Capital Structure (Chapter 15)
Formulas:
D = market value of debt
S = market value of stock
V = value of firm = S + D
V = EBIT(1-T) / WACC
n0 = original # shares
P1 = (V1 – D0) / n0 = [S1 + (D1 – D0)] / n0
n 1 = S1 / P1
Subscripts: U stands for unleveraged firm, with no debt.
L stands for leveraged firm.
rSU = required return on equity of unlevered firm
WACCU = rSU
WACCL = (wd)(rd)(1-T) + (wCE)(rSL)
wd = D/V
wCE = S/V
Hamada’s Equations:
L = U [1 + (1-T)(D/S)]
U = L / [1 + (1-T)(D/S)]
Analyzing the impact of stock repurchase with debt financing.
(A “recapitalization.”)
Capital structure decisions should be made with the goal of maximizing the value
of the firm, or minimizing the cost of capital. Suppose the firm is considering
issuing debt to repurchase outstanding shares of stock. We would like to
estimate the impact on a variety of things, including firm value, earnings per
share, price per share, and the WACC. Subscripts of 0 refer to the initial
situation, and subscripts of 1 refer to new situation.
Example: Firm has no debt initially (D0 = 0), and has 100,000 shares of stock
outstanding (n0 = 100,000). EBIT = $1,000,000. T = .30. rSU = .10.
The firm has 100% dividend payout, and hence zero growth.
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Net profit and eps can be calculated using a pro forma income statement:
Pro Forma Income Statement:
Debt:
EBIT
Int Exp
EBT
Tax
NPAT
0
1,000
0
_____
Once we have net profit, we can calculate eps, among other things:
eps = NPAT / n0 = _________________
Because debt = 0, WACC = rSU = .10. Because g = 0, FCF = NOPAT.
V0 = (EBIT)(1–T) / WACC = (1,000)(.7) / .10 = __________________
S0 = V0 – D0 = $7 mil – 0 = $7 mil
P0 = S0 / n0 = $7 mil / 100K = ________________
We can also calculate P0 using eps (remember, D = eps and g is zero).
P0 = div / rS = eps / rS
= $7 / .10 = ___________________
Our firm is considering a proposal to increase the debt ratio to 30% by selling
bonds and using the funds to repurchase stock. The interest rate on the debt is
8%. The new rSL will be .105.
We will calculate the new WACC (WACC1), total value of the firm (V1), the new
stock price (P1), the new # of shares (n1), and eps.
WACC1 = wd (rd) (1 – T) + wCE(rSL)
= .3(.08)(.7) + .7(.105)
= ________________
V1 = EBIT(1 – T) / WACC1
= 1,000(.7) / .0903
= ______________
D1 = .3(7,751,938) = __________________
S1 = V1 – D1 = 7,751,938 – 2,325,581 = _______________
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P1 = (V1 – D0) / n0
= 7,751,938 / 100,000 = $______________
n 1 = S1 / P1
= 5,426,357 / 77.52
= ______________
When we go from 0 debt to 30% debt financing, the # of shares will drop by 30%.
We can now calculate eps, once we have NPAT using a pro forma income
statement:
Pro Forma Income Statement with 30% debt ratio:
EBIT
Int Exp*
EBT
Tax
NPAT
n1
eps
1,000,000
186,046
________
70,000
*Interest expense = 2,325,581 x .08 = 186,046
We can double-check the new stock price:
P1 = div / rSL = eps / rS
= $8.14 / .105 = $__________
Initially, debt is zero, and:
WACC0 = rSU = .10
V = $7 mil
P0 = $70
After the recapitalization:
WACCL = .0903
V1 = $7,751,938
P1 = $77.52
Should the firm accept the proposal?
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Calculating the change in beta.
In the example above, the effect of a recapitalization on the firm’s r S has been
given. However, we can calculate the effect of a change in the debt ratio on beta
and rS.
Example:
Debt ratio = 25%
T = 30%
rRF = 4% and RPM = 8%
rd = 9%
Initially:
rS = rRF + (RPM) = 4 + 1.0 (8) = 12%
WACC0 = .25 (.09)(.7) + .75(.12) = .1058
 = 1.0
Proposal: Increase debt ratio to 40% with new rd = 10%.
Step 1: Calculate what beta would be in the absence of leverage (U).
U = L / [1 + (1-T)(D/S)]
= 1.0 / [1 + .7(25/75)] = 1 / 1.233 = ______________
Step 2: Calculate the new beta with higher leverage:
L = U [1 + (1-T)(D/S)]
= .81 [1 + .7(40/60)] = _______________
Now, we can calculate the new rS and new WACC:
New rS = 4 + 1.19(8) = _________________
New WACC1 = .4(10)(.7) + .6(13.5)
= ________________
WACC1 > WACC0
Even though the interest rate on debt is lower than rS, increasing the debt ratio
causes WACC to go up!
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Additional Problems on Effect of Issuing Debt to Repurchase Stock
Example #1. EBIT = $10 mil
Tax rate = 40%
100% dividend payout
g=0
Initial situation:
D0 = $20 mil at 9% interest rate; the market value of debt = BV
rS = 12%
n0 = 1 mil
Calculate the initial values of the following:
Net Profit
eps
Stock price (P0)
Total value of stock (S0)
Market Value of firm (V0)
WACC0
Proposal: Increase debt ratio to 50%, retire old debt and repurchase stock.
Interest rate on new debt is 11%. New rS is 14%.
Calculate values of the following if the proposal is accepted:
WACC1
V1
D1
S1
P1
n1
NPAT
Eps
Initial values:
Net Profit $4,920,000
eps $4.92
Stock price (P0) $41
Market Value of firm (V0) $61mil
WACC = 9.84%
After Change:
WACC1 = 10.3%
V1 = 58.252K
P1 = 38.25
NPAT = $4,078K
D1 = 29,126K
n1 = 761,412 shares
eps = $5.36
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Example #2.
Initially:
EBIT = $100,000
Tax rate = 40%
100% dividend payout
g=0
Debt = $250,000 @ 8% interest rate
n0 = 10,000
rS = .11
Calculate the initial values of:
Net Profit
eps
Stock price (P0)
Total value of stock (S0)
Market Value of firm (V0)
WACC0
Proposal: Increase debt ratio to 60%. New interest rate is 12%. Retire old debt,
repurchase stock. New rS = .15
Calculate the following post-recapitalization values:
WACC1
V1
D1
S1
P1
n1
NPAT
eps
Answers:
Initial situation:
NPAT = 48,000
eps = $4.80
P0 $43.64
V0 $686,365
WACC0 = 8.74%
After Change:
WACC1 = 10.32%
V1 = 581.4K
D1 = 348.8K
P1 = $33.14
n1 = 7,019
eps = $4.97
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