M13_MCDO8122_01_ISM_C13

# M13_MCDO8122_01_ISM_C13 - Chapter 13 Corporate Applications...

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Chapter 13 Corporate Applications n Question 13.1 One could first value equity ( E ) as a call option and value the debt by subtracting equity from the asset value (i.e., B = A - E ). We chose the “insurance” approach. We start with valuing default-free debt which is equal to 0 08 120 . T e - . × Insurance is the put option value: Insurance BSPut(100 120 0 30 0 08 0) T = , , . , . , , The debt value, denote it as 0 , T B , is the difference: 0 08 120 Insurance. T e - . × - The yield for a T maturity bond is 0 ln(120/ )/ . T T B T ρ , = Equity is the difference 0 0 100 . T T A B B , , - = - Doing this for each maturity, we arrive at: Maturity ( T ) 1 2 5 10 Default-Free Bond 110.7740 102.2573 80.4384 53.9195 Insurance 18.6705 18.1410 15.1037 10.1571 Debt 0 ( ) T B , 92.1034 84.1163 65.3347 43.7623 Yield ( ) T 0.2646 0.1776 0.1216 0.1009 Equity 7.8966 15.8837 34.6653 56.2377 Debt-to-Equity 11.6637 5.2957 1.8847 0.7782 n Question 13.2 Let B be the maturity value of the debt. We start with valuing default-free debt which is equal to 0 08 120 . T e - . × Insurance is the put option value: Insurance BSPut(100 0 30 0 08 0) B T = , , . , . , , The debt value, denote it as 0 , T B , is the difference: 0 08 Insurance. T Be - . × - The yield for a T maturity bond is 0 ln( / )/ . T T B B T , = Equity is the difference 0 0 100 . T T A B B , , - = - Doing this for each maturity, we arrive at: Maturity ( T ) 1 2 5 10 Face Value B 127.42 135.3 161.98 218.65 Default-Free Bond 117.6235 115.2951 108.5784 98.2458 Insurance 23.6211 26.7229 31.8825 35.2832 Debt 0 ( ) T B , 94.0023 88.5722 76.6959 62.9626 Yield ( ) T 0.3042 0.2118 0.1495 0.1245 Equity 5.9977 11.4278 23.3041 37.0374 Debt-to-Equity 15.6732 7.7506 3.2911 1.7000

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153 McDonald • Fundamentals of Derivatives Markets n Question 13.3 1. Let C ( K ) denote the Black-Scholes call option value with strike K with the other parameters understood (e.g., the underlying asset value is 100). The value of senior debt is 100 - C (30), intermediate A debt value (second in line) is C (30) - C (60), intermediate B debt value (third in line) is C (60) - C (90), and junior debt (last in line) value is C (90) - C (120). We find the following debt values and yields: Maturity 1 2 5 10 Values Senior 27.6935 25.5606 20.0293 13.2808 Intermediate A 27.4896 24.8431 18.5451 11.9417 Intermediate B 23.4449 20.3509 15.1765 10.1086 Junior 13.4754 13.3616 11.5838 8.4313 Yields Senior 8.00% 8.01% 8.08% 8.15% Intermediate A 8.74% 9.43% 9.62% 9.21% Intermediate B 24.65% 19.40% 13.63% 10.88% Junior 80.03% 40.44% 19.03% 12.69% 2. Varying volatility and interest rates with 5 years to maturity, we have: Volatility ( σ ) 0.1 0.2 0.3 0.4 0.5 Values Senior 20.1096 20.1088 20.0293 19.5700 18.5858 Intermediate A 20.1095 19.8965 18.5451 16.3672 14.0584 Intermediate B 20.0386 18.1148 15.1765 12.6273 10.5498 Junior 18.4306 14.3211 11.5838 9.6497 8.1386 Yields Senior 8.00% 8.00% 8.08% 8.54% 9.58% Intermediate A 8.00% 8.21% 9.62% 12.12% 15.16% Intermediate B 8.07% 10.09% 13.63% 17.31% 20.90% Junior 9.74% 14.79% 19.03% 22.69% 26.09% Risk-Free Rate ( r ) 0.02 0.05 0.08 0.11 0.14 Values Senior 26.8154 23.1979 20.0293 17.2713 14.8811 Intermediate A 22.7715 20.6975 18.5451 16.4261 14.4196 Intermediate B 16.2383 15.8984 15.1765 14.1670 12.9723 Junior 10.8714 11.4009 11.5838 11.4312 10.9866 Yields Senior 2.24% 5.14% 8.08% 11.04% 14.02% Intermediate A 5.51% 7.42% 9.62% 12.05% 14.65% Intermediate B 12.28% 12.70% 13.63% 15.01% 16.77% Junior 20.30% 19.35% 19.03% 19.30% 20.09%
Chapter 13 Corporate Applications 154 The higher the volatility on assets and risk-free rate, the higher the yield on the debt issues. The only exception is that lower interest rates could raise the yield on the most junior debt. We also find at low volatility and high interest rates, the yields are close to each other. High volatility and low interest rates lead to great variation in the yields.

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M13_MCDO8122_01_ISM_C13 - Chapter 13 Corporate Applications...

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