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Convertible Bonds with Call Notice PeriodsConvertible Bonds with Call Notice Periods

SONAD 2003 Friday, May 2

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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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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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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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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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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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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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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

S

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Time =T

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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

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100

125

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175

10 60 110 160S

V

Time = ti+1

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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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100

125

150

175

10 60 110 160

Time = ti+1

S

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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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75

100

125

150

175

10 60 110 160

Time = ti+1

S

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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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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

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100

125

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10 60 110 160

Time = ti+1

S

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100

125

150

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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

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100

125

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10 60 110 160

Time = ti+1

S

V

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10 60 110 160

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

ti

continue

call

ti

Time = ti

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75

100

125

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10 60 110 160

Time = ti+1

S

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10 60 110 160 S

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Time = ti

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

ti

ti continue

call

Time = ti

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75

100

125

150

175

10 60 110 160

S

V

Time = ti

S

V

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10 60 110 160

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

ti

continue

call

ti

Time = ti

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100

125

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10 60 110 160

Time = ti

S

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

call

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125

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10 60 110 160

Time = ti

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

ti

callstop

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100

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10 60 110 160

Time = ti

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

call

continueti

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10 60 110 160S

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Time = ti

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

continueticall

continueti

call

S*50

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100

125

150

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10 60 110 160S

V

Time = ti

V

Vc,ti

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

continueticall

continueti

call

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10 60 110 160S

V

Time = ti

V

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

continueticall

continueti

call

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100

125

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10 60 110 160

V

Time = ti

S

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continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

Notice periodsNotice periods

continueti

continueticall

continueti

call

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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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Notice periodsNotice periods

continueti

continue

continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

call

ti

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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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Notice periodsNotice periods

continueti

continue

continueti-2

call

call

continueti+1call

continueti

ti+1

stopstop

stop

call

ti

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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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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

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Notice period Tn = 30 days

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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

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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

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140

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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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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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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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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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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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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?!

?