Question

Gladstone Corporation is about to launch a new product. Depending on the success of the new product, Gladstone may have one of four values next year: $150 million, $135 million, $95 million, and $80 million. These outcomes are all equally likely, and this risk is diversifiable. Gladstone will not make any payouts to investors during next year. Suppose the risk-free interest rate is 5%. Assume there are no taxes. a. What is the initial value of Gladstone's equity without leverage? Now assume Gladstone has zero-coupon debt with a $100 million face value due next year. b. What is the initial value of Gladstone's debt? c. What is the yield-to-maturity of the debt? What is the expected return? d. What is the initial value of Gladstone's equity? What is Gladstone's total value with leverage?

Answer 1

The initial value of Gladstone's equity without leverage is $109.52 million. When leverage is introduced, the initial value of the debt is $95.24 million, the yield-to-maturity and expected return on the debt are both 5%, and the value of equity with leverage is $14.28 million. The total value of the firm with leverage remains at $109.52 million.

Step-by-step explanation:

To calculate the initial value of Gladstone's equity without leverage, we need to consider the expected value of the corporation next year. With outcomes of $150 million, $135 million, $95 million, and $80 million, each with an equal probability of occurring, the expected value is the average of these figures. Therefore, the expected value of the equity is ($150m + $135m + $95m + $80m) / 4 = $115 million. To determine the present value, we discount it at the risk-free rate of 5%, which gives us $115m / (1.05) = $109.52 million.

When Gladstone introduces leverage with zero-coupon debt of $100 million due next year, the initial value of the debt depends on the likelihood of default. Since the lowest value of the firm is $80 million, which is below the face value of the debt, there will be a default in one of the scenarios. However, since the question states that the risk is diversifiable and does not impact the debt's value, we will assume that the risk-free rate applies to the debt as well. Therefore, the value of the debt is discounted at the risk-free rate of 5%, resulting in a value of $100m / (1.05) = $95.24 million.

The yield-to-maturity (YTM) of the debt is the rate at which the present market price of the bond would make the investor indifferent between buying the bond and holding it to maturity versus investing in a risk-free bond. Because we have used a risk-free rate to value the debt, the YTM is 5%. The expected return is the YTM, and given no default, it is also 5%.

The value of Gladstone's equity with leverage is the expected value of the firm minus the value of the debt. Hence, the initial value of the equity is $109.52 million - $95.24 million = $14.28 million. The total value of Gladstone with leverage is the sum of the value of debt and the value of equity, i.e., $95.24 million (debt) + $14.28 million (equity) = $109.52 million, which interestingly is the same as the value without leverage.