Corporate Valuation and FinancingExercises Session 5 « Risky Debt, Convertible Bonds, Callable Bonds »

« Risky Debt » 

«  Convertible Bonds»

 «  Callable Bonds » 

Q1 – The Merton Model

• 100.000 shares• 20€/ shares• No taxes• Risk free rate 3%• u = 4• d = ¼ = 0,25

• Issue a zero-coupon maturing in 3 years – at maturity 1.000.000€

Q1 – The Merton Modela)

• What is the value of the bond if one uses a binomial tree with a one year step?

Q1 – The Merton Modela)

Q1 – The Merton Modelb), c), d)

• Should your boss take the offer? • What is the risk premium of the company? • Broadly speaking which kind of rating could they expect with such a

figure?

Q2 – Merton in continuous time (RWJ 22.27)a)

• Compute d1 and d2 to find the value of the bond:

• Then compute N(d1) and N(d2) :

1

1

1

ln

0.5

1000ln

700 exp 0.08 0.50.5 0.4 0.5

0.4 0.51.544

SPV K

d TT

d

d

2

2

1.544 0.4 0.5

1.261

d

d

1

2

0.9387

0.8964

N d

N d

Q2 – Merton in continuous time (RWJ 22.27)a)

• The value of equity is equal to the value of a call:

• The value of debt is:

1 2exp

335.85

fE S N d K r T N d

1000 335.85

664.15

D

Q2 – Merton in continuous time (RWJ 22.27)b)

• Debt yield is the solution of the following equation:

• This means that the spread is (in bp):

exp

ln

664.15ln

7000.5

0.105136

D F y T

DF

yT

y

y

251.36spread

Q2– Merton in continuous time (RWJ 22.27)c) & d)

• c) Debt’s decomposition :

• d) Delta of equity:

Risk-neutral probability of default:

1 2

D Risk-Free Debt + Put

exp exp

700 exp 0.08 0.5 1000 0.0613 700 exp 0.08 0.5 0.1036

672.55 8.40

f fF r T S N d K r T N d

1

0.9387

Delta N d

2Risk Neutral Probability of Default 1

0.1036

N d

Q2 – Merton in continuous time (RWJ 22.27)e)

• Debt’s decomposition:

rT 1

2Loss 2Prob. of default if no

Expected Amountrecoveryof Recovery given Default

Expected Loss given Default

0.08 0.5

1-N(d )[1 ( )] [ Ve ]

1-N(d )

700 1 0

rTD e F N d F

e

0.08 0.5

0.1036615.65

84.35

0.0613.8964 700 1000

0.1036e

Q2 – Merton in continuous time (RWJ 22.27)f)

• Beta equity?

• Beta debt?

• WACC?

(MM58 holds)0.14WACC

1 1

664.151 0.9387 1

335.85

2.795

e a

DN d

E

1 1 1

335.851 0.0613 1

664.15

0.092

d a

EN d

D

« Risky Debt » 

«  Convertible Bonds»

 «  Callable Bonds » 

Q3 – Zoubowskya)

• To build the binomial tree, compute u and d :

• And the risk neutral probability of an up movement:

• We also compute the value of q (the percentage of shares hold by debtholders if they decide to convert the convertible):

1 0.5

1.5

1 0.3333

0.6666

u

d

1

1.04 0.67

1.5 0.670.448

fr dp

u d

1000000

6000000 1000000

0.143

q

Q3 – Zoubowskya)

• Build the binomial tree for the value of the company:

Q3 – Zoubowskya)

• The value of the debt in 2009 is equal to:

• In 2009, in the up-up state, the bondholders will decide to convert as value of the stock they get if they convert is higher than the face value of the bond:

• In 2009, in the up-down state, they will not convert as:

,min ,Max V q F V

115714286

810000000 0.143 100000000

51685714

361800000 0.143 100000000

Q3 – Zoubowskyb)

• We go back in time to find the value in 2008 and 2009 :

Q3 – Zoubowskyb)

• In 2008, they will decide to convert if the value of shares they get is higher than the expected value if they decide to wait:

• Hence the value in 2009, in the up case is:

• The computation for 2008 down and 2007 are similar.

2009 20091

1

u d

f

p V p VV q

r

77142857

102923077

0.448 115714286 1 0.448 100000000540000000 0.143

1 0.04

Q3 – Zoubowskyc)

• To find the value of the callable convertible bond, we have to proceed sequentially. Firstly we determine if shareholders will call, then we determine the decision of the bondholder (convert or accept the call price)? The binomial tree is :

Q3 – Zoubowskyc)

• The shareholders decide to call if :

• In 2008 up state, the shareholder will decide to call as the value of the call price times the number of bonds is lower than the expected value for the bondholders.

• Once the bond is called, bond holders have the choice between accepting the call price or convert their shares, in 2008 up-state they decide to convert as:

70000000 77142857

102923077

0.448 115714286 1 0.448 100000000max 70 1000000,540000000 0.143

1 0.04

77142857 70000000

540000000 0.143 70 1000000

V q n C

Call Price

2009 20091max ,

1

u d

f

p V p Vn C V q

r

Q3 – Zoubowskyc) & d)

• In 2008 down state, shareholders decide to call as:

• And bondholders decide not to convert as:

• Similar computations leads to the value in 2007.• d) Mr Zoubowsky should to call in 2007.

34457143 70000000

241200000 0.143 70 1000000

70000000 34457143

96153846

0.448 100000000 1 0.448 100000000max 70 1000000,241200000 0.143

1 0.04

« Risky Debt » 

«  Convertible Bonds»

 «  Callable Bonds » 

Q4 – Freshwater

Interest rates tree

Year 0 1 =s 35%

5,03%

4,00%

2,50%

Q4 – Freshwater

Are you lucky?

Year 1 Year 2

Bond 006 cash-flows 6 106

DF @ node r1,H 1,0503

PV @ node r1,H 100,92

Bond 006 value @ node r1,H 106,92

DF @ node r1,L 1,0250

PV @ node r1,L 103,41

Bond 006 value @ node r1,L 109,41

Value in 0 104,01

= 0,5x(106,92/1,04) + 0,5x(109,41/1,04)

--> OK the tree generates a value for the on-the-run issue equal to its market price.

Q4 – Freshwater

A) Option free bond value

Year 0 1 2

100

4,5

99,49

4,5

101,17

101,95

4,5

100

4,5

Q4 – Freshwater

B) Callable bond value 101

Year 0 1 2

100

4,5

99,49

4,5

100,72

101,00

4,5

100

4,5

Q4 – Freshwater

C) Value of the call option 101,17 – 100,72 = 0,46

=Option free bond value - Callable bond value

Q4 – Freshwater

D) Arbitrage free model -->

the i rate tree is constructed so that the value produced by the model when applied to an on the run issue

is equal to its market price. It is also said to be 'calibrated to the market'.

E) Higher volatility --> higher option value --> lower callable bond value [Callable bond value = option-free bond value - call option value]