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A public company with no debt is contemplating issuing its first corporate bonds
A public company with no debt is contemplating issuing its first corporate bonds. The company generates quite a bit of revenue and could benefit from expensing interest debt to shield some of its revenue from taxes. It estimates that the value of the tax shields created by the debt issuance would be proportional to the ratio of debt to assets, x, and would be given by the following formula: D(x) = 0.2.x However, the company also knows that debt comes with certain unique costs, such as lost opportunities due to maxed out debt capacity. It estimates that these costs are generally small for small amounts of debt, but increase fast as the amount of debt increases. It believes that the following quadratic function would capture fairly well these costs: V(x) 0.2•x? + 0.1 •x The company also knows that the ratio of debt to assets cannot be larger than 1 or smaller than 0. We now want to determine the optimal amount of debt that the company should issue. Q1: Express the net benefit to the firm (benefit minus cost) of issuing debt as a function of x. Net benefit equals Q2: Find the debt-to-assets ratio x that maximizes the firm's net benefit function. The optimal debt-to-asset ratio is x =
Expert Solution
Given Information
x = Debt/Assets
Value of tax shield by debt issuance is given by D(x) = 0.2 * x
Costs of debt is given by V(x) = 0.2 * x2 + 0.1 * x
1. Net Benefit = Value of tax shield – Costs of debt = D(x) – V(x)
=> Net Benefit = 0.2 *x – (0.2 * x2 + 0.1 * x) = -0.2 * x2 + 0.1 * x
2. Calculation of debt-to-assets that maximizes the net benefit
Net Benefit = -0.2 * x2 + 0.1 * x
On comparing the equation with general quadratic equation ax2 + bx + c
We have, a=-0.2 b=0.1 c=0
Here, a (coefficient of x2) is negative, so equation would give a maximum and the maximum value is given by –
Maximum Net Benefit = c – (b2/4a) = 0 – (0.12/4*(-0.2)) = 0.0125
Putting it in the equation to find x, we have
0.0125 = -0.2 * x2 + 0.1 * x
Solving the equation*, we get x = ¼ or 0.25 (Optimal debt-to-asset ratio)
* For a quadratic equation, ax2 + bx + c
Solution is given by x =[ -b +/- (b2 – 4ac)0.5] / 2a
Here, we have x = [ -0.1 +/- (0.12 – 4(-0.2)(-0.0125)0.5] / 2*(-0.2) = ¼ or 0.25
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