Question

Problem 15-12 Equity Viewed as an Option A. Fethe Inc. is a custom manufacturer of guitars, mandolins, and other stringed instruments and is located near Knoxville, Tennessee. 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%, and the standard deviation of returns for companies similar to Fethe is 50%. Fethe's owners view their equity investment as an option and they would like to know the value of their investment. Using the Black-Scholes option pricing model, how much is Fethe's equity worth? In your calculations round normal distribution values to 4 decimal places. Round your answer to two decimal places. $ million How much is the debt worth today? What is its yield? Round your answer for the debt worth to two decimal places. Round your answer for the yield on the debt to one decimal place. Debt worth today: $ million Yield on the debt: % How would the equity value and the yield on the debt change if Fethe's managers could use risk management techniques to reduce its volatility to 30%? In your calculations round normal distribution values to 4 decimal places. Round your answer for the equity worth to two decimal places. Round your answer for the yield on the debt to one decimal place. New equity worth: $ million New yield on the debt: % Can you explain this? The value of the stock goes down and the value of the debt goes up because with lower risk, Fethe has of a chance of a "home run".

Answer 1

Using Black Scholes equation: C = S_t * N(d₁) - K * e^(-2*0.06) * N(d₂)

value of share, S_t = $5 million ( Total assets)

value of strike price, K = $2 million (Value of debt)

risk free rate, r = 6%; volatility, v = 50%; term, t = 2 years

d₁ = (log(S_t/k) + (r+(v^2/2))(t))/(v*t^(1/2)) = 1.8191 ; N(d₁) = 0.9656

d₂ = d₁ - v*t^(1/2) = 1.1120 ; N(d₂) = 0.8669

C = 5000000 * 0.9656 - 2000000 * e^(-0.12) * 0.8669

= 3289965

Therefore, the value of equity = $3289965

value of debt today = 5000000-3289965 = $1710035

Yield : 1710035 =2000000/(1+y)^2

y = (2000000/1710035)^(1/2) - 1 = 0.0814 or 8.14%

If the value of volatility is changed to 30%,

d₁ = (log(S_t/k) + (r+(v^2/2))(t))/(v*t^(1/2)) = 2.6547 ; N(d₁) = 0.9960

d₂ = d₁ - v*t^(1/2) = 2.2304 ; N(d₂) = 0.9871

C = 5000000 * 0.9960 - 2000000 * e^(-0.12) * 0.9871

= 3229125

Therefore, the value of equity = $3229125

value of debt today = 5000000-3229125 = $1770875

Yield : 1770875 =2000000/(1+y)^2

y = (2000000/1710035)^(1/2) - 1 = 0.0627 or 6.27%