Problem Set 3 - Solutions PDF

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FM212 - Principles of Finance Michaelmas Term: Asset Pricing Problem Set 3 Solutions - Introduction to Bonds and Common Stocks

1. Suppose you purchase a ten-year bond with 6% annual coupons and a par value of £100. You hold the bond for four years, and sell it immediately after receiving the fourth coupon. If the bond’s yield to maturity was 5% when you purchased and also sold the bond what are the cash flows from your investment in the bond over the four years? Solution: First, we compute the initial price of the bond by discounting its 10 annual coupons of £6 and final face value of £100 at the 5% yield to maturity:

P0 = £6

"

1 (1.05)10

1−

0.05

#

+

£100 (1.05)

10

= £107.72

Therefore the the first cash flow at t = 0 is -£107.72. At t = 1, t = 2 and t = 3 the cash flows from the coupons are £6. At t = 4 when you sell the the bond just after the fourth coupon has been paid the price of the bond is:

P4 = £6

"

1−

1 (1.05)6

0.05

#

+

£100 6 = £105.08 (1.05)

Therefore therefore the last cash flow at t = 4 is the price of the bond of £105.08 plus the last coupon of £6. In summary: t=0 Cash Flow

-£107.72

t=1 t=2 t=3 £6 1

£6

£6

t=4 £111.08

2. Crecimiento s.a. currently plows back 30% of its earnings and earns a return of 25% on this investment. The forward looking dividend yield on the stock is 5%. (a) Assuming that Crecimiento can continue to plow back this proportion of earnings and earn a return of 25% on the investment, how rapidly will earnings and dividends grow? What is the expected return on Crecimiento’s stock? (b) Suppose that management suddenly announces that future investment opportunities have dried up. Now Crecimiento intends to pay out all its earnings. How will the stock price change in percentage terms? (c) Suppose that management simply announces that the expected return on new investment will in the future be the same as the cost of capital. Now what is Crecimiento’s stock price? Solution (a): We know that g, the growth rate of dividends and earnings, is given by:

g = ROE × Plowback Ratio g = 0.25 × 0.3 = 0.075 We also know that from rearranging the Gordon Growth formula: E (r) =

D1 +g P0

E (r) = 0.05 + 0.075 = 0.125 Solution (b): Method 1 We know that: D1 = 0.05 P0 Therefore: D1 = 0.05 × P0 A plowback ratio of 0.3 implies a payout ratio of 0.7, and hence:

2

D1 = 0.7 EP S1 And therefore: D1 = 0.7 × EP S1 Equating these two expressions for D1 gives a relationship between price and earnings per share: 0.05 × P0 = 0.7 × EP S1 0.05 EP S1 = P0 0.7 Also, we know from the lecture notes that:   P V GO EP S1 = E (r) 1 − P0 P0 Therefore:  0.05  P V GO = 1 − 0.7 = 0.4286 P0 0.125 The ratio of the present value of growth opportunities to price is 42.86 percent. Thus, if suddenly there are no future investment opportunities, the stock price will decrease by 42.86 percent. Method 2 Remember from the Gordon Growth Model that: P0 =

D1 E (r) − g

Which we can also write as: P0 =

EP S1 (1 − Plowback Ratio) E (r) − g

3

We also know that the price with no growth, P0,NG is equal to: P0,NG =

EP S1 E (r)

Hence, the percentage change in price is: P0,NG P0,NG − P0 −1 = P0 P0 Plugging in for P0 and P0,N G : P0,NG −1= P0



EP S1 E(r)



EP S1 (1−Plowback Ratio) E(r)−g

−1

Canceling through EP S1 : E (r) − g P0,NG −1= −1 E (r) (1 − Plowback Ratio) P0 Plugging in the numbers: P0,NG 0.125 − 0.075 −1= − 1 = −0.4286 0.125 (1 − 0.3) P0 Thus, if suddenly there are no future investment opportunities, the stock price will decrease by 42.86 percent. Solution (c): In Part (b), all future investment opportunities are assumed to have a net present value of zero. If all future investment opportunities have a rate of return equal to the capitalization rate, this is equivalent to the statement that the net present value of these investment opportunities is zero. Hence, the impact on share price is the same as in Part (b). 3. (a) Complete the table below. Assume for Year 1 onwards any cash flows occur at the end of the year. (b) Assume that the opportunity cost of capital is 12%. Calculate the value of the company’s stock (assume that after year 4 the company grows at a constant rate, i.e. the growth rate stays constant at year 4 level). (c) What part of that value reflects the discounted value of P3 , the price forecasted for year 3? 4

Book equity per share Earnings per share (EPS) Return on Equity (ROE) Payout Ratio Dividends per share Growth rate of dividends

Today 10 -

Year 1 Year 2 Year 3 Year 4

0.25 0.20

0.25 0.20

0.16 0.50

0.16 0.50

-

(d) What part of P3 reflects the present value of growth opportunities (PVGO) after year 3? (e) Suppose that competition will catch up with the company by year 4, so that it can earn only its cost of capital on any investments made in year 4 or subsequently. What is the stock worth now under this assumption? (Make additional assumptions if necessary). Solution (a): Since we know the following: ROEt =

EP St Book Equity per Sharet−1

Dt = EP St × Payout Ratiot

Book Equity per Sharet = Book Equity per Sharet−1 + EP St (1 − Payout Ratiot ) | {z } Retained Earningst

We can then complete the table: Book equity per share Earnings per share (EPS) Return on Equity (ROE) Payout Ratio Dividends per share Growth rate of dividends

Today 10 -

Year 1 Year 2 Year 3 Year 4 12 14.40 15.55 16.79 2.50 3.00 2.30 2.49 0.25 0.25 0.16 0.16 0.20 0.20 0.50 0.50 0.50 0.60 1.15 1.24 0.20 0.92 0.08

Solution (b): The company’s stock price should be:   £1.15 1 £0.5 £1.24 £0.6 + + = £23.81 P0 = + 1.12 (1.12)2 (1.12)3 (1.12)3 0.12 − 0.08 Solution (c): The horizon value contributes: 5

P V (PH ) =

1 (1.12)3



 1 £1.24 = 3 (£31) = £22.07 0.12 − 0.08 (1.12)

Solution (d): Without PVGO, P3 would equal earnings for year 4 capitalized at 12 percent: P3NG =

£2.49 = £20.75 0.12

Therefore: PVGO = £31.00 − £20.75 = £10.25 Solution (e): The PVGO of £10.25 is lost at year 3. Therefore, the current stock price of £23.81 will decrease by: £10.25 (1.12)

3

= £7.30

The new stock price will be: £23.81 − £7.30 = £16.51

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