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U n iv ersity o f
Convertible Bonds with Call Notice PeriodsConvertible Bonds with Call Notice Periods
SONAD 2003 Friday, May 2
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U n iv ersity o f
Andreas Grau (agrau@uwaterloo.ca)
Peter Forsyth (paforsyth@elora.uwaterloo.ca)
Kenneth Vetzal (kvetzal@uwaterloo.ca)
Convertible Bonds with Notice PeriodsConvertible Bonds with Notice Periods
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GoalsGoals
Build a precise model of convertible bonds with notice periods
Evaluate the effect of a notice period for– the optimal call strategy– the value of the convertible bond
Evaluate the effect of suboptimal call strategies
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U n iv ersity o f
GoalsGoals
Build a precise model of convertible bonds with notice periods
Evaluate the effect of a notice period for– the optimal call strategy– the value of the convertible bond
Evaluate the effect of suboptimal call strategies
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U n iv ersity o f
GoalsGoals
Build a precise model of convertible bonds with notice periods
Evaluate the effect of a notice period for– the optimal call strategy– the value of the convertible bond
Evaluate the effect of suboptimal call strategies
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U n iv ersity o f
GoalsGoals
Build a precise model of convertible bonds with notice periods
Evaluate the effect of a notice period for– the optimal call strategy– the value of the convertible bond
Evaluate the effect of suboptimal call strategies
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CB BackgroundCB Background
CB is a bond with the option to convert it into shares
Interesting for issuers with poor credit rating (start-up company)
Same return, but lower risk as stocks
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CB BackgroundCB Background
CB is a bond with the option to convert it into shares
Interesting for issuers with poor credit rating (start-up company)
Same return, but lower risk as stocks
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U n iv ersity o f
CB BackgroundCB Background
CB is a bond with the option to convert it into shares
Interesting for issuers with poor credit rating (start-up company)
Same return, but lower risk as stocks
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U n iv ersity o f
CB BackgroundCB Background
CB is a bond with the option to convert it into shares
Interesting for issuers with poor credit rating (start-up company)
Same return, but lower risk as stocks
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CB featuresCB featuresCompany xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
Company xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
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Company xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
Company xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
CB featuresCB features Company xy-Stock-
Volatility Dividends Di
Company xy-Stock-
Volatility Dividends Di
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U n iv ersity o f
Company xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
Company xy- Convertible bond -
Amount issued
Face value F
Conversion ratio Coupon payments ci
Maturity T
OptionsPut price Bp
Call period starting at time ts
Call price Bcl
Trigger price,
Notice period Tn,
CB featuresCB features Company xy-Stock-
Volatility Dividends Di
Company xy-Stock-
Volatility Dividends Di
Risk free rate rRisk free rate r
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CB models – no defaultCB models – no default
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CB models – no defaultCB models – no default
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CB models – no defaultCB models – no default
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CB models – T&FCB models – T&F
Kostas Tsiveriotis and Chris Fernandes 1998: Valuing Convertible Bonds with Credit Risk
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CB models – T&FCB models – T&F
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CB models – AFVCB models – AFV
Elie Ayache, Peter A. Forsyth, and Kenneth R. Vetzal 2002: Next Generation Models for Convertible Bonds with Credit Risk
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CB models – AFVCB models – AFV
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Notice periodsNotice periods
ti-2continue
callstop
ti-1
call
continue
stop
call
continueti
stop
ti+1
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Notice periodsNotice periods
continueti-2
call
call
continueti-1call
continueti
ti+1
stopstop
stop
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Notice periodsNotice periods
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
ti continue
call
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continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
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continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
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continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
50
75
100
125
150
175
10 60 110 160S
V
Time = ti+1
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
50
75
100
125
150
175
10 60 110 160 S
V
Time = ti+Tn
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continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
call
ti
S
V
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
50
75
100
125
150
175
10 60 110 160
Time = ti+Tn
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continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continue
S
V
call
ti
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
50
75
100
125
150
175
10 60 110 160
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
ti
continue
call
ti
Time = ti
50
75
100
125
150
175
10 60 110 160
Time = ti+1
S
V
50
75
100
125
150
175
10 60 110 160 S
V
Time = ti
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
ti
ti continue
call
Time = ti
50
75
100
125
150
175
10 60 110 160
S
V
Time = ti
S
V
50
75
100
125
150
175
10 60 110 160
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
ti
continue
call
ti
Time = ti
50
75
100
125
150
175
10 60 110 160
Time = ti
S
V
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
call
50
75
100
125
150
175
10 60 110 160
Time = ti
S
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
ti
callstop
50
75
100
125
150
175
10 60 110 160
Time = ti
S
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
call
continueti
50
75
100
125
150
175
10 60 110 160S
V
Time = ti
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
continueticall
continueti
call
S*50
75
100
125
150
175
10 60 110 160S
V
Time = ti
V
Vc,ti
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
continueticall
continueti
call
50
75
100
125
150
175
10 60 110 160S
V
Time = ti
V
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
continueticall
continueti
call
50
75
100
125
150
175
10 60 110 160
V
Time = ti
S
V
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
continueticall
continueti
call
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U n iv ersity o f
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
Notice periodsNotice periods
continueti
continueticall
continueti
call
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Notice periodsNotice periods
continueti
continue
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
call
ti
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U n iv ersity o f
Notice periodsNotice periods
continueti
continue
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
call
ti
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U n iv ersity o f
Notice periodsNotice periods
continueti
continue
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
call
ti
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U n iv ersity o f
Notice periodsNotice periods
continueti
continue
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
call
ti
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U n iv ersity o f
Notice periodsNotice periods
continueti
continue
continueti-2
call
call
continueti+1call
continueti
ti+1
stopstop
stop
call
ti
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Mathematical modelMathematical model
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Example CBExample CB
Company xy- Convertible bond –
Amount issued 100M
Face value F 100
Conversion ratio 1Coupon payments ci 2, semi-annually
Maturity T 5 years
Options
Call period starting at year 1
Call price Bcl 140Notice period Tn 30 days
Company xy- Convertible bond –
Amount issued 100M
Face value F 100
Conversion ratio 1Coupon payments ci 2, semi-annually
Maturity T 5 years
Options
Call period starting at year 1
Call price Bcl 140Notice period Tn 30 days
Company xy- Stock –
Market capitalization 10,000MImplied volatility 20%Dividends 2 once a year,
immediately after the coupon
Company xy- Stock –
Market capitalization 10,000MImplied volatility 20%Dividends 2 once a year,
immediately after the coupon
Capital market- Bonds –
Term structure of risk free rate:Flat, 5% continuously compounded
Capital market- Bonds –
Term structure of risk free rate:Flat, 5% continuously compounded
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Example: Effect on valueExample: Effect on value
0
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125
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0 25 50 75 100 125
Notice period Tn = 30 days
S
V
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Example: Effect on valueExample: Effect on value
0.00
0.50
1.00
1.50
2.00
80 90 100 110 120 130 140
Tn= 10 daysTn= 20 daysTn= 30 days
Tn= 60 days
Tn= 90 days
S
V
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Example: Optimal callExample: Optimal call160
155
150
145
140
135
time t [years]
S*
No notice period (Tn = 0)
1 2 3 4 5
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Example: Optimal callExample: Optimal call160
155
150
145
140
1351 2 3 4 5
time t [years]
S*
Notice period Tn = 30 days
no dividends
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160
155
150
145
140
135
time t [years]
S*
Notice period Tn = 30 days
dividends (Di=2)
Example: Optimal callExample: Optimal call
1 2 3 4 5
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ApproximationsApproximations
Ingersoll:– No notice period
S*(t) = Bcl(t) + A(t)
Jonathan Ingersoll 1977: An examination of corporate call policies on convertible securities.
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ApproximationsApproximations
Ingersoll:– No notice period
S*(t) = Bc(t) + A(t)
160
155
150
145
140
135
time t [years]
S*
1 2 3 4 5
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ApproximationsApproximations
Butler
Alexander W. Butler 2002: Revisiting Optimal Call Policy for Convertible Bonds
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ApproximationsApproximations
Butler160
155
150
145
140
135
time t [years]
S*
1 2 3 4 5
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Effect of ApproximationsEffect of Approximations
0
0.1
0.2
0.3
0.4
0.5
80 90 100 110 120 130 140
S
V
Ingersoll
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U n iv ersity o f
Effect of ApproximationsEffect of Approximations
0
0.1
0.2
0.3
0.4
0.5
80 90 100 110 120 130 140
S
V
Ingersoll
Butler
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U n iv ersity o f
ConclusionConclusion
Notice periods in CB can be done by solving a one dimensional problem each time step of the solution
The result is the value of the CB as well as the optimal call strategy
Previously published approximations for the optimal call strategy were not very precise
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U n iv ersity o f
ConclusionConclusion
Notice periods in CB can be done by solving a one dimensional problem each time step of the solution
The result is the value of the CB as well as the optimal call strategy
Previously published approximations for the optimal call strategy were not very precise
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U n iv ersity o f
ConclusionConclusion
Notice periods in CB can be done by solving a one dimensional problem each time step of the solution
The result is the value of the CB as well as the optimal call strategy
Previously published approximations for the optimal call strategy were not very precise
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U n iv ersity o f
ConclusionConclusion
Notice periods in CB can be done by solving a one dimensional problem each time step of the solution
The result is the value of the CB as well as the optimal call strategy
Previously published approximations for the optimal call strategy were not very precise
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Questions?!Questions?!
?