17. 8/13/2006
Chapter Solution to Ch 17 P11 Build a Model
A. Fethe is a custom manufacturer of guitars, mandolins and other stringed instruments located near Knoxville, TN. Fethe's current value of operations, which is also its value of debt plus equity, is estimated to be $5 million. Fethe has $2 million face-value zero-coupon debt that is due in 2 years. The risk free rate is 6 percent, and the volatility of companies similar to Fethe is 50 percent. Fethe's owners view their equity investment as an option and would like to know the value of their investment. a. Using the Black-Scholes Option Pricing Model, how much is Fethe's equity worth? Black-Scholes Option Pricing Model Total Value of Firm Face Value of Debt Risk Free rate Maturity of debt (years) Standard Dev. d1 d2 N(d1) N(d2) Call Price = Equity Value
5.00 this is the current value of operations 2.00 0.06 2.00 0.50 this is sigma--also known as volatility 1.8191 use the formula from the text 1.1120 use the formula from the text 0.9656 use the Normsdist function in the function wizard 0.8669 $3.2900
b. How much is the debt worth today? What is its yield? Debt value = Total Value - Equity Value = Debt yield = $1.71 8.146%
c. How much would the equity value and the yield on the debt change if Fethe's management were able to use risk management techniques to reduce its volatility to 30 percent? Can you explain this? Equity value at 50% volatility Equity value at 30% volatility Percent change 3.29000 3.22910 -0.01851
a. Graph the cost of debt versus the face value of debt for values of the face value from $0.5 to $8 million. Cost of Debt Face Value of Debt 8.146% hint: use a data table 0.5 6.19% 9.00% 1 6.38% 8.00% 1.5 7.02%
Yield on Debt
2 2.5 3 3.5 4 4.5
8.15% 9.71% 11.62% 13.80% 16.18% 18.70%
7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 1 0.00% 3 of 0 1 2 3 4 5 Face Value of Debt 6 7 8 9
7.00% Yield on Debt 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% 0 1 2 3 4 5 Face Value of Debt 6 7 8 9
5 5.5 6 6.5 7 7.5 8
21.34% 24.06% 26.84% 29.66% 32.50% 35.37% 38.23%
b. Graph the values of debt and equity for volatilities from 0.10 to 0.90 when the face value of the debt is $2 million. Value of Debt Value of Equity Volatility Face Value of Debt Volatility Face Value of Debt $1.71 2 $3.29 2 0.1 1.7738 0.1 3.2262 Note: Use a 2-dimensional data t 0.2 1.7738 0.2 3.2262 can change the face value of debt 0.3 1.7709 0.3 3.2291 the base 'face value of debt for al 0.4 1.7523 0.4 3.2477 that gives the value of equity or d 0.5 1.7100 0.5 3.2900 "Volatility" and the column inpu 0.6 1.6462 0.6 3.3538 C18. Note that this looks differen 0.7 1.5663 0.7 3.4337 data table in part a. 0.8 1.4759 0.8 3.5241 0.9 1.3793 0.9 3.6207
Equity and Debt Values
3.5000 3.0000 2.5000 Value 2.0000 1.5000 1.0000 0.5000 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Volatility (sigma) Value of Debt Value of Equity
c. Repeat part b, but instead using a face value of debt of $5 million. What can you say about the difference between the graphs in part b and part c? Value of Debt Value of Equity Volatility Face Value of Debt Volatility Face Value of Debt $1.71 5 $3.29 5 0.1 4.3613 0.1 0.6387 See the note above for part b tha 0.2 4.1401 0.2 0.8599 0.3 3.8938 0.3 1.1062 0.4 3.6439 0.4 1.3561 0.5 3.3959 0.5 1.6041 0.6 3.1525 0.6 1.8475
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0.7 0.8 0.9
2.9156 2.6863 2.4656
0.7 0.8 0.9
2.0844 2.3137 2.5344
3.5000 3.0000 2.5000
Value
2.0000 1.5000 1.0000 0.5000 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 Volatility (sigma) 0.7 0.8 0.9 1 Value of Debt Value of Equity
The value of debt and equity change much more dramatically as the volatility changes when there is more debt.
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