Question

North Technology Inc. has a zero-coupon bond that matures in five years with a face value of $60,000. The current value of the company’s asset is $57,000 and the standard deviation of rate of return on assets is 50% per year. The continuously compounded risk-free rate of interest is 6%.

1. What are the market values of the company’s debt and equity?
2. What is the yield to maturity on North Technology’s debt?
3. What is the value of shareholders’ limited liability?
4. Suppose the company can re-structure its balance sheet so that the standard deviation of its return on assets increases to 60% per year. Assuming all other things remaining the same, who (shareholders vs. bondholders) benefits from the restructuring and why? What are the magnitudes of loss/gain to each type of stakeholders? Show your calculations.

Answer 1

(a) In this context, the equity of the firm can be seen as a call option on the underlying asset as the rationale given here: -

• The equity in a firm is a residual claim, i.e., equity holders lay claim to all cashflows left over after other financial claim-holders (debt, preferred stock etc.) have been satisfied.
• If a firm is liquidated, the same principle applies, with equity investors receiving whatever is leftover in the firm after all outstanding debts and other financial claims are paid off.
• The principle of limited liability protects equity investors in publicly traded firms if the value of the firm is less than the value of the outstanding debt, and they cannot lose more than their investment in the firm.

The payoff to equity on liquidation can be written as: -

Using Black-Scholes formulae (used for valuing call option) to value equity where

• the value of underlying asset S is the value of total assets $57,000,
• the exercise price K is the value of outstanding debt $60,000,
• the life of the option T is the life of zero-coupon bond 5 years,
• standard deviation in the value of the underlying asset is 50%, and
• riskless rate r is continuously compounded risk-free interest 6%

Using the formulae,

Calculating,

Putting in the values,

On solving, the value of C (in this case, the value of equity E): -

The market value of debt is the market value of current assets minus value of equity. In this case, the market value of debt is $(57,000 - 28,248.836) = $28,751.164

(b) Calculating the yield to maturity (in case it is R): -

Hence, the yield to maturity is 15.85%.

(c) The principle of limited liability protects equity investors in publicly traded firms if the value of the firm is less than the value of the outstanding debt, and they cannot lose more than their investment in the firm. The value of the limited liability is their investment in the firm and the total equity value at the beginning at t = 0.

(d) In this case, we do the calculations again. Using the formulae,

Calculating,

Putting in the values,

On solving, the value of C (in this case, the value of equity E): -

The market value of debt is the market value of current assets minus value of equity. In this case, the market value of debt is $(57,000 - 31,888.342) = $25,111.658

Clearly equity holders benefit from the restructuring. In case the volatility of assets increase, the value of call option increase because the likelihood of value of assets going over the face value of the bond increases. In this context, the value of equity increases, because when the volatility of assets increase they are more likely to go over the value of outstanding debt and have payoffs for equity holders. The magnitude of gain (for shareholders) and loss (for bondholders) is same that can be obtained by equity value when asset volatility is 60% minus equity value when asset volatility is 50%. The value is $(31,388.342 - 28,248.836) = $3,139.506.