Qu: a ) Cash flow (1 ): The CFS before the constant growth of If $ 100 $100 +1.06 $100+106 $100+1 06 1. 07 1.07+ 1.08 t - Jan 1 * = Janl * = Janl * = Jan1 * = Jan1 + = Janl * = Jan1 2010 2020 2021 2022 2023 2024 2025 Cash flow ( 2) : A perpetuity with constant growth of 9% that generates its first cash flow at * = Jan1, 2023 $100 +1. 06 $100+1 06 $100 +1.06 +1.07+168+1.09 +1.07+168+1.092 +4. 07 + 1 08+1.093 * = Jan1 * = Jan1 *= Jan1 += Jan1 * = Jan1 * = Jan1 * = Jan1 2019 2020 2021 2022 2023 2024 2025 To find the present value of Cash flow (1), discount each cash flow individually PV x -SamA, 2019 of CFC= $100+ $100+1.06, $100+ 1. 06+1.07 $100* 1.06 + 1. 07x 1.08- $372.25 1.12 1 . 12 2 1 123 To find the present value of Cash flow (2), use the perpetuity formula with growth c PK = Jam 2021 Of CFCDJ- $ 100+ 1.06* 1. 07#1 08 + 109 133.518 r-g - 4450.68 0 .12 - 0.09 0 . 02 Note that the formula given the value of perpetuity on Jan 1, 2022 So , we haveto discount PV x = JanA , 2029 of CF ( 2) to today's value. PV* = Jams , 20 19 Of ( F ( 2 ) 44:50. 6 $3, 167. 85 1 . 123 Finally , add two present values at *- Jan 1 , 2019 : $372 . 25+ $ 3,167 . 85 - $3 540 . 10 So , the value of Sparty's business on Jan 1, 2019 : $3,540. 10Q4. Sparty opened a local business in Costa Mesa. When Sparty opened business on Jan. 1, 2018, Sparty's projection for the growth of cash inflows was as follows: Year Growth Rate Jan. 1, 2019~Dec. 31, 2019 6% Jan. 1, 2020~Dec. 31, 2020 7% Jan. 1, 2021~Dec. 31, 2021 8% Jan. 1, 2022~co (forever) 9% Sparty's cash inflow on Dec. 31, 2018, was $100. Assume that Dec. 31 and Jan 1. are the same point in time. Use 12% as your discount rate. The timeline of the projected cash flows is as follows: +$100 * 1.06* +$100 * 1.06 +$100 * 1.06 1.07 * 1.08 * 1.07 * 1.08 * 1.07 * 1.08 * 1.09 * 1.09 * 1.09 +$100 +$100 * 1.06 +$100 * 1.06 * 1.07 t = Jan 1, 2019 t = Jan 1, 2020 t = Jan 1, 2021 t = Jan 1, 2022 t = Jan 1, 2023 t = Jan 1, 2024 a. Find the value of Sparty's business on Jan 1. 2019. b. Find the value of Sparty's business on Jan 1, 2020.Step 1: Calculate the cash inflows 1. Cash inflow on Dec 31, 2019 . Growth rate in 2019 = 6% . Initial cash inflow (on Dec 31, 2018) = $100 . Cash inflow at the end of 2019 (Dec 31, 2019): Cash inflow on Dec 31, 2019 = 100 x (1 + 0.06) = 100 x 1.06 = 106 2. Cash inflow on Dec 31, 2020 . Growth rate in 2020 = 7% . Cash inflow at the end of 2020: Cash inflow on Dec 31, 2020 = 106 x (1 + 0.07) = 106 x 1.07 = 113.42 3. Cash inflow on Dec 31, 2021 . Growth rate in 2021 = 8% . Cash inflow at the end of 2021: Cash inflow on Dec 31, 2021 = 113.42 x (1 + 0.08) = 113.42 x 1.08 = 122.494, Cash inflow from 2022 onwards (growing perpetually) e Growth rate from 2022 onward = 9% e Cash inflow at the end of 2022: Cash inflow on Dec 31, 2022 = 122.49 x (1 + 0.09) = 122.49 x 1.09 = 133.51 e Since the cash inflows grow at a constant rate of 9% forever, we will use the formula for the present value of a growing perpetuity. Step 2: Discount future cash flows to Jan 1, 2020 We now calculate the present value (PV) of all future cash flows by discounting them back to Jan 1, 2020, using the discount rate of 12%. 1. Present Value of cash inflow on Jan 1, 2021 e (Cash inflow on Dec 31, 2020 = $113.42 e Discounting it back by 1 year (from Jan 1, 2021 to Jan 1, 2020} using a 12% discount rate: 113.42 113.42 PVagyy = 02% 101.27 HT 14012 T L12 2. Present Value of cash inflow on Jan 1, 2022 . Cash inflow on Dec 31, 2021 = $122.49 . Discounting it back by 2 years (from Jan 1, 2022 to Jan 1, 2020); 122.49 122.49 122.49 PV2022 = (1 + 0.12)2 - 97.63 1.122 1.2544 3. Present Value of perpetuity starting from Jan 1, 2023 . Cash inflow from Jan 1, 2022 grows at 9% forever. . The present value of a growing perpetuity starting at $133.51 (the cash inflow at the end of 2022) with a discount rate of 12% and a growth rate of 9% can be calculated using the formula: C PV perpetuity = r - 9 Where: . C = 133.51 (cash inflow at the start of perpetuity), . = 0.12 (discount rate), . g = 0.09 (growth rate). Substituting the values: 133.51 133.51 PVperpetuity - 4450.33 0.12 - 0.09 0.03 Now, discount this perpetuity back to Jan 1, 2020 (2 years back);4450.33 4450.33 PVperpetuity at Jan 1, 2020 = 3547.71 (1 + 0.12)2 1.2544 Step 3: Total Present Value on Jan 1, 2020 Now, sum all the present values calculated above to get the total value of Sparty's business on Jan 1, 2020. PVtotal = PV2021 + PV2022 + PVperpetuity at Jan 1, 2020 Substituting the values: PVtotal = 101.27 + 97.63 + 3547.71 = 3746.61 Thus, the value of Sparty's business on January 1, 2020, is $3,746.61