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May 07, 2023 147 likes •922 views

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SLIDE 1

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

❲♦❧❢❣❛♥❣ ❑❛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❧✐♥ ❇r❛♥❞❡♥❜✉r❣ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❧✈❜✳✇✐✇✐✳❤✉✲❜❡r❧✐♥✳❞❡ ❝❛s❡✳❤✉✲❜❡r❧✐♥✳❞❡ ✐rt❣✶✼✾✷✳❤✉✲❜❡r❧✐♥✳❞❡ ❜✲t✉✳❞❡

slide-2
SLIDE 2

▼♦t✐✈❛t✐♦♥ ✶✲✶

▼♦t✐✈❛t✐♦♥

⊡ ❘✐s❦ ❊①♣♦s✉r❡

◮ ▼❡❛s✉r❡ t❛✐❧ ❡✈❡♥ts ◮ ❈♦♥❞✐t✐♦♥❛❧ ❛✉t♦r❡❣r❡ss✐✈❡ ❡①♣❡❝t✐❧❡ ✭❈❆❘❊✮ ♠♦❞❡❧

❊①♣❡❝t✐❧❡s

⊡ ❚✐♠❡✲✈❛r②✐♥❣ ♣❛r❛♠❡t❡rs

◮ ❚✐♠❡✲✈❛r②✐♥❣ ♣❛r❛♠❡t❡rs ✐♥ ❈❆❘❊

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

◮ ■♥t❡r✈❛❧ ❧❡♥❣t❤ r❡✢❡❝ts t❤❡ str✉❝t✉r❛❧ ❝❤❛♥❣❡s ✐♥ ❡❝♦♥♦♠②

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-3
SLIDE 3

▼♦t✐✈❛t✐♦♥ ✶✲✷

❖❜❥❡❝t✐✈❡s

⊡ ▲♦❝❛❧✐s✐♥❣ ❈❆❘❊ ▼♦❞❡❧s

◮ ▲♦❝❛❧ ♣❛r❛♠❡tr✐❝ ❛♣♣r♦❛❝❤ ✭▲P❆✮ ◮ ❇❛❧❛♥❝❡ ❜❡t✇❡❡♥ ♠♦❞❡❧❧✐♥❣ ❜✐❛s ❛♥❞ ♣❛r❛♠❡t❡r ✈❛r✐❛❜✐❧✐t②

⊡ ❚❛✐❧ ❘✐s❦ ❉②♥❛♠✐❝s

◮ ❊st✐♠❛t✐♦♥ ✇✐♥❞♦✇s ✇✐t❤ ✈❛r②✐♥❣ ❧❡♥❣t❤s ◮ ❚✐♠❡✲✈❛r②✐♥❣ ❡①♣❡❝t✐❧❡ ♣❛r❛♠❡t❡rs

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

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SLIDE 4

▼♦t✐✈❛t✐♦♥ ✶✲✸

❊❝♦♥♦♠❡tr✐❝s ❛♥❞ ❘✐s❦ ▼❛♥❛❣❡♠❡♥t

❊❝♦♥♦♠❡tr✐❝s ⊡ ▼♦❞❡❧❧✐♥❣ ❜✐❛s ✈s✳ ♣❛r❛♠❡t❡r ✈❛r✐❛❜✐❧✐t② ⊡ ■♥t❡r✈❛❧ ❧❡♥❣t❤ ❛♥❞ ❡❝♦♥♦♠✐❝ ✈❛r✐❛❜❧❡s ❘✐s❦ ▼❛♥❛❣❡♠❡♥t ⊡ P❛r❛♠❡t❡r ❞②♥❛♠✐❝s ❛♥❞ str✉❝t✉r❛❧ ❝❤❛♥❣❡s ⊡ ▼❡❛s✉r✐♥❣ t❛✐❧ r✐s❦

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-5
SLIDE 5

▼♦t✐✈❛t✐♦♥ ✶✲✹

❘✐s❦ ❊①♣♦s✉r❡

❆♥ ✐♥✈❡st♦r ♦❜s❡r✈❡s ❞❛✐❧② ❉❆❳ r❡t✉r♥s ❢r♦♠ ✷✵✵✺✵✶✵✸ t♦ ✷✵✶✹✶✷✸✶ ❛♥❞ ❡st✐♠❛t❡s t❤❡ ✉♥❞❡r❧②✐♥❣ r✐s❦ ❡①♣♦s✉r❡ ✈✐❛ ❡①♣❡❝t✐❧❡s ✭❡✳❣✳✱ ✶✪ ❛♥❞ ✺✪✮ ♦✈❡r ❛ ♦♥❡✲②❡❛r t✐♠❡ ❤♦r✐③♦♥✳ ▼♦❞❡❧❧✐♥❣ str❛t❡❣✐❡s ✭❛✮ ❉❛t❛ ✇✐♥❞♦✇s ✜①❡❞ ♦♥ ❛♥ ❛❞ ❤♦❝ ❜❛s✐s ✭❜✮ ❆❞❛♣t✐✈❡❧② s❡❧❡❝t❡❞ ❞❛t❛ ✐♥t❡r✈❛❧s✿ t✐♠❡✲✈❛r②✐♥❣ ♣❛r❛♠❡t❡rs

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

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SLIDE 6

▼♦t✐✈❛t✐♦♥ ✶✲✺

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

❆♥ ✐♥✈❡st♦r ❞❡❝✐❞❡s ❛❜♦✉t t❤❡ ❞❛✐❧② ❛❧❧♦❝❛t✐♦♥ ✐♥t♦ ❛ st♦❝❦ ♣♦rt❢♦❧✐♦ ✭❉❆❳✮✳ ●♦❛❧✿ ❛ ♣r♦♣♦rt✐♦♥ ♦❢ t❤❡ ✐♥✐t✐❛❧ ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ✭✶✵✵✮ ✐s ♣r❡s❡r✈❡❞ ❛t t❤❡ ❡♥❞ ♦❢ ❛ ❤♦r✐③♦♥✱ ✐✳❡✳✱ t❤❡ t❛r❣❡t ✢♦♦r ❡q✉❛❧s ✾✵✳ ❉❡❝✐s✐♦♥ ❛t ❞❛② t✿ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❤❡ ❞✐s❝♦✉♥t❡❞ ✢♦♦r ✉♣ t♦ t ✐s ✐♥✈❡st❡❞ ✐♥t♦ t❤❡ st♦❝❦ ♣♦rt❢♦❧✐♦ ✭❉❆❳✮✱ t❤❡ r❡st ✐♥t♦ ❛ r✐s❦❧❡ss ❛ss❡t✳ ▼✉❧t✐♣❧✐❡r m s❡❧❡❝t✐♦♥✿ ❝♦♥st❛♥t ♦r t✐♠❡✲✈❛r②✐♥❣ ✭❧❈❆❘❊✮

❈♦♥st❛♥t ♠

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-7
SLIDE 7

▼♦t✐✈❛t✐♦♥ ✶✲✻

❘❡s❡❛r❝❤ ◗✉❡st✐♦♥s

❍♦✇ t♦ ❛❝❝♦✉♥t ❢♦r t✐♠❡✲✈❛r②✐♥❣ ♣❛r❛♠❡t❡rs ✐♥ t❛✐❧ ❡✈❡♥t r✐s❦ ♠❡❛s✉r❡s ❡st✐♠❛t✐♦♥❄ ❲❤❛t ❛r❡ t❤❡ t②♣✐❝❛❧ ❞❛t❛ ✐♥t❡r✈❛❧ ❧❡♥❣t❤s ❛ss❡ss✐♥❣ r✐s❦ ♠♦r❡ ❛❝❝✉r❛t❡❧②✱ ✐✳❡✳✱ str✐❦✐♥❣ ❛ ❜❛❧❛♥❝❡ ❜❡t✇❡❡♥ ❜✐❛s ❛♥❞ ✈❛r✐❛❜✐❧✐t②❄ ❍♦✇ ✇❡❧❧ ❞♦❡s t❤❡ ❧❈❆❘❊ t❡❝❤♥✐q✉❡ ♣❡r❢♦r♠ ✐♥ ♣r❛❝t✐❝❡❄

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-8
SLIDE 8

❖✉t❧✐♥❡

✶✳ ▼♦t✐✈❛t✐♦♥

  • ✷✳ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡ ✭❈❆❘❊✮

✸✳ ▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✹✳ ❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✺✳ ❆♣♣❧✐❝❛t✐♦♥s ✻✳ ❈♦♥❝❧✉s✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-9
SLIDE 9

❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡ ✭❈❆❘❊✮ ✷✲✶

❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡

⊡ ❚❛②❧♦r ✭✷✵✵✽✮✱ ❑✉❛♥ ❡t ❛❧✳ ✭✷✵✵✾✮✱ ❊♥❣❧❡ ❛♥❞ ▼❛♥❣❛♥❡❧❧✐ ✭✷✵✵✹✮

❈❆❱✐❛❘

⊡ ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡ Y ✭❡✳❣✳ r❡t✉r♥s✮✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞✱ yt✱ t = ✶, ..., n ⊡ ❈❆❘❊ s♣❡❝✐✜❝❛t✐♦♥ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ✐♥❢♦r♠❛t✐♦♥ s❡t Ft−✶ yt = et,τ + εt,τ

ετ ∼ ❆◆❉

  • ✵, σ✷

ε,τ , τ

  • et,τ = α✵,τ + α✶,τyt−✶ + α✷,τ
  • y+

t−✶

✷ + α✸,τ

  • y−

t−✶

✷

◮ ❊①♣❡❝t✐❧❡ et,τ ❛t τ ∈ (✵, ✶)✱ θτ =

  • α✵,τ, α✶,τ, α✷,τ, α✸,τ, σ✷

ε,τ

⊤ ◮ ❘❡t✉r♥s✿ y +

t−✶ = ♠❛① {yt−✶, ✵}✱ y − t−✶ = ♠✐♥ {yt−✶, ✵}

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-10
SLIDE 10

❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡ ✭❈❆❘❊✮ ✷✲✷

P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥

⊡ ❉❛t❛ ❝❛❧✐❜r❛t✐♦♥ ✇✐t❤ t✐♠❡✲✈❛r②✐♥❣ ✐♥t❡r✈❛❧s ⊡ ❖❜s❡r✈❡❞ r❡t✉r♥s Y = {y✶, . . . , yn} ⊡ ◗✉❛s✐ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❡st✐♠❛t❡ ✭◗▼▲❊✮

  • θI,τ = ❛r❣ ♠❛①

θτ∈Θ ℓI (Y; θτ)

ℓI (·)

◮ I = [t✵ − v, t✵] ✲ ✐♥t❡r✈❛❧ ♦❢ (v + ✶) ♦❜s❡r✈❛t✐♦♥s ❛t t✵ ◮ ℓI (·) ✲ q✉❛s✐ ❧♦❣ ❧✐❦❡❧✐❤♦♦❞

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-11
SLIDE 11

❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡ ✭❈❆❘❊✮ ✷✲✸

❊st✐♠❛t✐♦♥ ◗✉❛❧✐t②

⊡ ▼❡r❝✉r✐♦ ❛♥❞ ❙♣♦❦♦✐♥② ✭✷✵✵✹✮✱ ❙♣♦❦♦✐♥② ✭✷✵✵✾✮ ⊡ ◗✉❛❧✐t② ♦❢ ❡st✐♠❛t✐♥❣ tr✉❡ ♣❛r❛♠❡t❡r ✈❡❝t♦r θ∗

τ ❜② ◗▼▲❊

θI,τ ✐♥ t❡r♠s ♦❢ ❑✉❧❧❜❛❝❦✲▲❡✐❜❧❡r ❞✐✈❡r❣❡♥❝❡❀ Rr (θ∗

τ) ✲ r✐s❦ ❜♦✉♥❞

❊θ∗

τ

  • ℓI(Y;

θI,τ) − ℓI(Y; θ∗

τ)

  • r

≤ Rr (θ∗

τ)

Rr

  • θ∗

τ

  • ❛✉ss✐❛♥ ❘❡❣r❡ss✐♦♥

⊡ ✬▼♦❞❡st✬ r✐s❦✱ r = ✵.✺ ✭s❤♦rt❡r ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t②✮ ⊡ ✬❈♦♥s❡r✈❛t✐✈❡✬ r✐s❦✱ r = ✶ ✭❧♦♥❣❡r ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t②✮ ❙♦❧♦♠♦♥ ❑✉❧❧❜❛❝❦ ❛♥❞ ❘✐❝❤❛r❞ ❆✳ ▲❡✐❜❧❡r ♦♥ ❇❇■✿

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-12
SLIDE 12

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✶

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮

⊡ ▲P❆✱ ❙♣♦❦♦✐♥② ✭✶✾✾✽✱ ✷✵✵✾✮

◮ ❚✐♠❡ s❡r✐❡s ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❧♦❝❛❧❧② ❛♣♣r♦①✐♠❛t❡❞ ◮ ❋✐♥❞✐♥❣ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❤♦♠♦❣❡♥❡✐t②

❉❡t❛✐❧s

◮ ❇❛❧❛♥❝❡ ❜❡t✇❡❡♥ ♠♦❞❡❧❧✐♥❣ ❜✐❛s ❛♥❞ ♣❛r❛♠❡t❡r ✈❛r✐❛❜✐❧✐t②

⊡ ❚✐♠❡ s❡r✐❡s ❧✐t❡r❛t✉r❡

◮

  • ❆❘❈❍(✶, ✶) ♠♦❞❡❧s ✲ ❷í➸❡❦ ❡t ❛❧✳ ✭✷✵✵✾✮

◮ ❘❡❛❧✐③❡❞ ✈♦❧❛t✐❧✐t② ✲ ❈❤❡♥ ❡t ❛❧✳ ✭✷✵✶✵✮ ◮ ▼✉❧t✐♣❧✐❝❛t✐✈❡ ❊rr♦r ▼♦❞❡❧s ✲ ❍är❞❧❡ ❡t ❛❧✳ ✭✷✵✶✺✮

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-13
SLIDE 13

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✷

■♥t❡r✈❛❧ ❙❡❧❡❝t✐♦♥

⊡ (K + ✶) ♥❡st❡❞ ✐♥t❡r✈❛❧s ✇✐t❤ ❧❡♥❣t❤ nk = |Ik| I✵ ⊂ I✶ ⊂ · · · ⊂ Ik ⊂ · · · ⊂ IK

  • θ✵
  • θ✶
  • θk
  • θK

❊①❛♠♣❧❡✿ ❉❛✐❧② ✐♥❞❡① r❡t✉r♥s ❋✐① t✵✱ Ik = [t✵ − nk, t✵]✱ nk =

  • n✵ck

✱ c > ✶ {nk}✶✶

k=✵ = {✷✵ ❞❛②s, ✷✺ ❞❛②s, . . . , ✷✺✵ ❞❛②s}✱ c = ✶.✷✺

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-14
SLIDE 14

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✸

▲♦❝❛❧ ❈❤❛♥❣❡ P♦✐♥t ❉❡t❡❝t✐♦♥

⊡ ❋✐① t✵✱ s❡q✉❡♥t✐❛❧ t❡st ✭k = ✶, . . . , K✮ H✵ : ♣❛r❛♠❡t❡r ❤♦♠♦❣❡♥❡✐t② ✇✐t❤✐♥ Ik H✶ : ∃ ❝❤❛♥❣❡ ♣♦✐♥t ✇✐t❤✐♥ Jk = Ik \ Ik−✶ Tk,τ = s✉♣

s∈Jk

  • ℓAk,s
  • Y,

θAk,s,τ

  • + ℓBk,s
  • Y,

θBk,s,τ

  • − ℓIk+✶
  • Y,

θIk+✶,τ

  • ✇✐t❤ Ak,s = [t✵ − nk+✶, s] ❛♥❞ Bk,s = (s, t✵]

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-15
SLIDE 15

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✹

❈r✐t✐❝❛❧ ❱❛❧✉❡s✱ zk,τ

⊡ ❙✐♠✉❧❛t❡ zk ✲ ❤♦♠♦❣❡♥❡✐t② ♦❢ t❤❡ ✐♥t❡r✈❛❧ s❡q✉❡♥❝❡ I✶, . . . , Ik ⊡ ✬Pr♦♣❛❣❛t✐♦♥✬ ❝♦♥❞✐t✐♦♥ ❊θ∗

τ

  • ℓIk
  • Y;

θIk,τ

  • − ℓIk
  • Y;

θτ

  • r

≤ ρkRr (θ∗

τ)

ρk = ρk K ❢♦r ❛ ❣✐✈❡♥ s✐❣♥✐✜❝❛♥❝❡ ❧❡✈❡❧ ρ

  • θτ ✲ ❛❞❛♣t✐✈❡ ❡st✐♠❛t❡

⊡ ❈❤❡❝❦ zk,τ ❢♦r ✭s✐①✮ ❞✐✛❡r❡♥t θ∗

τ

P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-16
SLIDE 16

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✺

❈r✐t✐❝❛❧ ❱❛❧✉❡s✱ zk,τ

20 40 60 120 10 20

Length in Days Values 20 40 60 120 10 20 Length in Days 20 40 60 120 10 20 Length in Days

❋✐❣✉r❡ ✶✿ ❙✐♠✉❧❛t❡❞ ❝r✐t✐❝❛❧ ✈❛❧✉❡s ❛❝r♦ss ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡r ❝♦♥st❡❧❧❛t✐♦♥s

P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s ❢♦r t❤❡ ♠♦❞❡st ❝❛s❡ r = ✵.✺✱ τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-17
SLIDE 17

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✻

❈r✐t✐❝❛❧ ❱❛❧✉❡s✱ zk,τ

20 40 60 120 200 400

Length in Days Values

20 40 60 120 200 400

Length in Days

20 40 60 120 200 400

Length in Days

❋✐❣✉r❡ ✷✿ ❙✐♠✉❧❛t❡❞ ❝r✐t✐❝❛❧ ✈❛❧✉❡s ❛❝r♦ss ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡r ❝♦♥st❡❧❧❛t✐♦♥s

P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ ❝❛s❡ r = ✶✱ τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-18
SLIDE 18

▲♦❝❛❧ P❛r❛♠❡tr✐❝ ❆♣♣r♦❛❝❤ ✭▲P❆✮ ✸✲✼

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

▲P❆ zk,τ ✲ ❈r✐t✐❝❛❧ ❱❛❧✉❡s

⊡ ❈♦♠♣❛r❡ Tk,τ ❛t ❡✈❡r② st❡♣ k ✇✐t❤ zk,τ ⊡ ❉❛t❛ ✇✐♥❞♦✇ ✐♥❞❡① ♦❢ t❤❡ ✐♥t❡r✈❛❧ ♦❢ ❤♦♠♦❣❡♥❡✐t② ✲ k ⊡ ❆❞❛♣t✐✈❡ ❡st✐♠❛t❡

  • θτ =

θIˆ

k,τ,

  • k = ♠❛①

k≤K {k : Tℓ,τ ≤ zℓ,τ, ℓ ≤ k}

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-19
SLIDE 19

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✶

❉❛t❛

⊡ ❙❡r✐❡s

◮ ❉❆❳✱ ❋❚❙❊ ✶✵✵ ❛♥❞ ❙✫P ✺✵✵ r❡t✉r♥s ✷✵✵✺✵✶✵✸✲✷✵✶✹✶✷✸✶ ✭✷✻✵✽ ❞❛②s✮ ◮ ❘❡s❡❛r❝❤ ❉❛t❛ ❈❡♥t❡r ✭❘❉❈✮ ✲ ❉❛t❛str❡❛♠

⊡ ❙❡t✉♣

◮ ❊①♣❡❝t✐❧❡ ❧❡✈❡❧s✿ τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶ ◮ ▼♦❞❡st ✭r = ✵.✺✮ ❛♥❞ ❝♦♥s❡r✈❛t✐✈❡ ✭r = ✶✮ r✐s❦ ❝❛s❡s ◮ {nk}✶✶

k=✵ = {✷✵ ❞❛②s, ✷✺ ❞❛②s, . . . , ✷✺✵ ❞❛②s}

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-20
SLIDE 20

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✷

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 ❋✐❣✉r❡ ✸✿ ❊st✐♠❛t❡❞ ❧❡♥❣t❤ nˆ

k ♦❢ ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t② ❢r♦♠ ✷✵✵✻✵✶✵✸✲

✷✵✶✹✶✷✸✶ ❢♦r t❤❡ ♠♦❞❡st r✐s❦ ❝❛s❡ r = ✵.✺✱ ❛t ❡①♣❡❝t✐❧❡ ❧❡✈❡❧ τ = ✵.✵✺✳ ❚❤❡ r❡❞ ❧✐♥❡ ♣r❡s❡♥ts t❤❡ ♦♥❡✲♠♦♥t❤ s♠♦♦t❤❡❞ ✈❛❧✉❡s✳

P❛r❛♠❡t❡r ❋❧❛❣

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-21
SLIDE 21

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✸

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 ❋✐❣✉r❡ ✹✿ ❊st✐♠❛t❡❞ ❧❡♥❣t❤ nˆ

k ♦❢ ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t② ❢r♦♠ ✷✵✵✻✵✶✵✸✲

✷✵✶✹✶✷✸✶ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ r✐s❦ ❝❛s❡ r = ✶✱ ❛t ❡①♣❡❝t✐❧❡ ❧❡✈❡❧ τ = ✵.✵✺✳ ❚❤❡ r❡❞ ❧✐♥❡ ♣r❡s❡♥ts t❤❡ ♦♥❡✲♠♦♥t❤ s♠♦♦t❤❡❞ ✈❛❧✉❡s✳

P❛r❛♠❡t❡r ❋❧❛❣

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-22
SLIDE 22

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✹

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 ❋✐❣✉r❡ ✺✿ ❊st✐♠❛t❡❞ ❧❡♥❣t❤ nˆ

k ♦❢ ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t② ❢r♦♠ ✷✵✵✻✵✶✵✸✲

✷✵✶✹✶✷✸✶ ❢♦r t❤❡ ♠♦❞❡st r✐s❦ ❝❛s❡ r = ✵.✺✱ ❛t ❡①♣❡❝t✐❧❡ ❧❡✈❡❧ τ = ✵.✵✶✳ ❚❤❡ r❡❞ ❧✐♥❡ ♣r❡s❡♥ts t❤❡ ♦♥❡✲♠♦♥t❤ s♠♦♦t❤❡❞ ✈❛❧✉❡s✳

P❛r❛♠❡t❡r ❋❧❛❣

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-23
SLIDE 23

❊♠♣✐r✐❝❛❧ ❘❡s✉❧ts ✹✲✺

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

2006 2010 2014 60 120 180 DAX Length 2006 2010 2014 60 120 180 FTSE 100 2006 2010 2014 60 120 180 S&P 500 ❋✐❣✉r❡ ✻✿ ❊st✐♠❛t❡❞ ❧❡♥❣t❤ nˆ

k ♦❢ ✐♥t❡r✈❛❧s ♦❢ ❤♦♠♦❣❡♥❡✐t② ❢r♦♠ ✷✵✵✻✵✶✵✸✲

✷✵✶✹✶✷✸✶ ❢♦r t❤❡ ❝♦♥s❡r✈❛t✐✈❡ r✐s❦ ❝❛s❡ r = ✶✱ ❛t ❡①♣❡❝t✐❧❡ ❧❡✈❡❧ τ = ✵.✵✶✳ ❚❤❡ r❡❞ ❧✐♥❡ ♣r❡s❡♥ts t❤❡ ♦♥❡✲♠♦♥t❤ s♠♦♦t❤❡❞ ✈❛❧✉❡s✳

P❛r❛♠❡t❡r ❋❧❛❣

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-24
SLIDE 24

❆♣♣❧✐❝❛t✐♦♥s ✺✲✶

❘✐s❦ ❊①♣♦s✉r❡

2006 2008 2010 2012 2014 −0.1 0.1 Returns Time ❋✐❣✉r❡ ✼✿ ❉❆❳ ✐♥❞❡① r❡t✉r♥s ✭∗✮ ❛♥❞ ❛❞❛♣t✐✈❡❧② ❡st✐♠❛t❡❞ ❡①♣❡❝t✐❧❡ et,τ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮ ❢r♦♠ ✷✵✵✻✵✶✵✸✲✷✵✶✹✶✷✸✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-25
SLIDE 25

❆♣♣❧✐❝❛t✐♦♥s ✺✲✷

❘✐s❦ ❊①♣♦s✉r❡

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ESet,τ

2006 2008 2010 2012 2014 −0.1 0.1 Returns Time ❋✐❣✉r❡ ✽✿ ❉❆❳ ✐♥❞❡① r❡t✉r♥s ✭∗✮✱ ❛❞❛♣t✐✈❡❧② ❡st✐♠❛t❡❞ ❡①♣❡❝t✐❧❡ et,τ ❛♥❞ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ESet,τ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮ ❢r♦♠ ✷✵✵✻✵✶✵✸✲✷✵✶✹✶✷✸✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-26
SLIDE 26

❆♣♣❧✐❝❛t✐♦♥s ✺✲✸

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

⊡ P♦rt❢♦❧✐♦ ♣r♦t❡❝t✐♦♥ str❛t❡❣②

❉❡t❛✐❧s

◮ ❆✐♠✿ ●✉❛r❛♥t❡❡ ❛ ♣r♦♣♦rt✐♦♥ ❧❡✈❡❧ ♦❢ ✇❡❛❧t❤ ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥✳ ◮ ❚❤❡ ✐♥✈❡st♦r ❝❛♥ r❡❞✉❝❡ t❤❡ ❞♦✇♥s✐❞❡ r✐s❦ ❛s ✇❡❧❧ ❛s ♣❛rt✐❝✐♣❛t✐♥❣ ✐♥ ❣❛✐♥s ♦❢ r✐s❦② ❛ss❡ts✳

❊①❛♠♣❧❡ ❉❡❝✐s✐♦♥ ❛t ❞❛② t✿ ♠✉❧t✐♣❧❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❤❡ ❞✐s❝♦✉♥t❡❞ ✢♦♦r ✉♣ t♦ t ✐s ✐♥✈❡st❡❞ ✐♥t♦ t❤❡ st♦❝❦ ♣♦rt❢♦❧✐♦ ✭❉❆❳✮✱ t❤❡ r❡st ✐♥t♦ ❛ r✐s❦❧❡ss ❛ss❡t

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-27
SLIDE 27

❆♣♣❧✐❝❛t✐♦♥s ✺✲✹

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

⊡ ❈r✉❝✐❛❧ ✐♥❣r❡❞✐❡♥t✿ t❤❡ ♠✉❧t✐♣❧✐❡r m

◮ m✿ t❤❡ ♣r♦♣♦rt✐♦♥ ✈❛❧✉❡ ✐♥✈❡st❡❞ ✐♥t♦ r✐s❦② ❛ss❡ts ◮ ❚❤❡ ❧❛r❣❡r m✱ t❤❡ ♠♦r❡ r✐s❦② ❡①♣♦s✉r❡

⊡ ❍♦✇ t♦ s❡❧❡❝t t❤❡ ♠✉❧t✐♣❧✐❡r❄

◮ ❙t❛♥❞❛r❞ ❝♦♥st❛♥t ✈❛❧✉❡

❈♦♥st❛♥t ♠

◮ ❇❛s❡❞ ♦♥ t❛✐❧ r✐s❦ ♠❡❛s✉r❡✱ ❱❛❘ ♦r ❊❙

❉❡t❛✐❧s

⊡ ▼✉❧t✐♣❧✐❡r s❡❧❡❝t✐♦♥ ✲ ❍❛♠✐❞✐ ❡t ❛❧✳ ✭✷✵✶✹✮✱ ❧❈❆❘❊ mt,τ =

  • ESet,τ
  • −✶

◮ Pr❛❝t✐❝❡✿ t❤r❡s❤♦❧❞ r❛♥❣❡ ❢♦r mt,τ✱ [✶, ✶✷]

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-28
SLIDE 28

❆♣♣❧✐❝❛t✐♦♥s ✺✲✺

▼✉❧t✐♣❧✐❡r ❉②♥❛♠✐❝s

2006 2008 2010 2012 2014 4 8 12 Multiplier Time ❋✐❣✉r❡ ✾✿ ❚✐♠❡✲✈❛r②✐♥❣ ♠✉❧t✐♣❧✐❡r mt,τ ❢♦r ❉❆❳ ✐♥❞❡① r❡t✉r♥s ❜❛s❡❞ ♦♥ ❧❈❆❘❊ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮ ❢r♦♠ ✷✵✵✻✵✶✵✸✲✷✵✶✹✶✷✸✶

▼✉❧t✐♣❧✐❡r ❉❡♥s✐t②

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-29
SLIDE 29

❆♣♣❧✐❝❛t✐♦♥s ✺✲✻

P❡r❢♦r♠❛♥❝❡

❖♥❡✲②❡❛r r♦❧❧✐♥❣ ❞❡t❛✐❧s ❈❆❱✐❛❘✲❜❛s❡❞ r♦❧❧✐♥❣ ❞❡t❛✐❧s ❚❛r❣❡t ✢♦♦r

2006 2008 2010 2012 2014 100 150 200 Price Index Time ❋✐❣✉r❡ ✶✵✿ P♦rt❢♦❧✐♦ ✈❛❧✉❡✿ ✭❛✮ ❉❆❳ ✐♥❞❡① ✭❜❧❛❝❦✮✱ ✭❜✮ m = ✺ ✱ ✭❝✮ ♦♥❡✲②❡❛r r♦❧❧✐♥❣ ✱ ✭❞✮ ❈❆❱✐❛❘ ♦♥❡✲②❡❛r r♦❧❧✐♥❣ ✭α = ✵.✵✻✺✮✱ ✭❡✮ mt,τ ✲ ❧❈❆❘❊ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮ ❢r♦♠ ✷✵✵✻✵✶✵✸✲✷✵✶✹✶✷✸✶✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-30
SLIDE 30

❆♣♣❧✐❝❛t✐♦♥s ✺✲✼

P❡r❢♦r♠❛♥❝❡

❋✐❣✉r❡ ✶✶✿ P♦rt❢♦❧✐♦ r❡t✉r♥ ♠♦♠❡♥ts ❝♦♠♣❛r✐s♦♥✳ ❘❡t✉r♥s ❛♥❞ ✈♦❧❛t✐❧✐t② ❛r❡ ❛♥♥✉❛❧✐③❡❞✳ ❚❤❡ ✐♥✈❡st♠❡♥t str❛t❡❣② ✐s ♦♥ ❛ ♦♥❡✲②❡❛r ✐♥✈❡st♠❡♥t ❜❛s✐s✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-31
SLIDE 31

❆♣♣❧✐❝❛t✐♦♥s ✺✲✽

P❡r❢♦r♠❛♥❝❡ ✲ ❙✉♠♠❛r②

⊡ ❧❈❆❘❊ ✈s ❡♠♣✐r✐❝❛❧ ❞❛t❛

◮ ❙❧✐❣❤t❧② ❧♦✇❡r r❡t✉r♥ ✭✼✳✸✻✪ ✈s ✽✳✼✾✪✮ ❧❛r❣❡❧② ❧♦✇❡r ✈♦❧❛t✐❧✐t② ✭✶✸✳✻✵✪ ✈s ✷✷✳✺✹✪✮ ◮

  • ✉❛r❛♥t❡❡ t❤❡ t❛r❣❡t ✢♦♦r ✈❛❧✉❡

⊡ ❧❈❆❘❊ ✈s ♦t❤❡r str❛t❡❣✐❡s

◮ ❤✐❣❤❡r r❡t✉r♥ t❤❛♥ t❤❡ ❝❛♥❞✐❞❛t❡s ✇✐t❤ ❈❆❱✐❛❘✲❜❛s❡❞ ♦r ❡①♣❡❝t✐❧❡ ♦♥❡✲②❡❛r r♦❧❧✐♥❣ ◮ ❖✉t♣❡r❢♦r♠ t②♣✐❝❛❧ ❝♦♥st❛♥t ♠✉❧t✐♣❧✐❡r ❜❡♥❝❤♠❛r❦s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-32
SLIDE 32

❈♦♥❝❧✉s✐♦♥s ✻✲✶

❈♦♥❝❧✉s✐♦♥s

⊡ ▲♦❝❛❧✐s✐♥❣ ❈❆❘❊ ▼♦❞❡❧

◮ ❇❛❧❛♥❝❡ ❜❡t✇❡❡♥ ♠♦❞❡❧❧✐♥❣ ❜✐❛s ❛♥❞ ♣❛r❛♠❡t❡r ✈❛r✐❛❜✐❧✐t② ◮ P❛r❛♠❡t❡r ❞②♥❛♠✐❝s

⊡ ❚❛✐❧ ❘✐s❦ ❉②♥❛♠✐❝s

◮ ❊①♣❡❝t✐❧❡ ❧❡✈❡❧s τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶ ◮ ❊①♣❡❝t✐❧❡ ❛♥❞ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

⊡ ❆ss❡t ❆❧❧♦❝❛t✐♦♥

◮ P♦rt❢♦❧✐♦ ✐♥s✉r❛♥❝❡ ♦♥ ❉❆❳ ❛t ❧❡✈❡❧ τ = ✵.✵✺ ◮ ❖✉t♣❡r❢♦r♠ ♦♥❡✲②❡❛r r♦❧❧✐♥❣ ✇✐♥❞♦✇ ❛♥❞ ♦t❤❡r ❜❡♥❝❤♠❛r❦s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-33
SLIDE 33

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

❲♦❧❢❣❛♥❣ ❑❛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❧✐♥ ❇r❛♥❞❡♥❜✉r❣ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❧✈❜✳✇✐✇✐✳❤✉✲❜❡r❧✐♥✳❞❡ ❝❛s❡✳❤✉✲❜❡r❧✐♥✳❞❡ ✐rt❣✶✼✾✷✳❤✉✲❜❡r❧✐♥✳❞❡ ❜✲t✉✳❞❡

2006 2008 2010 2012 2014 −0.1 0.1

slide-34
SLIDE 34

❘❡❢❡r❡♥❝❡s ✼✲✶

❘❡❢❡r❡♥❝❡s

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

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-35
SLIDE 35

❘❡❢❡r❡♥❝❡s ✼✲✷

❘❡❢❡r❡♥❝❡s

❈❤❡♥✱ ❨✳ ❛♥❞ ❍är❞❧❡✱ ❲✳ ❛♥❞ P✐❣♦rs❝❤✱ ❯✳ ▲♦❝❛❧✐③❡❞ ❘❡❛❧✐③❡❞ ❱♦❧❛t✐❧✐t② ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ❙t❛t✐st✐❝❛❧ ❆ss♦❝✐❛t✐♦♥ ✶✵✺✭✹✾✷✮✿ ✶✸✼✻✕✶✸✾✸✱ ✷✵✶✵ ❷í➸❡❦✱ P✳✱ ❍är❞❧❡✱ ❲✳ ❛♥❞ ❙♣♦❦♦✐♥②✱ ❱✳ ❆❞❛♣t✐✈❡ P♦✐♥t✇✐s❡ ❊st✐♠❛t✐♦♥ ✐♥ ❚✐♠❡✲■♥❤♦♠♦❣❡♥❡♦✉s ❈♦♥❞✐t✐♦♥❛❧ ❍❡t❡r♦s❝❡❞❛st✐❝✐t② ▼♦❞❡❧s ❊❝♦♥♦♠❡tr✐❝s ❏♦✉r♥❛❧ ✶✷✿ ✷✹✽✕✷✼✶✱ ✷✵✵✾

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-36
SLIDE 36

❘❡❢❡r❡♥❝❡s ✼✲✸

❘❡❢❡r❡♥❝❡s

❊st❡♣✱ ❚✳ ❛♥❞ ❑r✐t③♠❛♥✱ ▼✳ ❚✐♠❡✲✐♥✈❛r✐❛♥t ♣♦rt❢♦❧✐♦ ♣r♦t❡❝t✐♦♥✿✐♥s✉r❛♥❝❡ ✇✐t❤♦✉t ❝♦♠♣❧❡①✐t② ❏♦✉r♥❛❧ ♦❢ P♦rt❢♦❧✐♦ ▼❛♥❛❣❡♠❡♥t ✶✹✭✹✮✿ ✸✽✕✹✷✱ ✶✾✽✽ ❋ö❧❧♠❡r✱ ❍✳ ❛♥❞ ▲❡✉❦❡rt✱ P✳ ◗✉❛♥t✐❧❡ ❤❡❞❣✐♥❣ ❋✐♥❛♥❝❡ ❛♥❞ ❙t♦❝❤❛st✐❝s ✸✿ ✷✺✶✕✷✼✸✱ ✶✾✾✾

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-37
SLIDE 37

❘❡❢❡r❡♥❝❡s ✼✲✹

❘❡❢❡r❡♥❝❡s

  • ❡r❧❛❝❤✱ ❘✳❍✳✱ ❈❤❡♥✱ ❈✳❲✳❙✳ ❛♥❞ ▲✐♥✱ ▲✳❨✳

❇❛②❡s✐❛♥ ●❆❘❈❍ ❙❡♠✐✲♣❛r❛♠❡tr✐❝ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ❋♦r❡❝❛st✐♥❣ ✐♥ ❋✐♥❛♥❝✐❛❧ ▼❛r❦❡ts ❇✉s✐♥❡ss ❆♥❛❧②t✐❝s ❲♦r❦✐♥❣ P❛♣❡r ◆♦✳ ✵✶✴✷✵✶✷✱ ✷✵✶✷ ❍❛♠✐❞✐✱ ❇✳✱ ❏✉r❝③❡♥❦♦✱ ❊✳ ❛♥❞ ▼❛✐❧❧❡t✱ ❇✳ ❆ ❈❆❱✐❛❘ ♠♦❞❡❧❧✐♥❣ ❢♦r ❛ s✐♠♣❧❡ t✐♠❡✲✈❛r②✐♥❣ ♣r♦♣♦rt✐♦♥ ♣♦rt❢♦❧✐♦ ✐♥s✉r❛♥❝❡ str❛t❡❣② ❇❛♥❦❡rs✱ ▼❛r❦❡ts ✫ ■♥✈❡st♦rs ✶✵✷✿ ✹✕✷✶✱ ✷✵✵✾ ❍❛♠✐❞✐✱ ❇✳✱ ▼❛✐❧❧❡t✱ ❇✳ ❛♥❞ Pr✐❣❡♥t✱ ❏✳▲✳ ❆ ❞②♥❛♠✐❝ ❛✉t♦r❡❣r❡ss✐✈❡ ❡①♣❡❝t✐❧❡ ❢♦r t✐♠❡✲✐♥✈❛r✐❛♥t ♣♦rt❢♦❧✐♦ ♣r♦t❡❝t✐♦♥ str❛t❡❣✐❡s ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ ❉②♥❛♠✐❝s ✫ ❈♦♥tr♦❧ ✹✻✿ ✶✕✷✾✱ ✷✵✶✹

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-38
SLIDE 38

❘❡❢❡r❡♥❝❡s ✼✲✺

❘❡❢❡r❡♥❝❡s

❍är❞❧❡✱ ❲✳ ❑✳✱ ❍❛✉ts❝❤✱ ◆✳ ❛♥❞ ▼✐❤♦❝✐✱ ❆✳ ▲♦❝❛❧ ❆❞❛♣t✐✈❡ ▼✉❧t✐♣❧✐❝❛t✐✈❡ ❊rr♦r ▼♦❞❡❧s ❢♦r ❍✐❣❤✲❋r❡q✉❡♥❝② ❋♦r❡❝❛sts ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ ❊❝♦♥♦♠❡tr✐❝s✱ ✷✵✶✹ ❏✐❛♥❣✱ ❈✳❍✳✱ ▼❛✱ ❨✳❑✳ ❛♥❞ ❆♥✱ ❨✳❇✳ ❚❤❡ ❡✛❡❝t✐✈❡♥❡ss ♦❢ t❤❡ ❱❛❘✲❜❛s❡❞ ♣♦rt❢♦❧✐♦ ✐♥s✉r❛♥❝❡ str❛t❡❣②✿ ❆♥ ❡♠♣✐r✐❝❛❧ ❛♥❛❧②s✐s ■♥t❡r♥❛t✐♦♥❛❧ ❘❡✈✐❡✇ ♦❢ ❋✐♥❛♥❝✐❛❧ ❆♥❛❧②s✐s✱ ✶✽✭✹✮✿ ✶✽✺✕✶✾✼✱ ✷✵✵✾ ❑✉❛♥✱ ❈✳▼✳✱ ❨❡❤✱ ❏✳❍✳ ❛♥❞ ❍s✉✱ ❨✳❈✳ ❆ss❡ss✐♥❣ ✈❛❧✉❡ ❛t r✐s❦ ✇✐t❤ ❈❆❘❊✱ t❤❡ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡ ♠♦❞❡❧s ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠❡tr✐❝s ✶✺✵✭✷✮✿ ✷✻✶✕✷✼✵✱ ✷✵✵✾

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-39
SLIDE 39

❘❡❢❡r❡♥❝❡s ✼✲✻

❘❡❢❡r❡♥❝❡s

▼❡r❝✉r✐♦✱ ❉✳ ❛♥❞ ❙♣♦❦♦✐♥②✱ ❱✳ ❙t❛t✐st✐❝❛❧ ✐♥❢❡r❡♥❝❡ ❢♦r t✐♠❡✲✐♥❤♦♠♦❣❡♥❡♦✉s ✈♦❧❛t✐❧✐t② ♠♦❞❡❧s ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s ✸✷✭✷✮✿ ✺✼✼✕✻✵✷✱ ✷✵✵✹ ❙♣♦❦♦✐♥②✱ ❱✳ ❊st✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✇✐t❤ ❞✐s❝♦♥t✐♥✉✐t✐❡s ✈✐❛ ❧♦❝❛❧ ♣♦❧②♥♦♠✐❛❧ ✜t ✇✐t❤ ❛♥ ❛❞❛♣t✐✈❡ ✇✐♥❞♦✇ ❝❤♦✐❝❡ ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s ✷✻✭✹✮✿ ✶✸✺✻✕✶✸✼✽✱ ✶✾✾✽

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-40
SLIDE 40

❘❡❢❡r❡♥❝❡s ✼✲✼

❘❡❢❡r❡♥❝❡s

❙♣♦❦♦✐♥②✱ ❱✳ ▼✉❧t✐s❝❛❧❡ ▲♦❝❛❧ ❈❤❛♥❣❡ P♦✐♥t ❉❡t❡❝t✐♦♥ ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s t♦ ❱❛❧✉❡✲❛t✲❘✐s❦ ❚❤❡ ❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s ✸✼✭✸✮✿ ✶✹✵✺✕✶✹✸✻✱ ✷✵✵✾ ❚❛②❧♦r✱ ❏✳❲✳ ❊st✐♠❛t✐♥❣ ❱❛❧✉❡ ❛t ❘✐s❦ ❛♥❞ ❊①♣❡❝t❡❞ ❙❤♦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 ✻✭✷✮✿ ✷✼✸✕✷✾✷✱ ✶✾✾✻

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-41
SLIDE 41

❆♣♣❡♥❞✐① ✽✲✶

❲❤② ❊①♣❡❝t✐❧❡s❄ ◗✉❛♥t✐❧❡ ❱❛❘

▼♦t✐✈❛t✐♦♥

❋✐❣✉r❡ ✶✷✿ ❉✐str✐❜✉t✐♦♥ ♦❢ r❡t✉r♥s✱ t❤❡ ✺✪ q✉❛♥t✐❧❡ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞ ✉♥❞❡r t❤❡ ❝❤❛♥❣✐♥❣ t❛✐❧ str✉❝t✉r❡

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-42
SLIDE 42

❆♣♣❡♥❞✐① ✽✲✷

❊①♣❡❝t✐❧❡ ✈✳s✳ ◗✉❛♥t✐❧❡

▼♦t✐✈❛t✐♦♥

⊡ ❚❛✐❧ ✐♥❢❡r❡♥❝❡

◮ ◗✉❛♥t✐❧❡✿ ③❡r♦✲♠♦♠❡♥t ♦❢ t❛✐❧ str✉❝t✉r❡ ✲ ♣r♦❜❛❜✐❧✐t② ❈❡♥tr❛❧ q✉❛♥t✐❧❡✿ ♠❡❞✐❛♥ ◮ ❊①♣❡❝t✐❧❡✿ ✜rst ♠♦♠❡♥t ♦❢ t❛✐❧ str✉❝t✉r❡ ❈❡♥tr❛❧ ❡①♣❡❝t✐❧❡✿ ♠❡❛♥

⊡ ❊①♣❡❝t✐❧❡s ❛r❡ s❡♥s✐t✐✈❡ t♦ ❡①tr❡♠❡ ♠❛❣♥✐t✉❞❡✱ ♦✉t❧✐❡rs ⊡ ❊①♣❡❝t✐❧❡s ❧✐♥❦ t♦ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✭❊❙✮ ♥✐❝❡❧②

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-43
SLIDE 43

❆♣♣❡♥❞✐① ✽✲✸

▼✲◗✉❛♥t✐❧❡s

▼♦t✐✈❛t✐♦♥

⊡ ▲♦ss ❢✉♥❝t✐♦♥✱ ❇r❡❝❦❧✐♥❣ ❛♥❞ ❈❤❛♠❜❡rs ✭✶✾✽✽✮ zα = ❛r❣ ♠✐♥

θ ❊ ρα,γ (Y − θ)

✇❤❡r❡ ρα,γ (u) = |α − {u < ✵}| |u|γ✱ γ ≥ ✶

◮ ◗✉❛♥t✐❧❡ ✲ ❆▲❉ ❧♦❝❛t✐♦♥ ❡st✐♠❛t❡ qα = ❛r❣ ♠✐♥

θ ❊ ρα,✶ (Y − θ)

◮ ❊①♣❡❝t✐❧❡ ✲ ❆◆❉ ❧♦❝❛t✐♦♥ ❡st✐♠❛t❡ eα = ❛r❣ ♠✐♥

θ ❊ ρα,✷ (Y − θ)

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-44
SLIDE 44

❆♣♣❡♥❞✐① ✽✲✹

▲♦ss ❋✉♥❝t✐♦♥

▼♦t✐✈❛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✮

▲◗❘❝❤❡❝❦

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-45
SLIDE 45

❆♣♣❡♥❞✐① ✽✲✺

❊①♣❡❝t✐❧❡s ❛♥❞ ◗✉❛♥t✐❧❡s

▼♦t✐✈❛t✐♦♥

⊡ ▼✲◗✉❛♥t✐❧❡ α ✶ − α = eα

−∞ |y − eα|γ−✶ dF(y)

∞

eα |y − eα|γ−✶ dF(y)

◮ ❊①♣❡❝t✐❧❡ ✲ ●❧♦❜❛❧ ✐♥✢✉❡♥❝❡✱ ♦❜t❛✐♥❡❞ ❢r♦♠ γ = ✷, α ✶ − α = eα

−∞ |y − eα| dF(y)

∞

eα |y − eα| dF(y)

◮ ◗✉❛♥t✐❧❡ ✲ ▲♦❝❛❧ ✐♥✢✉❡♥❝❡✱ ♦❜t❛✐♥❡❞ ❢r♦♠ γ = ✶, α ✶ − α = P (Y ≤ qα) P (Y > qα)

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-46
SLIDE 46

❆♣♣❡♥❞✐① ✽✲✻

❈❆❱✐❛❘ ✲ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦r❡❣r❡ss✐✈❡ ❱❛❧✉❡ ❛t ❘✐s❦ ❜② ❘❡❣r❡ss✐♦♥ ◗✉❛♥t✐❧❡s

❈❆❘❊ ❈❆❱✐❛❘ ♣❡r❢♦r♠❛♥❝❡

⊡ ❊♥❣❧❡ ❛♥❞ ▼❛♥❣❛♥❡❧❧✐ ✭✷✵✵✹✮ ⊡ ❆s②♠♠❡tr✐❝ s❧♦♣❡ s♣❡❝✐✜❝❛t✐♦♥✱ ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ✐♥❢♦r♠❛t✐♦♥ s❡t Ft−✶ ❛t t✐♠❡ t yt = qt,α + εt,α Quantα(εt,α|Ft−✶) = ✵ qt,α = β✵ + β✶qt−✶,α + β✷y+

t−✶ + β✸y− t−✶

◮ ◗✉❛♥t✐❧❡ ✭❱❛❘✮ qt,α ❛t α ∈ (✵, ✶)✱ Quantα(εt,α|Ft−✶) ✐s t❤❡ α✲q✉❛♥t✐❧❡ ♦❢ εt,α ❝♦♥❞✐t✐♦♥❛❧ ♦♥ ✐♥❢♦r♠❛t✐♦♥ s❡t Ft−✶ ◮ ❲✐t❤ ❆◆❉✱ s❡t α = ✵.✵✻✺ s✉❝❤ t❤❛t eτα = qα ✇❤❡♥ τα = ✵.✵✺

❉❡t❛✐❧s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-47
SLIDE 47

❆♣♣❡♥❞✐① ✽✲✼

❆s②♠♠❡tr✐❝ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥ ✭❆◆❉✮

❈❆❘❊

⊡ ❆◆❉

  • µ, σ✷, τ
  • ♣❞❢✿

f (w) = ✷ σ

  • π

|τ − ✶| + π τ −✶ ❡①♣

  • −ρτ

w − µ σ

  • ◮

❈❤❡❝❦ ❢✉♥❝t✐♦♥✿ ρτ (u) = |τ − ■ {u ≤ ✵}| u✷ ◮ ❆◆❉

  • µ, σ✷, ✶/✷
  • = ◆(µ, σ✷)✱ ●❡r❧❛❝❤ ❡t ❛❧✳ ✭✷✵✶✷✮

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-48
SLIDE 48

❆♣♣❡♥❞✐① ✽✲✽

P❉❋

−10 −5 5 10 15 20 0.1 0.2 0.3 0.4 PDF w

❋✐❣✉r❡ ✶✹✿ ❉❡♥s✐t② ❢✉♥❝t✐♦♥ ❢♦r s❡❧❡❝t❡❞ ❆◆❉s✿ ✭❛✮ µ = ✵, τ = ✵.✺ ✭❜✮ µ = −✶, τ = ✵.✷✺ ✭❝✮ µ = −✷, τ = ✵.✵✺ ✭❞✮ µ = −✸, τ = ✵.✵✶✱ ✇✐t❤ σ✷

ετ = ✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-49
SLIDE 49

❆♣♣❡♥❞✐① ✽✲✾

◗✉❛s✐ ▲♦❣ ▲✐❦❡❧✐❤♦♦❞ ❋✉♥❝t✐♦♥

P❛r❛♠❡t❡r ❊st✐♠❛t✐♦♥

⊡ ■❢ ετ ∼ ❆◆❉

  • µ, σ✷

ε, τ

  • ✇✐t❤ ♣❞❢ fε (·)

t❤❡♥ Y ∼ ❆◆❉

  • eτ + µ, σ✷

ε, τ

  • ⊡ ◗✉❛s✐ ❧♦❣ ❧✐❦❡❧✐❤♦♦❞ ❢✉♥❝t✐♦♥ ❢♦r ♦❜s❡r✈❡❞ ❞❛t❛

Y = {y✶, . . . , yn} ♦✈❡r ❛ ✜①❡❞ ✐♥t❡r✈❛❧ I ℓI (Y; θτ) =

  • t∈I

❧♦❣ fε (yt − et,τ)

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-50
SLIDE 50

❆♣♣❡♥❞✐① ✽✲✶✵

  • ❛✉ss✐❛♥ ❘❡❣r❡ss✐♦♥

❊st✐♠❛t✐♦♥ ◗✉❛❧✐t②

Yi = f (Xi) + εi, i = ✶, . . . , n✱ ✇❡✐❣❤ts W = {wi}n

i=✶

L (W , θ) =

n

  • i=✶

ℓ {Yi, fθ (Xi)} wi✱ ❧♦❣✲❞❡♥s✐t② ℓ (·)✱ θ = ❛r❣ ♠❛①

θ∈Θ L (W , θ)

✶✳ ▲♦❝❛❧ ❝♦♥st❛♥t✱ f (Xi) ≈ θ∗✱ εi ∼ ◆

  • ✵, σ✷

❊θ∗

  • L(W ,

θ) − L(W , θ∗)

  • r

≤ ❊ |ξ|✷r , ξ ∼ ◆ (✵, ✶) ✷✳ ▲♦❝❛❧ ❧✐♥❡❛r✱ f (Xi) ≈ θ∗⊤Ψi✱ εi ∼ ◆

  • ✵, σ✷

✱ ❜❛s✐s ❢✉♥❝t✐♦♥s Ψ = {ψ✶ (X✶) , . . . , ψp (Xp)}✱ ♠✉❧t✐✈❛r✐❛t❡ ξ ❊θ∗

  • L(W ,

θ) − L(W , θ∗)

  • r

≤ ❊ |ξ|✷r , ξ ∼ ◆ (✵, Ip)

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-51
SLIDE 51

❆♣♣❡♥❞✐① ✽✲✶✶

❘✐s❦ ❇♦✉♥❞

❊st✐♠❛t✐♦♥ ◗✉❛❧✐t②

τ = ✵.✵✺ τ = ✵.✵✶ ▲♦✇ ▼✐❞ ❍✐❣❤ ▲♦✇ ▼✐❞ ❍✐❣❤ r = ✵.✺ ✵✳✷✹ ✵✳✸✸ ✵✳✷✺ ✵✳✸✽ ✵✳✸✽ ✵✳✶✺ r = ✶.✵ ✷✳✹✵ ✹✳✻✷ ✷✳✼✺ ✺✳✾✵ ✺✳✽✶ ✶✳✶✺

❚❛❜❧❡ ✶✿ ❙✐♠✉❧❛t❡❞ Rr (θ∗

τ)✱ ✇✐t❤ ❡①♣❡❝t✐❧❡ ❧❡✈❡❧s τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶✱

❢♦r s✐① s❡❧❡❝t❡❞ ♣❛r❛♠❡t❡r ❝♦♥st❡❧❧❛t✐♦♥ ❣r♦✉♣s

P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-52
SLIDE 52

❆♣♣❡♥❞✐① ✽✲✶✷

P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s

❘✐s❦ ❇♦✉♥❞ ❈r✐t✐❝❛❧ ❱❛❧✉❡s

τ = ✵.✵✺ τ = ✵.✵✶ ▲♦✇ ▼✐❞ ❍✐❣❤ ▲♦✇ ▼✐❞ ❍✐❣❤

  • α✵,τ

✲✵✳✵✵✵✸ ✵✳✵✵✵✸ ✵✳✵✵✵✼ ✲✵✳✵✵✵✸ ✵✳✵✵✵✸ ✵✳✵✵✵✼

  • α✶,τ

✲✵✳✶✵✺✽ ✲✵✳✵✸✵✻ ✵✳✵✺✷✹ ✲✵✳✶✵✸✺ ✲✵✳✵✸✶✷ ✵✳✵✺✹✼

  • α✷,τ

✲✵✳✺✽✵✵ ✲✵✳✺✷✽✽ ✵✳✷✹✸✽ ✲✵✳✺✽✵✽ ✲✵✳✺✷✻✻ ✵✳✷✵✽✾

  • α✸,τ

✵✳✺✵✺✵ ✵✳✺✽✺✷ ✷✳✶✷✶✸ ✵✳✺✶✸✹ ✵✳✺✽✼✶ ✷✳✷✵✻✻

  • σ✷

ε,τ

✵✳✵✵✵✶ ✵✳✵✵✵✶ ✵✳✵✵✵✷ ✵✳✵✵✵✶ ✵✳✵✵✵✶ ✵✳✵✵✵✷

❚❛❜❧❡ ✷✿ ◗✉❛rt✐❧❡s ♦❢ ❡st✐♠❛t❡❞ ❈❆❘❊ ♣❛r❛♠❡t❡rs ❜❛s❡❞ ♦♥ ♦♥❡✲②❡❛r ❡st✐✲ ♠❛t✐♦♥ ✇✐♥❞♦✇✱ ✐✳❡✳✱ ✷✺✵ ♦❜s❡r✈❛t✐♦♥s✱ ❢♦r t❤❡ t❤r❡❡ st♦❝❦ ♠❛r❦❡t r❡t✉r♥s ✲ ❉❆❳✱ ❋❚❙❊ ✶✵✵✱ ❙✫P ✺✵✵ ✲ ❢r♦♠ ✷✵✵✺✵✶✵✸✲✷✵✶✹✶✷✸✶ ✭✷✻✵✽ tr❛❞✐♥❣ ❞❛②s✮

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-53
SLIDE 53

❆♣♣❡♥❞✐① ✽✲✶✸

❙❡❧❡❝t❡❞ P❛r❛♠❡t❡r ❙❝❡♥❛r✐♦s

❆❞❛♣t✐✈❡ ❊st✐♠❛t✐♦♥

Low Mid High 1000 2000 3000 DAX Low Mid High 1000 2000 3000 FTSE 100 Low Mid High 1000 2000 3000 S&P 500 ❋✐❣✉r❡ ✶✺✿ ❍✐st♦❣r❛♠ ♦❢ t❤❡ s❡❧❡❝t❡❞ ♣❛r❛♠❡t❡r s❝❡♥❛r✐♦s ✭▲♦✇✱ ▼✐❞ ❛♥❞ ❍✐❣❤✮ ❢♦r ❛❞❛♣t✐✈❡ ❡st✐♠❛t✐♦♥ ✇✐t❤ τ = ✵.✵✺ ❛♥❞ τ = ✵.✵✶✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-54
SLIDE 54

❆♣♣❡♥❞✐① ✽✲✶✹

❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧

❘✐s❦ ❊①♣♦s✉r❡ ❈❆❱✐❛❘

⊡ ❊①♣❡❝t✐❧❡ ❧❡✈❡❧ τα s✉❝❤ t❤❛t eτα = qα ✭α✲q✉❛♥t✐❧❡✮✱ ❨❛♦ ❛♥❞ ❚♦♥❣ ✭✶✾✾✻✮✱ ❆❝❡r❜✐ ❛♥❞ ❚❛s❝❤❡ ✭✷✵✵✷✮ τα = α · qα − qα

−∞

ydF(y) ❊[Y ] − ✷ qα

−∞

ydF(y) − (✶ − ✷α) qα ✇❤❡r❡ Y ∼ AND✳ ⊡ ❊①♣❡❝t❡❞ ❙❤♦rt❢❛❧❧ ✭❊❙✮✱ ❑✉❛♥ ❡t ❛❧✳ ✭✷✵✵✾✮ ESeτα =

  • ✶ + τα (✶ − ✷τα)−✶ α−✶
  • eτα

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-55
SLIDE 55

❆♣♣❡♥❞✐① ✽✲✶✺

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥ ❙tr❛t❡❣②

❙tr❛t❡❣② ▼✉❧t✐♣❧✐❡r

⊡ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧✱ ✇❡ ❛✐♠ t♦ ♠❛✐♥t❛✐♥✿ ❊st❡♣ ❛♥❞ ❑r✐t③♠❛♥ ✭✶✾✽✽✮ Vt ≥ k × ♠❛①

  • F ∗ e−rf ∗(T−t), s✉♣

p≤t

Vp

  • = F s

t

◮ Vt✿ ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛t t✐♠❡ t✱ t ∈ (✵, T] F s

t ✿ ♣r♦t❡❝t✐♦♥ ✈❛❧✉❡ ✭t❛r❣❡t ✢♦♦r✮

◮ k ❡①♦❣❡♥♦✉s ♣❛r❛♠❡t❡r (✵, ✶)✱ s❡t k = ✵.✾ rf r✐s❦② ❢r❡❡ r❛t❡✱ ✐♥✐t✐❛❧ ✈❛❧✉❡ F = ✶✵✵ ◮ ❈✉s❤✐♦♥ ✈❛❧✉❡ Ct = Vt − F s

t ≥ ✵

⊡ ❆❧❧♦❝❛t❡ Gt = m · Ct ♣r♦♣♦rt✐♦♥ ✐♥t♦ st♦❝❦ ♣♦rt❢♦❧✐♦ ✭❉❆❳✮✱ ❛♥❞ t❤❡ r❡♠❛✐♥✐♥❣ Vt − Gt ✐♥t♦ r✐s❦❧❡ss ❛ss❡t✱ ♠✉❧t✐♣❧✐❡r m ≥ ✵✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-56
SLIDE 56

❆♣♣❡♥❞✐① ✽✲✶✻

❊①❛♠♣❧❡✿ ❈PP■ ✲ ❈♦♥st❛♥t ♣r♦♣♦rt✐♦♥ ♣♦rt❢♦❧✐♦ ✐♥s✉r❛♥❝❡ ❈♦♥s✐❞❡r ❛♥ ✐♥s✉r❛♥❝❡ str❛t❡❣② ✉♥❞❡r ❈PP■ ✇✐t❤ ❝♦♥st❛♥t ✢♦♦r F = ✶✵✵✱ ❝♦♥st❛♥t m = ✺✱ ❛♥❞ r✐s❦❧❡ss ❛ss❡t r❛t❡ rf = ✵ ✭❝❛s❤✮✳ ❚❤❡ ✐♥✐t✐❛❧ r✐s❦② ❛ss❡t ✈❛❧✉❡ ✐s ✶✵✵✱ ❛♥❞ ❛t ❡❛❝❤ st❡♣ ❣♦❡s ✉♣✭❞♦✇♥✮ ✶✺✳ ✐♥✐t✐❛❧ r✐s❦② ❛ss❡t ✈❛❧✉❡ F ✶✵✵ ♣r♦♣♦rt✐♦♥ k ✵✳✾ r✐s❦❧❡ss r❛t❡ rf ✵ ❝♦♥st❛♥t ♠✉❧t✐♣❧✐❡r m ✺ st❡♣s ✹

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-57
SLIDE 57

❆♣♣❡♥❞✐① ✽✲✶✼

❘✐s❦② ❛ss❡t ✈❛❧✉❡ ✶✻✵ ✶✹✺ ✭✵✳✶✵✮ ✶✸✵ ✭✵✳✶✷✮ ✶✸✵ ✶✶✺ ✭✵✳✶✸✮ ✶✶✺ ✭✵✳✶✸✮ ✶✵✵ ✭✵✳✶✺✮ ✶✵✵ ✭✵✳✶✺✮ ✶✵✵ ✽✺ ✭✵✳✶✽✮ ✽✺ ✭✵✳✶✽✮ ✭✲✵✳✶✺✮ ✼✵ ✭✵✳✷✶✮ ✼✵ ✭✲✵✳✶✽✮ ✺✺ ✭✵✳✷✼✮ ✭✲✵✳✷✶ ✮ ✹✵ ✭✲✵✳✷✼ ✮

❚❛❜❧❡ ✸✿ ❘✐s❦② ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❤❡ ✈❛❧✉❡ ✐♥ ❧♦✇ ❜r❛❝❦❡t ❞❡♥♦t❡s t❤❡ ❛ss❡t r❡t✉r♥✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-58
SLIDE 58

❆♣♣❡♥❞✐① ✽✲✶✽

P♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❤❡ ❝✉s❤✐♦♥ ✶✺✾✳✶✽ ✶✸✺✳✺✾ ✭✻✾✳✶✽✮ ✶✶✽✳✾✶ ✭✹✺✳✻✵✮ ✶✵✸✳✻✶ ✶✵✼✳✺✵ ✭✷✽✳✾✶✮ ✾✽✳✷✹ ✭✶✸✳✻✶✮ ✶✵✵ ✭✶✼✳✺✮ ✾✹✳✼✶ ✭✽✳✷✹✮ ✾✶✳✶✺ ✭✶✵✮ ✾✷✳✺✵ ✭✹✳✼✶✮ ✾✵✳✻✶ ✭✶✳✶✺✮ ✭✷✳✺✮ ✾✵✳✷✾ ✭✵✳✻✶✮ ✽✾✳✾✼✾ ✭✵✳✷✾✮ ✽✾✳✾✼✾ ✭✵ ✮ ✭✲✵✳✵✷ ✮ ✽✾✳✾✼✾ ✭✵ ✮

❚❛❜❧❡ ✹✿ P♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❤❡ ✈❛❧✉❡ ✐♥ ❧♦✇ ❜r❛❝❦❡t ❞❡♥♦t❡s t❤❡ ❝✉s❤✐♦♥✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-59
SLIDE 59

❆♣♣❡♥❞✐① ✽✲✶✾

▼✉❧t✐♣❧✐❡r

▼✉❧t✐♣❧✐❡r

⊡ P♦rt❢♦❧✐♦ ✈❛❧✉❡ Vt Vt+✶ = Vt + Gtrt+✶ + (Vt − Gt) rft+✶ ✇✐t❤ rt st♦❝❦ ✐♥❞❡① r❡t✉r♥ ❛♥❞ rft r✐s❦❧❡ss r❛t❡ ⊡ ❈✉s❤✐♦♥ ✈❛❧✉❡ Ct = Vt − F s

t ≥ ✵

Ct+✶ = Ct{✶ + m · rt+✶ + (✶ − m) rft+✶} ⊡ ∀t ≤ T✱ s✐♥❝❡ t❤❡ ✈❛❧✉❡ Ct ≥ ✵ m · rt+✶ + (✶ − m) rft+✶ ≥ −✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-60
SLIDE 60

❆♣♣❡♥❞✐① ✽✲✷✵

▼✉❧t✐♣❧✐❡r

▼✉❧t✐♣❧✐❡r

  • ❛♣ r✐s❦

⊡ ■❢ rft ✐s r❡❧❛t✐✈❡❧② s♠❛❧❧✱ ❛♥❞ ✇❤❡♥ rt+✶ < ✵✱ ②✐❡❧❞ t❤❡ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ♠✉❧t✐♣❧✐❡r✿ Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❣✉❛r❛♥t❡❡ ✐s s❛t✐s✜❡❞ ❛t ❛♥② t✐♠❡ ♦❢ t❤❡ ♠❛♥❛❣❡♠❡♥t ♣❡r✐♦❞ ✇✐t❤ ❛ ♣r♦❜❛❜✐❧✐t② ❡q✉❛❧ t♦ ✶ ≺ ≻ ∀t ≤ T − ✶, m ≤

  • −r−

t+✶

−✶ ✇❤❡r❡ r−

t+✶ = ♠✐♥ {rt+✶, ✵}✳

◮ ▼✉❧t✐♣❧✐❡r mt✱ t❤❡ ❧❡✈❡r❛❣❡ ✈❛❧✉❡ ♦♥ r✐s❦② ❛ss❡ts✱ ✐s ♥❡❣❛t✐✈❡❧② r❡❧❛t❡❞ t♦ t❤❡ ♠❛①✐♠✉♠ ❡①tr❡♠❡ ❧♦ss ♦❢ r✐s❦② ❛ss❡ts✳ ◮ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ rt+✶ = −✶✵%✱ m ≤ ✶✵❀ ■❢ rt+✶ = −✷✵%✱ m ≤ ✺✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-61
SLIDE 61

❆♣♣❡♥❞✐① ✽✲✷✶

  • ❛♣ ❘✐s❦

▼✉❧t✐♣❧✐❡r

⊡ ■♥ ♣r❛❝t✐❝❡✱ ❞✉❡ t♦ t❤❡ ❞✐s❝r❡t❡✲t✐♠❡ r❡❜❛❧❛♥❝✐♥❣✱ t❤❡ ♥♦♥♥❡❣❛t✐✈❡ ❝✉s❤✐♦♥ ✈❛❧✉❡ ❝❛♥ ♥♦t ❜❡ ❣✉❛r❛♥t❡❡❞ ♣❡r❢❡❝t❧②✳

❉❡t❛✐❧s

⊡ ●❛♣ r✐s❦✿ t❤❡ r✐s❦ ♦❢ ✈✐♦❧❛t✐♥❣ t❤❡ ✢♦♦r ♣r♦t❡❝t✐♦♥✱ ✐✳❡✳✱ t❤❡ t✐♥② ❧❡✈❡❧ ♦❢ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ❝✉s❤✐♦♥ ✈❛❧✉❡s ❛r❡ ♥♦♥✲♣♦s✐t✐✈❡✳ ⊡ ❍♦✇ t♦ ❞❡✜♥❡ t❤❡ ❣❛♣ r✐s❦✿

◮ ❝♦♥tr♦❧ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ♣♦t❡♥t✐❛❧ ❧♦ss ✲ ❱❛❘ ❜❛s❡❞ ♠✉❧t✐♣❧✐❡r ◮ ❝♦♥tr♦❧ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❧♦ss s✐③❡ ✲ ❊❙ ❜❛s❡❞ ♠✉❧t✐♣❧✐❡r

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-62
SLIDE 62

❆♣♣❡♥❞✐① ✽✲✷✷

  • ❛♣ ❘✐s❦ ✲ ❝♦♥tr♦❧ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛

♣♦t❡♥t✐❛❧ ❧♦ss ✲ ❱❛❘ ❜❛s❡❞ ♠✉❧t✐♣❧✐❡r

▼✉❧t✐♣❧✐❡r

⊡ ●✐✈❡♥ ❛ ❝♦♥✜❞❡♥❝❡ ❧❡✈❡❧ ✶ − α✱ t❤❡ ✐♥s✉r❛♥❝❡ ❝♦♥❞✐t✐♦♥✱ ✐✳❡✳✱ ♣♦rt❢♦❧✐♦ ✈❛❧✉❡ ✐s ❛❜♦✈❡ ✢♦♦r✱ ✐s ❣✉❛r❛♥t❡❡❞✱ ❋ö❧❧♠❡r ❛♥❞ ▲❡✉❦❡rt ✭✶✾✾✾✮✱ P (Ct ≥ ✵, ∀t ≤ T) ≥ ✶ − α ⊡ ❊q✉✐✈❛❧❡♥t❧②✱ ✭s❡t t✐♠❡✲✈❛r②✐♥❣ ♠✉❧t✐♣❧✐❡r✮ P

  • mt ≤
  • −r−

t+✶

−✶ , ∀t ≤ T − ✶

  • ≥ ✶ − α

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-63
SLIDE 63

❆♣♣❡♥❞✐① ✽✲✷✸

▼✉❧t✐♣❧✐❡r

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

  • ❛♣ r✐s❦

⊡ ●❛♣ r✐s❦✿ ❝♦♥tr♦❧ ♦❢ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛ ♣♦t❡♥t✐❛❧ ❧♦ss

❉❡t❛✐❧s

▼✉❧t✐♣❧✐❡r mt ✇✐t❤ q✉❛♥t✐❧❡ ✲ ❆♠❡✉r ❛♥❞ Pr✐❣❡♥t ✭✷✵✶✹✮ mt,qα = |VaRα(rt+✶)|−✶ ⊡ ●❛♣ r✐s❦✿ ❝♦♥tr♦❧ ♦❢ t❤❡ ♣♦t❡♥t✐❛❧ ❧♦ss s✐③❡ ▼✉❧t✐♣❧❡ mt ✇✐t❤ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ✲ ❍❛♠✐❞✐ ❡t ❛❧✳ ✭✷✵✶✹✮ mt,τ =

  • ESet,τ
  • −✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-64
SLIDE 64

❆♣♣❡♥❞✐① ✽✲✷✹

▼✉❧t✐♣❧✐❡r ❉❡♥s✐t②

▼✉❧t✐♣❧✐❡r ❉②♥❛♠✐❝s

2 4 6 8 10 12 0.1 0.2 0.3 Density Multiplier ❋✐❣✉r❡ ✶✻✿ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t❡ ♦❢ t❤❡ ♠✉❧t✐♣❧✐❡r mt,τ ❢♦r ❉❆❳ ✐♥❞❡① r❡t✉r♥s ❜❛s❡❞ ♦♥ ❧❈❆❘❊ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮ ❢r♦♠ ✷✵✵✻✵✶✵✸✲✷✵✶✹✶✷✸✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-65
SLIDE 65

❆♣♣❡♥❞✐① ✽✲✷✺

❈❆❘❊✲❜❛s❡❞ ♦♥❡✲②❡❛r r♦❧❧✐♥❣

P❡r❢♦r♠❛♥❝❡

2006 2008 2010 2012 2014 −0.1 0.1 Returns 2006 2008 2010 2012 2014 4 8 12 Multiplier Time

❋✐❣✉r❡ ✶✼✿ ❊st✐♠❛t❡❞ ❡①♣❡❝t✐❧❡ ❛♥❞ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ❜② ❈❆❘❊ ❜❛s❡❞ ♦♥❡✲ ②❡❛r ✜①❡❞ r♦❧❧✐♥❣ ✇✐♥❞♦✇ ✭✉♣♣❡r ♣❛♥❡❧✮✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠✉❧t✐♣❧✐❡r ✭❧♦✇❡r ♣❛♥❡❧✮ ❢♦r ❉❆❳ ✐♥❞❡① r❡t✉r♥s ❢r♦♠ ✷✵✵✻✵✶✵✸ t♦ ✷✵✶✹✶✷✸✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-66
SLIDE 66

❆♣♣❡♥❞✐① ✽✲✷✻

❈❆❱✐❛❘✲❜❛s❡❞ ♦♥❡✲②❡❛r r♦❧❧✐♥❣

P❡r❢♦r♠❛♥❝❡ ❈❆❱✐❛❘

2006 2008 2010 2012 2014 −0.1 0.1 Returns 2006 2008 2010 2012 2014 4 8 12 Multiple Time ❋✐❣✉r❡ ✶✽✿ ❊st✐♠❛t❡❞ ❱❛❘ ✭α = ✵.✵✻✺✮ ❛♥❞ ❡①♣❡❝t❡❞ s❤♦rt❢❛❧❧ ❜② ❈❆❱✐❛❘ ✲ ❜❛s❡❞ ♦♥❡✲②❡❛r r♦❧❧✐♥❣ ✭✉♣♣❡r ♣❛♥❡❧✮✱ ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠✉❧t✐♣❧✐❡r ✭❧♦✇❡r ♣❛♥❡❧✮ ❢♦r ❉❆❳ ❢r♦♠ ✷✵✵✻✵✶✵✸ t♦ ✷✵✶✹✶✷✸✶

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-67
SLIDE 67

❆♣♣❡♥❞✐① ✽✲✷✼

P♦rt❢♦❧✐♦ ✈❛❧✉❡ ❛♥❞ t❛r❣❡t ✢♦♦r

P❡r❢♦r♠❛♥❝❡

2006 2008 2010 2012 2014 100 150 200 Price Index Time

❋✐❣✉r❡ ✶✾✿ P♦rt❢♦❧✐♦ ✈❛❧✉❡✿ ✭❛✮ ❉❆❳ ✐♥❞❡① ✭❜❧❛❝❦✮✱ ✭❜✮ mt,τ ✲ ❧❈❆❘❊ ✭r = ✶ ❛♥❞ τ = ✵.✵✺✮✱ ✭❝✮ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❛r❣❡t ✢♦♦r F s

t ✱ ❢r♦♠ ✷✵✵✻✵✶✵✸✲

✷✵✶✹✶✷✸✶✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-68
SLIDE 68

❆♣♣❡♥❞✐① ✽✲✷✽

P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

▼♦t✐✈❛t✐♦♥ P♦rt❢♦❧✐♦ Pr♦t❡❝t✐♦♥

08/07 01/08 06/08 11/08 03/09 96 98 100 Value Time 08/07 01/08 06/08 11/08 03/09 40 60 80 100 120 Dax Index 03/09 09/09 03/10 09/10 03/11 100 150 200 Time Value

❋✐❣✉r❡ ✷✵✿ P♦rt❢♦❧✐♦ ✈❛❧✉❡✿ ✭❛✮ ❉❆❳ ✐♥❞❡①✱ ✭❜✮ m = ✸✱ ✭❝✮ m = ✻✱ ✭❞✮ m = ✾✱ ✭❡✮ m = ✶✷ ♦♥ ❉❆❳ ✐♥❞❡① ✐♥ ❛ ❜✉❧❧ ♠❛r❦❡t ❢r♦♠ ✷✵✵✾✵✸✵✾✲✷✵✶✶✵✺✶✵ ✭❧❡❢t ♣❛♥❡❧✱ ✺✻✼ ♦❜s❡r✈❛t✐♦♥s✮ ❛♥❞ ✐♥ ❛ ❜❡❛r ♠❛r❦❡t ❢r♦♠ ✷✵✵✼✵✼✶✻✲✷✵✵✾✵✸✵✻ ✭r✐❣❤t ♣❛♥❡❧✱ ✹✸✶ ♦❜s❡r✈❛t✐♦♥s✮✳

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-69
SLIDE 69

❆♣♣❡♥❞✐① ✽✲✷✾

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

▼♦t✐✈❛t✐♦♥

2005 2010 2014 −4 −2 2 4

  • α1

Time 1 year 2005 2010 2014 −4 −2 2 4

  • α1

Time 1 month

❋✐❣✉r❡ ✷✶✿ ❊st✐♠❛t❡❞ α✶,✵.✵✺ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

♠♦r❡ ♣❛r❛♠❡t❡rs

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-70
SLIDE 70

❆♣♣❡♥❞✐① ✽✲✸✵

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

▼♦t✐✈❛t✐♦♥

2005 2010 2014 −4 −2 2 4

  • α1

Time 1 year 2005 2010 2014 −4 −2 2 4

  • α1

Time 1 month

❋✐❣✉r❡ ✷✷✿ ❊st✐♠❛t❡❞ α✶,✵.✵✶ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

♠♦r❡ ♣❛r❛♠❡t❡rs

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-71
SLIDE 71

❆♣♣❡♥❞✐① ✽✲✸✶

P❛r❛♠❡t❡r ❉✐str✐❜✉t✐♦♥s

▼♦t✐✈❛t✐♦♥

❋✐❣✉r❡ ✷✸✿ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t❡s ♦❢ α✶,✵.✵✺ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵✱ ✻✵✱ ✶✷✺ ♦r ✷✺✵ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-72
SLIDE 72

❆♣♣❡♥❞✐① ✽✲✸✷

P❛r❛♠❡t❡r ❉✐str✐❜✉t✐♦♥s

▼♦t✐✈❛t✐♦♥

❋✐❣✉r❡ ✷✹✿ ❑❡r♥❡❧ ❞❡♥s✐t② ❡st✐♠❛t❡s ♦❢ α✶,✵.✵✶ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵✱ ✻✵✱ ✶✷✺ ♦r ✷✺✵ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-73
SLIDE 73

❆♣♣❡♥❞✐① ✽✲✸✸

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

2005 2010 2014 −200 −100 100 200

  • α2

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α2

Time 1 month

❋✐❣✉r❡ ✷✺✿ ❊st✐♠❛t❡❞ α✷,✵.✵✺ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-74
SLIDE 74

❆♣♣❡♥❞✐① ✽✲✸✹

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

2005 2010 2014 −200 −100 100 200

  • α2

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α2

Time 1 month

❋✐❣✉r❡ ✷✻✿ ❊st✐♠❛t❡❞ α✷,✵.✵✶ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-75
SLIDE 75

❆♣♣❡♥❞✐① ✽✲✸✺

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

2005 2010 2014 −200 −100 100 200

  • α3

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α3

Time 1 month

❋✐❣✉r❡ ✷✼✿ ❊st✐♠❛t❡❞ α✸,✵.✵✺ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1

slide-76
SLIDE 76

❆♣♣❡♥❞✐① ✽✲✸✻

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

P❛r❛♠❡t❡r ❉②♥❛♠✐❝s

2005 2010 2014 −200 −100 100 200

  • α3

Time 1 year 2005 2010 2014 −400 −200 200 400

  • α3

Time 1 month

❋✐❣✉r❡ ✷✽✿ ❊st✐♠❛t❡❞ α✸,✵.✵✶ ❢♦r ❉❆❳ ❛♥❞ ❋❚❙❊✶✵✵ ✉s✐♥❣ ✷✵ ✭✶ ♠♦♥t❤✮ ♦r ✷✺✵ ✭✶ ②❡❛r✮ ♦❜s❡r✈❛t✐♦♥s

❧❈❆❘❊ ✲ ❧♦❝❛❧✐s✐♥❣ ❈♦♥❞✐t✐♦♥❛❧ ❆✉t♦❘❡❣r❡ss✐✈❡ ❊①♣❡❝t✐❧❡s

2006 2008 2010 2012 2014 −0.1 0.1