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# Chapter 7

Download Presentation ## Chapter 7

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1. ### Chapter 7

Equity Markets & Stock Valuation
2. Outline: Common Stock Valuation Features of Common and Preferred Stocks The Stock Markets Concepts and Skills: Stock prices depend on: Future dividends The growth rate of those dividends Compute stock prices using the dividend growth model Understand how corporate directors are elected Understand how stock markets work Understand how stock prices are quoted
3. 7.1 Common Stock Valuation Share of Ownershipentitles you to: Share of Profits Share of Equity (Equity = Assets – Liabilities) If the company is liquidated Which means the assets are sold And liabilities (bonds and other debts) are paid What is left (the residual) goes to the shareholders So shares of stock are sometimes called Residual Claims Share of vote for Board of Directors A share of stock is worth the PV the money received: Dividends paid while the stock is held Sale Price of the stock
4. Example: A stock is held for 1 year: It will pay a \$2 dividend in one year It can be sold for \$14 in one year The required return is 20% on investments with this risk Calculate the most you are willing to pay. Notation: P0 = Price at time 0 D1 = Dividend at time 1 P1 = Price at time 1
5. Example: Same stock is held for 2 years: Div at time 2 = \$2.10 D1= \$2.00 and D2 = \$2.10 Price at time 2 = P2 = \$14.70 Still require a 20% return Cash flow diagram:
6. Clicker Question: A share of stock will pay a \$1 dividend in one year and a \$1.25 dividend in two years The price of the stock in two years will be \$25 Calculate the current price of the stock assuming a required return of 10% \$20.00 \$21.69 \$22.60 \$24.77 \$27.25
7. Clicker Answer: D1 = \$1 D2 = \$1.25 P2 = \$25 R = 10% P0 = D1/(1 + R) + (P2 + D2)/(1 + R)2 = \$1/(1.1) + (\$25 + \$1.25)/(1.1)2 = \$0.91 + \$21.69 = \$22.60 The answer is C.
8. General Formula For a Stock’s Price: The price at any time (including now, P0) is the present value of all future cash flows For a stock the CFs are called dividends But if we sell the stock… Then we get the Price at the time it is sold How do we calculate the price at the time of sale?
9. Derivation of the General Formula (Part 1): P0 is a function of D1 and P1 But P1 is a function of D2 and P2. So Sub for P1 in P0 equation: So now P0 is a function of D1, D2 and P2(but not P1 anymore)
10. Derivation of the General Formula (Part 2): So now P0 is a function of D1, D2 and P2 But P2 can be written as a function of D3 and P3 Sub for P2in the first equation to get P0 as a function D1, D2, D3 and P3: But P3 can be written as a function of D4and P4and so So get P0 as a function D1, D2,D3 , D4…
11. Discussion of the General Formula The price at any time (for example, now) equals: The present value of all future cash flows! A stock’s CFs are called “dividends” Its as if we just keep pushingback the sale timeto include the nextdividendand then the next dividendand so on… So instead of estimating a future sale price… all we have to do is EstimateAll Future Dividends
12. How can we estimate All Future Dividends? To calculate the current price (P0), we need to know the sale price at some future time… OR Since a future sale price is a function of all the dividends that come after that time We can push back the sale price forever! We just need to estimate all future dividends
13. All we need is an estimate of all future dividends… So the price now is the PV of all future dividends… How can we estimate all future dividends? We cant! But we can make some simplifying assumptions We’ll look at Three Assumptionsabout dividends And these will be our “Base Cases” Then we will develop formulas to calculate a stock’s price for the each of these three dividend assumption…
14. Three Dividend Assumptions Our “Base Cases” Dividends are Constantforever Dividends are \$0.50 each year forever… Dividends have Constant Growthforever The first dividend is \$0.50, and it grows by 5% per year forever… Dividends Grow at a High Rate for a few years then slow to Constant Growthforever The first dividend is \$0.50, it grows by 15% for 3 years, then growth decreases to 5% per year forever…
15. Dividend Assumption One: Dividends are the constant forever: D0= D1 = D2 = … So all dividends are \$1.00 forever Since all the D’s are the same, we don’t need subscripts… D0 = D1 = D2 = D = \$1.00 Call this “Constant Dividends” or “Zero Growth”
16. Dividend Assumption Two: Dividends growat a constant rate(g): D1 = D0(1 + g) D2 = D1(1 + g) D3 = D2(1 + g) In general: Dt +1 = Dt(1 + g) If the dividend at time 0 is \$1.00 (D0 = 1.00) And grow at 5% forever (g= 0.05): D1 = 1.00(1.05) = \$1.05 D2 = 1.05(1.05) = 1.00(1.05)2 = \$1.1025 D3 = 1.1025(1.05) = 1.00(1.05)3 = 1.331 Call this “Constant Dividend Growth”
17. Dividend Assumption Three: Dividends grow at a high rate for a few years (g1) Then slow to a Constant Growth Rate (g2) forever Dividends start at \$1, grow at 20%for 2 years and then grow at 5% forever: g1 = 20% and g2 = 5% D1 = D0(1 + g1) = \$1.00(1.20) = \$1.20 D2 = D1(1 + g1) = \$1.20(1.20) = \$1.44 D3 = D2(1 + g2) = \$1.44(1.05) = \$1.512 D4 = D3(1 + g2) = \$1.512(1.05) = \$1.5876 Call this “Super-Normal Growth” (20% growth can’t last for very long!)
18. Calculating P0 under Assumption 1: Dividendsare Constant Forever: D0= D1 = D2 = D3 = D (So D with no subscript) This is just the Present Value of a Perpetuity: So the value of a stock with CONSTANT DIVIDENDSis:
19. The Constant Dividend Model: Dividends are constant forever Zero Dividend Growth Does this seem silly? Maybe not. If dividends grow just at the rate of inflation then in real terms, dividends are constant: Let NominalDividend Growth = g = 3% Let Expected Inflation= h = 3% RealDividends: D0= D1 = D2 = D
20. Constant Dividends (Zero Growth) Example 1 A stock will pay a dividend of \$1.50 per year forever The required annual discount is 8% P0 = D/R = \$1.50/.08 = \$18.75 Note that we could also has written: The Inflation-Adjusted Dividend is \$1.50 per year The Real Discount Rate is 8%
21. Constant Dividends Example 2 A stock will pay a no dividends until time 4 Then it will pay a constant dividend of \$2 per year forever The required annual discount is 5% Calc the price at t = 3 (P3) using the first dividend (D4): P3 = D/R = \$2/0.05 = \$40.00 Then take the present value of P3 back to P0: P0= P3/(1+R)3 = \$40/(1.05)3 = \$34.55 Why do we value at time 3 first? Because the dividends start at time 4, and we have an easy formula to value at time 3
22. Clicker Question: A stock will pay no dividends until 10 years from now It’s first dividend will be at time 10 It will then pay an annual dividend of \$3 per year forever Calculate the current price (P0) of the stock assuming a required return of 10% \$12.72 \$27.56 \$30.00 \$32.56 \$72.12
23. Clicker Answer: First: Value the stock at time 9 (P9) Using the time dividend at time 10 (D10) : P9 = D10/R = \$3/0.10 = \$30 Second: Value the stock at time 0 (P0) by taking the PV the time 9 value(the PV or P9): P0 = P9/(1 + R)9 = \$30/(1.10)9 = \$12.72 OR: N=9 R=10 FV=30 PV = -12.72 The Answer is A Why do we first value the stock at time 9? Because the CFs start at time 10 and we have an easy formulato value the stock at time 9
24. Constant Dividends Example 3 Quarterly Dividends: Starting next quarter, a stock will pay a \$0.30 dividend per quarter forever. The required return is 10% APR-Quarterly So we nee to use the periodic rate = 0.10/4 = 0.025 Calculate the current price: P0 = D/R D = \$0.30 R = 0.10/4 = 0.025 P0 = D/R = 0.30/0.025 = \$12.00
25. Clicker Question: Starting next quarter, a stock will pay a dividend of \$0.50 per quarter forever. Calculate the current price of the stock assuming a required return of 16% APR Quarterly. \$3.13 \$12.00 \$12.50 \$16.00 \$16.25
26. Clicker Answer: R = 0.16/4 = 0.04 P0 = D/R = \$0.50/0.04 = \$12.50 The Answer is C Note that the question could have read: A stock will pay an annual dividend of \$2.00 in equal quarterly installments forever… Instead of: A stock will pay a dividend of \$0.50 per quarter forever…
27. Now to Dividend Assumption Two: Dividends grow at a constant rate (g) forever: D1 = D0(1 + g) D2 = D1(1 + g) D3 = D2(1 + g) In general: Dt +1 = Dt(1 + g) We need a good formula for calculating the price (P0) under this assumption: Here’s the Constant Dividend Growth Rate Formula Using the dividend just paid (D0 ) & the nextdividend (D1):
28. Derivation of the Constant Dividend Growth Rate Formula: Start with the general price formula: Since D1 is a function of D0 and D2 is a function of D1 and so on… We can substitute D0for all the future dividends: D1= D0(1 + g) D2= D1(1 + g) = D0(1 + g)2 D3= D2(1 + g) = D0(1 + g)3
29. Derivation of the Constant Dividend Growth Rate Formula: Substitute D0 for all of the future dividends… Factor out the D0: Rewrite the ratios with a single exponent
30. Derivation of the Constant Dividend Growth Rate Formula: It must be the case that g is less then R. Why? A company that grows faster than its discount rate will have infinite value! We’ll talk more about this can’t happen later… If g < R then (1 + g)/(1 + R) < 1 Increasing the exponent decreases the value: If g = 0.05 and R = 0.10 then (1.05)/(1.10) = 0.9545 (0.9545)2 = 0.91 (0.9545)3 = 0.87 (0.9545)4 = 0.83…
31. Derivation of the Constant Dividend Growth Rate Formula: Increasing the exponent decreases the value: If g = 0.05 and R = 0.10 then (1.05)/(1.10) = 0.9545 (0.9545)2 = 0.91 (0.9545)3 = 0.87 (0.9545)4 = 0.83… Adding more terms of decreasing value means the sum of the terms in in the brackets “converges” to a finite value:
32. Constant Dividend Growth Model Also called the Gordon Growth Model (GGM) or the Dividend Growth Model (DGM) We have two versions: The price at time 0 (P0) as a function of the current dividend (D0) Which was just paid an instant ago, so we don’t get that dividend! The price at time 0 (P0) as a function of the dividend in 1 period (D1) We can also value the stock at any time t in the future (Pt) Either as a function of the dividend at time t (Dt) or time t+1 (Dt+1):
33. Constant Dividend Growth Model Is constant growth at growth rate g a silly assumption? Maybe. Maybe not. If the company is in a mature industry, growth maybe at the rate of population growth Think about the Sustainable Growth Rate: (ROE x b)/(1 – ROE x b) This growth rate assumes constant efficiency and leverage ROE = ROA x EM = Efficiency x Leverage Constant dividend policy Plowback Ratio = b If these are true, then growth will be constant…
34. Example: Constant Dividend Growth A company just paid an annual dividend of \$0.50 The dividend will increase by 2% per year (forever) The required return on this stock is 15% Calculate the price: Note that the question states the dividend was “just paid” This means we are given D0 D0 = \$0.50 g = 2% R = 15% P0 = D0(1+g)/(R-g) = 0.50(1.02)/(0.15 - 0.02)= \$3.92
35. Example 2: Constant Dividend Growth A company will pay a dividend of \$0.40 in one year The dividend will increase by 3% per year (forever) The required return on this stock is 13% Calculate the price: Note that the question states “will pay” This means we are given D1 D1= \$0.40 g = 3% R = 13% P0 = D1/(R - g) = 0.40/(0.13 - 0.03) = \$4.00
36. Clicker Question: A company just paid a dividend of \$1.00 and it will pay its next dividend in exactly one year The dividends will increase by 4% per year (forever) The required return on this stock is 10% Calculate the price. \$14.00 \$14.33 \$16.67 \$17.33 \$18.33
37. Clicker Answer: D0 = \$1.00 g = 4% R = 10% D1 = D0(1 + g) = \$1.00(1.04) = \$1.04 P0 = D1/(R - g) = 1.04/(0.10 - 0.04) = \$17.33 The answer is D
38. Example 3: Gordon Growth Company (Text Example 7.3, page 212) GGC will pay a dividend of \$4 next period Dividends will to grow at 6% per year The required return is 16%. What is the current price? P0 = D1/(R - g) = 4/(0.16 - 0.06) = \$40
39. GGC Continued: (Same Example) What is the price expected to be in year 4? (Use D4 to calculate P4) D4 = D1(1 + g)3 = \$4(1+.06)3 = \$4.76 P4= D4(1 + g) /(R – g) P4= \$4.76(1.06) /(0.16 – 0.06) = \$50.50 You can also use D5 to calculate P4 D5 = D1(1 + g)4 = \$4(1+.06)4 = \$5.05 P4= D5/(R – g) P4= \$5.05/(0.16 – 0.06) = \$50.50
40. GGC Continued: What is the 4 yr return on the price of the stock? P0 = \$40 and P4 = \$50.50 FV = PV(1 + R)t Return = (FV/PV)(1/t) – 1 = (P4/P0)(1/t) – 1 = (\$50.50/\$40)1/4 – 1 = 0.06 =6% PV = -40 FV = 50.50 N = 4 Solve for R = 6% Recall g = 6% Price grows at the same rate as dividend growth!
41. GGC Continued: What is the Return from t = 4 to t = 5? First we need to calculate P5: P5= D6/(R – g) = \$5.05(1.06)/(0.16 – 0.06) = \$5.353/0.1 = \$53.53 Return = P5/P4 – 1 Return = \$53.53/\$50.50 – 1 = 0.06 = 6% Let’s see why 
42. GGC Continued: What is the four-year implied price return? P0 = \$40 and P4 = \$50.50 FV = PV(1 + R)t Or: PV = -40 FV = 50.50 N = 4 I/Y = 6% Recall g = 6% So the price grows at the dividend growth rate or The Capital Gain Rate = The Dividend Growth Rate!
43. GGC Continued: At what rate do prices grow? In other words: What is the Capital Gain Ratein any year? (some algebra): So the dividend growth rate (g) is both: The Dividend Growth Rate The Capital Gain Rate
44. More about the Constant Dividend Growth Model: P0 = D1/(R – g) If g > R in the above equation, then P0is negative So for the model to work, it must be the case that R > g Note: If g > R, then (1+g)/(1+R) > 1 And then we couldn’t have simplified the equation on slide 29 But what if growth does exceed the discount rate? At least for a short period of time… Then we will have to use another model (Assumption 3) Eventually growth will have to slow… Because if g > R forever, the stock has infinite value: If dividends (and therefore prices) grow faster than the discount rate Then the PV of each successive term is larger than the last The 100th dividend is worth more in PV terms than the 1st dividend
45. Non-Constant Growth Dividend Growth will be high for a while Maybe even higher than R Then growth will decrease and become constant forever Example: A company with a required return of 10% just paida dividend of \$0.47 (So D0 = \$0.47 and R = 10%) Dividends will grow 16% for the next five years Call this g1 = 16% After which dividend growth will slow to 6% forever Call this g2 = 6% (Note that g2 < R) We’ll calculate the price in pieces 
46. Calculate P0 for Non-Constant Growth – Step 1 First assume NO dividends until \$1.00 at t = 5 Then constant dividend growth at 6% forever Calculate P0 if the correct discount rate is 10%: D5 = \$1.00 g = 0.06 R = 0.10 If we have D5, we can calculate P4 Then discount P4 back 4 periods to time 0 to get P0: P4 = D5/(R – g) = \$1.00(0.10 – 0.06) = \$25.00 P0 = P4/(1 + R)4 = \$25.00/(1.10)4 = \$17.08
47. Calculate P0 for Non-Constant Growth – Step 2 Now add D1, D2, D3 and D4 and discount these back to t = 0 The company just paid a dividend of \$0.47 (D0 = \$0.47) Dividends will grow 16% for the next five periods (After which dividend growth will slow to 6% forever) D1 = D0(1.16) = \$0.47(1.16) = \$0.55 D2 = D1(1.16) = \$0.55(1.16) = \$0.63 D3 = D2(1.16) = \$0.64(1.16) = \$0.73 D4 = D3(1.16) = \$0.74(1.16) = \$0.85 D5 = D4(1.16) = \$0.86(1.16) = \$1.00 Note that D5 was already included in the calculation of P4
48. Now calculate the price of the stock: The company just paid a \$0.47 dividend Dividends will grow at 16% for 5 years (g1 = 0.16) Then Dividends will grow at 6% forever (g2 = 0.06) Calculate the price if R = 10%: P0 = D1/(1 + R) + D2/(1 + R)2+ D3/(1 + R)3+ D4/(1 + R)4+ P4/(1 + R)4 = D1/(1 + R) + D2/(1 + R)2+ D3/(1 + R)3+ D4/(1 + R)4+ [D5/(R – g2)]/(1 + R)4 = .55/(1.10) + .63/(1.10)2 + .73/(1.10)3 +.85/(1.10)4 + [1.00/(.10 – .06)]/(1.10)4 = \$0.50 + \$0.53 + \$0.56 + \$0.59 + \$17.08 = \$19.26
49. Another Example: A company just paid a \$5m total dividend (not per share) (D0 = 5.00) Dividends will grow at 30% for 3 years (g1 = 0.30) Then Dividends will grow at 10% forever (g2 = 0.10) Calculate the price if R = 20%: Since growth is constant after time 3, calculate P2 using D3 D1 = 6.5; D2 = 8.45; D3 = 10.985 P0 = D1/(1 + R) + D2/(1 + R)2+ P2/(1 + R)2 P2 = D3/(R – g2)] = 10.985/(0.2 – 0.1) = 109.85 P0 = 6.5/(1.2) + 8.45/(1.2)2 + 109.85/(1.2)2 P0 = 5.42 + 5.87 + 76.28 = \$87.56 This is the company’s total value, not per share
50. Another Way To Do The Same Example Example 7.4, Page 214 Recall: D0 = 5, g1 = 0.30 for 3 years, g2 = 0.10 for ever, R = 20% Use D4 to calculate P3: P3 = D4/(R – g2) But D4 = D3(1 + g2) so rewrite P3: P3 = D3(1 + g2)/(R – g2) = \$10.985(1.10)/(0.2 – 0.1) = \$120.835 P0 = D1/(1 + R) + D2 /(1 + R)2 + D3/(1 + R)3 + D3/(1 + R)3 + P3/(1 + R)3 P0 = 6.5/(1.2) + 8.45/(1.2)2 + 10.985/(1.2)3 + 120.835/(1.2)3 P0 = 5.42 + 5.87 + 6.36 + 69.93 = \$87.57 P0 = 5.42 + 5.87 + 76.28 = \$87.56 (previous slide) \$76.28 includes D3 and all future dividends. \$69.93 includes D4 and all future dividends.
51. Clicker Question: A stock’s just paid a dividend of \$2.50. The dividend will grow at 20% for 2 years and then grow at 5% forever. The required discount rate is 10%. Calculate the price of the stock. Hint: D1 = \$3.00, D2 = \$3.60 and D3 = \$3.78 \$8.54 \$25.00 \$68.18 \$75.00 \$85.10
52. Clicker Answer: g1 = 20%, g2 = 5%, R = 10% D1 = \$3.00, D2 = \$3.60 and D3 = \$3.78 The first dividend to which g2 is applied is D2 So calculate P1 using D2 : P1 = D2/(R – g2) = \$3.60/(0.10 – 0.05) = \$72 Calculate P0 as the PV of D1 and P1: P0 = \$3/(1.1) + \$72/(1.1) = \$68.18 Why don’t I need to use D3? Because D3 is included in P1. What if I want to use D3 to calculate P2? P2 = D3/(R – g2) = \$3.78/(0.10 – 0.05) = \$75.60 P0 = \$3.00/(1.1) + \$3.60/(1.1)2 + 75.60/(1.1)2 = \$68.18 Either way we get the same answer. The answer is C.
53. Look at a Stock’s Required Return The required return is also called the discount rate So solve for R and see what the components are: P0= D1/(R – g) R = D1/P0 + g = Dividend Yield + Growth Rate D1/P0 = Dividend Yield g = Growth Rate So the Required Return is the sum of the Dividend Yield and the Growth Rate Note that if D is quarterly, then g is quarterly and R is quarterly
54. Look at a Stock’s Required Return (Cont) Recall the growth rate (g) is also the rate at which prices increase So g is also the Capital Gain Rate R = D1/P0 + g = Dividend Yield + Capital Gain Rate So the Required Return is also the sum of the Dividend Yield and the Capital Gain Rate
55. Required Return Example: The market price for a stock is \$40 (P0 = \$40) The next (annual) dividend will be \$1.40 (D1 = \$1.40) You expect the dividends to grow by 7% forever (g = 0.07) Calculate the Required Rate of Return: ` R = D1/P0 + g = \$1.40/\$40 + 0.07 = 0.105 = 10.50% R is also the return from holding the stock. Verify this: Calculating the price in one year : P1 = D1(1 + g)/(R – g) = \$1.40(1.07)/(0.105 – 0.07) = \$42.80 The Total Return Cap Gain plus the Dividend Total Return = (P1 + D1)/P0 – 1 = (\$42.80 + 1.40)/\$40 – 1 = 10.50% = R
56. Required Return Example 2: Same stock but now the market price for the stock is \$35 (P0 = \$35) The next (annual) dividend will still be \$1.40 (D1 = \$1.40) You still expect the dividends to grow by 7% forever (g = 0.07) What has happened to the return from holding the stock? Calculate the Required Rate of Return: R = D1/P0 + g = \$1.40/\$35 + 0.07 = 0.11 = 11.00% Price decreased, the return from holding the stock increased Verify: P1 = D1(1 + g)/(R – g) = \$1.40(1.07)/(0.11 – 0.07) = \$37.45 Total Return = (\$37.45 + 1.40)/\$35 – 1 = 0.11 = 11% = R Now you are only paying \$35 for the same \$1.40 dividend. Dividend yield was \$1.40/\$40 = 3.5% Dividend yield is now \$1.40/\$35 = 4.0%
57. Clicker Question: A stock’s dividend yield is 6% Its dividend growth rate is 4% forever. Calculate the required return of the stock. 6% 7% 8% 9% 10%
58. Clicker Answer: Dividend Yield = D1/P0 = 0.06 Growth Rate = g = 0.04 R = D1/P0 + g = 0.06 + 0.04 = 0.10 = 10% The Answer is E.
59. Recap: Table 7.1, Page 217
60. 7.2 Features of Common and Preferred Stock Preferred Stock (usually) has: No Voting Rights Preference over common stock in Receiving Dividends Preference over common in Distribution of Assets in the Event of Liquidation Preferred stock holders are after bond holders and other creditors but ahead of common stock holders Common Stock (usually) has: The Right to Vote for Board of Directors (one share, one vote) Recall the Board Hires the Management No Preference in Receiving Dividends No Preference in Distribution of Assets in the Event of Liquidation