# Capital Structure Question Solution

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• Published : November 3, 2011

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FINE 3100
Problems for Midterm – Additional Capital Structure Problems

Question 1 Belgarion Enterprises

Asset beta, the riskiness of the firm, can be found as the weighted average of the betas of its debt and equity, where the weights are fraction of the firm financed by debt and equity:

β A = D/V βD + E/V βE = .5 × 0 + .5 × 1.4 = .7

To find the beta of the firm with no debt, find βo or βu using the formula for levered equity: βE,L = βo + [βo – βD] D/E ( 1 – TC)

Rearrange to find βo = βE,L + βD D/E ( 1 – TC)
1 + D/E ( 1 – TC)

Since the debt beta is zero, the equation simplifies to:
βo = βE,L = 1.4 / ( 1 + (.5/.5) × (1 - .4) ) = .875 1 + D/E ( 1 – TC)

The asset beta is higher if the firm has NO DEBT, in the otherwise perfect financial markets world. The firm with debt has an asset that the firm no debt does not: the interest tax shield. The riskiness of the tax shield is lower than the riskiness of the firm’s operating assets (its business risk). In fact, in this case, the interest tax shield is riskless because the debt is riskless. The beta of the levered firm’s assets is lower than beta of the unlevered firm’s assets. Remember, bankruptcy is costless in this problem. (If bankruptcy is not costless, the result may not hold – by increasing leverage, the probability of bankruptcy goes up and therefore the expected costs of bankruptcy increase. In this case, the firm’s riskiness may well increase with leverage).

Question 2 Little Industries

a) Current market values
EL = 300,000 × \$3 = \$900,000
Value per bond: (.05 × 1000)/.1 = 50/.1 = \$500
Total bonds: D= (.05×100,000)/.1 = \$50,000

VL = D + EL = 50,000 + 900,000 = \$950,000

b) Current required rates of return
Debt: rD = 10 % (given)
Equity: rE,L = (EBIT – I) × (1 - TC) = (270,000 – 5,000) × (1 - .4) = .1766666 = 17.7% EL 900,000

WACC = (D/VL) × rD × (1-TC) + (EL/VL) × rE.L
= (50,000/950,000) × .1 × (1-.4) + 900,000/950,000 × .177 = .1708

c) For case of perpetual debt:
VL = Vu + Tc D
Therefore: Vu = VL - Tc D = 950,000 - .4 × 50,000 = 950,000 – 20,000 = 930,000

NOTE: another way to solve for the unlevered firm value is to first calculate the unlevered cost of equity and then use it to discount the unlevered firm’s cash flows 1. Unlevered cost of equity

Recall: rE.L = r0 + (r0 – rD) D/E (1 – Tc)
Rearrange the formula for r0:
r0 = [rE,L + rD D/E (1 – Tc) ]/ [1 + D/E (1 – Tc)]
= (.177 + .1×50,000/900,000 ×.6)/(1+50,000/900,000×.6) = .1741935

VU = EU = EBIT × (1- TC)/r0  = 270,000 × .6/.1741935 = 930,000

d)

(i) After restructuring, the firm will be 30% debt financed. Let D* be the total debt after refinancing and VL* be the total firm value after refinancing. It must be true that:

D* = .3 × VL*

Since VL = Vu + Tc D, then VL* = Vu + Tc D*
Substituting for D*

VL* = Vu + Tc .3 × VL*
Solve for VL*

(1 - .3×TC) VL* = Vu
VL* = Vu/ (1 - .3×TC) = 930,000/ ( 1 - .3 ×.4) = 1,056,818.2

And D* = . 3 × VL* = .3 × 1,056,818.2 = 317,045.5
EL* = . 7 × VL* = .7 × 1,056,818.2 = 739,772.7

(ii) By issuing new debt and retiring equivalent value of equity, total firm value increases

VOLD = 950,000
VNEW = 1,056,818.2

Increase in firm value = 1,056,818.2 - 950,000 = 106,818.2

Since the required rate of return to debt is unchanged, we can assume that all of the benefit of the restructuring is captured by the shareholders. On the announcement of the proposed restructuring, the total value of equity will increase by the increase in firm value:

Value of existing equity on the announcement = 900,000 + 106,818.2 = 1,006,818.2 New share price = 1,006,818.2/300,000 = \$3.356

To figure out the number of shares repurchased, first figure out the dollar value of the new debt issued:...