❡r❛❣❡ ❞✐str✐❜✉t✐♦♥✱ ✇❡ ❤❛✈❡ ∂pe ∂h ❛s ♥❡❣❛t✐✈❡✳ ❚❤❡r❡❢♦r❡✱ E(h) ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ h✳ • ❲❡ s❤♦✇ t❤❛t p E(h), h ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❜♦t❤ E ❛♥❞ h ❆s s❤♦✇♥ ❛❜♦✈❡✱ t❤❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ pe ✇✐t❤ r❡s♣❡❝t t♦ E ✐s ♥❡❣❛t✐✈❡✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t p E(h), h ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ E ✳ ◆♦✇✱ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ t❤❡ t♦t❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ pe ✇✐t❤ r❡s♣❡❝t t♦ h ❛s ❢♦❧❧♦✇s✿ ∂pe dE ∂pe dpe = + dh ∂h ∂E dh ✭❆✸✺✮ ❯s✐♥❣ ❊①♣r❡ss✐♦♥ ✭❆✸✹✮✱ ✇❡ ♦❜t❛✐♥✿ dpe ∂pe = dh ∂h (1 − α)(θyL − pe ) − ❍❡♥❝❡✱ dpe dh ∂pe ∂E ❤❛s t❤❡ s❛♠❡ s✐❣♥ ❛s (1 − α)(θyL − pe ) N P V − (1 − α)θyL + (1 − α)(E + 2pe − ρ∗ ) ∂pe ∂h ✭❆✸✻✮ ✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t p E(h), h ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ h✳ • P❛rt ✭✐✐✮ ♦❢ Pr♦♣♦s✐t✐♦♥ ✹ ✐s t❤❡ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❢❛❝t t❤❛t p E(h), h ✐s ❞❡❝r❡❛s✐♥❣ ✇✐t❤ h✳ ✸✾ Electronic copy available at: https://ssrn.com/abstract=2696126 ❘❡❢❡r❡♥❝❡s ❬✶❪ ❆❝❤❛r②❛✱ ❱✳❱✳✱ ❍✳❙✳ ❙❤✐♥ ❛♥❞ ❚✳ ❨♦r✉❧♠❛③❡r ✭✷✵✶✶✮✿ ✧❈r✐s✐s ❘❡s♦❧✉t✐♦♥ ❛♥❞ ❇❛♥❦ ▲✐q✉✐❞✐t②✧✱ ❘❡✈✐❡✇ ♦❢ ❋✐♥❛♥❝✐❛❧ ❙t✉❞✐❡s✱ ✷✹✱ ✷✶✻✻ ✲ ✷✷✵✺✳ ❬✷❪ ❆❝❤❛r②❛✱ ❱✳ ❱✳ ❛♥❞ ❙✳ ❱✐s✇❛♥❛t❤❛♥ ✭✷✵✶✶✮✿ ✧▲❡✈❡r❛❣❡✱ ▼♦r❛❧ ❍❛③❛r❞✱ ❛♥❞ ▲✐q✉✐❞✲ ✐t②✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝❡✱ ✻✻✱ ✾✾ ✲ ✶✸✽✳ ❬✸❪ ❆❝❤❛r②❛✱ ❱✳❱✳✱ ❆✳▼✳ ■②❡r ❛♥❞ ❘✳ ❑✳ ❙✉♥❞❛r❛♠ ✭✷✵✶✺✮✿ ✧❘✐s❦✲❙❤❛r✐♥❣ ❛♥❞ t❤❡ ❈r❡✲ ❛t✐♦♥ ♦❢ ❙②st❡♠✐❝ ❘✐s❦✧✱ ❯♥♣✉❜❧✐s❤❡❞ ❲♦r❦✐♥❣ P❛♣❡r✳ ❬✹❪ ❆❧❧❡♥✱ ❋✳ ❛♥❞ ❉✳ ●❛❧❡ ✭✶✾✾✹✮✿ ✧▲✐♠✐t❡❞ ▼❛r❦❡t P❛rt✐❝✐♣❛t✐♦♥ ❛♥❞ ❱♦❧❛t✐❧✐t② ♦❢ ❆ss❡t Pr✐❝❡s✧✱ ❆♠❡r✐❝❛❧ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇✱ ✽✹✱ ✾✸✸ ✲ ✾✺✺✳ ❬✺❪ ❆❧❧❡♥✱ ❋✳ ❛♥❞ ❉✳ ●❛❧❡ ✭✷✵✵✹✮✿ ✧❋✐♥❛♥❝✐❛❧ ❋r❛❣✐❧✐t②✱ ▲✐q✉✐❞✐t②✱ ❛♥❞ ❆ss❡t Pr✐❝❡s✧✱ ❏♦✉r♥❛❧ ♦❢ ❊✉r♦♣❡❛♥ ❊❝♦♥♦♠✐❝ ❆ss♦❝✐❛t✐♦♥✱ ✷✱ ✶✵✶✺ ✲ ✶✵✹✽✳ ❬✻❪ ❆❧❧❡♥✱ ❋✳ ❛♥❞ ❉✳ ●❛❧❡ ✭✷✵✵✺✮✿ ✧❋r♦♠ ❈❛s❤✲■♥✲❚❤❡✲▼❛r❦❡t✲Pr✐❝✐♥❣ t♦ ❋✐♥❛♥❝✐❛❧ ❋r❛❣✐❧✐t②✧✱ ❏♦✉r♥❛❧ ♦❢ ❊✉r♦♣❡❛♥ ❊❝♦♥♦♠✐❝ ❆ss♦❝✐❛t✐♦♥✱ ✸✱ ✺✸✺ ✲ ✺✹✻✳ ❬✼❪ ❇♦❧t♦♥✱ P✳✱ ❚✳ ❙❛♥t♦s ❛♥❞ ❏✳❆✳ ❙❝❤❡✐♥❦♠❛♥ ✭✷✵✶✶✮✿ ✧❖✉ts✐❞❡ ❛♥❞ ■♥s✐❞❡ ▲✐q✉✐❞✐t②✧✱ ◗✉❛rt❡r❧② ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s✱ ✶✷✻✱ ✷✺✾ ✲ ✸✷✶✳ ❬✽❪ ❇❡s❛♥❦♦✱ ❉✳ ❛♥❞ ●✳ ❑❛♥❛t❛s ✭✶✾✾✻✮✿ ✧❚❤❡ ❘❡❣✉❧❛t✐♦♥ ♦❢ ❇❛♥❦ ❈❛♣✐t❛❧✿ ❉♦ ❈❛♣✐t❛❧ ❙t❛♥❞❛r❞s Pr♦♠♦t❡ ❇❛♥❦ ❙❛❢❡t②❄✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ■♥t❡r♠❡❞✐❛t✐♦♥✱ ✺✱ ✶✻✵ ✲ ✶✽✸✳ ❬✾❪ ❇❧✉♠✱ ❏✳ ✭✶✾✾✾✮✿ ✧❉♦ ❈❛♣✐t❛❧ ❆❞❡q✉❛❝② ❘❡q✉✐r❡♠❡♥ts ❘❡❞✉❝❡ ❘✐s❦s ✐♥ ❇❛♥❦✐♥❣✧✱ ❏♦✉r♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡✱ ✷✸✱ ✼✺✺ ✲ ✼✼✶✳ ❬✶✵❪ ❈❛❧♦♠✐r✐s✱ ❈✳ ❲✳✱ ❋✳ ❍❡✐❞❡r ❛♥❞ ▼✳ ❍♦❡r♦✈❛ ✭✷✵✶✹✮✿ ✧❆ ❚❤❡♦r② ♦❢ ❇❛♥❦ ▲✐q✉✐❞✐t② ❘❡q✉✐r❡♠❡♥ts✧✱ ❈♦❧✉♠❜✐❛ ❇✉s✐♥❡ss ❙❝❤♦♦❧ ❘❡s❡❛r❝❤ P❛♣❡r ◆♦ ✶✹✲✸✾✳ ❬✶✶❪ ❋r❡✐①❛s✱ ❳✳ ❛♥❞ ❏✲❈✳ ❘♦❝❤❡t ✭✷✵✵✽✮✿ ✧▼✐❝r♦❡❝♦♥♦♠✐❝s ♦❢ ❇❛♥❦✐♥❣✧✱ ▼■❚ Pr❡ss✳ ❬✶✷❪ ❋r❡✐①❛s✱ ❳✳✱ ❆✳ ▼❛rt✐♥ ❛♥❞ ❉✳ ❙❦❡✐❡ ✭✷✵✶✶✮✿ ✧❇❛♥❦ ❧✐q✉✐❞✐t②✱ ■♥t❡r❜❛♥❦ ▼❛r❦❡ts✱ ❛♥❞ ▼♦♥❡t❛r② P♦❧✐❝②✧✱ ❘❡✈✐❡✇ ♦❢ ❋✐♥❛♥❝✐❛❧ ❙t✉❞✐❡s✱ ✷✹✱ ✷✻✺✻ ✲ ✷✻✾✷✳ ✹✵ Electronic copy available at: https://ssrn.com/abstract=2696126 ❬✶✸❪ ●❛❧❡✱ ❉✳ ❛♥❞ ❚✳ ❨♦r✉❧♠❛③❡r ✭✷✵✶✸✮✿ ✧▲✐q✉✐❞✐t② ❍♦❛r❞✐♥❣✧✱ ❚❤❡♦r❡t✐❝❛❧ ❊❝♦♥♦♠✐❝s✱ ✽✱ ✷✾✶ ✲ ✸✷✹✳ ❬✶✹❪ ❍❡✐❞❡r✱ ❋✳ ❛♥❞ ▼✳ ❍♦❡r♦✈❛ ❛♥❞ ❈✳ ❍♦❧t❤❛✉s❡♥ ✭✷✵✶✺✮✿ ✧▲✐q✉✐❞✐t② ❍♦❛r❞✐♥❣ ❛♥❞ ■♥t❡r❜❛♥❦ ▼❛r❦❡t ❙♣r❡❛❞s✿ ❚❤❡ ❘♦❧❡ ♦❢ ❈♦✉♥t❡r♣❛rt② ❘✐s❦✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ❊❝♦♥♦♠✐❝s✱ ✶✶✽✱ ✸✸✻ ✲ ✸✺✹✳ ❬✶✺❪ ❍♦❧♠strö♠✱ ❇✳ ❛♥❞ ❏✳ ❚✐r♦❧❡ ✭✶✾✾✽✮✿ ✧Pr✐✈❛t❡ ❛♥❞ P✉❜❧✐❝ ❙✉♣♣❧② ♦❢ ▲✐q✉✐❞✐t②✧✱ ❏♦✉r♥❛❧ ♦❢ P♦❧✐t✐❝❛❧ ❊❝♦♥♦♠②✱ ✶✵✻✱ ✶ ✲ ✹✵✳ ❬✶✻❪ ❑❛s❤②❛♣✱ ❆✳❑✳✱ ❉✳P✳ ❚s♦♠♦❝♦s ❛♥❞ ❆✳ P✳ ❱❛r❞♦✉❧❛❦✐s ✭✷✵✶✼✮✿ ✧❖♣t✐♠❛❧ ❇❛♥❦ ❘❡❣✲ ✉❧❛t✐♦♥ ✐♥ t❤❡ Pr❡s❡♥❝❡ ♦❢ ❈r❡❞✐t ❛♥❞ ❘✉♥ ❘✐s❦✧✱ ❙❛✐❞ ❇✉s✐♥❡ss ❙❝❤♦♦❧ ❘P ✷✵✶✼✲✶✼✳ ❬✶✼❪ ▼❛❧❤❡r❜❡✱ ❋✳ ✭✷✵✶✹✮✿ ✧❙❡❧❢✲❋✉❧✜❧❧✐♥❣ ▲✐q✉✐❞✐t② ❉r②✲❯♣s✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝❡✱ ✻✾✱ ✾✹✼ ✲ ✾✼✵✳ ❬✶✽❪ ❘❡♣✉❧❧♦✱ ❘✳ ✭✷✵✵✹✮✿ ✧❈❛♣✐t❛❧ ❘❡q✉✐r❡♠❡♥ts✱ ▼❛r❦❡t P♦✇❡r✱ ❛♥❞ ❘✐s❦✲❚❛❦✐♥❣ ✐♥ ❇❛♥❦✐♥❣✧✱ ❏♦✉r♥❛❧ ♦❢ ❋✐♥❛♥❝✐❛❧ ■♥t❡r♠❡❞✐❛t✐♦♥✱ ✶✸✱ ✶✺✻ ✲ ✶✽✷✳ ❬✶✾❪ ❘♦❝❤❡t✱ ❏✲❈✳ ✭✶✾✾✷✮✿ ✧❈❛♣✐t❛❧ ❘❡q✉✐r❡♠❡♥ts ❛♥❞ ❚❤❡ ❇❡❤❛✈✐♦r ♦❢ ❈♦♠♠❡r❝✐❛❧ ❇❛♥❦s✧✱ ❊✉r♦♣❡❛♥ ❊❝♦♥♦♠✐❝ ❘❡✈✐❡✇✱ ✸✻✱ ✶✶✸✼ ✲ ✶✶✼✽✳ ❬✷✵❪ ❚✐r♦❧❡✱ ❏✳ ✭✷✵✶✵✮✿ ✧■❧❧✐q✉✐❞✐t② ❛♥❞ ❆❧❧ ✐ts ❋r✐❡♥❞s✧✱ ❇■❙ ❲♦r❦✐♥❣ P❛♣❡rs ◆♦ ✸✵✸✳ ❬✷✶❪ ❱❛♥❍♦♦s❡✱ ❉✳ ✭✷✵✵✼✮✿ ✧❚❤❡♦r✐❡s ♦❢ ❇❛♥❦ ❇❡❤❛✈✐♦r ✉♥❞❡r ❈❛♣✐t❛❧ ❘❡❣✉❧❛t✐♦♥✧✱ ❏♦✉r✲ ♥❛❧ ♦❢ ❇❛♥❦✐♥❣ ❛♥❞ ❋✐♥❛♥❝❡✱ ✸✶✱ ✸✻✽✵ ✲ ✸✻✾✼✳ ❬✷✷❪ ❲❛❧t❤❡r✱ ❆✳ ✭✷✵✶✻✮✿ ✧❏♦✐♥t❧② ❖♣t✐♠❛❧ ❘❡❣✉❧❛t✐♦♥ ♦❢ ❇❛♥❦ ❈❛♣✐t❛❧ ❛♥❞ ▲✐q✉✐❞✐t②✧✱ ❏♦✉r♥❛❧ ♦❢ ▼♦♥❡②✱ ❈r❡❞✐t ❛♥❞ ❇❛♥❦✐♥❣✱ ✹✽✱ ✹✶✺ ✲ ✹✹✽✳ ✹✶ Electronic copy available at: https://ssrn.com/abstract=2696126 ... ∂E ✭❆✸✵✮ < ❛♥❞ pe > ρ∗ ❢♦r E ≤ E ≤ E ✱ ✇❡ ❤❛✈❡✿ ∂g(E) > ∀E ≤ E ≤ E ∂E and lim g(E) = +∞ E−→E ✭❆✸✶✮ ❍❡♥❝❡✱ lim − G(E) = +∞ and E−→E lim + G(E) = −∞ ✭❆✸✷✮ E−→E ▼♦r❡♦✈❡r✱ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t... ❧✐q✉✐❞✐t② ❤♦❧❞✐♥❣s ♦❢ ❜❛♥❦ i ✐s✿ ▼❛① Πi ci ∈[0,1] ✭❆✷✮ s✉❜❥❡❝t t♦ αD1i + (1 − α)D1i 1ρi ≤ρ∗ + (1 − α)min D1i , (1 − ci )βi p + (1 − ci ) (1 − βi )ρ∗ + ci ) 1ρi >ρ∗ = − Ei ✭❆✸✮ D1i − ci = ρi − ci ✸✸... ✭❆✷✺✮ f (E, h)dE ❍❡♥❝❡✱ pe ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ E ✳ ❉❡✜♥❡ E ❛♥❞ E ❛s ❢♦❧❧♦✇s✿ pe E = θyL and pe E = ρ∗ ✭❆✷✻✮ ❙✐♥❝❡ pe ✐s ❛ ❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ E ✱ ✇❡ ❤❛✈❡ E < E ✳ ●✐✈❡♥ t❤❛t t❤❡ ♥❛t✉r❛❧