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ey Section 8 Shares and Their Valuation LEARNING OBJECTIVES After looking over this chapter, students should be able to: • Identify some of the more important legal rights that come with stock ownership and define the subsequent terms: web proxy, proxy deal with, takeover, and preemptive correct. • In brief explain so why classified inventory might be used by a corporation and what founders’ shares are. • Differentiate between strongly held and publicly owned or operated corporations and list the three distinct types of stock exchange transactions.

Determine the value of a share of common inventory when: (1) dividends are expected to grow at some continuous rate, (2) dividends are expected to remain regular, and (3) dividends are required to grow at some super-normal, or nonconstant, growth price. • Compute the anticipated rate of return on a constant growth stock. • Apply the overall company (corporate value) unit to worth a firm in case of when the company does not pay dividends or can be privately held. • Explain why a stock’s intrinsic benefit might differ between the total company style and the gross growth model. Explain the next terms: balance, marginal entrepreneur, and Efficient Markets Speculation (EMH), identify among the three levels of industry efficiency, quickly explain the implications with the EMH on financial decisions, and discuss the benefits of empirical studies about market effectiveness and the inference of behavioral finance about those effects. • Read and be familiar with stock market page given in the daily newspapers. • Explain the reasons pertaining to investing in international stocks and identify the “bets” a buyer is producing when he will invest abroad. Define desired stock, decide the value of a share of preferred stock, or offered its value, calculate its expected come back. 1 . SPIEL SUGGESTIONS This chapter delivers important and useful information concerning common and preferred shares. Moreover, the valuation of stocks reinforces the principles covered in both Chapters 6 and 7, thus Chapter eight extends and reinforces these chapters. All of us begin the lecture using a discussion of the functions of prevalent stocks, and we discuss how stocks and shares are highly valued in the market and exactly how stock prices are reported in the press. We consider the address with a exploration of preferred shares.

The details of what we cover, and the method we cover it, is visible by scanning services Blueprints Chapter 8. Pertaining to other suggestions about the lecture, you should see the “Lecture Suggestions” in Chapter 2, where all of us describe the way we conduct each of our classes. DAYS ON PART: 3 OF 58 DAYS (50-minute periods) ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1True. The significance of a share of inventory is the PHOTOVOLTAIC of it is expected foreseeable future dividends. In the event the two traders expect similar future dividend stream, and in addition they agree on the stock’s riskiness, then they should certainly reach related conclusions regarding the stock’s benefit. -2A perpetual bond is just like a no-growth stock and a talk about of desired stock in the following methods: 1 . All derive their very own values from a series of money inflows, promotion payments in the perpetual bond, and payouts from equally types of stock. installment payments on your All three happen to be assumed to obtain indefinite lives with no maturity value (M) for the perpetual connect and no capital gains produce for the stocks. 8-3Yes. If a organization decides to increase its payout ratio, then a dividend produce component can rise, nevertheless the expected long lasting capital profits yield will decline. 8-4No. The correct equation has D1 in the numerator and a minus sign in the denominator. -5a. The regular investor within a listed firm is not really considering maintaining his proportionate share of title and control. If he wanted to boost his control, he could simply purchase more share on the open market. As a result, most buyers are not concerned with whether new shares are sold directly (at about market prices) or through rights offerings. Nevertheless , if a legal rights offering will be used to effect a stock break up, or when it is being used to minimize the underwriting cost of a problem (by considerable underpricing), the preemptive correct may well be good for the firm and to their stockholders.. The preemptive proper is evidently important to the stockholders of closely organised firms whose owners are curious about maintaining all their relative control positions. SOLUTIONS TO END-OF-CHAPTER COMPLICATIONS 8-1D0 sama dengan $1. 50, g1-3 = 5%, gn = 10%, D1 through D5 sama dengan? D1 sama dengan D0(1 + g1) = $1. 50(1. 05) = $1. 5750. D2 sama dengan D0(1 & g1)(1 + g2) sama dengan $1. 50(1. 05)2 sama dengan $1. 6538. D3 sama dengan D0(1 & g1)(1 + g2)(1 + g3) sama dengan $1. 50(1. 05)3 = $1. 7364. D4 sama dengan D0(1 + g1)(1 & g2)(1 + g3)(1 & gn) sama dengan $1. 50(1. 05)3(1. 10) = $1. 9101. D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = $1. 50(1. 05)3(1. 10)2 = $2. 1011. 8-2D1 = $0. 60, g = 7%, ks = 15%, [pic] sama dengan? [pic] -3P0 = 20 dollars, D0 sama dengan $1. 00, g = 10%, [pic] =?, ks =? [pic] = P0(1 + g) = $20(1. 10) sama dengan $22. ks= [pic] & g = [pic] + 0. 15 = [pic] + 0. 10 = 15. fifty percent. ks sama dengan 15. fifty percent. 8-4Dp sama dengan $5. 00, Vp = $60, kp =? kp = [pic] = [pic] = eight. 33%. 8-5a. The fatal, or intervalle, date is a date if the growth price becomes constant. This arises at the end of Year 2 . b. zero 1 2 3 | | | | 1 ) 25 1 ) 50 1 . 80 1 ) 89 thirty seven. 80 = [pic] The horizon, or terminal, benefit is the worth at the horizon date coming from all dividends expected thereafter. In this problem it really is calculated the following: pic] c. The firm’s innate value can be calculated while the total of the present value coming from all dividends throughout the supernormal progress period and also present value of the airport terminal value. Utilizing your financial calculator, enter the subsequent inputs: CF0 = 0, CF1 = 1 . 55, CF2 = 1 . 70 + thirty seven. 80 = 39. 70, I sama dengan 10, then solve intended for NPV sama dengan $34. 09. 6. The firm’s free cash flow is usually expected to grow at a consistent rate, consequently we can apply a constant progress formula to determine the total value of the firm. Firm Worth = FCF1/(WACC – g) Firm Value = $150, 000, 000/(0. 10 , 0. 05) Firm Worth = $3, 000, 000, 000.

To find the value of an equity declare upon the business (share of stock), we have to subtract out your market value of debt and preferred stock. This company happens to be totally equity funded, and this stage is pointless. Hence, to find the value of any share of stock, we all divide collateral value (or in this case, company value) by the number of shares outstanding. Equity Value every share sama dengan Equity Value/Shares outstanding Fairness Value per share sama dengan $3, 1000, 000, 000/50, 000, 500 Equity Value per discuss = $60. Each talk about of common stock may be worth $60, in line with the corporate value model. 8-7a. 0 one particular 2 three or more 4 | | | | 3, 000, 500 6, 1000, 000 twelve, 000, 000 15, 000, 000 By using a financial calculator, enter the pursuing inputs: CF0 = 0, CF1 sama dengan 3000000, CF2 = 6000000, CF3 sama dengan 10000000, CF4 = 15000000, I sama dengan 12, and then solve for NPV sama dengan $24, 112, 308. w. The firm’s terminal worth is determined as follows: [pic] c. The firm’s total value is definitely calculated the following: 0 1 2 3 4 5 | | | | | | 3, 500, 000 six, 000, 000 10, 000, 000 15, 000, 000 16, 050, 000

PV =? 321, 000, 000 = [pic] Using your financial calculator, enter the following advices: CF0 sama dengan 0, CF1 = 3000000, CF2 = 6000000, CF3 = 10000000, CF4 sama dengan 15000000 & 321000000 = 336000000, We = 12, and then solve for NPV = $228, 113, 612. d. To look for Barrett’s share price, you need to first locate the value of its equity. The cost of Barrett’s value is equal to the value of the overall firm less the market value of their debt and preferred inventory. Total organization value$228, 113, 612 The true market value, debt & preferred 70, 000, 1000 (given in problem) The true market value of equity$168, 113, 612

Barrett’s selling price per talk about is calculated as: [pic] 8-8FCF = EBIT(1 – T) & Depreciation – [pic] , ([pic] = $500, 500, 000 + $100, 000, 000 , $200, 500, 000 , $0 sama dengan $400, 1000, 000. Organization value = [pic] = [pic] sama dengan [pic] sama dengan $10, 500, 000, 000. This is the total firm value. Now find the market worth of the equity. MVTotal= MVEquity + MVDebt $10,50, 000, 1000, 000= MVEquity + $3, 000, 500, 000 MVEquity= $7, 1000, 000, 1000. This is the market value of all the equity. Divide by the number of stocks and shares to find the selling price per talk about. $7, 500, 000, 000/200, 000, 1000 = 35 dollars. 00. 8-9a. Terminal worth = [pic] = [pic]= $713. 33 million.. 0 1 a couple of 3 5 | | | | | -15 30 forty five 42. eighty ($ 17. 70) twenty three. 49 522. 10 753. 33 $527. 89 Using a financial calculator, enter the next inputs: CF0 = 0, CF1 sama dengan -20, CF2 = 40, CF3 sama dengan 753. 33, I = 13, and after that solve pertaining to NPV sama dengan $527. fifth there�s 89 million. c. Total valuet=0 = $527. 89 , 000, 000. Value of common value = $527. 89 , $100 = $427. 89 million. Cost per share = [pic] = $42. 79. 8-10The problem requests you to identify the value of [pic], presented the following details: D1 sama dengan $2, w = zero. 9, kRF = a few. %, RPM = 6%, and P0 = $25. Proceed as follows: Step 1 : Estimate the required price of return: ks sama dengan kRF + (kM , kRF)b sama dengan 5. 6% + (6%)0. 9 = 11%. 2: Use the continuous growth price formula to calculate g: [pic] Step three: Calculate [pic]: [pic] = P0(1 + g)3 = $25(1. 03)3 sama dengan $27. 3182 ( $27. 32. Otherwise, you could calculate D4 then use the constant growth level formula to solve for [pic]: D4 = D1(1 + g)3 = $2. 00(1. 03)3 = $2. 1855. [pic] = $2. 1855/(0. eleven – 0. 03) sama dengan $27. 3182 ( $27. 32. 8-11Vp = Dp/kp, therefore , kp = Dp/Vp. a. kp = $8/$60 = 13. 3%. n. kp sama dengan $8/$80 sama dengan 10. 0%. c. l = $8/$100 = almost eight. 0%. m. kp sama dengan $8/$140 sama dengan 5. seven percent. 8-12[pic] 8-13a. ki sama dengan kRF + (kM , kRF)bi. kC = 9% + (13% , 9%)0. 4 sama dengan 10. 6%. kD = 9% + (13% , 9%)(-0. 5) = seven percent. Note that in pieces is under the risk-free price. But since this stock is similar to an insurance policy since it “pays off” when anything bad takes place (the marketplace falls), the lower return can be not irrational. b. Through this situation, the expected charge of return is as follows: [pic] sama dengan D1/P0 + g = $1. 50/$25 + 4% = 10%. However , the mandatory rate of return can be 10. 6th percent. Traders will seek to sell the stock, shedding its price to the pursuing: pic] At this point, [pic], plus the stock will probably be in balance. 8-14Calculate the dividend funds flows and place them on a time line. As well, calculate the stock selling price at the end from the supernormal progress period, including it, along with the dividend to become paid by t sama dengan 5, because CF5. In that case, enter the funds flows since shown around the time line into the cash flow signup, enter the required rate of return as I = 15, and then discover the value of the stock using the NPV calculations. Be sure to get into CF0 = 0, or else your solution will be incorrect. D0 = 0, D1 = 0, D2 = 0, D 3 = 1 ) 0, D4 = 1 ) 00(1. 5) = 1 . 5, D5 = 1 ) 00(1. 5)2 = 2 . 25, D6 = 1 . 00(1. 5)2(1. 08) = $2. 43. [pic] sama dengan? 0 1 2 3 4 five 6 | | | | | | | 1 . 00 1 . 60 2 . twenty-five 2 . 43 0. 658 +34. 71 = 0. 858 18. 378 thirty six. 96 $19. 894 sama dengan [pic] [pic] = D6/([pic] , g) = $2. 43/(0. 12-15 , zero. 08) = $34. 71. This is the inventory price towards the end of 12 months 5.

CF0 = zero, CF1-2 sama dengan 0, CF3 = 1 ) 0, CF4 = 1 . 5, CF5 = thirty eight. 96, I = 15%. With these kinds of cash runs in the CFLO register, press NPV to have the value in the stock today: NPV sama dengan $19. fifth 89. 8-15a. The most liked stock pays $8 yearly in returns. Therefore , their nominal level of return would be: Nominal rate of return = $8/$80 sama dengan 10%. Or perhaps alternatively, you could determine the security’s periodic return and multiply by simply 4. Routine rate of return sama dengan $2/$80 = 2 . five per cent. Nominal charge of go back = installment payments on your 5% ( 4 = 10%. n. EAR = (1 & NOM/4)4 , 1 EAR = (1 + zero. 10/4)4 , 1 EAR = 0. 103813 = 10. 3813%. -16The worth of virtually any asset is the present value of all long term cash runs expected to end up being generated from your asset. Consequently, if we will find the present benefit of the returns during the period preceding long-run constant development and take away that total from the current stock value, the remaining value would be the present value in the cash runs to be received during the period of long-run constant expansion. D1 sama dengan $2. 00 ( (1. 25)1 sama dengan $2. 50PV(D1) = $2. 50/(1. 12)1= $2. 2321 D2 = $2. 00 ( (1. 25)2 = $3. 125PV(D2) = $3. 125/(1. 12)2= $2. 4913 D3 sama dengan $2. 00 ( (1. 25)3 sama dengan $3. 90625PV(D3) = $3. 0625/(1. 12)3= $2. 7804 ( PV(D1 to D3)= $7. 5038 Therefore , the PV of the remaining payouts is: $58. 8800 – $7. 5038 = $51. 3762. Compounding this benefit forward to 12 months 3, we find that the worth of all payouts received during constant expansion is $72. 18. [$51. 3762(1. 12)3 = $72. 18. ] Applying the constant growth formula, we can fix for the constant growth price: [pic]= D3(1 + g)/(ks – g) $72. 1807= $3. 90625(1 + g)/(0. 12 – g) $8. 6616 , $72. 18g= $3. 90625 + $3. 90625g $4. 7554= $76. 08625g 0. 0625= g 6. 25%= g. 8-17First, solve intended for the current selling price. P0 sama dengan D1/(ks – g) P0 = $0. 50/(0. two , 0. 07) P0 = $12. 00. In the event the stock is at a constant expansion state, the dividend development rate is likewise the capital gains yield to get the inventory and the stock price expansion rate. Therefore, to find the price of the inventory four years from today: [pic] sama dengan P0(1 + g)4 [pic] = $12. 00(1. 07)4 [pic] sama dengan $13. 10796? $13. 11. [pic] 8-18a. [pic] m. [pic] 8-19 0 you 2 3 4 | | | | | D0 = 2 . 00 D1 D2 D3 D4 g = 5% [pic] a. D1 = $2(1. 05) sama dengan $2. twelve, D2 sama dengan $2(1. 05)2 = $2. 21, D 3 = $2(1. 5)3 sama dengan $2. 32. b. Economical Calculator Option: Input zero, 2 . 10, 2 . 21, and 2 . 32 in the cash flow register, input My spouse and i = doze, PV =? PV = $5. 29. c. Financial Calculator Answer: Input zero, 0, zero, and thirty four. 73 into the cash flow register, I sama dengan 12, PHOTOVOLTAIC =? PHOTO VOLTAIC = $24. 72. m. $24. seventy two + $5. 29 = $30. 01 = Optimum price you must pay for the stock. at the. [pic] farreneheit. No . The value of the share is certainly not dependent upon the holding period. The value determined in Parts a through m is the worth for a 3-year holding period. It is comparable to the value calculated in Part e except for a small rounding mistake.

Any other having period might produce the same value of [pic], that is, [pic] = $30. 00. 8-20a. 1 . [pic] 2 . [pic] = $2/0. 15 sama dengan $13. 33. 3. [pic] 4. [pic] b. 1 . [pic] = $2. 30/0 = Undefined. 2 . [pic] = $2. 40/(-0. 05) = -$48, which is nonsense. These results show the formula will not make sense if the required level of returning is corresponding to or below the predicted growth level. c. Number 8-21The solution depends on when one works the problem. We used the February a few, 2003, concern of The Wsj: a. $16. 81 to $36. 72. b. Current dividend sama dengan $0. seventy five. Dividend yield = $0. 75/$19. 8 ( several. 9%. You might want to use ($0. 75)(1 & g)/$19. twenty four, with g estimated in some manner. c. The $19. twenty four close was up $0. 98 through the previous day’s close. d. The return on the inventory consists of a dividend yield of approximately 3. being unfaithful percent and several capital gains yield. We might expect the total rate of return upon stock being in the 15 to 12 percent selection. 8-22a. End of Yr: 02 goal 04 05 06 ’07 08 | | | | | | | D0 = 1 . 75 D1 D2 D3 D4 D5 D6 Dt= D0(1 + g)t D2003= $1. 75(1. 15)1 = $2. 01. D2004= $1. 5(1. 15)2 sama dengan $1. 75(1. 3225) = $2. 31. D2005= $1. 75(1. 15)3 = $1. 75(1. 5209) = $2. 66. D2006= $1. 75(1. 15)4 = $1. 75(1. 7490) sama dengan $3. 06. D2007= $1. 75(1. 15)5 = $1. 75(2. 0114) = $3. 52. b. Step 1 : PV of returns = [pic]. PHOTOVOLTAIC D2003 = $2. 01/(1. 12)= $1. 79 PHOTOVOLTAIC D2004 = $2. 31/(1. 12)2= $1. 84 PHOTOVOLTAIC D2005 = $2. 66/(1. 12)3= $1. 89 PHOTO VOLTAIC D2006 = $3. 06/(1. 12)4= $1. 94 PHOTO VOLTAIC D2007 = $3. 52/(1. 12)5= $2. 00 PV of dividends= $9. 46 Step 2: [pic] This is the value of the stock 5 years from today. The PHOTO VOLTAIC of this cost, discounted back again 5 years, is as uses: PV of [pic] = $52. 80/(1. 12)5 = $29. 6. Step 3: The buying price of the share today is really as follows: [pic]sama dengan PV dividends Years 2003-2007 + PHOTOVOLTAIC of [pic] = $9. 46 & $29. 96 = $39. 42. This problem could also be resolved by substituting the proper principles into the following equation: [pic]. Calculator solution: Suggestions 0, 2 . 01, installment payments on your 31, installment payments on your 66, a few. 06, 56. 32 (3. 52 & 52. 80) into the cashflow register, insight I sama dengan 12, PV =? PV = $39. 43. c. 2003 D1/P0 = $2. 01/$39. 43= 5. 10% Capital increases yield= six. 90* Anticipated total return= 12. 00% 2008 D6/P5 = $3. 70/$52. 80= 7. 00% Capital increases yield= 5. 00 Anticipated total return= 12. 00% We know that ks is 12 percent, and the dividend produce is 5. 10 percent, therefore , the capital benefits yield has to be 6. 90 percent. The main points to note here are the following: 1 . The whole yield is often 12 percent (except intended for rounding errors). 2 . The main city gains deliver starts fairly high, in that case declines as the supernormal growth period approaches its end. The dividend yield rises. several. After 12/31/07, the inventory will increase at a 5 percent charge. The gross yield will equal six percent, the administrative centre gains yield will equal 5 percent, and the total return will be doze percent. m.

People in high income tax brackets is often more inclined to purchase “growth” shares to take the capital gains and so delay the payment of taxes until a later date. The firm’s inventory is “mature” at the end of 2007. at the. Since the business’s supernormal and normal expansion rates are lower, the dividends and, hence, the present value with the stock price will be lower. The total come back from the share will still be doze percent, nevertheless the dividend yield will be larger and the capital gains yield will be smaller than they were with all the original progress rates. This result takes place because we assume the same last gross but a much lower current stock selling price.. As the necessary return improves, the price of the stock falls off, but the capital benefits and gross yields enhance initially. Of course , the long-term capital profits yield remains to be 4 percent, so the long lasting dividend produce is 10 percent. 8-23a. Component 1: Visual representation in the problem: Supernormal Normal expansion growth zero 1 two 3 ( | | | | ••• | D0 D1 (D2 + [pic]) D 3 D( PVD1

PVD2 [pic] P0 D1 = D0(1 + gs) = $1. 6(1. 20) = $1. 92. D2 = D0(1 + gs)2 = $1. 60(1. 20)2 = $2. 304. [pic] [pic]= PV(D1) + PV(D2) + PV([pic]) = [pic] = $1. 92/1. twelve + $2. 304/(1. 10)2 + $61. 06/(1. 10)2 = $54. 11. Economical Calculator solution: Input 0, 1 . 92, 63. 364(2. 304 & 61. 06) into the income register, suggestions I = 10, PHOTO VOLTAIC =? PV = $54. 11. Component 2: Anticipated dividend produce: D1/P0 = $1. 92/$54. 11 sama dengan 3. 57%. Capital profits yield: Initial, find [pic], which will equals the sum from the present values of D2 and [pic] discounted for just one year. [pic] Financial Calculator solution: Type 0, 63. 364(2. 304 + sixty one. 6) into the cash flow enroll, input I = 10, PV =? PV = $57. sixty. Second, locate the capital benefits yield: [pic] Dividend yield = 3. 55% Capital gains yield = 6. 45 15. 00% = ks. b. Due to the for a longer time period of supernormal growth, the value of the share will be bigger for each year. Although the total return will remain the same, ks = 10%, the syndication between gross yield and capital increases yield is going to differ: The dividend yield will start off lower plus the capital profits yield will start off bigger for the 5-year supernormal growth state, relative to the 2-year supernormal growth point out.

The dividend yield increases and the capital gains deliver will drop over the 5-year period till dividend deliver = 4% and capital gains deliver = 6%. c. Over the supernormal development period, the entire yield will be 10 percent, nevertheless the dividend produce is relatively low during the early years of the supernormal growth period and the capital gains yield is relatively excessive. As we close to the end with the supernormal progress period, the administrative centre gains produce declines and the dividend produce rises. Following your supernormal expansion period is finished, the capital increases yield is going to equal gn = 6%.

The total produce must similar ks sama dengan 10%, hence the dividend yield must the same 10% , 6% sama dengan 4%. d. Some traders need cash dividends (retired people), while others would prefer progress. Also, shareholders must spend taxes every year on the dividends received in the past year, while taxes on capital gains may be delayed before the gain is definitely realized. 8-24a. ks = kRF & (kM , kRF)b sama dengan 11% + (14% , 11%)1. 5 = 15. 5%. [pic] = D1/(ks , g) = $2. 25/(0. one hundred fifty five , zero. 05) sama dengan $21. 43. b. ks = 9% + (12% , 9%)1. 5 sama dengan 13. five per cent. [pic] = $2. 25/(0. 135 , 0. 05) = $26. 47. c. ks = 9% + (11% , 9%)1. 5 = 12. 0%. [pic] = $2. 25/(0. 12 , zero. 5) sama dengan $32. 13. d. New data offered: kRF sama dengan 9%, kM = 11%, g = 6%, b = 1 . 3. ks = kRF + (kM , kRF)b = 9% + (11% , 9%)1. 3 sama dengan 11. 6%. [pic] sama dengan D1/(ks , g) sama dengan $2. 27/(0. 116 , 0. 06) = $40. 54. 8-25a. Old ks = kRF + (kM , kRF)b = 9% + (3%)1. 2 sama dengan 12. 6%. New ks = 9% + (3%)0. 9 sama dengan 11. 7%. Old price: [pic] Fresh price: [pic] Since the fresh price is lower than the old value, the growth in customer products must be rejected. The decrease in risk is not sufficient to offset the decline in profitability as well as the reduced expansion rate. n. POld = $38. 21 years old. PNew = [pic]. Solving for ks we now have the following: $38. 1= [pic] $2. 10= $38. 21(ks) , $1. 9105 $4. 0105= $38. 21(ks) ks= 0. 10496. Solving intended for b: 15. 496% = 9% & 3%(b) 1 ) 496% = 3%(b) n = zero. 49865. Examine: ks = 9% & (3%)0. 49865 = 15. 496%. [pic] = [pic] = $38. 21. Consequently , only if management’s analysis proves that risk can be decreased to w = 0. 49865, or perhaps approximately zero. 5, should the new plan be put in to effect. SCHEDULE PROBLEM 8-26The detailed answer for the spreadsheet is actually available both on the instructor’s resource CD-ROM and on the instructor’s part of South-Western’s web site, http://brigham. swlearning. com. INTEGRATED CIRCUMSTANCE

Mutual of Chicago Insurance provider Stock Valuation 8-27ROBERT BALIK AND CAROL KIEFER HAPPEN TO BE SENIOR VICE-PRESIDENTS OF THE SHARED OF CHICAGO INSURANCE COMPANY. THEY ARE REALLY CO-DIRECTORS OF THE COMPANY’S PENSION PLAN FUND MANAGEMENT DIVISION, WITH BALIK HAVING RESPONSIBILITY TO GET FIXED CASH FLOW SECURITIES (PRIMARILY BONDS) AND KIEFER GETTING RESPONSIBLE FOR FAIRNESS INVESTMENTS. AN IMPORTANT NEW CONSUMER, THE A BUNCH OF STATES LEAGUE OF CITIES, FEATURES REQUESTED THAT MUTUAL OF CHICAGO PRESENT AN INVESTMENT SEMINAR TO THE MAYORS OF THE DISPLAYED CITIES, AND BALIK AND KIEFER, WHO WILL MAKE THE REAL PRESENTATION, POSSESS ASKED ONE TO HELP THEM.

TO ILLUSTRATE THE NORMAL STOCK VALUE PROCESS, BALIK AND KIEFER HAVE ASKED YOU TO EXAMINE THE EXCELLENT TEMPS FIRM, AN EMPLOYMENT FIRM THAT SUPPLIES WORD CPU OPERATORS AND COMPUTER PROGRAMMERS TO BUSINESSES WITH BRIEFLY HEAVY WORK LOADS. YOU SHOULD BE ANSWER THE NEXT QUESTIONS. A. DESCRIBE QUICKLY THE PROTECTION UNDER THE LAW AND BENEFITS OF PREVALENT STOCKHOLDERS. SOLUTION:[SHOW S8-1 THROUGH S8-5 HERE. ] THE COMMON STOCKHOLDERS ARE DEFINITELY THE OWNERS OF A CORPORATION, AND THEREFORE THEY HAVE PARTICULAR RIGHTS AND PRIVILEGES BECAUSE DESCRIBED LISTED BELOW. 1 . POSSESSION IMPLIES CONTROL.

THUS, A FIRM’S PREVALENT STOCKHOLDERS HAVE THE RIGHT TO DECIDE ITS FIRM’S DIRECTORS, WHO HAVE IN TURN DECIDE THE OFFICERS WHO MANAGE THE BUSINESS. installment payments on your COMMON STOCKHOLDERS OFTEN HAVE THE RIGHT, CALLED THE PREEMPTIVE RIGHT, TO PURCHASE ANY EXTRA SHARES SOLD BY THE COMPANY. IN SOME CLAIMS, THE PREEMPTIVE RIGHT IS USUALLY AUTOMATICALLY INCORPORATED INTO EVERY BUSINESS CHARTER, IN OTHERS, YOU NEED TO INSERT THIS SPECIFICALLY INTO THE CHARTER. W. 1 . WRITE OUT A FORMULA THAT CAN BE USED TO VALUE ANY STOCK, IN SPITE OF ITS DIVIDEND PATTERN. SOLUTION:[SHOW S8-6 HERE. ] THE COST OF ANY SHARE IS THE PRESENT VALUE OF ITS ANTICIPATED DIVIDEND STREAM: [pic] sama dengan [pic]

HOWEVER , SOME STOCKS AND OPTIONS HAVE DIVIDEND GROWTH PATTERNS THAT ALLOW THEM BE VALUED USING SHORT-CUT FORMULAS. N. 2 . JUST WHAT CONSTANT PROGRESS STOCK? EXACTLY HOW ARE CONSTANT PROGRESS STOCKS RESPECTED? ANSWER:[SHOW S8-7 AND S8-8 HERE. ] A CONSTANT GROWTH STOCK IS DEFINITELY ONE IN WHOSE DIVIDENDS ARE EXPECTED TO GROW AT A CONTINUING RATE PERMANENTLY. “CONSTANT GROWTH” MEANS THAT THE BEST ESTIMATE FOR THE FUTURE GROWTH LEVEL IS SOME CONSTANT QUANTITY, NOT THAT INDIVIDUALS REALLY ANTICIPATE GROWTH TO BE THE SAME EACH AND EVERY YEAR. MANY COMPANIES POSSESS DIVIDENDS WHICH MIGHT BE EXPECTED TO INCREASE STEADILY INTO THE FORESEEABLE FUTURE, AND SUCH COMPANIES ARE HIGHLY VALUED AS REGULAR GROWTH STOCKS AND OPTIONS.

FOR A CONTINUOUS GROWTH SHARE: D1 = D0(1 + g), D2 = D1(1 + g) = D0(1 + g)2, AND SO ON. WITH THIS FREQUENT DIVIDEND DESIGN, THE GENERAL SHARE VALUATION VERSION CAN BE SIMPLIFIED TO THE FOLLOWING VERY IMPORTANT EQUATION: [pic] sama dengan [pic] = [pic]. THIS IS THE POPULAR “GORDON, ” OR “CONSTANT-GROWTH” MODEL FOR VALUING STOCKS AND OPTIONS. HERE D1 IS THE NEXT EXPECTED DIVIDEND, WHICH IS PRESUMED TO BE PAID OUT 1 YEAR VIA NOW, kS IS THE NEEDED RATE OF RETURN AROUND THE STOCK, AND g MAY BE THE CONSTANT PROGRESS RATE. B. 3. WHAT HAPPENS IF A BUSINESS HAS A CONTINUOUS g THAT EXCEEDS ITS ks? CAN MANY STOCKS AND SHARES HAVE PREDICTED g >, ks INSIDE THE SHORT RUN (THAT IS, FOR FEW YEARS)?

IN THE LONG RUN (THAT IS, FOREVER)? ANSWER:[SHOW S8-9 IN THIS ARTICLE. ] THE STYLE IS DERIVED MATHEMATICALLY, AND THE DERIVATION REQUIRES THAT ks >, g. IF PERHAPS g CAN BE GREATER THAN ks, THE VERSION GIVES A ADVERSE STOCK COST, WHICH IS non-sensical. THE STYLE SIMPLY CANNOT PROVIDE UNLESS (1) ks >, g, (2) g IS DEFINITELY EXPECTED TO END UP BEING CONSTANT, AND (3) g CAN FAIRLY BE EXPECTED TO CONTINUE INDEFINITELY. STOCKS AND OPTIONS MAY POSSESS PERIODS OF SUPERNORMAL EXPANSION, WHERE gS >, ks, HOWEVER , THIS KIND OF GROWTH RATE CANNOT BE CONTINUAL INDEFINITELY. IN THE LONG-RUN, g <, ks. C. IMAGINE BON TEMPERATURE HAS A BETA COEFFICIENT OF JUST ONE., THAT THE RISK-FREE RATE (THE YIELD UPON T-BONDS) CAN BE 7 PERCENT, AND THAT THE ESSENTIAL RATE OF RETURN OUT THERE IS 12 PERCENT. WHAT IS THE REQUIRED PRICE OF COME BACK ON THE BUSINESS STOCK? RESPONSE:[SHOW S8-10 HERE. ] HERE WE UTILIZE SML TO CALCULATE BON TEMPS’ NEEDED RATE OF RETURN: ks= kRF & (kM – kRF)bBon Temperature = seven percent + (12% , 7%)(1. 2) = 7% + (5%)(1. 2) = 7% + 6% = 13%. D. ASSUME THAT BON TEMPERATURES IS A REGULAR GROWTH COMPANY WHOSE PREVIOUS DIVIDEND (D0, WHICH WAS PAID YESTERDAY) WAS $2. 00 AND IN WHOSE DIVIDEND CAN BE EXPECTED TO INCREASE INDEFINITELY FOR A six PERCENT LEVEL. 1 .

WHAT IS THE BUSINESS’S EXPECTED DIVIDEND STREAM IN THE NEXT THREE YEARS? ANSWER:[SHOW S8-11 BELOW. ] BON TEMPERATURE IS A CONSTANT GROWTH SHARE, AND ITS DIVIDEND IS ANTICIPATED TO GROW FOR A CONSTANT CHARGE OF 6 PERCENT ANNUALLY. EXPRESSED AS A TIME LINE, WE NOW HAVE THE FOLLOWING CREATE. JUST GET INTO 2 IN THE CALCULATOR, AFTER THAT KEEP SPREADING BY 1 + g = 1 ) 06 TO GET D1, D2, AND D3: 0 1 2 3 | | | | D0 = installment payments on your 00 installment payments on your 12 2 . 247 2 . 382 1 . 88 1 . 76 1 . 65… D. 2 . PRECISELY WHAT IS THE FIRM’S CURRENT INVENTORY PRICE? RESPONSE:[SHOW S8-12 HERE. WE COULD EXTEND THE TIME LINE IN OUT PERMANENTLY, FIND THE COST OF BON TEMPS’ DIVIDENDS FOR EACH AND EVERY YEAR ON OUT FORWARD6171, AND THEN THE PV OF EACH DIVIDEND CHEAPER AT t = 13%. FOR EXAMPLE , THE PV OF D1 IS DEFINITELY $1. 8761, THE PHOTOVOLTAIC OF D2 IS $1. 7599, AND SO FORTH. NOTE THAT THE DIVIDEND REPAYMENTS INCREASE WITH TIME, BUT AS LENGTHY AS ks >, g, THE PRESENT VALUES DECREASE OVER TIME. IF WE EXPANDED THE CHART ON AWAY FOREVER AND AFTER THAT SUMMED THE PVs OF THE DIVIDENDS, WE WOULD HAVE THE WORTH OF THE INVENTORY. HOWEVER , SINCE THE STOCK KEEPS GROWING AT A CONSTANT RATE, IT IS VALUE COULD BE ESTIMATED MAKING USE OF THE CONSTANT EXPANSION MODEL: pic] sama dengan [pic] sama dengan [pic] sama dengan [pic] = $30. 30. D. several. WHAT IS THE STOCK’S ANTICIPATED VALUE ONE FULL YEAR FROM TODAY? ANSWER:[SHOW S8-13 IN THIS ARTICLE. ] AFTER ONE FULL YEAR, D1 MAY HAVE BEEN PAID OUT, SO THE PREDICTED DIVIDEND STREAM WILL THEN BE D2, D3, D4, AND SO ON. THEREFORE, THE EXPECTED VALUE ONE FULL YEAR FROM NOW COULD BE $32. 15: [pic] sama dengan [pic] sama dengan [pic] sama dengan [pic] = $32. 12. D. 4. WHAT ARE THE EXPECTED DIVIDEND YIELD, THE CAPITAL GAINS DELIVER, AND THE TOTAL RETURN THROUGHOUT THE FIRST SEASON? ANSWER:[SHOW S8-14 IN THIS ARTICLE. ] THE PREDICTED DIVIDEND YIELD IN ANY SEASON n CAN BE DIVIDEND DELIVER = [pic], EVEN THOUGH THE EXPECTED CAPITAL GAINS PRODUCE IS

CAPITAL GAINS DELIVER = [pic] = t , [pic]. THUS, THE GROSS YIELD INSIDE THE FIRST 12 MONTHS IS six PERCENT, AS THE CAPITAL INCREASES YIELD IS 6 PERCENT: TOTAL GO BACK = 13. 0% DIVIDEND YIELD sama dengan $2. 12/$30. 29 = 7. 0% CAPITAL INCREASES YIELD = 6. 0% E. RIGHT NOW ASSUME THAT THE STOCK IS CURRENTLY SELLING IN $30. twenty nine. WHAT IS THE EXPECTED PRICE OF RETURN ON THE SHARE? ANSWER: THE CONSTANT GROWTH UNIT CAN BE REARRANGED TO THIS FORM: [pic] = [pic]. HERE THE LATEST PRICE IN THE STOCK IS FAMOUS, AND WE RESOLVE FOR THE EXPECTED GO BACK. FOR EXCELLENT TEMPS: pic] sama dengan $2. 12/$30. 29 + 0. 060 = zero. 070 + 0. 060 = 13%. F. WHAT WOULD THE STOCK PRICE BE IF PERHAPS ITS RETURNS WERE ANTICIPATED TO HAVE NO GROWTH? RESPONSE:[SHOW S8-15 HERE. ] IN THE EVENT THAT BON TEMPS’ DIVIDENDS WEREN’T EXPECTED TO DEVELOP AT ALL, AFTER THAT ITS GROSS STREAM WOULD BE A PERPETUITY. PERPETUITIES ARE VALUED AS DISPLAYED BELOW: 0 1 a couple of 3 | | | | installment payments on your 00 installment payments on your 00 installment payments on your 00 1 ) 77 1 . 57 1 . 39… P0 = 12-15. 38 P0 = D/kS = $2. 00/0. 13 = $15. 38. BE AWARE THAT IF A RECOMMENDED STOCK IS KNOWN AS A PERPETUITY, IT COULD BE VALUED WITH THIS FORMULA. G.

TODAY ASSUME THAT BON TEMPS CAN BE EXPECTED TO EXPERIENCE SUPERNORMAL GROWTH OF 30 PERCENT FOR THE NEXT 3 YEARS, AFTER THAT TO RETURN TO THE LONG-RUN REGULAR GROWTH CHARGE OF six PERCENT. PRECISELY WHAT IS THE STOCK’S VALUE UNDER THESE CIRCUMSTANCES? WHAT IS THE EXPECTED GROSS YIELD AND CAPITAL GAINS YIELD IN YEAR 1? YEAR some? ANSWER:[SHOW S8-16 THROUGH S8-18 RIGHT HERE. ] BON TEMPERATURE IS NO LONGER A CONSISTENT GROWTH SHARE, SO THE FREQUENT GROWTH VERSION IS CERTAINLY NOT APPLICABLE. NOTE, HOWEVER , WHICH THE STOCK IS DEFINITELY EXPECTED TO BE A CONSTANT PROGRESS STOCK IN 3 YEARS. THEREFORE, IT HAS A NONCONSTANT GROWTH PERIOD FOLLOWED BY CONTINUOUS GROWTH.

THE BEST WAY TO VALUE SUCH NONCONSTANT DEVELOPMENT STOCKS IS USUALLY TO SET THE SITUATION UP ON A FB TIMELINE AS DEMONSTRATED BELOW: zero 1 two 3 4 | | | | | 2 . 600 a few. 380 some. 394 four. 65764 installment payments on your 301 installment payments on your 647 3. 045 46. 114 fifty four. 107 JUST ENTER $2 AND INCREASE IN NUMBERS BY (1. 30) TO GET D1 = $2. 60, INCREASE THAT CONSEQUENCE BY 1 . 3 TO GET D2 = $3. 38, AND ETC .. THEN KNOW THAT AFTER YEAR 3, BELEG TEMPS BECOMES A CONSTANT DEVELOPMENT STOCK, WITH THAT POINT [pic] CAN BE FOUND USING THE CONSTANT DEVELOPMENT MODEL. pic] MAY BE THE PRESENT WORTH AS OF capital t = 3 OF THE RETURNS IN YEAR 4 AND BEYOND AND IT IS ALSO CALLED THE TERMINAL VALUE. WITH THE MONEY FLOWS TO GET D1, D2, D3, AND [pic] DEMONSTRATED ON THE TIME LINE, WE DISCOUNT EACH BENEFIT BACK TO YEAR 0, AND THE SUM OF THOSE FOUR PVs IS THE VALUE OF THE INVENTORY TODAY, P0 = $54. 107. THE DIVIDEND DELIVER IN 12 MONTHS 1 IS 4. 80 PERCENT, AND THE CAPITAL GAINS PRODUCE IS almost eight. 2 PERCENT: DIVIDEND DELIVER = [pic] = zero. 0480 sama dengan 4. 8%. CAPITAL BENEFITS YIELD sama dengan 13. 00% , four. 8% sama dengan 8. 2%. DURING THE NONCONSTANT GROWTH PERIOD, THE GROSS YIELDS AND CAPITAL PROFITS YIELDS ARE NOT CONSTANT, AS WELL AS THE CAPITAL PROFITS YIELD WILL NOT EQUAL g.

HOWEVER , AFTER YEAR three or more, THE INVENTORY BECOMES A CONSTANT GROWTH STOCK, WITH g = CAPITAL GAINS PRODUCE = 6th. 0% AND DIVIDEND DELIVER = 13. 0% , 6. 0% = several. 0%. L. SUPPOSE BON TEMPS IS EXPECTED TO EXPERIENCE ZERO GROWTH DURING THE INITIAL 3 YEARS AND AFTER THAT TO RESUME ITS STEADY-STATE GROWTH OF six PERCENT IN THE FOURTH SEASON. WHAT IS THE STOCK’S WORTH NOW? PRECISELY WHAT IS ITS EXPECTED DIVIDEND PRODUCE AND ITS CAPITAL GAINS YIELD IN YEAR 1? YR 4? SOLUTION:[SHOW S8-19 AND S8-20 HERE. ] NOW WE HAVE THIS SITUATION: 0 1 a couple of 3 5 | | | | | 2 . 00 2 . 0 installment payments on your 00 2 . 00 2 . 12 1 . 77 1 ) 57 1 . 39 20. 99 25. 72 sama dengan [pic] DURING YEAR you: DIVIDEND YIELD = [pic] = zero. 0778 = 7. 78%. CAPITAL INCREASES YIELD = 13. 00% , several. 78% sama dengan 5. 22%. AGAIN, IN YEAR some BON TEMPERATURES BECOMES A CONSTANT GROWTH INVENTORY, HENCE g = CAPITAL GAINS YIELD = 6. 0% AND DIVIDEND YIELD = 7. 0%. We. FINALLY, IMAGINE BON TEMPS’ EARNINGS AND DIVIDENDS ARE EXPECTED TO DECLINE BY A FREQUENT 6 PERCENT PER YEAR, THAT IS, g = -6%. FOR WHAT REASON WOULD ANY PERSON BE HAPPY TO BUY THIS SORT OF A STOCK, WITH WHAT PRICE OUGHT IT TO SELL? WHAT WOULD BE THE DIVIDEND YIELD AND CAPITAL PROFITS YIELD IN EACH YEAR?

ANSWER:[SHOW S8-21 AND S8-22 HERE. ] THE COMPANY IS GENERATING SOMETHING AND PAYING SOME DIVIDENDS, SO IT CLEARLY CONTAINS A VALUE MORE THAN ZERO. THAT VALUE CAN BE FOUND WITH THE REGULAR GROWTH SOLUTION, BUT EXACTLY WHERE g CAN BE NEGATIVE: [pic] = [pic] = [pic] = [pic] = [pic] = $9. 89. MAINLY BECAUSE IT IS A CONSTANT GROWTH SHARE: g = CAPITAL GAINS YIELD = -6. 0%, HENCE: GROSS YIELD = 13. 0% , (-6. 0%) = 19. 0%. AS A EXAMINE: DIVIDEND DELIVER = [pic] = zero. 190 sama dengan 19. 0%. THE DIVIDEND AND CAPITAL GAINS PRODUCES ARE REGULAR OVER TIME, ALTHOUGH A HIGH (19. 0 PERCENT) DIVIDEND DELIVER IS NEEDED TO BALANCE THE UNFAVORABLE CAPITAL GAINS YIELD.

L. BON TEMPERATURES EMBARKS WITH AN AGGRESSIVE GROWTH THAT REQUIRES ADDITIONAL CAPITAL. ADMINISTRATION DECIDES TO FINANCE THE EXPANSION SIMPLY BY BORROWING $40 MILLION THROUGH HALTING DIVIDEND PAYMENTS TO BOOST RETAINED INCOME. THE PROJECTED FREE MONEY FLOWS FOR THREE YEARS WILL BE -$5 MILLION, $10 MILLION, AND $20 MILLION. FOLLOWING YOUR THIRD 12 MONTHS, FREE CASH FLOW IS EXPECTED TO EXPAND AT A CONTINUING 6 PERCENT. THE OVERALL EXPENSE OF CAPITAL IS USUALLY 10 PERCENT. PRECISELY WHAT IS BON TEMPS’ TOTAL WORTH? IF IT HAS 10 MIL SHARES OF STOCK AND $40 , 000, 000 TOTAL DEBT, WHAT IS THE PRICE PER DISCUSS? ANSWER:[SHOW S8-23 THROUGH S8-28 RIGHT HERE. 0 one particular 2 three or more 4 | | | | | -5 10 20 21. 20 dollar -4. 545 8. 264 15. 026 398. 197 $416. 942 = TOTAL VALUE WORTH OF EQUITY = TOTAL VALUE , DEBT sama dengan $416. 94 , $40 = $376. 94 MILLION. PRICE EVERY SHARE = $376. 94/10 = $37. 69. K. WHAT DOES MARKET EQUILIBRIUM SUGGEST? ANSWER:[SHOW S8-29 AND S8-30 IN THIS ARTICLE. ] EQUILIBRIUM MEANS STABLE, ZERO TENDENCY TO IMPROVE. MARKET SENSE OF BALANCE MEANS THAT RATES ARE STABLE, AT ITS CURRENT PRICE, THERE IS ABSOLUTELY NO GENERAL TREND FOR PEOPLE TO ACTUALLY WANT TO BUY AS WELL AS TO SELL SECURITIES THAT IS IN EQUILIBRIUM.

LIKEWISE, WHEN EQUILIBRIUM EXISTS, THE EXPECTED LEVEL OF COME BACK WILL BE CORRESPONDING TO THE REQUIRED RATE OF RETURN: [pic] = D1/P0 + g sama dengan k = kRF + (kM , kRF)b. T. IF BALANCE DOES NOT EXIST, HOW WILL THAT BE ESTABLISHED? ANSWER:[SHOW S8-31 AND S8-32 RIGHT HERE. ] SECURITIES WILL PROBABLY BE BOUGHT AND SOLD BEFORE THE EQUILIBRIUM COST IS ESTABLISHED. M. WHAT IS THE EFFICIENT MARKETPLACES HYPOTHESIS, EXACTLY WHAT ARE ITS THREE FORMS, AND WHAT ARE IT IS IMPLICATIONS? ANSWER:[SHOW S8-33 THROUGH S8-37 HERE. ] THE EMH IN GENERAL IS THE SPECULATION THAT INVESTMENTS ARE NORMALLY IN EQUILIBRIUM AND THEREFORE ARE “PRICED PRETTY, ” SO THAT IT IS IMPOSSIBLE TO “BEAT THE MARKETPLACE. WEAK-FORM PRODUCTIVITY SAYS THAT INVESTORS ARE NOT ABLE TO PROFIT FROM TAKING A LOOK AT PAST MOVEMENTS IN STOCK PRICES, THE FACT THAT STOCKS AND SHARES WENT DOWN FOR THE LAST FEW DAYS IS NOT A REASON TO CONSIDER THAT THEY WILL RISE (OR DOWN) IN THE FUTURE. THIS TYPE HAS BEEN VERIFIED PRETTY WELL BY EMPIRICAL ASSESSMENTS, EVEN THOUGH PEOPLE STILL UTILIZE “TECHNICAL RESEARCH. ” SEMISTRONG-FORM EFFICIENCY SAYS THAT ALL WIDELY AVAILABLE INFORMATION IS SHOWN IN STOCK PRICES, ACCORDINGLY IT WON’T DO MUCH GREAT TO PORE OVER GROSS ANNUAL REPORTS TRYING TO FIND UNDERVALUED STOCKS.

THIS ONE IS (WE THINK) LARGELY ACCURATE, BUT SUPERIOR ANALYSTS CAN STILL OBTAIN AND PROCESS NEW INFORMATION QUICKLY ENOUGH TO ACHIEVE A SMALL EDGE. STRONG-FORM EFFICIENCY SAYS THAT EVERY INFORMATION, ACTUALLY INSIDE INFO, IS INLAYED IN STOCK PRICES. THIS TYPE DOES NOT HOLD, INSIDERS KNOW MORE, AND COULD BENEFIT FROM THAT INFO TO MAKE ABNORMAL PROFITS INSIDE THE MARKETS. TRADING ON THE BASIS OF INSIDER INFORMATION IS USUALLY ILLEGAL. D. PHYFE FIRM RECENTLY GIVEN PREFERRED SHARE. IT PAYS A DIVIDEND OF $5, PLUS THE ISSUE COST WAS $50 PER DISCUSS. WHAT IS THE EXPECTED GO BACK TO AN INVESTOR ON THIS PREFERRED SHARE?

ANSWER:[SHOW S8-38 AND S8-39 BELOW. ] [pic]= [pic] = [pic] = 10%. , , , , , , , , ks = 15% gn = 6% ( 1/(1. 15)3 ( 1/(1. 13)3 ( 1/(1. 13)2 ( 1/1. 13 gs sama dengan 50% gn = 8% [pic] ks = 12% gs sama dengan 15% gn = five per cent WACC = 10% [pic] = 35. 29 = [pic] g = 0% g = 0% g = 0% gn sama dengan 6% ks = 13% [pic] sama dengan $66. fifty four = [pic] gs sama dengan 30% gs = thirty percent gs sama dengan 30% gn = 6% ks sama dengan 13% g = 0% ks sama dengan 13% g = 6% ks = 13% ks = 10% gs sama dengan 20% gs = 20% gn = 5% WACC = 12% WACC = 12% gn = seven percent [pic] WACC = 13% gn = 7% 530 = [pic] ( 1/(1. 15)4 ( 1/(1. 15)5 ks sama dengan 12% ( 1/1. 13 ( 1/(1. 13)2 ( 1/(1. 13)3 ( 1/(1. 13)2 ( 1/(1. 13)2 ( 1/1. 13 ( 1/(1. 13)2 ( 1/(1. 13)3 ( 1/(1. 13)3 ( 1/1. 13 ( 1/1. 13 (%89

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