Suppose a zero-coupon bond is selling for $614.00 today. It promises to pay $1,000 in exactly 10 years with annual compounding. What is the firm’s after-tax cost of debt if this is its sole debt outstanding (assuming the firm is in the 20% tax bracket)?

To calculate the firm's after-tax cost of debt, we need to consider the discount rate and the tax bracket. Here's how to do it:

1. First, let's calculate the discount rate. We know that the zero-coupon bond is selling for $614 today and promises to pay $1,000 in exactly 10 years. This represents the future value (FV) and present value (PV) of the bond, respectively.

Using the future value formula, we can calculate the discount rate (r):

FV = PV * (1 + r)^n

Where:
FV = $1,000
PV = $614
n = number of periods = 10

By rearranging the formula, we get:

r = (FV / PV)^(1/n) - 1

Plugging in the values, we have:

r = ($1,000 / $614)^(1/10) - 1

Calculating this, we find that the discount rate is approximately 4.48%.

2. Next, let's calculate the after-tax cost of debt. Since the firm is in the 20% tax bracket, we can assume the tax rate (Tc) is 20%. The after-tax cost of debt (rc) is given by:

rc = r * (1 - Tc)

Plugging in the values, we get:

rc = 4.48% * (1 - 20%)

Calculating this, we find that the firm's after-tax cost of debt is approximately 3.58%.