BUAD Fall 2015 PP 4 solutions 1 The two components are the dividend yield and the capital gains yield For most companies the capital gains yield is larger This is easy to see for companies that pay no dividends For companies that do pay dividends the dividend yields are rarely over 5 percent and are often much less 2 The price of any financial instrument is the PV of the future cash flows The future dividends of this stock are an annuity for 11 years so the price of the stock is the PVA which will be P0 2 50 PVIFA10 11 16 24 3 This stock has a constant growth rate of dividends but the required return changes twice To find the value of the stock today we will begin by finding the price of the stock at Year 6 when both the dividend growth rate and the required return are stable forever The price of the stock in Year 6 will be the dividend in Year 7 divided by the required return minus the growth rate in dividends So P6 D6 1 g R g D0 1 g 7 R g 3 20 1 05 7 11 05 75 05 Now we can find the price of the stock in Year 3 We need to find the price here since the required return changes at that time The price of the stock in Year 3 is the PV of the dividends in Years 4 5 and 6 plus the PV of the stock price in Year 6 The price of the stock in Year 3 is P3 3 20 1 05 4 1 13 3 20 1 05 5 1 132 3 20 1 05 6 1 133 75 05 1 133 P3 61 62 Finally we can find the price of the stock today The price today will be the PV of the dividends in Years 1 2 and 3 plus the PV of the stock in Year 3 The price of the stock today is P0 3 20 1 05 1 15 3 20 1 05 2 1 15 2 3 20 1 05 3 1 15 3 61 62 1 15 3 P0 48 54 4 We can use the constant dividend growth model which is Pt Dt 1 g R g So the price of each company s stock today is Red stock price 2 65 15 08 37 86 Yellow stock price 2 65 15 11 66 25 Blue stock price 2 65 15 14 265 00 As the growth rate increases the stock price increases This is a function of the time value of money It is also important to note that relatively small changes in the growth rate can have a dramatic impact on the stock price 5 To calculate the payback period we need to find the time that the project has recovered its initial investment The cash flows in this problem are an annuity so the calculation is simpler If the initial cost is 1 700 the payback period is Payback 2 530 585 2 91 years There is a shortcut to calculate the future cash flows are an annuity Just divide the initial cost by the annual cash flow For the 3 300 cost the payback period is Payback 3 300 765 5 64 years The payback period for an initial cost of 4 900 is a little trickier Notice that the total cash inflows after eight years will be Total cash inflows 8 585 4 680 If the initial cost is 4 900 the project never pays back Notice that if you use the shortcut for annuity cash flows you get Payback 4 900 585 8 38 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 6 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 56 400 7 100 1 IRR 8 400 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 35 24 7 The NPV of a project is the PV of the inflows minus the PV of the outflows 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 56 400 7 100 8 400 6 900 34 000 The NPV at a 10 percent required return is NPV 56 400 7 100 1 1 8 400 1 12 6 900 1 13 37 819 23 The NPV at a 20 percent required return is NPV 56 400 7 100 1 2 8 400 1 22 6 900 1 23 40 656 94 And the NPV at a 30 percent required return is NPV 56 400 7 100 1 3 8 400 1 32 6 900 1 33 42 827 40 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 8 The profitability index is defined as the PV of the cash inflows divided by the PV of the cash outflows The equation for the profitability index at a required return of 10 percent is PI 10 300 1 1 9 200 1 12 5 700 1 13 18 000 1 181 The equation for the profitability index at a required return of 15 percent is PI 10 300 1 15 9 200 1 152 5 700 1 153 18 000 1 092 The equation for the profitability index at a required return of 22 percent is PI 10 300 1 22 9 200 1 222 5 700 1 223 18 000 0 987 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