Question

Liu Industries is a highly levered firm. Suppose there is a large probability that Liu
will default on its debt. The value of Liu’s operations is $4 million. The firm’s debt
consists of 1-year, zero coupon bonds with a face value of $2 million. Liu’s volatility,
σ, is 0.60, and the risk-free rate rRF is 6%.
Because Liu’s debt is risky, its equity is like a call option and can be valued
with the Black-Scholes Option Pricing Model (OPM). (See Chapter 8 for details
of the OPM.)
(1) What are the values of Liu’s stock and debt? What is the yield on the debt?
(2) What are the values of Liu’s stock and debt for volatilities of 0.40 and 0.80?
What are yields on the debt?
(3) What incentives might the manager of Liu have if she understands the relationship
between equity value and volatility? What might debtholders do in response?

Answer 1

As the probability of default on its debt is large for Liu Industries, their current stake in equity of business is equivalent to holding a call option with the exercise price same as company's debt.

Spot value of the asset (S_t) = current value of business = $ 4,000,000

Exercise Price (K) = Company's Debt = $ 2,000,000

Volatility (σ) = 0.6

Time to expirty = 1 year

Risk Free Rate of Interest = 6%

1) Value of call option as per Black-Scholes Model is C = S_t​*N(d₁​)−Ke^−rtN(d₂​)

where d₁ = (ln(S_t​/K) + (r+σ²/2)*t)/σ√t ; d₂ = d₁ - σ√t

Substituting the values, d₁ = 1.555; d₂ = 0.955. Therefore, N(d₁​) = 0.94 ; N(d₂​) = 0.83 as taken from Normal Distribution Table

C = 4,000,000*0.94-2,000,000*e^−.06*0.83 = 2,196,381

Therefore, Equity Value = 2,196,381. Debt Value = Value of firm - Equity = 4,000,000 - 2,196,381 = 1,803,619.

Debt Yield = (Face Value of Bond/Value of Debt) - 1 = (2,000,000/1,803,619) - 1 = 10.89%

2) a) Considering σ = 0.4 & everything else remains the same,

Value of call option as per Black-Scholes Model is C = S_t​*N(d₁​)−Ke^−rtN(d₂​)

where d₁ = (ln(S_t​/K) + (r+σ²/2)*t)/σ√t ; d₂ = d₁ - σ√t

Substituting the values, d₁ = 2.083; d₂ = 1.683. Therefore, N(d₁​) = 0.98 ; N(d₂​) = 0.95 as taken from Normal Distribution Table

C = 4,000,000*0.98-2,000,000*e^−.06*0.95 = 2,128,964

Therefore, Equity Value = 2,128,964. Debt Value = Value of firm - Equity = 4,000,000 - 2,128,964 = 1,871,036.

Debt Yield = (Face Value of Bond/Value of Debt) - 1 = (2,000,000/1,871,036) - 1 = 6.89%

b) Considering σ = 0.8 & everything else remains the same,

Value of call option as per Black-Scholes Model is C = S_t​*N(d₁​)−Ke^−rtN(d₂​)

where d₁ = (ln(S_t​/K) + (r+σ²/2)*t)/σ√t ; d₂ = d₁ - σ√t

Substituting the values, d₁ = 1.341; d₂ = 0.541. Therefore, N(d₁​) = 0.91 ; N(d₂​) = 0.71 as taken from Normal Distribution Table

C = 4,000,000*0.91-2,000,000*e^−.06*0.71 = 2,310,865.65

Therefore, Equity Value = 2,310,865.65. Debt Value = Value of firm - Equity = 4,000,000 - 2,310,865.65 = 1,689,134.35

Debt Yield = (Face Value of Bond/Value of Debt) - 1 = (2,000,000/1,689,134.35) - 1 = 18.4%

3) As it can be seen from the workings in (2), as the volatility increases, equity value of the company increases. Hence, as the company takes more risks, the value of the equity increases. So, Liu will have incentives to take up more risky projects. Understanding the same, debtholders might place covenants in such a way that covenants restrain this risk seeking behavior of Liu.