# Finance Revision 019

1.    Increasing the time to expiration increases the value of an option. The reason is that the option gives the holder the right to buy or sell. The longer the holder has that right, the more time there is for the option to increase (or decrease in the case of a put) in value. For example, imagine an out-of-the-money option that is about to expire. Because the option is essentially worthless, increasing the time to expiration would obviously increase its value.

2.    An increase in volatility acts to increase both call and put values because the greater volatility increases the possibility of favorable in-the-money payoffs.

3.     Interest rate increases are good for calls and bad for puts. The reason is that if a call is exercised in the future, we have to pay a fixed amount at that time. The higher the interest rate, the lower the present value of that fixed amount. The reverse is true for puts in that we receive a fixed amount.

4.    If you buy a put option on a stock that you already own you guarantee that you can sell the stock for the exercise price of the put. Thus, you have effectively insured yourself against a stock price decline below this point. This is the protective put strategy.

5.    The intrinsic value of a call is Max[S – E, 0]. The intrinsic value of a put is Max[E – S, 0]. The intrinsic value of an option is the value at expiration.

6.    The time value of both a call option and a put option is the difference between the price of the option and the intrinsic value. For both types of options, as maturity increases, the time value increases since you have a longer time to realize a price increase (decrease). A call option is more sensitive to the maturity of the contract.

7.    Since you have a large number of stock options in the company, you have an incentive to accept the second project, which will increase the overall risk of the company and reduce the value of the firm’s debt. However, accepting the risky project will increase your wealth, as the options are more valuable when the risk of the firm increases.

8.    Rearranging the put-call parity formula, we get: S – PV(E) = C – P. Since we know that the stock price and exercise price are the same, assuming a positive interest rate, the left hand side of the equation must be greater than zero. This implies the price of the call must be higher than the price of the put in this situation.

9.    Rearranging the put-call parity formula, we get: S – PV(E) = C – P. If the call and the put have the same price, we know C – P = 0. This must mean the stock price is equal to the present value of the exercise price, so the put is in-the-money.

10.   A stock can be replicated using a long call (to capture the upside gains), a short put (to reflect the downside losses) and a T-bill (to reflect the time value component – the “wait” factor).

Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.

Basic

1.    With continuous compounding, the FV is:

FV = \$1,000 ´ e.08(6) = \$1,616.07

2.    With continuous compounding, the PV is:

PV = \$10,000 ´ e–.09(3) = \$7,633.79

3.    Using put-call parity and solving for the put price, we get:

\$61 + P = \$65e–(.026)(.25) + \$4.12

P = \$7.70

4.    Using put-call parity and solving for the call price we get:

\$53 + \$4.89 = \$50e–(.036)(.5) + C

C = \$8.78

5.    Using put-call parity and solving for the stock price we get:

S + \$2.87 = \$70e–(.048)(3/12) + \$4.68

S = \$70.98

6.    Using put-call parity, we can solve for the risk-free rate as follows:

\$65.80 + \$2.86 = \$65e–R(2/12) + \$4.08

\$64.58 = \$65e–R(2/12)

0.9935 = e–R(2/12)

ln(0.9935) = ln(e–R(2/12))

–0.0065 = –R(2/12)

Rf = 3.89%

7.    Using put-call parity, we can solve for the risk-free rate as follows:

\$83.12 + \$4.80 = \$80e–R(5/12) + \$9.30

\$78.62 = \$80e–R(5/12)

0.9828 = e–R(5/12)

ln(0.9828) = ln(e–R(5/12))

–0.0174 = –R(5/12)

Rf = 4.18%

8.    Using the Black-Scholes option pricing model to find the price of the call option, we find:

d1 = [ln(\$38/\$35) + (.06 + .542/2) ´ (3/12)] / (.54 ´ ) = .4951

d2 = .4951 – (.54 ´ ) = .2251

N(d1) = .6897

N(d2) = .5891

Putting these values into the Black-Scholes model, we find the call price is:

C = \$38(.6897) – (\$35e–.06(.25))(.5891) = \$5.90

Using put-call parity, the put price is:

Put = \$35e–.06(.25) + 5.90 – 38 = \$2.38

9.    Using the Black-Scholes option pricing model to find the price of the call option, we find:

d1 = [ln(\$86/\$90) + (.04 + .622/2) ´ (8/12)] / (.62 ´ ) = .2160

d2 = .2160 – (.62 ´ ) = –.2902

N(d1) = .5855

N(d2) = .3858

Putting these values into the Black-Scholes model, we find the call price is:

C = \$86(.5855) – (\$90e–.04(8/12))(.3858) = \$16.54

Using put-call parity, the put price is:

Put = \$90e–.04(8/12) + 16.54 – 86 = \$18.18

10.  The delta of a call option is N(d1), so:

d1 = [ln(\$87/\$85) + (.05 + .562/2) ´ .75] / (.56 ´ ) = .3678

N(d1) = .6435

For a call option the delta is .64. For a put option, the delta is:

Put delta = .64 – 1 = –.36

The delta tells us the change in the price of an option for a \$1 change in the price of the underlying asset.

11.  Using the Black-Scholes option pricing model, with a ‘stock’ price is \$1,600,000 and an exercise price is \$1,750,000, the price you should receive is:

d1 = [ln(\$1,600,000/\$1,750,000) + (.05 + .202/2) ´ (12/12)] / (.20 ´ ) = –.0981

d2 = –.0981 – (.20 ´ ) = –.2981

N(d1) = .4609

N(d2) = .3828

Putting these values into the Black-Scholes model, we find the call price is:

C = \$1,600,000(.4609) – (\$1,750,000e–.05(1))(.3828) = \$100,231.18

12.   Using the call price ew found in the previous problem and put-call parity, you would need to pay:

Put = \$1,750,000e–.05(1) + 100,231.18 – 1,600,000 = \$164,882.67

You would have to pay \$164,882.67 in order to guarantee the right to sell the land for \$1,750,000.

13.  Using the Black-Scholes option pricing model to find the price of the call option, we find:

d1 = [ln(\$86/\$90) + (.06 + .532/2) ´ (6/12)] / (.53 ´ ) = .1461

d2 = .1461 – (.53 ´ ) = –.2286

N(d1) = .5581

N(d2) = .4096

Putting these values into the Black-Scholes model, we find the call price is:

C = \$86(.5581) – (\$90e–.06(.50))(.4096) = \$12.22

Using put-call parity, we find the put price is:

Put = \$90e–.06(.50) + 12.22 – 86 = \$13.56

a. The intrinsic value of each option is:

Call intrinsic value = Max[S – E, 0] = \$0

Put intrinsic value = Max[E – S, 0] = \$4

b. Option value consists of time value and intrinsic value, so:

Call option value = Intrinsic value + Time value

\$12.22 = \$0 + TV

TV = \$12.22

Put option value = Intrinsic value + Time value

\$13.56 = \$4 + TV

TV = \$9.56

c. The time premium (theta) is more important for a call option than a put option, therefore, the time

premium is, in general, larger for a call option.

14.  Using put-call parity, the price of the put option is:

\$20.26 + P = \$20e–.05(1/3) + \$3.81

P = \$3.22

Intermediate

15.  If the exercise price is equal to zero, the call price will equal the stock price, which is \$85.

16.   If the standard deviation is zero, d1 and d2 go to +∞, so N(d1) and N(d2) go to 1. This is the no risk call option formula we discussed in an earlier chapter, so:

C = S – Ee–rt

C = \$84 – \$80e–.05(6/12) = \$5.98

17.  If the standard deviation is infinite, d1 goes to positive infinity so N(d1) goes to 1, and d2 goes to negative infinity so N(d2) goes to 0. In this case, the call price is equal to the stock price, which is \$35.

18.  We can use the Black-Scholes model to value the equity of a firm. Using the asset value of \$12,500 as the stock price, and the face value of debt of \$10,000 as the exercise price, the value of the firm’s equity is:

d1 = [ln(\$10,500/\$10,000) + (.05 + .382/2) ´ 1] / (.38 ´ ) = .4500

d2 = .4500 – (.38 ´ ) = .0700

N(d1) = .6736

N(d2) = .5279

Putting these values into the Black-Scholes model, we find the equity value is:

Equity = \$10,500(.6736) – (\$10,000e–.05(1))(.5279) = \$2,051.70

The value of the debt is the firm value minus the value of the equity, so:

D = \$10,500 – 2,051.70 = \$8,448.30

19.  a.  We can use the Black-Scholes model to value the equity of a firm. Using the asset value of \$11,200 as the stock price, and the face value of debt of \$10,000 as the exercise price, the value of the firm if it accepts project A is:

d1 = [ln(\$11,200/\$10,000) + (.05 + .552/2) ´ 1] / (.55 ´ ) = .5720

d2 = .5720 – (.55 ´ ) = .0220

N(d1) = .7163

N(d2) = .5088

Putting these values into the Black-Scholes model, we find the equity value is:

EA = \$11,200(.7163) – (\$10,000e–.05(1))(.5088) = \$3,183.37

The value of the debt is the firm value minus the value of the equity, so:

DA = \$11,200 – 3,183.37 = \$8,016.63

And the value of the firm if it accepts Project B is:

d1 = [ln(\$11,500/\$10,000) + (.05 + .342/2) ´ 1] / (.34 ´ ) = .7281

d2 = .7281 – (.34 ´ ) = .3881

N(d1) = .7667

N(d2) = .6510

Putting these values into the Black-Scholes model, we find the equity value is:

EB = \$11,500(.7667) – (\$10,000e–.05(1))(.6510) = \$2,624.55

The value of the debt is the firm value minus the value of the equity, so:

DB = \$11,500 – 2,624.55 = \$8,875.45

b.  Although the NPV of project B is higher, the equity value with project A is higher. While NPV represents the increase in the value of the assets of the firm, in this case, the increase in the value of the firm’s assets resulting from project B is mostly allocated to the debtholders, resulting in a smaller increase in the value of the equity. Stockholders would, therefore, prefer project A even though it has a lower NPV.

c.  Yes. If the same group of investors have equal stakes in the firm as bondholders and stock-holders, then total firm value matters and project B should be chosen, since it increases the value of the firm to \$11,500 instead of \$11,200.

d.  Stockholders may have an incentive to take on more risky, less profitable projects if the firm is leveraged; the higher the firm’s debt load, all else the same, the greater is this incentive.

20.  We can use the Black-Scholes model to value the equity of a firm. Using the asset value of \$22,000 as the stock price, and the face value of debt of \$20,000 as the exercise price, the value of the firm’s equity is:

d1 = [ln(\$22,000/\$20,000) + (.05 + .532/2) ´ 1] / (.53 ´ ) = .5392

d2 = .5392 – (.53 ´ ) = .0092

N(d1) = .7051

N(d2) = .5037

Putting these values into the Black-Scholes model, we find the equity value is:

Equity = \$22,000(.70512) – (\$20,000e–.05(1))(.5037) = \$5,930.64

The value of the debt is the firm value minus the value of the equity, so:

D = \$22,000 – 5,930.64 = \$16,069.36

The return on the company’s debt is:

\$16,069.36 = \$20,000e–R(1)

.803468 = e–R

RD = –ln(.803468) = 21.88%

21.  a.  The combined value of equity and debt of the two firms is:

Equity = \$2,051.70 + 5,930.64 = \$7,982.34

Debt = \$8,448.30 + 16,069.36 = \$24,517.66

b.  For the new firm, the combined market value of assets is \$32,500, and the combined face value of debt is \$30,000. Using Black-Scholes to find the value of equity for the new firm, we find:

d1 = [ln(\$32,500/\$30,000) + (.05 + .312/2) ´ 1] / (.31 ´ ) = .5745

d2 = .5745 – (.31 ´ ) = .2645

N(d1) = .7172

N(d2) = .6043

Putting these values into the Black-Scholes model, we find the equity value is:

E = \$32,500(.7172) – (\$30,000e–.05(1))(.6043) = \$6,063.61

The value of the debt is the firm value minus the value of the equity, so:

D = \$32,500 – 6,063.61 = \$26,436.39

c. The change in the value of the firm’s equity is:

Equity value change = \$6,063.61 – 7,982.34 = –\$1,918.73

The change in the value of the firm’s debt is:

Debt = \$26,436.39 – 24,517.66 = \$1,918.73

d.  In a purely financial merger, when the standard deviation of the assets declines, the value of the

equity declines as well. The shareholders will lose exactly the amount the bondholders gain. The bondholders gain as a result of the coinsurance effect. That is, it is less likely that the new company will default on the debt.

22.   a.  Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$22,000,000/\$30,000,000) + (.06 + .392/2) ´ 10] / (.39 ´ ) = .8517

d2 = .8517 – (.39 ´ ) = –.3816

N(d1) = .8028

N(d2) = .3514

Putting these values into Back-Scholes:

E = \$22,000,000(.8028) – (\$30,000,000e–.06(10))(.3514) = \$11,876,514.69

b.  The value of the debt is the firm value minus the value of the equity, so:

D = \$22,000,000 – 11,876,514.69 = \$10,123,485.31

c.  Using the equation for the PV of a continuously compounded lump sum, we get:

\$10,123,485.31 = \$30,000,000e–R(10)

.33745 = e–R10

RD = –(1/10)ln(.33745) = 10.86%

d.  Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$22,750,000/\$30,000,000) + (.06 + .392/2) ´ 10] / (.39 ´ ) = .8788

d2 = .8788 – (.39 ´ ) = –.3544

N(d1) = .8103

N(d2) = .3615

Putting these values into Back-Scholes:

E = \$22,750,000(.8103) – (\$30,000,000e–.06(10))(.3615) = \$12,481,437.06

e.  The value of the debt is the firm value minus the value of the equity, so:

D = \$22,750,000 – 12,481,437.06 = \$10,268,562.94

Using the equation for the PV of a continuously compounded lump sum, we get:

\$10,268,562.94 = \$30,000,000e–R(10)

.35429 = e–R10

RD = –(1/10)ln(.35429) = 10.72%

When the firm accepts the new project, part of the NPV accrues to bondholders. This increases the present value of the bond, thus reducing the return on the bond. Additionally, the new project makes the firm safer in the sense it increases the value of assets, thus increasing the probability the call will end in-the-money and the bondholders will receive their payment.

Challenge

23.  a.  Using the equation for the PV of a continuously compounded lump sum, we get:

PV = \$30,000 ´ e–.05(2) = \$27,145.12

b.  Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$13,000/\$30,000) + (.05 + .602/2) ´ 2] / (.60 ´ ) = –.4434

d2 = –.4344 – (.60 ´ ) = –1.2919

N(d1) = .3287

N(d2) = .0982

Putting these values into Back-Scholes:

E = \$13,000(.3287) – (\$30,000e–.05(2))(.0982) = \$1,608.19

And using put-call parity, the price of the put option is:

Put = \$30,000e–.05(10) + 1,608.19 – 13,000 = \$15,753.31

c.  The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so:

Value of risky bond = \$27,145.12 – 15,753.31 = \$11,391.81

Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get:

\$11,391.81 = \$30,000e–R(2)

.37973 = e–R2

RD = –(1/2)ln(.37973) = 48.42%

d.  The value of the debt with five years to maturity at the risk-free rate is:

PV = \$30,000 ´ e–.05(5) = \$23,364.02

Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$13,000/\$30,000) + (.05 + .602/2) ´ 5] / (.60 ´ ) = .2339

d2 = .2339 – (.60 ´ ) = –1.1078

N(d1) = .5925

N(d2) = .1340

Putting these values into Back-Scholes:

E = \$13,000(.5925) – (\$30,000e–.05(2))(.1340) = \$4,571.62

And using put-call parity, the price of the put option is:

Put = \$30,000e–.05(10) + \$4,571.62 – \$13,000 = \$14,935.64

The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so:

Value of risky bond = \$23,364.02 – 14,935.64 = \$8,428.38

Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get:

Return on debt: \$8,428.38 = \$30,000e–R(5)

.28095 = e–R5

RD = –(1/5)ln(.28095) = 25.39%

The value of the debt declines because of the time value of money, i.e., it will be longer until shareholders receive their payment. However, the required return on the debt declines. Under the current situation, it is not likely the company will have the assets to pay off bondholders. Under the new plan where the company operates for five more years, the probability of increasing the value of assets to meet or exceed the face value of debt is higher than if the company only operates for two more years.

24.  a.  Using the equation for the PV of a continuously compounded lump sum, we get:

PV = \$60,000 ´ e–.06(5) = \$44,449.09

b.  Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$57,000/\$60,000) + (.06 + .502/2) ´ 5] / (.50 ´ ) = .7815

d2 = .7815 – (.50 ´ ) = –.3366

N(d1) = .7827

N(d2) = .3682

Putting these values into Back-Scholes:

E = \$57,000(.7827) – (\$60,000e–.06(5))(.3682) = \$28,248.84

And using put-call parity, the price of the put option is:

Put = \$60,000e–.06(5) + 28,248.84 – 57,000 = \$15,697.93

c.  The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so:

Value of risky bond = \$44,449.09 – 15,697.93 = \$28,751.16

Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get:

Return on debt: \$28,751.16 = \$60,000e–R(5)

.47919 = e–R(5)

RD = –(1/5)ln(.47919) = 14.71%

d.  Using the equation for the PV of a continuously compounded lump sum, we get:

PV = \$60,000 ´ e–.06(5) = \$44,449.09

Using Black-Scholes model to value the equity, we get:

d1 = [ln(\$57,000/\$60,000) + (.06 + .602/2) ´ 5] / (.60 ´ ) = .8562

d2 = .8562 – (.50 ´ ) = –.4854

N(d1) = .8041

N(d2) = .3137

Putting these values into Black-Scholes:

E = \$57,000(.8041) – (\$60,000e–.06(5))(.3137) = \$31,888.34

And using put-call parity, the price of the put option is:

Put = \$60,000e–.06(5) + 31,888.34 – 57,000 = \$19,337.44

The value of a risky bond is the value of a risk-free bond minus the value of a put option on the firm’s equity, so:

Value of risky bond = \$44,449.09 – 19,337.44 = \$25,111.66

Using the equation for the PV of a continuously compounded lump sum to find the return on debt, we get:

Return on debt: \$25,111.66 = \$60,000e–R(5)

.41853 = e–R(5)

RD = –(1/5)ln(.41853) = 17.42%

The value of the debt declines. Since the standard deviation of the company’s assets increases, the value of the put option on the face value of the bond increases, which decreases the bond’s current value.

e.  From c and d, bondholders lose: \$25,111.66 – 28,751.16 = –\$3,639.51

From c and d, stockholders gain: \$31,888.34 – 28,248.84 = \$3,639.51

This is an agency problem for bondholders. Management, acting to increase shareholder wealth in this manner, will reduce bondholder wealth by the exact amount that shareholder wealth is increased.

25.    a.  Going back to the chapter on dividends, the price of the stock will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a dividend is paid, reducing the upside potential of the call by the amount of the dividend. The price of a call option will decrease when the dividend yield increases.

b.  Using the Black-Scholes model with dividends, we get:

d1 = [ln(\$84/\$80) + (.05 – .02 + .502/2) ´ .5] / (.50 ´ ) = .3572

d2 = .3572 – (.50 ´ ) = .0036

N(d1) = .6395

N(d2) = .5015

C = \$84e–(.02)(.5)(.6395) – (\$80e–.02(.5))(.5015) = \$14.06

26. a.   Going back to the chapter on dividends, the price of the stock will decline by the amount of the dividend (less any tax effects). Therefore, we would expect the price of the stock to drop when a dividend is paid. The price of put option will increase when the dividend yield increases.

b.  Using put-call parity to find the price of the put option, we get:

\$84e–.02(.5) + P = \$80e–.05(.5) + 14.06

P = \$8.92

27.  N(d1) is the probability that “z” is less than or equal to N(d1), so 1 – N(d1) is the probability that “z” is greater than N(d1). Because of the symmetry of the normal distribution, this is the same thing as the probability that “z” is less than N(–d1). So:

N(d1) – 1 = N(–d1).

28.  From put-call parity:

P    = E × eRt + C – S

Substituting the Black-Scholes call option formula for C and using the result in the previous question produces the put option formula:

P    = E × eRt + C – S

P    = E × eRt + S ×N(d1) – E × eRt ×N(d2) – S

P    = S ×(N(d1) – 1) + E × eRt ×(1 – N(d2))

P    = E × eRt ×N(–d2) – S × N(–d1)

29.  Based on Black-Scholes, the call option is worth \$50! The reason is that present value of the exercise price is zero, so the second term disappears. Also, d1 is infinite, so N(d1) is equal to one. The problem is that the call option is European with an infinite expiration, so why would you pay anything for it since you can never exercise it? The paradox can be resolved by examining the price of the stock. Remember that the call option formula only applies to a non-dividend paying stock. If the stock will never pay a dividend, it (and a call option to buy it at any price) must be worthless.

30.  The delta of the call option is N(d1) and the delta of the put option is N(d1) – 1. Since you are selling a put option, the delta of the portfolio is N(d1) – [N(d1) – 1]. This leaves the overall delta of your position as 1. This position will change dollar for dollar in value with the underlying asset. This position replicates the dollar “action” on the underlying asset.

1.    In the purchase method, assets are recorded at market value, and goodwill is created to account for the excess of the purchase price over this recorded value. In the pooling of interests method, the balance sheets of the two firms are simply combined; no goodwill is created. The choice of accounting method has no direct impact on the cash flows of the firms. EPS will probably be lower under the purchase method because reported income is usually lower due to the required amortization of the goodwill created in the purchase.

2.    a.     Greenmail refers to the practice of paying unwanted suitors who hold an equity stake in the firm a premium over the market value of their shares, to eliminate the potential takeover threat.

b.     A white knight refers to an outside bidder that a target firm brings in to acquire it, rescuing the firm from a takeover by some other unwanted hostile bidder.

c.     A golden parachute refers to lucrative compensation and termination packages granted to management in the event the firm is acquired.

d.     The crown jewels usually refer to the most valuable or prestigious assets of the firm, which in the event of a hostile takeover attempt, the target sometimes threatens to sell.

e.     Shark repellent generally refers to any defensive tactic employed by the firm to resist hostile takeover attempts.

f.      A corporate raider usually refers to a person or firm that specializes in the hostile takeover of other firms.

g.     A poison pill is an amendment to the corporate charter granting the shareholders the right to purchase shares at little or no cost in the event of a hostile takeover, thus making the acquisition prohibitively expensive for the hostile bidder.

h.     A tender offer is the legal mechanism required by the SEC when a bidding firm goes directly to the shareholders of the target firm in an effort to purchase their shares.

i.      A leveraged buyout refers to the purchase of the shares of a publicly-held company and its subsequent conversion into a privately-held company, financed primarily with debt.

3.     Diversification doesn’t create value in and of itself because diversification reduces unsystematic, not systematic, risk. As discussed in the chapter on options, there is a more subtle issue as well. Reducing unsystematic risk benefits bondholders by making default less likely. However, if a merger is done purely to diversify (i.e., no operating synergy), then the NPV of the merger is zero. If the NPV is zero, and the bondholders are better off, then stockholders must be worse off.

4.     A firm might choose to split up because the newer, smaller firms may be better able to focus on their particular markets. Thus, reverse synergy is a possibility. An added advantage is that performance evaluation becomes much easier once the split is made because the new firm’s financial results (and stock prices) are no longer commingled.

5.     It depends on how they are used. If they are used to protect management, then they are not good for stockholders. If they are used by management to negotiate the best possible terms of a merger, then they are good for stockholders.

6.    One of the primary advantages of a taxable merger is the write-up in the basis of the target firm’s assets, while one of the primary disadvantages is the capital gains tax that is payable. The situation is the reverse for a tax-free merger.

The basic determinant of tax status is whether or not the old stockholders will continue to participate in the new company, which is usually determined by whether they get any shares in the bidding firm. An LBO is usually taxable because the acquiring group pays off the current stockholders in full, usually in cash.

7.    Economies of scale occur when average cost declines as output levels increase. A merger in this particular case might make sense because Eastern and Western may need less total capital investment to handle the peak power needs, thereby reducing average generation costs.

8.    Among the defensive tactics often employed by management are seeking white knights, threatening to sell the crown jewels, appealing to regulatory agencies and the courts (if possible), and targeted share repurchases. Frequently, antitakeover charter amendments are available as well, such as poison pills, poison puts, golden parachutes, lockup agreements, and supermajority amendments, but these require shareholder approval, so they can’t be immediately used if time is short. While target firm shareholders may benefit from management actively fighting acquisition bids, in that it encourages higher bidding and may solicit bids from other parties as well, there is also the danger that such defensive tactics will discourage potential bidders from seeking the firm in the first place, which harms the shareholders.

9.     In a cash offer, it almost surely does not make sense. In a stock offer, management may feel that one suitor is a better long-run investment than the other, but this is only valid if the market is not efficient. In general, the highest offer is the best one.

10.   Various reasons include: (1) Anticipated gains may be smaller than thought; (2) Bidding firms are typically much larger, so any gains are spread thinly across shares; (3) Management may not be acting in the shareholders’ best interest with many acquisitions; (4) Competition in the market for takeovers may force prices for target firms up to the zero NPV level; and (5) Market participants may have already discounted the gains from the merger before it is announced.

### Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.

Basic

1.    For the merger to make economic sense, the acquirer must feel the acquisition will increase value by at least the amount of the premium over the market value, so:

Minimum economic value = \$740M – 650M = \$90M

2.                        a)         Since neither company has any debt, using the pooling method, the asset value of the combined must equal the value of the equity, so:

Assets = Equity = 20,000(\$20) + 20,000(\$7) = \$540,000

b)     With the purchase method, the assets of the combined firm will be the book value of Firm X, the acquiring company, plus the market value of Firm Y, the target company, so:

Assets from X = 20,000(\$20) = \$400,000 (book value)

Assets from Y = 20,000(\$18) = \$360,000 (market value)

The purchase price of Firm Y is the number of shares outstanding times the sum of the current stock price per share plus the premium per share, so:

Purchase price of Y = 20,000(\$18 + 5) = \$460,000

The goodwill created will be:

Goodwill = \$460,000 – 360,000 = \$100,000.

And the total asset of the combined company will be:

Total assets XY = Total equity XY = \$400,000 + 360,000 + 100,000 = \$860,000

3.    In the pooling method, all accounts of both companies are added together to total the accounts in the new company, so the post-merger balance sheet will be:

Meat Co., post-merger

Current assets         \$13,400             Current liabilities        \$  4,700

Fixed assets              19,600             Long-term debt               2,800

Equity                           25,500

Total               \$33,000                                               \$33,000

4.    Since the acquisition is funded by long-term debt, the post-merger balance sheet will have long-term debt equal to the original long-term debt of Meat’s balance sheet plus the new long-term debt issue, so:

Post-merger long-term debt = \$1,900 + 17,000 = \$18,900

Goodwill will be created since the acquisition price is greater than the book value. The goodwill amount is equal to the purchase price minus the market value of assets. Generally, the market value of current assets is equal to the book value, so:

Goodwill created = \$17,000 – (\$12,000 market value FA) – (\$3,400 market value CA) = \$1,600

Current liabilities and equity will remain the same as the pre-merger balance sheet of the acquiring firm. Current assets will be the sum of the two firm’s pre-merger balance sheet accounts, and the fixed assets will be the sum of the pre-merger fixed assets of the acquirer and the market value of fixed assets of the target firm. The post-merger balance sheet will be:

Meat Co., post-merger

Current assets         \$13,400             Current liabilities        \$  3,100

Fixed assets              26,000             Long-term debt             18,900

Goodwill                    1,600             Equity                           19,000

Total               \$41,000                                               \$41,000

5.    In the pooling method, all accounts of both companies are added together to total the accounts in the new company, so the post-merger balance sheet will be:

Silver Enterprises, post-merger

Current assets         \$  3,700             Current liabilities        \$  2,700

Other assets                1,150             Long-term debt                  900

Net fixed assets          6,700             Equity                             7,950

Total               \$11,550                                               \$11,550

6.    Since the acquisition is funded by long-term debt, the post-merger balance sheet will have long-term debt equal to the original long-term debt of Silver’s balance sheet plus the new long-term debt issue, so:

Post-merger long-term debt = \$900 + 8,400 = \$9,300

Goodwill will be created since the acquisition price is greater than the book value. The goodwill amount is equal to the purchase price minus the market value of assets. Since the market value of fixed assets of the target firm is equal to the book value, and the book value of all other assets is equal to market value, we can subtract the total assets from the purchase price, so:

Goodwill created = \$8,400 – (\$4,250 market value TA) = \$4,150

Current liabilities and equity will remain the same as the pre-merger balance sheet of the acquiring firm. Current assets and other assets will be the sum of the two firm’s pre-merger balance sheet accounts, and the fixed assets will be the sum of the pre-merger fixed assets of the acquirer and the market value of fixed assets of the target firm. Note, in this case, the market value and the book value of fixed assets are the same. The post-merger balance sheet will be:

Silver Enterprises, post-merger

Current assets         \$  3,700             Current liabilities        \$  1,800

Other assets                1,150             Long-term debt               9,300

Net fixed assets          6,700             Equity                             4,600

Goodwill                    4,150

Total               \$15,700                                               \$15,700

7.    a.     The cash cost is the amount of cash offered, so the cash cost is \$94 million.

To calculate the cost of the stock offer, we first need to calculate the value of the target to the acquirer. The value of the target firm to the acquiring firm will be the market value of the target plus the PV of the incremental cash flows generated by the target firm. The cash flows are a perpetuity, so

V* = \$78,000,000 + \$3,100,000/.12 = \$103,833,333

The cost of the stock offer is the percentage of the acquiring firm given up times the sum of the market value of the acquiring firm and the value of the target firm to the acquiring firm. So, the equity cost will be:

Equity cost = .40(\$135M + 103,833,333) = \$95,533,333

b.     The NPV of each offer is the value of the target firm to the acquiring firm minus the cost of acquisition, so:

NPV cash = \$103,833,333 – 94,000,000 = \$9,833,333

NPV stock = \$103,833,333 – 95,533,333 = \$8,300,000

c.     Since the NPV is greater with the cash offer the acquisition should be in cash.

8.    a.     The EPS of the combined company will be the sum of the earnings of both companies divided by the shares in the combined company. Since the stock offer is one share of the acquiring firm for three shares of the target firm, new shares in the acquiring firm will increase by one-third. So, the new EPS will be:

EPS = (\$300,000 + 675,000)/[180,000 + (1/3)(60,000)] = \$4.875

The market price of Pitt will remain unchanged if it is a zero NPV acquisition. Using the PE ratio, we find the current market price of Pitt stock, which is:

P = 21(\$675,000)/180,000 = \$78.75

If the acquisition has a zero NPV, the stock price should remain unchanged. Therefore, the new PE will be:

P/E = \$78.75/\$4.875 = 16.15

b.     The value of Aniston to Pitt must be the market value of the company since the NPV of the acquisition is zero. Therefore, the value is:

V* = \$300,000(5.25) = \$1,575,000

The cost of the acquisition is the number of shares offered times the share price, so the cost is:

Cost = (1/3)(60,000)(\$78.75) = \$1,575,000

So, the NPV of the acquisition is:

NPV = 0 = V* + DV – Cost = \$1,575,000 + DV – 1,575,000

DV =  \$0

Although there is no economic value to the takeover, it is possible that Pitt is motivated to purchase Aniston for other than financial reasons.

9.    a.     The NPV of the merger is the market value of the target firm, plus the value of the synergy, minus the acquisition costs, so:

NPV = 900(\$24) + \$3,000 – 900(\$27) = \$300

b.     Since the NPV goes directly to stockholders, the share price of the merged firm will be the market value of the acquiring firm plus the NPV of the acquisition, divided by the number of shares outstanding, so:

Share price = [1,500(\$34) + \$300]/1,500 = \$34.20

c.     The merger premium is the premium per share times the number of shares of the target firm outstanding, so the merger premium is:

Merger premium = 900(\$27 – 24) = \$2,700

d.     The number of new shares will be the number of shares of the target times the exchange ratio, so:

New shares created = 900(3/5) = 540 new shares

The value of the merged firm will be the market value of the acquirer plus the market value of the target plus the synergy benefits, so:

VBT = 1,500(\$34) + 900(\$24) + 3,000 = \$75,600

The price per share of the merged firm will be the value of the merged firm divided by the total shares of the new firm, which is:

P = \$75,600/(1,500 + 540) = \$37.06

e.     The NPV of the acquisition using a share exchange is the market value of the target firm plus synergy benefits, minus the cost. The cost is the value per share of the merged firm times the number of shares offered to the target firm shareholders, so:

NPV = 900(\$24) + \$3,000 – 540(\$37.06) = \$4,588.24

Intermediate

10.  The cash offer is better for given the target firm shareholders receive \$27 per share. In the share offer, the target firm’s shareholders will receive:

Equity offer value = (3/5)(\$24) = \$14.40 per share

The shareholders of the target firm would prefer the cash offer. The exchange ratio which would make the target firm shareholders indifferent between the two offers is the cash offer price divided by the new share price of the firm under the cash offer scenario, so:

Exchange ratio = \$27/\$34.20 = .7895

11.  The cost of the acquisition is:

Cost = 220(\$20) = \$4,400

Since the stock price of the acquiring firm is \$40, the firm will have to give up:

Shares offered = \$4,400/\$40 = 110 shares

a.     The EPS of the merged firm will be the combined EPS of the existing firms divided by the new shares outstanding, so:

EPS = (\$900 + 600)/(550 + 110) = \$2.27

b.     The PE of the acquiring firm is:

Original P/E = \$40/(\$900/550) = 24.44 times

Assuming the PE ratio does not change, the new stock price will be:

New P = \$2.27(24.44) = \$55.56

c.     If the market correctly analyzes the earnings, the stock price will remain unchanged since this is a zero NPV acquisition, so:

New P/E = \$40/\$2.27 = 17.60 times

d.     The new share price will be the combined market value of the two existing companies divided by the number of shares outstanding in the merged company. So:

P = [(550)(\$40) + 220(\$15)]/(550 + 110) = \$38.33

And the PE ratio of the merged company will be:

P/E = \$38.33/\$2.27 = 16.87 times

At the proposed bid price, this is a negative NPV acquisition for A since the share price declines. They should revise their bid downward until the NPV is zero.

12.  Beginning with the fact that the NPV of a merger is the value of the target minus the cost, we get:

NPV   = VB* – Cost

NPV   = DV + VB – Cost

NPV   = DV – (Cost – VB)

NPV   = DV – Merger premium

Challenge

13.  a.     To find the value of the target to the acquirer, we need to find the share price with the new growth rate. We begin by finding the required return for shareholders of the target firm. The earnings per share of the target are:

EPSP = \$580,000/550,000 = \$1.05 per share

The price per share is:

PP = 9(\$1.05) = \$9.49

And the dividends per share are:

DPSP = \$290K/550K = \$0.527

The current required return for Pulitzer shareholders, which incorporates the risk of the company is:

RE = [\$0.527(1.05)/\$9.49] + .05 = .1083

The price per share of Pulitzer with the new growth rate is:

PP = \$0.527(1.07)/(.1083 – .07) = \$14.72

The value of the target firm t0 the acquiring firm is the number of shares outstanding times the price per share under the new growth rate assumptions, so:

VT* = 550,000(\$14.72) = \$8,094,782.61

b.     The gain to the acquiring firm will be the value of the target firm to the acquiring firm minus the market value of the target, so:

Gain = \$8,094,782.61 – 550,000(\$9.49) = \$2,874,782.61

c.     The NPV of the acquisition is the value of the target firm to the acquiring firm minus the cost of the acquisition, so:

NPV = \$8,094,782.61 – 550,000(\$18) = –\$1,805,217.39

d.     The most the acquiring firm should be willing to pay per share is the offer price per share plus the NPV per share, so:

Maximum bid price = \$18 + (–\$1,805,217.39/550,000) = \$14.72

Notice, this is the same value we calculated earlier in part a as the value of the target to the acquirer.

e.     The price of the stock in the merged firm would be the market value of the acquiring firm plus the value of the target to the acquirer, divided by the number of shares in the merged firm, so:

PFP = (\$25,000,000 + 8,094,782.61)/(1,000,000 + 100,000) = \$30.09

The NPV of the stock offer is the value of the target to the acquirer minus the value offered to the target shareholders. The value offered to the target shareholders is the stock price of the merged firm times the number of shares offered, so:

NPV = \$8,094,782.61 – 100,000(\$30.09) = \$5,086,166.01

f.      Yes, the acquisition should go forward, and Foxy should offer the 100,000 shares since the NPV is higher.

g.     Using the new growth rate in the dividend growth model, along with the dividend and required return we calculated earlier, the price of the target under these assumptions is:

PP = \$0.527(1.06)/(.1083 – .06) = \$11.56

And the value of the target firm to the acquiring firm is:

VP* = 550,000(\$11.56) = \$6,360,000.00

The gain to the acquiring firm will be:

Gain = \$6,360,000 – 550,000(\$9.49) = \$1,140,000.00

The NPV of the cash offer is now:

NPV cash = \$6,360,000 – 550,000(\$18) = –\$3,540,000

And the new price per share of the merged firm will be:

PFP = [\$25M + 6,360,000]/(1,000,000 + 100,000) = \$28.51

And the NPV of the stock offer under the new assumption will be:

NPV stock = \$6,360,000 – 100,000(\$28.51) = \$3,509,090.91

Even with the lower projected growth rate, the stock offer still has a positive NPV. Foxy should purchase Pulitzer with a stock offer of 100,000 shares.

LEASING

1.     Some key differences are: (1) Lease payments are fully tax-deductible, but only the interest portion of the loan is; (2) The lessee does not own the asset and cannot depreciate it for tax purposes; (3) In the event of a default, the lessor cannot force bankruptcy; and (4) The lessee does not obtain title to the asset at the end of the lease (absent some additional arrangement).

2.     The less profitable one because leasing provides, among other things, a mechanism for transferring tax benefits from entities that value them less to entities that value them more.

3.     Potential problems include: (1) Care must be taken in interpreting the IRR (a high or low IRR is preferred depending on the setup of the analysis); and (2) Care must be taken to ensure the IRR under examination is not the implicit interest rate just based on the lease payments.

4.    a.     Leasing is a form of secured borrowing. It reduces a firm’s cost of capital only if it is cheaper than other forms of secured borrowing. The reduction of uncertainty is not particularly relevant; what matters is the NAL.

b.     The statement is not always true. For example, a lease often requires an advance lease payment or security deposit and may be implicitly secured by other assets of the firm.

c.     Leasing would probably not disappear, since it does reduce the uncertainty about salvage value and the transactions costs of transferring ownership. However, the use of leasing would be greatly reduced.

5.    A lease must be disclosed on the balance sheet if one of the following criteria is met:

1.     The lease transfers ownership of the asset by the end of the lease. In this case, the firm essentially owns the asset and will have access to its residual value.

2.     The lessee can purchase the asset at a price below its fair market value (bargain purchase option) when the lease ends. The firm essentially owns the asset and will have access to most of its residual value.

3.     The lease term is for 75% or more of the estimated economic life of the asset. The firm basically has access to the majority of the benefits of the asset, without any responsibility for the consequences of its disposal.

4.     The present value of the lease payments is 90% or more of the fair market value of the asset at the start of the lease. The firm is essentially purchasing the asset on an installment basis.

6.    The lease must meet the following IRS standards for the lease payments to be tax deductible:

1.     The lease term must be less than 80% of the economic life of the asset. If the term is longer, the lease is considered to be a conditional sale.

2.     The lease should not contain a bargain purchase option, which the IRS interprets as an equity interest in the asset.

3.     The lease payment schedule should not provide for very high payments early and very low payments late in the life of the lease. This would indicate that the lease is being used simply to avoid taxes.

4.     Renewal options should be reasonable and based on the fair market value of the asset at renewal time. This indicates that the lease is for legitimate business purposes, not tax avoidance.

7.    As the term implies, off-balance sheet financing involves financing arrangements that are not required to be reported on the firm’s balance sheet. Such activities, if reported at all, appear only in the footnotes to the statements. Operating leases (those that do not meet the criteria in problem 2) provide off-balance sheet financing. For accounting purposes, total assets will be lower and some financial ratios may be artificially high. Financial analysts are generally not fooled by such practices. There are no economic consequences, since the cash flows of the firm are not affected by how the lease is treated for accounting purposes.

8.    The lessee may not be able to take advantage of the depreciation tax shield and may not be able to obtain favorable lease arrangements for “passing on” the tax shield benefits. The lessee might also need the cash flow from the sale to meet immediate needs, but will be able to meet the lease obligation cash flows in the future.

9.    Since the relevant cash flows are all aftertax, the aftertax discount rate is appropriate.

10.   Skymark’s financial position was such that the package of leasing and buying probably resulted in the overall best aftertax cost. In particular, Skymark may not have been in a position to use all of the tax credits and also may not have had the credit strength to borrow and buy the plane without facing a credit downgrade and/or substantially higher rates.

11.   There is the tax motive, but, beyond this, Royal Brunei knows that, in the event of a default, Skymark would relinquish the planes, which would then be re-leased. Fungible assets, such as planes, which can be readily reclaimed and redeployed are good candidates for leasing.

12.   They will be re-leased to Skymark or another air transportation firm, used by Royal Brunei, or they will simply be sold. There is an active market for used aircraft.

Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability constraints, when these intermediate steps are included in this solutions manual, rounding may appear to have occurred. However, the final answer for each problem is found without rounding during any step in the problem.

Basic

1.    We will calculate cash flows from the depreciation tax shield first. The depreciation tax shield is:

Depreciation tax shield = (\$3,000,000/4)(.35) = \$262,500

The aftertax cost of the lease payments will be:

Aftertax lease payment = (\$895,000)(1 – .35) = \$581,750

So, the total cash flows from leasing are:

OCF = \$262,500 + 581,750 = \$844,250

The aftertax cost of debt is:

Aftertax debt cost = .08(1 – .35) = .052

Using all of this information, we can calculate the NAL as:

NAL = \$3,000,000 – \$844,250(PVIFA5.20%,4) = \$20,187.17

The NAL is positive so you should lease.

2.    If we assume the lessor has the same cost of debt and the same tax rate, the NAL to the lessor is the negative of our company’s NAL, so:

NAL = – \$20,187.17

3.    To find the maximum lease payment that would satisfy both the lessor and the lessee, we need to find the payment that makes the NAL equal to zero. Using the NAL equation and solving for the OCF, we find:

NAL = 0 = \$3,000,000 – OCF(PVIFA5.20%,4)

OCF = \$849,969.49

The OCF for this lease is composed of the depreciation tax shield cash flow, as well as the aftertax lease payment. Subtracting out the depreciation tax shield cash flow we calculated earlier, we find:

Aftertax lease payment = \$849,969.49 – 262,500 = \$587,469.49

Since this is the aftertax lease payment, we can now calculate the breakeven pretax lease payment as:

Breakeven lease payment = \$587,469.49/(1 – .35) = \$903,799.22

4.    If the tax rate is zero, there is no depreciation tax shield foregone. Also, the aftertax lease payment is the same as the pretax payment, and the aftertax cost of debt is the same as the pretax cost. So:

Cost of debt = .08

Annual cost of leasing = leasing payment = \$895,000

The NAL to leasing with these assumptions is:

NAL = \$3,000,000 – \$895,000(PVIFA8%,4) = \$35,646.48

5.    We already calculated the breakeven lease payment for the lessor in Problem 3. Since the assumption about the lessor concerning the tax rate have not changed. So, the lessor breaks even with a payment of \$903,799.22.

For the lessee, we need to calculate the breakeven lease payment which results in a zero NAL. Using the assumptions in Problem 4, we find:

NAL = 0 = \$3,000,000 – PMT(PVIFA8%,4)

PMT = \$905,762.41

So, the range of lease payments that would satisfy both the lessee and the lessor are:

Total payment range = \$903,799.22 to \$905,762.41

6.    The appropriate depreciation percentages for a 3-year MACRS class asset can be found in Chapter 10. The depreciation percentages are .3333, .4444, .1482, and 0.0741. The cash flows from leasing are:

Year 1: (\$3,000,000)(.3333)(.35) + \$581,750 = \$931,715

Year 2: (\$3,000,000)(.4444)(.35) + \$581,750 = \$1,048,370

Year 3: (\$3,000,000)(.1482)(.35) + \$581,750 = \$737,360

Year 4: (\$3,000,000)(.0741)(.35) + \$581,750 = \$659,555

NAL = \$3,000,000 – \$931,715/1.052 – \$1,048,370/1.0522 – \$737,360/1.0523 – \$659,555/1.0524

NAL = -\$4,787.24

The machine should not be leased. This is because of the accelerated tax benefits due to depreciation, which represents a cost in the decision to lease compared to an advantage of the decision to purchase.

Intermediate

7.    The pretax cost savings are not relevant to the lease versus buy decision, since the firm will definitely use the equipment and realize the savings regardless of the financing choice made. The depreciation tax shield is:

Depreciation tax shield lost = (\$6M/5)(.34) = \$408,000

And the aftertax lease payment is:

Aftertax lease payment = \$1,400,000(1 – .34) = \$924,000

The aftertax cost of debt is:

Aftertax debt cost = .09(1 – .34) = .0594 or 5.94%

With these cash flows, the NAL is:

NAL = \$6M – 924,000 – \$924,000(PVIFA5.94%,4) – \$408,000(PVIFA5.94%,5) = \$148,385.28

The equipment should be leased.

To find the maximum payment, we find where the NAL is equal to zero, and solve for the payment. Using X to represent the maximum payment:

NAL = 0 = \$6M – X(1.0594)(PVIFA5.94%,5) – \$408,000(PVIFA5.94%,5)

X = \$957,196.77

So the maximum pretax lease payment is:

Pretax lease payment = \$957,196.77/(1 – .34) = \$1,450,298.13

8.    The aftertax residual value of the asset is an opportunity cost to the leasing decision, occurring at the end of the project life (year 5). Also, the residual value is not really a debt-like cash flow, since there is uncertainty associated with it at year 0. Nevertheless, although a higher discount rate may be appropriate, we’ll use the aftertax cost of debt to discount the residual value as is common in practice. Setting the NAL equal to zero:

NAL = 0 = \$6M – X(1.0594)(PVIFA5.94%,5) – 408,000(PVIFA5.94%,5) – 500,000/1.05945

X = \$873,371.47

So, the maximum pretax lease payment is:

Pretax lease payment = \$873,371.47/(1 – .34) = \$1,323,290.10

9.    The security deposit is a cash outflow at the beginning of the lease and a cash inflow at the end of the lease when it is returned. The NAL with these assumptions is:

NAL  = \$6M – 200,000 – 924,000 – \$924,000(PVIFA5.94%,4) – \$408,000(PVIFA5.94%,5)

+ \$200,000/1.05945

NAL  = \$98,260.61

With the security deposit, the firm should still lease the equipment rather than buy it, because the NAL is greater than zero. We could also solve this problem another way. From Problem 7, we know that the NAL without the security deposit is \$148,385.28, so, if we find the present value of the security deposit, we can simply add this to \$148,385.28. The present value of the security deposit is:

PV of security deposit = –\$200,000 + \$200,000/1.05945 = –\$50,124.67

So, the NAL with the security deposit is:

NAL = \$148,385.28 – 50,124.67 = \$98,260.61

Challenge

10.  With a four-year loan, the annual loan payment will be

\$3,000,000 = PMT(PVIFA8%,4)

PMT = \$905,762.41

The aftertax loan payment is found by:

Aftertax payment = Pretax payment – Interest tax shield

So, we need to find the interest tax shield. To find this, we need a loan amortization table since the interest payment each year is the beginning balance times the loan interest rate of 8 percent. The interest tax shield is the interest payment times the tax rate. The amortization table for this loan is:

So, the total cash flows each year are:

Aftertax loan payment                    OCF                  Total cash flow

Year 1: \$905,762.41 – (\$240,000)(.35)       = \$821,762.41     –     844,250       =        –\$22,487.59

Year 2: \$905,762.41 – (\$186,739.01)(.35)   = \$840,403.76     –     844,250       =            –3,846.24

Year 3: \$905,762.41 – (\$129,217.13)(.35)   = \$860,536.42     –     844,250       =            16,286.42

Year 4: \$905,762.41 – (\$67,093.51)(.35)     = \$882,279.68     –     844,250       =            38,029.68

So, the NAL with the loan payments is:

NAL = 0 – \$22,487.59/1.052 – \$3,846.24/1.0522 + \$16,286.42/1.0523 + \$38,029.68/1.0524

NAL = \$20,187.17

The NAL is the same because the present value of the aftertax loan payments, discounted at the aftertax cost of capital (which is the aftertax cost of debt) equals \$3,000,000.