Trigger Event (Core Tier 1) Assets Contingent

Capital

Senior Debt

Equity

Assets

Senior Debt

Equity

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››››

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Probability to default density = equation for first exit time used in

barrier option pricing under Black-Scholes

Coco = Zero Coupon Corporate Bond (bond part)

+ knock-in forward (share in case of conversion)

- Binary Down and In options (coupon)

Assets 𝐴

Deposits 𝐷

Coco-Bond 𝐵

Equity 𝐸

Bank Balance Sheet

𝐴 𝐷 𝐵 𝐸

𝐴 =  𝐷 + 𝐵 + 𝐸

   

The risk free rate

𝑑𝐴 = 𝑟 − 𝜆𝑘 𝐴 − 𝑟 + ℎ 𝐷 − 𝑐 𝐵 𝑑𝑡 + 𝜎𝐴 𝑑𝑧 + 𝑌 𝑑𝑞

𝜆𝑘𝑟

ℎ𝑐 𝜎

𝑌

𝑑𝑞𝑡

𝐷 : 𝐴 : 𝑑𝑧~

𝑘 = 𝐸[𝑌 ]:

𝑑𝑞

𝑑𝑞 =𝑃 𝑑 = 1 = 𝜆

 𝑤𝑖𝑡ℎ  𝑃 𝑑 = 0 = 1 − 𝜆

𝜅

�̅�𝜎

𝑑𝜁 = 𝜌

𝑑𝑟 = 𝜅 �̅� − 𝑟 𝑑𝑡 + 𝜎 𝑟 𝑑𝜁

200 400 600 800 1000 12000

2

4

6

Time

Inte

rest

Rat

e [%

]

Evolution of Interest Rate Payments

𝑔 > 0

𝑥 > 1

𝑥 =

𝑑𝐷𝐷

= 𝑔 𝑥 − 𝑥 𝑑𝑡

200 400 600 800 1000 120080

85

90

95

100

105

110

Time

Bala

nce

Shee

t Com

pone

nts

Size

[-]

Evolution of the Balance Sheet Components

AssetsDeposits

200 400 600 800 1000 1200

2

4

6

8

10

Time Step [Days]C

ore

Tier

1 R

atio

[%]

Trigger Ratio and Conversion

Trigger RatioTrigger Limit

200 400 600 800 1000 12000

20

40

60

80

100

Time Step [days]

Bala

nce

Shee

t Com

pone

nts

Size

[-]

Evolution of the Balance Sheet Components

AssetsEquitiesContingent CapitalDeposits

Trigger event

 

 

< 0

0%

5%

10%

15%

20%

25%

30%

35%

Core Tier 1 Core Tier 2 Without

Bank

rupt

cy L

ikel

ihoo

d

Trigger Type

Bankruptcy Likelihood

1 Crisis per year

0.5 Crisis per year

0.1 Crisis per year

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›

𝑐𝑉 = 𝐹

𝑉

𝑉 = 𝐸 𝑁𝑃𝑉 = 𝐸 [∫ 𝑒 ∫ 𝑐 𝐹𝑑𝑡] 𝑐

𝑟

› 𝜎 = 2› 𝜆››

Assets [bio  CHF]

Deposits [bio  CHF]

Coco-Bond [bio  CHF]

Equity [bio  CHF]

1,000 909.1 2 88.9

Conversion Share

Issuane Date

Expiration Date

Coupon rate Trigger Trigger

Limit

80% January

2011 2041 7.875% Core Tier 1 7%

Asset Volatility [% p.a.]O

ccur

ence

of C

rash

p.a

. [%

]

0.5 1 1.5 2 2.5 3 3.5 4

20

40

60

80

100

120

140

160

90

95

100

105

110

115

120

125

130

135

140

The NPV cannot go lower than the nominal times the

conversion share (80)

Without risk of conversion, the Net Present Value is the one of

a non-convertible bond with negligible risk of default

Cu

rrent P

rice

(Sept 1

3th

2

01

2)

60 80 100 120 140 160 180 2000

5

10

15

20

25

30

Fair Issue PricePe

rcen

tage

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›

200 400 600 800 1000 1200

2

4

6

8

10

Time Step [days]

Trig

ger R

atio

[%]

Trigger Ratio and Conversions

Trigger RatioCoco TriggeredTrigger Limit

The Coco is fully converted and the ratio Core Tier 1 Ratio can now go under the trigger limit

The CoCo is triggered as many times as required to keep the ratio above the trigger limit

CoCo Type Bankruptcy [%] CoCo Fully Converted [%]

Core Tier 1 Full Conversion

3.7% 75.1%

Core Tier 1 Partial Conversion

3.4% 17.3%

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Forward looking simulation (ex ante)

Backward real data (ex post)

Volatility p.a. 16% 20%

1 3 5 7 9 11 13 15 17 19 21 23 25 27 2970

80

90

100

110

120

130

140

150

Year

Pric

e

Price scenarios of a CoCo bond

0 200 400 600 800 1000 12000

2

4

6

8

10

12

Time Step [days]

Bala

nce

Shee

t Com

pone

nts

[-]

Share PriceEquitiesNb Of Share

𝐶 = 𝐶 𝐵 𝐸

Jump in Equity No jump in Share Price

tR E

BC

200 400 600 800 1000 12000

2

4

6

8

10

12

Time Step [days]

Bala

nce

Shee

t Com

pone

nts

[-]

Share PriceEquitiesTotal Nb Of Share

0 200 400 600 800 1000 12000

2

4

6

8

10

Time Step [days]

Bala

nce

Shee

t Com

pone

nts

Share PriceEquityTotal Nb of Shares

𝐶 >

Jump in Equity Drop in Share Price: Old shareholders lose money with the conversion Sells shares before conversion Forces conversion

𝐶 <

Jump in Equity Jump in Share Price Value Transfer from CoCo holders to shareholders Shareholders are interested in conversion Shareholers are interested in «health» decrease of the bank

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›–

–

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Parameter Value Description ∆𝑡 1 month Simulation time step

𝜆 1/12 Probability that a crash occurs be between 𝑡 and 𝑡 + Δ𝑡

𝜎 0.02 The volatility of the lognormal distributed amplitude of the crash return

𝜇 -0.01 The mean of the lognormal distributed amplitude of the crash return

𝜅 0.114 The speed factor for the mean reversion adjustment

�̅� 0.069 The long term target interest rate

𝜎 0.07 The volatility of the interest rate

𝜌 -0.2 The correlation between interest rate and asset volatility

𝑔 0.5 The speed factor for the mean reversion of the asset to deposit ratio

𝑥 1.1 The target asset to deposit ratio

Parameters from: Pennachi G., 2011, A Structural Model of Contingent Bank Capital

𝑑𝐷𝐷

= 𝑟 + ℎ 𝑑𝑡 − max  𝐷 − 𝑌 𝐴

𝐷, 0 𝑑𝑞

ℎ = 𝜆𝐸 max  𝐷 − 𝑌 𝐴

𝐷, 0

ℎ = 𝜆 𝑁 −𝑑 − 𝑥 𝑒 𝑁(−𝑑 )

𝑑 =ln 𝑥 + 𝜇

𝜎𝑑 = 𝑑 + 𝜎