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Unformatted text preview: Introduction to derivatives, Fall 2011 BUSI 588, Homework 5 solutions Homework 5 solutions 1. Please refer to hw5.xls for details on the calculations. (a) The solution to the first part proceeds by solving for F from the pricing equation D = 30 = 100- C ( S = 100 ,K = F, . 1 , 10 , . 2) Using Black-Scholes one can find that the face value is F = 78 . 82. See hw5.xls for details in the calculations. (b) The value of the equity is $70m if they do nothing. If they take the action once the debt has been issued, then the value of equity goes up to $71.08m, since the value of debt drops by $2.08m (note that the other million is lost through the action). Therefore, even though the action destroys $1m in value, the shareholders would have an incentive to take this action. (c) If the debtholders anticipated this action, they would demand a face value of F = 85 . 85. This is given by the solution to 30 = 99- C ( S = 99 ,K = F, . 1 , 10 , . 3) The value of the equity position is E = 69 m (how could you have anticipated this?). (d) The value-destroying action is only good in an ex-post sense. If debtholders anticipate this conflict of interest they will demand a face value that incorporates the volatility of 30% (and the loss in value). Note that this anticipation would be correct: the equityholders will have an incentive to take this action even with F = 85. Therefore, the equityholders would like to “tie their hands” by say including ex-ante a covenant that does not allow them to take on the action. With this covenant the firm will not be able to invest in the project, and therefore the value of the equity will be $70m (instead of $69m). 2. Please refer to hw5.xls for details on the calculations. (a) As discussed in class, the risky debt’s value is given by D = 250 1 . 08- P ( S = 200 ,K = 250 ,r f = 0 . 08 ,t = 1 ,σ = 0 . 25); where P ( · ) denotes the put value given all Black-Scholes parameters. Using Black-Scholes one can estimate the value of the put at $40.74, so the value of the bond is $190.74. 1 Note the implied credit spread on the bond is 250 / 190 . 74- 1 . 08 = 23 . 07%! Obviously, it should not surprise us that a firm that is close to bankruptcy (200 asset value versus 250 that is due next year) would carry a high credit spread. (b) If the project gets undertaken, the firm value is worth $220 (after the $12m injection). The firm’s equityholders will decide on whether to fund this project themselves or not based on whether the value of their stake increases by $12 or more (or not). The value of the equity stake in the previous part was $9.26: the value of a call option with a strike of $250. We need to compare that with the value of a call option when the underlying asset is $220 instead of $200. Using Black-Scholes one can verify that changing the valueasset is $220 instead of $200....
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