Solution of Derivative

# 3 100 1 7453 117 65335 2 see figures one two

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Unformatted text preview: ption is that lower interest rates could raise the yield on the most junior debt. Table Two (Problem 16.3b) Volatility 0.2 0.3 Senior 8.00% 8.08% IntermA 8.21% 9.62% IntermB 10.09% 13.63% Junior 14.79% 19.03% Risk Free Rate 0.05 0.08 Senior 5.14% 8.08% IntermA 7.42% 9.62% IntermB 12.70% 13.63% Junior 19.35% 19.03% 113 0.4 8.54% 12.12% 17.31% 22.69% 0.11 11.04% 12.05% 15.01% 19.30% Part 4 Financial Engineering and Applications Question 16.4. D = 100 − 34.6653 = 65.335, rA = 10%, and (from problem 4) rE = 12.3%: .10 = 65.335 34.665 rD + .123 =⇒ rD = 8.78%. 100 100 Since D = A − E the volatility of debt satisﬁes σD D = σA A − σD = σA A (1 − D ) = (1) σA A hence .3 (100) (1 − .7453) = 11.7% 65.335 (2) See Figures One, Two, and Three for the effect on expected returns when we vary A, σ , and r . For changes in r we assume that rA changes by the same amount (i.e. the risk premium on assets is constant at rA − r = 4%). Figure One (Problems 16.4 & 16.5) 0.22 0.2 H- Expected Returns 0.18 0.16 0.14 0.12 H, 0.1 0.08 50 100 ) 114 150 200 Chapter 16 Corporate Applications Figure Two (Problems 16.4 & 16.5) 0.3 Expected Returns 0.25 0.2 H0.15 H, 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 σ 0.6 0.7 0.8 0.9 1 0.14 0.16 0.18 0.2 Figure Three (Problems 16.4 & 16.5) 0.35 0.3 Expected Returns 0.25 0.2 rE 0.15 rD 0.1 0.05 0 0 0.02 0.04 0.06 0.08 0.1 r 115 0.12 Part 4 Financial Engineering and Applications a) y = ln (100/D) /5 = 0.1585, 0.0853, 0.0678, 0.0627, and 0.0600. b) See Table Three. Table Three (Problem 16.6b) Volatility Yield on Debt 0.10 0.0653 0.15 0.0732 0.20 0.0825 0.25 0.0926 0.30 0.1034 0.35 0.1148 0.40 0.1266 0.45 0.1390 0.50 0.1519 0.55 0.1652 0.60 0.1791 0.65 0.1934 0.70 0.2082 0.75 0.2236 0.80 0.2394 0.85 0.2557 0.90 0.2725 0.95 0.2898 1.00 0.3077 Question 16.6. Let V be the market value of the new project and let k be the cost of the project (V − k would be the NPV). If the project is paid for with junior debt of face value F , then the junior debt must be worth k ; this implies Junior Debt Value = C (100 + V , 150) − C (100 + V , 150 + F ) = k. (3) Shareholders wealth, if they do the project, will be C (100 + V , 150 + F ). Hence the change in their wealth is C (100 + V , 150 + F ) − C(100, 150) = C (100 + V , 150) − C(100, 150) − k (4) C (100, 150) = 26.0672 is the initial value of shareholders wealth. When k = 1, shareholders’ wealth would rise if C (100 + V , 150) > 26.0672 + 1 =⇒ V > 1.5806. When k = 10, shareholders wealth would rise if V > 14.9817. When k = 25, shareholders wealth would rise if V > 35.1288. These are solved numerically; you can use (and ) to approximate them algebraically. 116 Chapter 16 Corporate Applications Question 16.8. Using equation (16.12): a) (20/22)BSCall(5, 15, .3, .08, 5, 0) = .3209 per share which is .6418 in total. b) tal. (20/35)BSCall(5, 20, .3, .08, 10, 0) = .5601 per share which is 15 (.5601) = 8.4015 in to- Question 16.10. The debt is worth D = A − BSCall (100, 200, .30, .08, 10, 0) = 60.1035. The warrants are worth 20 100 28 BSCall 20 , 25, .30, .08, 10, 0 = .5191 per share which is 8 (.5191) = 4.1528 in total. Equity is the residual, E = 100 − 60.1035 − 4.1528 = 35.744 or 35.744/20 = 1.7872 per share. Question 16.12. A risk free bond would be 100e−.06(5) = 74.082. The 3 call options would be worth 3BSCall (22.278, 33.333, .30, .06, 5, 0) = 15.094 for a total of 74.082 + 15.094 = 89.176. This is (slightly) underestimates the value due to default risk. Question 16.14. Historical volatilities of MSFT are typically around 30% so their 10-K estimate is not bad if we use historical volatilities. Implied volatilities (the volatility that make Black Scholes agree with actual option values) are typically higher (approx. 40%). Overall, MSFT’s is fairly reasonable. One should ﬁnd LVLT’s historical and implied volatility in the ball park of 100%. Their 25% (used in their annual report) seriously underestimates volatility. a) 55.6929 b) The option can be modeled as holding two barrier (H = 60) options: a down-and-out (100 strike) and a down-and-in (60-strike). The value would be 48.4395 + 11.9491 = 60.389 which adds 8.43%. c) The value of the two options is maximized by setting H to approximately 68.5 which yields a value of 42.6285 + 18.0395 = 60.668. 117 Chapter 17 Real Options Question 17.2. If invest at time T you receive the (at time T ) an “NPV” .8 (1.02)T +1 .8 (1.02)T +2 − 1.5 + 1.05 1.052 = (1.02)T X − 1.5 NP VT = (1) (2) where X = .8 (1.02/1.05) + .8 (1.02/1.05)2 . This is growing at a decreasing rate; we can show this by looking at the growth rate gt = NP Vt +1 − NP Vt .02 = 15 NP Vt 1 − X(1..02)t (3) Notice g0 = .02/ (1 − 1.5/X) = .955 and as t gets very large gt approaches .02. The key insight is that if we invest in the machine, you will be receiving the NPV and this cash grows at 5% (the risk free rate). Therefore, it is not optimal for you to invest if the NPV is growing at a higher rate; i.e. if gt > 5% then you should not...
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## This document was uploaded on 03/11/2014 for the course FIN 402 at FPT University.

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