DUKE UNIVERSITYFuqua School of Business

FINANCE 351 - CORPORATE FINANCEProblem Set #8

Prof. Simon Gervais Fall 2011 – Term 2

Questions

1. Hors d’Age Cheeseworks has been paying a regular cash dividend of $4 per share each year forover a decade. The company is paying out all its earnings as dividends and is not expectedto grow. There are 100,000 shares outstanding selling for $80 per share. The company hassufficient cash on hand to pay the next annual dividend.

Suppose that Hors d’Age decides to cut its cash dividend to zero and announces that it willrepurchase shares instead.

(a) What is the immediate stock price reaction? Ignore taxes, and assume that the repur-chase program conveys no information about operating profitability or business risk.

(b) How many shares will Hors d’Age purchase?

(c) Project and compare future stock prices for the old and new policies. Do this for at leastyears 1, 2, and 3.

2. Formaggio Vecchio has just announced its regular quarterly cash dividend of $1 per share.

(a) When will the stock price fall to reflect dividend payment—on the record date, theex-dividend date, or the payment date?

(b) Assume that there are no taxes. By how much is the stock price likely to fall?

(c) Now assume that all investors pay tax of 30% on dividends and nothing on capital gains.What is the likely fall in the stock price?

(d) Suppose, finally, that everything is the same as in part (c), except that security dealerspay tax on both dividends and capital gains. How would you expect your answer to (c)to change? Explain.

3. Refer back to the last question. Assume no taxes and a stock price immediately after thedividend announcement of $100.

(a) If you own 100 shares, what is the value or your investment? How does the dividendpayment affect your wealth?

(b) Now suppose that Formaggio Vecchio cancels the dividend payment and announces thatit will repurchase 1% of its stock at $100. Do you rejoice or yawn? Explain.

4. The expected pretax return on three stocks is divided between dividends and capital gains inthe following way:

Expected ExpectedStock Dividend Capital Gain

A $0 $10B $5 $5C $10 $0

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(a) If each stock is priced at $100, what are the expected net returns on each stock to

(i) a (tax-exempt) pension fund;

(ii) a corporation paying tax at 35% (recall that corporations are taxed on only 30% ofthe dividends that they receive; they are fully taxed on the capital gains);

(iii) an individual paying tax at 39.6% on investment income and 28% on capital gains;

(iv) a security dealer paying tax at 35% on investment income and capital gains?

(b) Suppose that before the 1986 Tax Reform Act stocks A, B and C were priced to yieldan 8% after-tax return to individual investors paying 50% tax on dividends and 20% taxon capital gains. What would A, B and C each sell for?

5. The net income of Novis Corporation, which has 10,000 outstanding shares and a 100% payoutpolicy, is $32,000 today. The expected value of the firm one year hence is $1,545,600. Theappropriate discount rate for Novis is 12%.

(a) What is the current value of the firm?

(b) What is the ex-dividend price of Novis’ stock if the board follows its current dividendpolicy?

(c) At the dividend declaration meeting, several board members claimed that the dividendis too meager and is probably depressing Novis’ stock price. They propose that Novissell enough new shares to finance a $4.25 dividend. Assume that the new shareholdersare not entitled to this dividend (i.e., assume that their shares are issued ex-dividend).

(i) Comment on the claim that the low dividend is depressing the stock price. Supportyour argument with calculations.

(ii) If the proposal is adopted, at what price will the new shares sell and how many willbe sold?

6. Payall Inc., Payless Inc., and Paynone Inc. have identical operations. They follow a large,medium, and zero (no dividend) payout policy respectively. Paynone Inc.’s shares currentlytrade at $100, and are expected to trade at $125 in one year. The expected dividends per share(in one year) for Payall and Payless are $25 and $12.50 respectively, and their ex-dividendstock prices are expected to be $100 and $112.50 respectively. The market prices are setso that their after-tax expected returns are equal. What should the current share prices ofPayless Inc. and Payall Inc. be? Assume that the marginal personal tax rate on dividends is25%, and the effective tax rate on capital gains is zero.

7. The Sharpe Co. has a period 0 dividend of $1.25. Its target payout ratio is 40%. The period 1EPS is expected to be $4.5.

(a) If the adjustment rate is 0.3 as defined in the Lintner model, what will be the SharpeCo. dividend in period 1?

(b) If the adjustment rate is 0.6 instead, what is the dividend in period 1?

8. The Nilpoj corporation has 1 million shares outstanding with a total market value of $20 mil-(Difficult)

lion. Nilpoj is expected to pay $1 million of dividends at the end of the year (i.e. one year from

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now), and thereafter the amount paid out is expected to grow by 5% a year in perpetuity.Thus the expected dividend at the end of the second year is $1.05 million, and so on.

However, the company has heard that the value of a share depends on the flow of dividends.Therefore it announces that this year’s dividend will be increased to $2 million (from $1 mil-lion), and that the extra cash will be raised at the end of the year by an issue of shares.After that, the total amount paid out in dividends each year will be as previously forecast,i.e. $1.05 million at the end of year 2 and increasing at 5% a year in each subsequent year.

(a) What is Nilpoj’s cost of capital.

(b) What will the total value of the firm be at the end of the year (after the extra cash israised and after the $2 million dividend is paid out)?

(c) How many shares will the firm need to issue, and what is the price per share at the endof the year (after the extra cash is raised and after the $2 million dividend is paid out)?

(d) What fraction of future dividends (starting with the second-year dividends) belongs tothe original shareholders?

(e) Show that the original shareholders are not made better off by this decision (i.e. showthat the present value of the cash flows to the original shareholders remains $20 million).

9. The Government in Dukeraine imposes a flat tax rate of 20% on realized capital gains and(Optional)

losses. The tax rate on personal income is 30%. The corporate tax rate is 35%.

The stock of a firm in Dukeraine is currently priced at $100 per share, and is about to goex-dividend, paying $1 as dividend to holders of record. Assume that the (after-tax) interestrate is 10%, that the dividend is paid immediately and that all taxes are paid one year fromnow.

What should be the ex-dividend price for an investor who is planning to hold the stock forone year to be indifferent between buying the stock immediately before or immediately afterit goes ex-dividend? How do you explain the difference between the cum-dividend (withdividend) and ex-dividend (without dividend) prices?

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Solutions

1. (a) There should be no reaction.

(b) Solution 1: The total dividend is $4(100,000) = $400,000. If Hors d’Age repurchasesn shares at a price of P each, then we need

nP = 400,000. (1)

The remaining 100,000 − n shares should be worth P each as well, and so

8,000,000

100,000 − n= P (2)

Solving for n and P in (1) and (2), we find n = 4,762 and P = 84.00.

Solution 2: Without the dividend payment, the firm is worth V = $80(100,000) +

$4(100,000) = $8,400,000, which is $8,400,000100,000

= $84 per share. With $400,000, the firm

can therefore repurchase $400,000$84

= 4,762 shares.

(c) Old policy:

Year 1 Year 2 Year 3

Total Assetsbeginning of year 8.0M 8.0M 8.0Mend of year 8.4M 8.4M 8.4M

Dividends (= Earnings) 0.4M 0.4M 0.4MNumber of shares 100,000 100,000 100,000Price per share (ex-dividend) 80 80 80

New policy:

Year 1 Year 2 Year 3

Total Assetsbeginning of year 8.0M 8.0M 8.0Mend of year 8.4M 8.4M 8.4M

Earnings 0.4M 0.4M 0.4MBeginning of yearnumber of shares 100,000 95,238 90,703price per share 80 84.00 88.20

Number of shares repurchased 4,762 4,535 4,319End of yearnumber of shares 95,238 90,703 86,384price per share (ex-dividend) 84.00 88.20 92.61

2. (a) On the ex-dividend date.

(b) The stock price will fall by $1.

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(c) The stock price will fall by the after-tax dividend, i.e., by $1(1−0.3) = $0.70, so that theafter-tax return on dividends and capital gains are the same. To see this more clearly,suppose the stock price is 10 before the dividend is paid. If you buy the stock rightbefore the dividend is paid and sell it right after, your net profit should be essentiallyzero since you’ve held the stock for a small amount of time (a second, say). Here areyour cash flows:

• buy now: -10.00;

• dividend: +1.00;

• tax on dividend: -0.30;

• sell after dividend: S (to be found).

So the sum of all these cash flows should be zero:

−10 + 1− 0.30 + S = 0 ⇒ S = 9.30.

The stock price fell by $10.00 − $9.30 = $0.70.

(d) In this case, there should be no tax effects, i.e., the stock price will fall by $1. To bettersee this, suppose you buy the stock (for $10, say) right before the dividend is paid, andsell it right after the dividend is paid. Because you hold the stock for a small amountof time (a second, say), your net profit should be zero. You pay $10 for the stock, youreceive $1 as a dividend which is taxed at 30%; then you sell it for S (to be found) andpay taxes of 30%× (S − 10) on your capital gains (which are going to be negative here).So you net profit is

−10 + (1− 0.30) + S − 0.30(S − 10) = 0.

Solving for S, we find S = 9. So the change in S is $10− $9 = $1.

3. (a) After the dividend announcement, the value of your investment is

100× $100 = $10,000.

After the dividend payment, the stock price drops to $99, and you will have received adividend of $1 for each share you own. Hence, your wealth will remain the same:

(100 × $99) + (100 × $1) = $10,000.

(b) You yawn. After Formaggio Vecchio repurchases 1% of your shares (i.e., one share) at$100 each, you will be left with 99 shares worth $100 each. Your total wealth after therepurchase will then be:

(99 × $100) + (1× $100) = $10,000.

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4. (a) Here are the net expected returns on each stock for each of the four investors:

Investor Stock A Stock B Stock C

(i) Pension fund 10 10 10

(ii) Corporation10(1 − 0.35)

= 6.5

5[1− (0.3)(0.35)]+5(1− 0.35)

= 7.725

10[1 − (0.3)(0.35)]= 8.95

(iii) Individual10(1 − 0.28)

= 7.2

5(1− 0.396)+5(1− 0.28)

= 6.62

10(1 − 0.396)= 6.04

(iv) Security dealer10(1 − 0.35)

= 6.510(1 − 0.35)

= 6.510(1 − 0.35)

= 6.5

(b) The yearly after-tax payoff of stock A is 10(1 − 0.2) = 8, so that the price of stock Ashould be

PA =8

0.08= 100.

The yearly after-tax payoff of stock B is 5(1 − 0.5) + 5(1 − 0.2) = 6.5, so that the priceof stock B should be

PB =6.5

0.08= 81.25.

The yearly after-tax payoff of stock C is 10(1 − 0.5) = 5, so that the price of stock Cshould be

PC =5

0.08= 62.50.

5. (a) The value of the firm is the present value of its dividends and future value:

V0 = 32,000 +1,545,600

1.12= 1,412,000.

(b) Before the dividend is paid, each share is worth

P0 =1,412,000

10,000= 141.20.

After the dividend is paid, the shares will be worth

P ′

0 =1,545,600/1.12

10,000=

1,412,000 − 32,000

10,000= 138.00.

Notice that the difference between P0 and P ′

0 is the dividend per share, 32,000/10,000 =3.20.

(c) (i) According to Modigliani and Miller, it cannot be true that the low dividend isdepressing the price. In fact, since the dividend policy is irrelevant, the level of thedividend should not matter: any funds not distributed as dividends add to the valueof the firm through the stock price (capital gains). These directors merely want to

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change the timing of the dividends (more now, less in the future). As shown below,the current shareholders’ wealth is unaffected by this dividend increase, i.e., theshareholders are not made better off.To pay the $4.25 dividend per share (for a total dividend of $42,500), new sharesmust be sold. These new shares must have a value of $10,500 ($42,500 − $32,000).This means that some of the $1,545,600 firm value in one year will belong to thenew shareholders. How much? Well, these new shareholders will also demand a12% return on their investment, i.e., their $10,500 should then be worth $11,760($10,500 × 1.12). So the current shareholders’ wealth at time 0 must be

W0 = 42,500 +1,545,600 − 11,760

1.12= 1,412,000,

the same as before.

(ii) Let n denote the number of new shares that have to be issued, and P ′

0 the priceafter the $4.25 dividend has been paid.Solution 1: Since the new shareholders will not be fooled and will require their$10,500 investment to be worth exactly that after the dividend is paid, we musthave

nP ′

0 = 10,500. (3)

Also, after the dividend is paid, the shareholders (both old and new) will be sharingthe future value of the firm, that is

P ′

0 =1,545,600/1.12

10,000 + n. (4)

Solving (3) and (4) for n and P ′

0 yields n = 76.67 and P ′

0 = 136.95.Solution 2: After the dividend is paid and the money is raised from the new equityissue, the firm is worth

V =1,545,600

1.12= 1,380,000.

For the new shareholders to be willing to pay $10,500 for their share in the firm, itmust be that their claim is worth $10,500. This implies that 1,380,000 − 10,500 =1,369,500 belongs to the old shareholders, that is,

P ′

0 =1,369,500

10,000= 136.95 per share.

The number of new shares that must be issued is therefore

n =10,500

136.95= 76.67.

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6. The following table shows the after-tax return calculations for the three companies.

Paynone Payless Payall

Next year’s stock price 125.00 112.50 100.00Dividend 0.00 12.50 25.00Total pre-tax payoff 125.00 125.00 125.00Today’s stock price 100.00 Pless Pall

Capital gains 25.00 112.50 − Pless 100.00 − Pall

Tax on dividend (@25%) 0.00 3.13 6.25Tax on capital gains (@0%) 0.00 0.00 0.00Total after-tax income 25.00 12.50 − 3.13 25.00 − 6.25

+(112.50 − Pless) +(100.00 − Pall)

After-tax rate of return 25% 25% 25%

Since we would like the three after-tax expected returns to be the same, we must have

25% =12.50 − 3.13 + (112.50 − Pless)

Pless

, and

25% =25.00 − 6.25 + (100.00 − Pall)

Pall

.

These imply Pless = 97.50 and Pall = 95.00.

7. We know that the Lintner dividend model is given by

D1 = a(p∗E1) + (1− a)D0,

where, in this case, p∗ = 0.4, E1 = 4.50, and D0 = 1.25.

(a) If a = 0.3, we have

D1 = 0.3(0.4 × 4.50) + (1− 0.3)1.25 = 1.415.

(b) If a = 0.6, we have

D1 = 0.6(0.4 × 4.50) + (1− 0.6)1.25 = 1.58.

Notice that the increase in the dividend is more conservative in part (a) since the ad-justment rate is lower (i.e., more weight is put on last year’s dividend).

8. (a) We know that the current value of Nilpoj is $20 million, and that this number representsthe present value of discounted dividends, i.e.,

20,000,000 =1,000,000

1 + r+

1,000,000(1.05)

(1 + r)2+

1,000,000(1.05)2

(1 + r)3+ · · · =

1,000,000

r − 0.05.

This implies that the cost of capital for Nilpoj is r = 10%.

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(b) At the end of the year, the value of the firm will be

V1 = 1,000,000

[

1.05

1.1+

(1.05)2

(1.1)2+ · · ·

]

= 1,000,000

[

1.05

0.1 − 0.05

]

= 21,000,000.

(c) Let n denote the number of new shares that will need to be issued, and P1 the price ofeach share (old and new) at the end of the first year. Since the total number of sharesafter the new issue will be 1,000,000 + n, we have

P1 =V1

1,000,000 + n=

21,000,000

1,000,000 + n. (5)

It also has to be the case that the extra dividend of $1 million paid at the end of thefirst year is financed by the new issue of shares (i.e., the new investors get what theypay for):

nP1 = 1,000,000. (6)

Using this last equation in (5) yields:

(1,000,000 + n)P1 = 21,000,000

(6)⇔ 1,000,000P1 + 1,000,000 = 21,000,000

⇔ P1 = 20.

We can now use this value in (6) to obtain

n =1,000,000

20= 50,000 shares.

(d) The new shareholders will be getting a fraction

50,000

1,000,000 + 50,000=

1

21

of future dividends, whereas the original shareholders will be getting a fraction 1− 1

21= 20

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of these dividends.

(e) The present value of the cash flows to the original shareholders is

PV = 1,000,000

{

2

1.1+

20

21

[

1.05

(1.1)2+

(1.05)2

(1.1)3+

(1.05)3

(1.1)4+ · · ·

]}

= 1,000,000

{

2

1.1+

20

21

[

1.05

0.1 − 0.05

1

1.1

]}

= 20,000,000.

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9. Let Sed denote the ex-dividend stock price. We then have the following cash flows associatedwith buying cum-dividend or ex-dividend:

Cash Flow Today Cash Flow in 1 Year

Buy cum-dividend −100 + 1 S1 − 0.20(S1 − 100) − 0.30(1)Buy ex-dividend −Sed S1 − 0.20(S1 − Sed)

Difference in CF’s −99 + Sed 19.7 − 0.20Sed

Investors would be indifferent between buying cum-dividend or ex-dividend if and only if thepresent value of the differential cash flow is zero, i.e., if

−99 + Sed +19.7 − 0.20Sed

1.10= 0.

Solving, gives Sed = 99.111. Therefore, after the dividend payment, the stock price drops byless than $1. This reflects the fact that capital gains are taxed at a lower rate than dividends.

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