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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 10K 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 downandout (100
strike) and a downandin (60strike). 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.
 Spring '14
 NguyenThiMaiLan
 Derivatives

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