Dde 321 - Solutions Exercise 7

Advanced Corporate Finance

Leonidas Rompolis

EXERCISES -7 (SOLUTIONS) ⎛S ⎞ ln ⎜ t ⎟ + r ( T − t ) 1 K + σ T − t and d 2 = d1 − σ T − t . 1. a. d1 = ⎝ ⎠ 2 σ T−t For St = $200, K = $180, r = 3%, σ = 20% and T – t = 3/12 we have that: d1 = 1.1786 and d2 = 1.0786 Thus N(d1) = 0.8807 and N(d2) = 0.8596 N(-d1) = 0.1192 and N(-d2) = 0.1903 Therefore, − r T−t C t ( K, T ) = St N ( d1 ) − Ke ( ) N ( d 2 ) = $22.56

Pt ( K, T ) = Ke ( ) N ( −d 2 ) − St N ( −d1 ) = $1.2243 b. For St = $200, K = $180, r = 3%, σ = 20% and T – t = 2 we have that: d1 = 0.7260 and d2 = 0.4432 Thus N(d1) = 0.7660 and N(d2) = 0.6711 N(-d1) = 0.2339 and N(-d2) = 0.3288 Therefore, −r T−t C t ( K, T ) = St N ( d1 ) − Ke ( ) N ( d 2 ) = $39.44 −r T−t

Pt ( K, T ) = Ke ( ) N ( −d 2 ) − St N ( −d1 ) = $8.9572 c. The longer the time to maturity, the higher the value of the option is. The reason for that is that with more time to maturity there is a greater chance that either the stock price will climb higher above the strike price (call option) or it will be much less (put option). − r T−t

110 100 120 200 200 + + + + = −66.41 2 3 4 1.10 1.10 1.10 1.10 1.105 b. The firm has the option to expand its business at year 4. This is a call option with time to maturity 4 years, strike price the initial cost of the new project, i.e. K = 500 and underlying asset the value of the new project. In order to apply the Black-Scholes we must calculate the value of the new project the year 0. The value of the new project the year 4 is: 100 100 120 120 200 S4 = + + + + = 469.85 2 3 4 1.10 1.10 1.10 1.10 1.105 The present value is S St = 4 4 = 320.19 1.10 Thus,

2. a. NPV = −600 +

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Advanced Corporate Finance

Leonidas Rompolis

⎛S ⎞ ⎛ 320.19 ⎞ ln ⎜ t ⎟ + r ( T − t ) ln ⎜ ⎟ + 0.03 × 4 1 1 K⎠ 500 ⎠ ⎝ ⎝ d1 = + σ T−t = + 0.4 × 4 = −0.0071 2 2 σ T−t 0.4 × 4 and

d 2 = d1 − σ T − t = −0.0071 − 0.4 × 4 = −0.8071

Then we calculate N ( d1 ) and N ( d 2 ) . We have that:

N ( d1 ) = 0.4971 and

N ( d 2 ) = 0.2098 Therefore, the value of the option is: C = 320.19 × 0.4971 − 500 × e −0.03×4 × 0.2098 = 66.14 Therefore, the NPV of the initial project with the option to expand at year 3 is: NPV = −66.41 + 66.14 = −0.27

3.

a. The option to abandon is a European put option. If the future expected cash flows are less than the abandonment value the investor exercise the put option and obtain the abandonment value. If not, he continues the project. The time to maturity is 1 year, the strike price is K = $150 and the underlying asset is the value of the project after the year 1. In order to apply the Black-Scholes formula we must estimate the present value of the underlying asset and its volatility. We have that: 0.6 [ 0.8 × 410 + 0.2 ×180] + 0.4 [ 0.4 × 220 + 0.6 × 100] = $229.42 St = 1.102 For the volatility we start by calculating the possible internal rates of return. 200 410 0 = −250 + + ⇒ R 1 = −2.09 or R 1 = 0.49 1 + R 1 (1 + R 1 ) 2 We select R1 = 0.49. 100 180 0 = −250 + + ⇒ R 2 = −1.67 or R 2 = 0.07 1 + R 2 (1 + R 2 ) 2 We select R2 = 0.07. 50 220 0 = −250 + + ⇒ R 3 = −1.87 or R 3 = 0.04 1 + R 3 (1 + R 3 ) 2 We select R3 = 0.04. 50 100 0 = −250 + + ⇒ R 4 = −1.54 or R 4 = −0.25 1 + R 4 (1 + R 4 ) 2 We select R4 = -0.25. The corresponding probabilities are:

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Advanced Corporate Finance

Leonidas Rompolis p1 = 0.6 × 0.8 = 0.48 p 2 = 0.6 × 0.2 = 0.12 p3 = 0.4 × 0.4 = 0.16 p 4 = 0.4 × 0.6 = 0.24

The average IRR is: 4

R = ∑ pi R i = 0.19 i =1

and the variance 4

Var = ∑ pi ( R i − R ) = 0.094 2

i =1

The standard deviation is σ = Var = 0.31 . Therefore, ⎛S ⎞ ⎛ 229.42 ⎞ ln ⎜ t ⎟ + r ( T − t ) ln ⎜ ⎟ + 0.03 ×1 1 1 K⎠ 150 ⎠ ⎝ ⎝ + σ T−t = + 0.31× 1 = 1.6224 d1 = 2 2 σ T−t 0.31× 1 and d 2 = d1 − σ T − t = 1.6224 − 0.31× 1 = 1.3124 Then we calculate N ( −d1 ) and N ( −d 2 ) . We have that:

N ( −d1 ) = 0.094 and

N ( −d 2 ) = 0.052

The value of the put is: P = 150 × e −0.03×1 × 0.094 − 229.42 × 0.052 = $1.77 b. The NPV of the project without the option to abandon is: 100 × 0.6 + 50 × 0.4 NPV = −250 + + 1.10 0.6 ( 410 × 0.8 + 180 × 0.2 ) 0.4 ( 220 × 0.4 + 100 × 0.6 ) + + = $52 1.102 1.102 Therefore the NPV of the project with the option to abandon is: NPV = $52 + $1.77 = $53.77 ⎛V ⎞ ⎛ 25 ⎞ ln ⎜ 0 ⎟ + rT ln ⎜ ⎟ + 0.06 × 1 B 1 1 10 4. a. d1 = ⎝ T ⎠ + σ T= ⎝ ⎠ + 1.3 × 1 = 1.4 2 2 σ T 1.3 × 1 and d 2 = d1 − σ T = 1.4 − 1.3 × 1 = 0.1 Thus, the value of the firm shares is: S0 = 25 × 0.919 − 10 × e −0.06×1 × 0.54 = $17.89

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Advanced Corporate Finance

Leonidas Rompolis

The value of the dept equals: B0 = V0 − S0 = 25 − 17.89 = $7.11 b. B0 = BT e− rBT ⇒ rB = −

1 ⎛ B0 ln ⎜ T ⎝ BT

⎞ ⎛ 7.11 ⎞ ⎟ = − ln ⎜ ⎟ = 0.34 ⎝ 10 ⎠ ⎠

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