Question

a. Assume that Firms U and L are in the same risk class, and that both have EBIT = $500,000. Firm U uses no debt financing, and its cost of equity is rsU = 14%. Firm L has $1 million of debt outstanding at a cost of rd = 8%. There are no taxes. Assume that the MM assumptions hold, and then:

(1) Find V, S, rs, and WACC for Firms U and L.

(2) Graph (a) the relationships between capital costs and leverage as measured by D/V, and (b) the relationship between value and D.

b. Using the data given in Part b, but now assuming that Firms L and U are both subject to a 40 percent corporate tax rate, repeat the analysis called for in b(1) and b(2) under the MM with-tax model.

c. Suppose that Firms U and L are growing at a constant rate of 7% and that the investment in net operating assets required to support this growth is 10% of EBIT. Use the compressed adjusted present value (APV) model to estimate the value of U and L. Also estimate the levered cost of equity and the weighted average cost of capital.

 

d. Suppose the expected free cash flow
 for Year 1 is $250,000 but it is expected to grow faster than 7% during the next 3 years: FCF2 = $290,000 and FCF3 = $320,000, after which it will grow at a constant rate of 7%. The expected interest expense at Year 1 is $80,000, but it is expected to grow over the next couple of years before the capital structure becomes constant: Interest expense at Year 2 will be $95,000, at Year 3 it will be $120,000 and it will grow at 7% thereafter. What is the estimated horizon unlevered value of operations (i.e., the value at Year 3 immediately after the FCF at Year 3)? What is the current unlevered value of operations? What is the horizon value of the tax shield at Year 3? What is the current value of the tax shield? What is the current total value? The tax rate and unlevered cost of equity remain at 40% and 14%, respectively.

e. Suppose there is a large probability that L will default on its debt. For the purpose of this example, assume that the value of L's operations is $4 million (the value of its debt plus equity). Assume also that its debt consists of 1-year, zero coupon bonds with a face value of $2 million. Finally, assume that L's volatility, σ is 0.60 and that the risk-free rate rRF is 6%.

f. What is the value of L's stock for volatilities between 0.20 and 0.95?

David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm's level of debt financing. The company uses short-term debt to finance its temporary working capital needs, but it does not use any permanent (long-term) debt. Other solar technology companies average about 30% debt, and Mr. Lyons wonders why they use so much more debt and how it affects stock prices. To gain some insights into the matter, he poses the following questions to you, his recently hired assistant.

 

Answer 1

a. (1)
Proposition I.
1. The weighted average cost of capital is independent of the firm's capital structure.
2. The WACC of a firm with debt is equal to the unlevered cost of equity.
Proposition II.
The cost of equity, rsL = rsU + Risk premium = rsU + (rsU -rd)(D/S)

 

MM without Taxes
D	V	S	D/V	rd	rs	WACC
0.0	$3.50	$3.50	0.00%	8.00%	14.00%	14.00%
0.5	$3.50	$3.00	14.29%	8.00%	15.00%	14.00%
1.0	$3.50	$2.50	28.57%	8.00%	16.40%	14.00%
1.5	$3.50	$2.00	42.86%	8.00%	18.50%	14.00%
2.0	$3.50	$1.50	57.14%	8.00%	22.00%	14.00%

a. (2)

 

b.
Modigliani and Miller with CorporateTaxes
The MM results are different once corporate taxes are added in.
Proposition I ( with corporate taxes)
Value of levered firm is the unlevered value plus the debt tax shield: VL = VU + TD
Proposition II (with corporate taxes)
The cost of equity to a levered firm is the unlevered cost of equity plus a risk premium:
rsL = rsU + (rsU - rd)(1-T)(D/S)

 

Effects of Leverage: MM Models
MM with Corporate Taxes
Tc =	40.00%
D	V	S	D/V	rd	rd x (1-T)	rs	WACC
0.0	$2.14	$2.14	0.00%	8.00%	4.80%	14.00%	14.00%
0.5	$2.34	$1.84	21.37%	8.00%	4.80%	14.98%	12.80%
1.0	$2.54	$1.54	39.37%	8.00%	4.80%	16.34%	11.80%
1.5	$2.74	$1.24	54.74%	8.00%	4.80%	18.35%	10.93%
2.0	$2.94	$0.94	68.03%	8.00%	4.80%	21.66%	10.19%
2.5	$3.14	$0.64	79.62%	8.00%	4.80%	28.06%	9.54%
3.0	$3.34	$0.34	89.82%	8.00%	4.80%	45.76%	8.97%
3.5	$3.54	$0.04	98.87%	8.00%	4.80%	329.00%	8.46%

c.
Relevant information from part c.
EBIT		500,000
Tax rate		40%
Unlevered cost of equity		14%	= WACC if there is no debt
Cost of debt		8%
Additional information
Required reinvestment		50,000
growth rate		7%
FCF Calculation
NOPAT	=	EBIT	x	(1-T)
NOPAT	=	500,000	x	60%
NOPAT	=	300,000
FCF--7% growth		=	NOPAT	-	Required net reinvestment at 7% growth
FCF--7% growth		=	300,000	-	50,000
FCF--7% growth		=	250,000
FCF -- 0% growth		=	NOPAT
FCF -- 0% growth		=	300,000

APV with growth: rTS = rsU.			growth =	7.00%
T =	40.00%
D	V	S	D/V	Tax shield	rsL	WACC
	$4,028,571	$3,028,571	24.823%	$457,143	15.981%	13.206%
-	3,571,429	3,571,429	0.00%	-	14.00%	14.00%
500,000	3,800,000	3,300,000	13.16%	228,571	14.91%	13.58%
1,000,000	4,028,571	3,028,571	24.82%	457,143	15.98%	13.21%
1,500,000	4,257,143	2,757,143	35.23%	685,714	17.26%	12.87%
2,000,000	4,485,714	2,485,714	44.59%	914,286	18.83%	12.57%
2,500,000	4,714,286	2,214,286	53.03%	1,142,857	20.77%	12.30%
3,000,000	4,942,857	1,942,857	60.69%	1,371,429	23.26%	12.06%
3,500,000	5,171,429	1,671,429	67.68%	1,600,000	26.56%	11.83%

1.

The gain from the tax shield will be lower using the APV model than under MM because the APV model discounts the interest tax shield at the unlevered cost of equity, which is  larger than the cost of debt. The MM model discounts the tax shield at the cost of debt.

2.

The gain from debt is larger with growth than without growth.

3.

The value of the firm, whether levered or not, will be larger with growth, provided ROIC is greater than WACC. Although we don't show it here, ROIC is greater than WACC, so the value of the firm increases with growth.

The increase in the firm's value as a result of $1,000,000 in debt over its unlevered value is:
Increase in value =		12.80%	of the unlevered value
MM versus APV rsL and WACC
rd	8.0%
rsU	14.00%
Tax Rate	40%
D/V	50%
D/S	1
MM rsL	17.60%
MM WACC	11.20%
APV rsL	20.00%
APV WACC	12.40%
	MM rsL	MM WACC	APV rsL	APV WACC
D/V	17.60%	11.20%	20.00%	12.40%
0%	14.00%	14.00%	14.00%	14.00%
10%	14.40%	13.44%	14.67%	13.68%
20%	14.90%	12.88%	15.50%	13.36%
30%	15.54%	12.32%	16.57%	13.04%
40%	16.40%	11.76%	18.00%	12.72%
50%	17.60%	11.20%	20.00%	12.40%
60%	19.40%	10.64%	23.00%	12.08%
70%	22.40%	10.08%	28.00%	11.76%
80%	28.40%	9.52%	38.00%	11.44%

 

d.
Inputs:			1	2	3
Tax Rate =	40%	Free Cash Flow	$250.00	$290.00	$320.00
rsU =	14.00%	Interest expense	$80.00	$95.00	$120.00
gL =	7%
Estimate the unlevered value of operations
Unlevered Horizon Value =		(Free Cash Flow)(1+g)
		rsU - g
	Unlevered Horizon Value =	$4,891.43	thousand
			1	2	3
Free Cash Flow			$250.00	$290.00	$320.00
Unlevered Horizon Value					$4,891.43
Total			$250.00	$290.00	$5,211.43
Unlevered Value = PV at rsU =		$3,960.01	thousand
Estimate the value of the tax shield
Tax Shield Horizon Value =		( Tax Shield)(1+g)
		rsU - g
TS. Horizon Value =		$733.71	thousand
			1	2	3
Interest tax shield			$32.00	$38.00	$48.00
Tax shield horizon value					$733.71
Total			$32.00	$38.00	$781.71
Tax Shield Value = PV at rsU =		$584.94	thousand
Estimate the total value of operations
		Vops = Tax shield value + Unlevered value =	$4,544.95	thousand
e.
If L's debt is risky, then its equity is like a call option and can be valued with the Black-Scholes Option
Pricing Model (OPM). See Chapter 6 for details of the OPM.
Black-Scholes Option Pricing Model
Total Value of Firm		4.00	Analogous to the stock price from the BSOPM
Face Value of Debt		2.00	Analogous to the exercise price
Risk Free rate		0.06
Maturity of debt (years)		1.00	Analogous to time to expiration of option
Standard Dev.		0.60	This is the standard dev. of the total value of the firm, not just the stock.
d1		1.5552
d2		0.9552
N(d1)		0.9401
N(d2)		0.8303
Call Price = Equity Value		$ 2.1964
Debt Value	=	Total Value	-	Equity Value
	=	4.00	-	$ 2.1964	(all in millions)
	=	$ 1.8036
Debt yield	=	(Face Value	/Market Value)(1/N)-1
Debt yield	=	10.888%
The value of L's equity must be $2.20 million. The value of its debt must be what is left over: $1.80 million.
This gives a yield of 10.88% for the debt.
f.
Value of Stock and Debt for Different Volatilities
	Equity	Debt	Debt yield
Volatility	$ 2.20	$ 1.80	10.888%
0.20	2.12	1.88	6.18%
0.25	2.12	1.88	6.20%
0.30	2.12	1.88	6.27%
0.35	2.12	1.88	6.48%
0.40	2.13	1.87	6.89%
0.45	2.14	1.86	7.53%
0.50	2.16	1.84	8.41%
0.55	2.17	1.83	9.54%
0.60	2.20	1.80	10.89%
0.65	2.22	1.78	12.46%
0.70	2.25	1.75	14.24%
0.75	2.28	1.72	16.23%
0.80	2.31	1.69	18.40%
0.85	2.34	1.66	20.77%
0.90	2.38	1.62	23.33%
0.95	2.41	1.59	26.08%