# Corporate Valuation and Financing Exercises Session 5 « Risky Debt, Convertible Bonds, Callable Bonds »

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Corporate Valuation and Financing Exercises 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 Model a) What is the value of the bond if one uses a binomial tree with a one year step?

Q1 – The Merton Model a)

Q1 – The Merton Model b), 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 d 1 and d 2 to find the value of the bond: Then compute N(d 1 ) and N(d 2 ) :

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:

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):

Q2– Merton in continuous time (RWJ 22.27) c) & d) c) Debt’s decomposition : d) Delta of equity: Risk-neutral probability of default:

Q2 – Merton in continuous time (RWJ 22.27) e) Debt’s decomposition:

Q2 – Merton in continuous time (RWJ 22.27) f) Beta equity? Beta debt? WACC? (MM58 holds)

« Risky Debt » « Convertible Bonds» « Callable Bonds »

Q3 – Zoubowsky a) 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):

Q3 – Zoubowsky a) Build the binomial tree for the value of the company:

Q3 – Zoubowsky a) 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:

Q3 – Zoubowsky b) We go back in time to find the value in 2008 and 2009 :

Q3 – Zoubowsky b) 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.

Q3 – Zoubowsky c) 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 – Zoubowsky c) 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:

Q3 – Zoubowsky c) & 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.

« Risky Debt » « Convertible Bonds» « Callable Bonds »

Q4 – Freshwater Interest rates tree Year01  35% 5,03% 4,00% 2,50%

Q4 – Freshwater Are you lucky? Year 1Year 2 Bond 006 cash-flows6106 DF @ node r1,H1,0503 PV @ node r1,H100,92 Bond 006 value @ node r1,H106,92 DF @ node r1,L1,0250 PV @ node r1,L103,41 Bond 006 value @ node r1,L109,41 Value in 0104,01 = 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 Year012 100 4,5 99,49 4,5 101,17 101,95 4,5 100 4,5

Q4 – Freshwater B)Callable bond value101 Year012 100 4,5 99,49 4,5 100,72 101,00 4,5 100 4,5

Q4 – Freshwater C) Value of the call option101,17 – 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]

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