BF Quiz #3
Price of stock is really just....
PV of all expected future dividends
Stock value =
PV of dividends
Metallica Bearings, Inc., is a young start-up company. No dividends will be paid on the stock over the next nine years because the firm needs to plow back its earnings to fuel growth. The company will then pay a dividend of $23 per share 10 years from today and will increase the dividend by 5 percent per year thereafter. If the required return on this stock is 12 percent, what is the current share price?
Here, we have a stock that pays no dividends for nine years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. It is important to remember that the general constant dividend growth formula is: Pt = [Dt × (1 + g)]/(R - g) This means that since we will use the dividend in Year 10, we will be finding the stock price in Year 9. The dividend growth model is similar to the present value of an annuity and the present value of a perpetuity: The equation gives you the present value one period before the first payment. So, the price of the stock in Year 9 will be: P9 = D10/(R - g) P9 = $23/(.12 - .05) P9 = $328.57 The price of the stock today is the PV of the stock price in the future. We discount the future stock price at the required return. The price of the stock today will be: P0 = $328.57/1.129 P0 = $118.49
Poulter Corporation will pay a dividend of $3.25 per share next year. The company pledges to increase its dividend by 5.1 percent per year, indefinitely. If you require a return of 11 percent on your investment, how much will you pay for the company's stock today?
Using the constant growth model, we find the price of the stock today is: P0 = D1/(R - g) P0 = $3.25/(.11 - .051) P0 = $55.08
The stock price of Alps Co. is $67. Investors require a return of 10.5 percent on similar stocks. If the company plans to pay a dividend of $4.25 next year, what growth rate is expected for the company's stock price?
We need to find the growth rate of dividends. Using the constant growth model, we can solve the equation for g. Doing so, we find: g = R - (D1/P0) g = .105 - ($4.25/$67) g = .0416, or 4.16%
The next dividend payment by Hoffman, Inc., will be $2.65 per share. The dividends are anticipated to maintain a growth rate of 4.5 percent forever. If the stock currently sells for $43.15 per share, what is the required return?
We need to find the required return of the stock. Using the constant growth model, we can solve the equation for R. Doing so, we find: R = (D1/P0) + g R = ($2.65/$43.15) + .045 R = .1064, or 10.64%
An investment project provides cash inflows of $865 per year for eight years. a.What is the project payback period if the initial cost is $3,100? b.What is the project payback period if the initial cost is $4,300? c.What is the project payback period if the initial cost is $7,900?
a. To calculate the payback period, we need to find the time it takes to recover the initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $3,100, the payback period is: Payback = 3 + $505/$865 Payback = 3.58 years There is a shortcut to calculate payback period when the future cash flows are an annuity. Just divide the initial cost by the annual cash flow. For the $3,100 cost, the payback period is: Payback = $3,100/$865 Payback = 3.58 years b.For an initial cost of $4,300, the payback period is: Payback = $4,300/$865 Payback = 4.97 years c. The payback period for an initial cost of $7,900 is a little trickier. Notice that the total cash inflows after eight years will be: Total cash inflows = 8($865) Total cash inflows = $6,920 If the initial cost is $7,900, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = $7,900/$865 Payback = 9.13 years This answer does not make sense since the cash flows stop after eight years, so again, we must conclude the payback period is never.
constant dividend growth
firm will increase dividend by constant % every period
Constant growth model conditions
1) Dividend expected to grow at g forever 2) Stock price expected to grow at g forever 3) Expected dividend yield is constant 4) Expected capital gains yield is constant and equal to g 5) Expected total return, R, must be > g 6) Expected total return (R): = expected dividend yield (DY) + expected growth rate (g) = dividend yield + g
If own share of stock, can receive cash in 2 ways...
1. company pays dividends 2. you sell shares, either to another investor in market or back to the company
E-Eyes.com has a new issue of preferred stock it calls 20/20 preferred. The stock will pay a $20 dividend per year, but the first dividend will not be paid until 20 years from today. If you require a return of 7.3 percent on this stock, how much should you pay today?
Here we have a stock that pays no dividends for 20 years. Once the stock begins paying dividends, it will have the same dividends forever. We value the stock at that point, using the preferred stock equation. It is important to remember that the price we find will be the price one year before the first dividend, so: P19 = D20/RP19 = $20/.073P19 = $273.97 The price of the stock today is the present value of the stock price in the future. We discount the future stock price at the required return. The price of the stock today will be: P0 = $273.97/1.07319P0 = $71.83
Consider the following cash flows: Year. Cash Flow 0 -$19,400 1 10,400 2 9,320 3 6,900 What is the IRR of the above set of cash flows?
The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = -$19,400 + $10,400/(1 + IRR) + $9,320/(1 + IRR)2 + $6,900/(1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 19.05%
Bon Chance, Inc., has an odd dividend policy. The company has just paid a dividend of $3 per share and has announced that it will increase the dividend by $5 per share for each of the next four years, and then never pay another dividend. If you require a return of 9.7 percent on the company's stock, how much will you pay for a share today?
The price of a stock is the PV of the future dividends. This stock is paying four dividends, so the price of the stock is the PV of these dividends discounted at the required return. So, the price of the stock is: P0 = $8/1.097 + $13/1.0972 + $18/1.0973 + $23/1.0974 P0 = $47.61
Synovec Corporation is expected to pay the following dividends over the next four years: $7, $13, $18, and $3.25. Afterward, the company pledges to maintain a constant 5 percent growth rate in dividends forever. If the required return on the stock is 10.4 percent, what is the current share price?
With nonconstant dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the present value of the future stock price, plus the present value of all dividends during the nonconstant growth period. The stock begins constant growth after the fourth dividend is paid, so we can find the price of the stock at Year 4, when the constant dividend growth begins, as: P4 = D4(1 + g)/(R - g) P4 = $3.25(1.05)/(.104 - .05) P4 = $63.19 The price of the stock today is the present value of the first four dividends, plus the present value of the Year 4 stock price. So, the price of the stock today will be: P0 = $7/1.104 + $13/1.1042 + $18/1.1043 + $3.25/1.1044 + $63.19/1.1044 P0 = $75.11
perpetuity
a time period lasting through the ages; eternity
Piercy, LLC, has identified the following two mutually exclusive projects: Year. Cash Flow (A). Cash Flow (B) 0 −$77,500 −$77,500 1 43,000 21,500 2 29,000 28,000 3 23,000 34,000 4 21,000 41,000 a-1. What is the IRR for each of these projects? a-2. If you apply the IRR decision rule, which project should the company accept? b-1. Assume the required return is 11 percent. What is the NPV for each of these projects? b-2. Which project will you choose if you apply the NPV decision rule? c-1. Over what range of discount rates would you choose Project A? c-2. Over what range of discount rates would you choose Project B? c-3. At what discount rate would you be indifferent between these two projects?
a-1. The IRR is the interest rate that makes the NPV of the project equal to zero. The equation for the IRR of Project A is: 0 = -$77,500 + $43,000/(1 + IRR) + $29,000/(1 + IRR)2 + $23,000/(1 + IRR)3 + $21,000/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 21.50% The equation for the IRR of Project B is: 0 = -$77,500 + $21,500/(1 + IRR) + $28,000/(1 + IRR)2 + $34,000/(1 + IRR)3 + $41,000/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 19.58% a-2. Examining the IRRs of the projects, we see that the IRRA is greater than the IRRB, so the IRR decision rule implies accepting Project A. b-1. The NPV of Project A is: NPVA = -$77,500 + $43,000/1.11 + $29,000/1.112 + $23,000/1.113 + $21,000/1.114 NPVA = $15,426.54 And the NPV of Project B is: NPVB = -$77,500 + $21,500/1.11 + $28,000/1.112 + $34,000/1.113 + $41,000/1.114 NPVB = $16,463.27 b-2. The NPVB is greater than the NPVA, so we should accept Project B. c. To find the crossover rate, we subtract the cash flows from one project from the cash flows of the other project. Here, we will subtract the cash flows for Project B from the cash flows of Project A. Once we find these differential cash flows, we find the IRR. The equation for the crossover rate is: 0 = $21,500/(1 + R) + $1,000/(1 + R)2 - $11,000/(1 + R)3 - $20,000/(1 + R)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: R = 13.18% At discount rates above 13.18% choose Project A; for discount rates below 13.18% choose Project B; indifferent between A and B at a discount rate of 13.18%.
Cape Corp. will pay a dividend of $2.64 next year. The company has stated that it will maintain a constant growth rate of 4.5 percent a year forever. a.If you want a return of 12 percent, how much will you pay for the stock? b.If you want a return of 8 percent, how much will you pay for the stock?
a. Here we need to value a stock with two different required returns. Using the constant growth model and a required return of 12 percent, the stock price today is: P0 = D1/(R - g) P0 = $2.64/(.12 - .045) P0 = $35.20 b.And the stock price today with a return of 8 percent will be: P0 = D1/(R - g) P0 = $2.64/(.08 - .045) P0 = $75.43
Stenson, Inc., imposes a payback cutoff of three years for its international investment projects. Assume the company has the following two projects available. Year Cash Flow (A) Cash Flow (B) 0. -$75,000 -$ 125,000 1 33,000 29,000 2 36,000 32,000 3 19,000 35,000 4 9,000 240,000 a.What is the payback period for each project? b.Which, if either, of the projects should the company accept?
a. Project A has cash flows of: Cash flows = $33,000 + 36,000Cash flows = $69,000 during the first two years. The cash flows are still short by $6,000 of recapturing the initial investment, so the payback for Project A is: Payback = 2 + ($6,000/$19,000) Payback = 2.32 years Project B has cash flows of: Cash flows = $29,000 + 32,000 + 35,000 Cash flows = $96,000 during the first three years. The cash flows are still short by $29,000 of recapturing the initial investment, so the payback for Project B is: Payback = 3 + ($29,000/$240,000) Payback = 3.12 years b.Using the payback criterion and a cutoff of 3 years, accept Project A and reject Project B.
Bausch Company is presented with the following two mutually exclusive projects. The required return for both projects is 15 percent. Year. Project M. Project N 0. -$140,000 -$359,000 1. 61,500 159,300 2. 73,400 168,400 3. 68,100 154,800 4. 40,500 110,400 a. What is the IRR for each project? b. What is the NPV for each project? c. Which, if either, of the projects should the company accept?
a. The IRR for each project is: M: $140,000 = $61,500/(1 + IRR) + $73,400/(1 + IRR)2 + $68,100/(1 + IRR)3 + $40,500/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 28.18% N: $359,000 = $159,300/(1 + IRR) + $168,400/(1 + IRR)2 + $154,800/(1 + IRR)3 + $110,400/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 25.11% IRR decision rule implies we accept Project M because the IRR for M is greater than the IRR for N. b. The NPV for each project is: M: NPV = -$140,000 + $61,500/1.15 + $73,400/1.152 + $68,100/1.153 + $40,500/1.154 NPV = $36,912.07 N: NPV = -$359,000 + $159,300/1.15 + $168,400/1.152 + $154,800/1.153 + $110,400/1.154 NPV = $71,761.40 NPV criterion implies we accept Project N because Project N has a higher NPV than Project M. c. Accept Project N since the NPV is higher. IRR cannot be used to rank mutually exclusive projects.
A firm evaluates all of its projects by applying the IRR rule. Year. Cash Flow 0 −$157,300 1 74,000 2 87,000 3 46,000 a. What is the project's IRR? b.If the required return is 11 percent, should the firm accept the project?
a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = -$157,300 + $74,000/(1 + IRR) + $87,000/(1 + IRR)2 + $46,000/(1 + IRR)3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 16.26% b. Since the cash flows are conventional and the IRR is greater than the required return, we would accept the project.
Consider the following cash flows: Year. Cash Flow 0 -$19,400 1 10,400 2 9,320 3 6,900 a.What is the NPV at a discount rate of zero percent? b.What is the NPV at a discount rate of 10 percent? c.What is the NPV at a discount rate of 20 percent? d.What is the NPV at a discount rate of 30 percent?
a. The NPV of a project is the PV of the outflows plus by the PV of the inflows. At a zero discount rate (and only at a zero discount rate), the cash flows can be added together across time. So, the NPV of the project at a zero percent required return is: NPV = -$19,400 + 10,400 + 9,320 + 6,900 NPV = $7,220 b.The NPV at a 10 percent required return is: NPV = -$19,400 + $10,400/1.10 + $9,320/1.102 + $6,900/1.103 NPV = $2,941.10 c.The NPV at a 20 percent required return is: NPV = -$19,400 + $10,400/1.20 + $9,320/1.202 + $6,900/1.203 NPV = -$268.06 d.And the NPV at a 30 percent required return is: NPV = -$19,400 + $10,400/1.30 + $9,320/1.302 + $6,900/1.303 NPV = -$2,744.56 Notice that as the required return increases, the NPV of the project decreases. This will always be true for projects with conventional cash flows. Conventional cash flows are negative at the beginning of the project and positive throughout the rest of the project.
A project that provides annual cash flows of $2,620 for eight years costs $9,430 today. a.At a required return of 8 percent, what is the NPV of the project? b.At a required return of 24 percent, what is the NPV of the project? c.At what discount rate would you be indifferent between accepting the project and rejecting it?
a. The NPV of a project is the PV of the outflows plus the PV of the inflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 8 percent required return is: NPV = -$9,430 + $2,620(PVIFA8%, 8) NPV = $5,626.19 At an 8 percent required return, the NPV is positive, so we would accept the project. b.The equation for the NPV of the project at a 24 percent required return is: NPV = -$9,430 + $2,620(PVIFA24%, 8) NPV = -$466.40 At a 24 percent required return, the NPV is negative, so we would reject the project. c. We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. The IRR of the project is: 0 = -$9,430 + $2,620(PVIFAIRR, 8) IRR = .2219, or 22.19%
Fowler, Inc., just paid a dividend of $2.55 per share on its stock. The dividends are expected to grow at a constant rate of 3.9 percent per year, indefinitely. If investors require a return of 10.4 percent on this stock, what is the current price? What will the price be in three years? In 15 years?
a. The constant dividend growth model is: Pt = Dt × (1 + g)/(R - g) So, the price of the stock today is: P0 = D0(1 + g)/(R - g)P0 = $2.55(1.039)/(.104 - .039) P0 = $40.76 b. The dividend at Year 4 is the dividend today times the FVIF for the growth rate in dividends for four years, so: P3 = D3(1 + g)/(R - g) P3 = D0(1 + g)4/(R - g) P3 = $2.55(1.039)4/(.104 - .039) P3 = $45.72 We can do the same thing to find the dividend in Year 16, which gives us the price in Year 15, so: P15 = D15(1 + g)/(R - g) P15 = D0(1 + g)16/(R - g) P15 = $2.55(1.039)16/(.104 - .039)P15 = $72.36 There is another feature of the constant dividend growth model: The stock price grows at the dividend growth rate. So, if we know the stock price today, we can find the future value for any time in the future we want to calculate the stock price. In this problem, we want to know the stock price in three years, and we have already calculated the stock price today. The stock price in three years will be: P3 = P0(1 + g)3P3 = $40.76(1 + .039)3P3 = $45.72 And the stock price in 15 years will be: P15 = P0(1 + g)15P15 = $40.76(1 + .039)15 P15 = $72.36
The next dividend payment by Hoffman, Inc., will be $2.65 per share. The dividends are anticipated to maintain a growth rate of 4.5 percent forever. Assume the stock currently sells for $43.15 per share. What is the dividend yield? What is the expected capital gains yield?
a. The dividend yield is the dividend next year divided by the current price, so the dividend yield is: Dividend yield = D1/P0 Dividend yield = $2.65/$43.15 Dividend yield = .0614, or 6.14% b. The capital gains yield, or percentage increase in the stock price, is the same as the dividend growth rate, so: Capital gains yield = 4.5%
Consider the following two mutually exclusive projects: Year. Cash Flow (A). Cash Flow (B) 0. -$245,000 -$53,000 1. 34,000 31,900 2. 49,000 21,800 3. 51,000 17,300 4. 325,000 16,200 The required return on these investments is 13 percent. a. What is the payback period for each project? b. What is the NPV for each project? c. What is the IRR for each project? d. What is the profitability index for each project? e. Based on your answers in (a) through (d), which project will you finally choose?
a. The payback period for each project is: A: 3 + ($111,000/$325,000) = 3.34 years B: 1 + ($21,100/$21,800) = 1.97 years The payback criterion implies accepting Project B, because it pays back sooner than Project A. b. The NPV for each project is: A: NPV = -$245,000 + $34,000/1.13 + $49,000/1.132 + $51,000/1.133 + $325,000/1.134 NPV = $58,136.83 B: NPV = -$53,000 + $31,900/1.13 + $21,800/1.132 + $17,300/1.133 + $16,200/1.134 NPV = $14,228.22 NPV criterion implies we accept Project A because Project A has a higher NPV than Project B. c. The IRR for each project is: A: $245,000 = $34,000/(1 + IRR) + $49,000/(1 + IRR)2 + $51,000/(1 + IRR)3 + $325,000/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 20.54% B: $53,000 = $31,900/(1 + IRR) + $21,800/(1 + IRR)2 + $17,300/(1 + IRR)3 + $16,200/(1 + IRR)4 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 27.38% IRR decision rule implies we accept Project B because the IRR for B is greater than the IRR for A. d. The profitability index for each project is: A: PI = ($34,000/1.13 + $49,000/1.132 + $51,000/1.133 + $325,000/1.134)/$245,000 PI = 1.237 B: PI = ($31,900/1.13 + $21,800/1.132 + $17,300/1.133 + $16,200/1.134)/$53,000 PI = 1.268 Profitability index criterion implies we accept Project B because its PI is greater than Project A's. e. In this instance, the NPV criterion implies that you should accept Project A, while payback period, IRR, and the profitability index imply that you should accept Project B. The final decision should be based on the NPV since it does not have the ranking problem associated with the other capital budgeting techniques. Therefore, you should accept Project A.
Consider the following cash flows: Year. Cash Flow 0. −$29,500 1. 16,900 2. 13,600 3. 8,300 a. What is the profitability index for the above set of cash flows if the relevant discount rate is 10 percent? b.What is the profitability index if the discount rate is 15 percent? c.What is the profitability index if the discount rate is 22 percent?
a. The profitability index is defined as the PV of the future cash flows divided by the initial investment. The equation for the profitability index at a required return of 10 percent is: PI = ($16,900/1.10 + $13,600/1.102 + $8,300/1.103)/$29,500 PI = 1.113 b. The equation for the profitability index at a required return of 15 percent is: PI = ($16,900/1.15 + $13,600/1.152 + $8,300/1.153)/$29,500 PI = 1.032 c. The equation for the profitability index at a required return of 22 percent is: PI = ($16,900/1.22 + $13,600/1.222 + $8,300/1.223)/$29,500 PI = .934 We would accept the project if the required return were 10 percent or 15 percent since the PI is greater than one. We would reject the project if the required return were 22 percent since the PI is less than one.
Selling ------>
capital gains
Dividends ----->
cash income
supernormal growth
dividend growth isn't consistent initially, but settles down to constant growth eventually
constant dividend/ zero growth
firm will pay constant dividend forever; like preferred stock. price is computed using perpetuity formula
