Time Value of Money - Solutions

APPENDIX C Time Value of Money SOLUTIONS TO BRIEF EXERCISES BRIEF EXERCISE C-1 (a) Interest = p X i X n I = $5,000 X .08 X 12 years I = $4,800 Accumulated amount = $5,000 + $4,800 = $9,800 (b) Future value factor for 12 periods at 8% is 2.51817 (from Table 1) Accumulated amount = $5,000 X 2.51817 = $12,590.85 BRIEF EXERCISE C-2 1. a. b. 6% 2.5% 5 periods 6 periods 2. a. b. 5% 2% BRIEF EXERCISE C-3 FV = p X FV of 1 factor = $10,000 X 1.59385 = $15,938.50 BRIEF EXERCISE C-4 FV of an annuity of 1 = p X FV of an annuity factor = $60,000 X 12.57789 = $754,673.40 Appendix C-1 10 periods 12 periods BRIEF EXERCISE C-5 FV = p X FV of 1 factor + (p X FV of an annuity factor) = $7,000 X 3.99602 + ($1,000 X 37.45024) = $27,972.14 + $37,450.24 = $65,422.38 BRIEF EXERCISE C-6 FV = p X FV of 1 factor = $20,000 X 1.53862 = $30,772.40 BRIEF EXERCISE C-7 1. 12% 10% 4% 6 15 16 2. 12% 10% 4% 20 5 8 BRIEF EXERCISE C-8 (a) i = 10% ? 0 1 2 3 4 $10,000 5 6 7 8 Discount rate from Table 3 is .46651 (8 periods at 10%). Present value of $10,000 to be received in 8 years discounted at 10% is therefore $4,665.10 ($10,000 X .46651). Appendix C-2 BRIEF EXERCISE C-8 (Continued) (b) i = 8% ? $10,000 $10,000 $10,000 $10,000 $10,000 $10,000 0 1 2 3 4 5 6 Discount rate from Table 4 is 4.62288 (6 periods at 8%). Present value of 6 payments of $10,000 each discounted at 8% is therefore $46,228.80 ($10,000 X 4.62288). BRIEF EXERCISE C-9 i = 12% ? 0 1 2 $500,000 3 4 5 Discount rate from Table 3 is .56743 (5 periods at 12%). Present value of $500,000 to be received in 5 years discounted at 12% is therefore $283,715 ($500,000 X .56743). Smolinski Company should therefore invest $283,715 to have $500,000 in five years. BRIEF EXERCISE C-10 i = 15% ? 0 1 2 3 4 $875,000 5 6 7 8 Discount rate from Table 3 is .32690 (8 periods at 15%). Present value of $875,000 to be received in 8 years discounted at 15% is therefore $286,037.50 ($875,000 X .32690). Pizzeria Company should invest $286,037.50 to have $875,000 in eight years. Appendix C-3 BRIEF EXERCISE C-11 i = 6% ? $25,000 $25,000 $25,000 $25,000 0 1 2 3 $25,000 $25,000 4 14 15 Discount rate from Table 4 is 9.71225. Present value of 15 payments of $25,000 each discounted at 6% is therefore $242,806.25 ($25,000 X 9.71225). Kilarny Company should pay $242,806.25 for this annuity contract. BRIEF EXERCISE C-12 i = 9% ? 0 $110,000 $110,000 $110,000 $110,000 $110,000 $110,000 1 2 3 4 5 6 Discount rate from Table 4 is 4.48592. Present value of 6 payments of $110,000 each discounted at 9% is therefore $493,451.20 ($110,000 X 4.48592). Zarita Enterprises invested $493,451.20 to earn $110,000 per year for six years. Appendix C-4 BRIEF EXERCISE C-13 Diagram for Principal i = 4% ? 0 1 2 3 4 $100,000 19 20 i = 4% Diagram for Interest ? $5,500 $5,500 $5,500 $5,500 $5,500 $5,500 0 1 2 3 4 19 20 Present value of principal to be received at maturity: $100,000 X 0.45639 (PV of $1 due in 20 periods at 4% from Table 3) .............................................................. Present value of interest to be received periodically over the term of the bonds: $5,500 X 13.59033 (PV of $1 due each period for 20 periods at 4% from Table 4) ........................................................................ Present value of bonds ............................................................... $ 45,639* 74,747* $120,386* *Rounded. BRIEF EXERCISE C-14 The bonds will sell at a discount (for less than $100,000). This may be proven as follows: Present value of principal to be received at maturity: $100,000 X .31180 (PV of $1 due in 20 periods at 6% from Table 3) .............................................................. Present value of interest to be received periodically over the term of the bonds: $5,500 X 11.46992 (PV of $1 due each period for 20 periods at 6% from Table 4) ........................................................................ Present value of bonds ............................................................... *Rounded. Appendix C-5 $31,180* 63,085* $94,265* BRIEF EXERCISE C-15 Diagram for Principal i = 10% ? 0 1 2 3 $50,000 4 5 6 i = 10% Diagram for Interest ? $4,000 $4,000 $4,000 $4,000 $4,000 $4,000 0 1 2 3 4 5 6 Present value of principal to be received at maturity: $50,000 X .59627 (PV of $1 due in 6 periods at 9% from Table 3) ............................................................. Present value of interest to be received annually over the term of the note: $4,000 X 4.48592 (PV of $1 due each period for 6 periods at 9% from Table 4) ................................................................. Present value of note received .................................................. Appendix C-6 $29,813.50 17,943.68 $47,757.18 BRIEF EXERCISE C-16 Diagram for Principal i = 6% ? 0 1 2 3 $2,000,000 4 14 15 16 i = 6% Diagram for Interest ? $90,000 $90,000 $90,000 $90,000 0 1 2 3 $90,000 $90,000 $90,000 4 14 15 Present value of principal to be received at maturity: $2,000,000 X 0.39365 (PV of $1 due in 16 periods at 6% from Table 3) ............................................................. Present value of interest to be received periodically over the term of the bonds: $90,000 X 10.10590 (PV of $1 due each period for 16 periods at 6% from Table 4) ....................................................................... Present value of bonds and cash proceeds ............................. 16 $ 787,300 909,531 $1,696,831 BRIEF EXERCISE C-17 i = 11% ? 0 $2,690 $2,690 $2,690 $2,690 $2,690 $2,690 $2,690 $2,690 1 2 3 4 5 6 7 8 Discount rate from Table 4 is 5.14612. Present value of 8 payments of $2,690 each discounted at 11% is therefore $13,843.06 ($2,690 X 5.14612). Barney Googal should not purchase the tire retreading machine because the present value of the future cash flows is less than the purchase price of the retreading machine. Appendix C-7 BRIEF EXERCISE C-18 i = 5% ? 0 $112,825 $112,825 $112,825 $112,825 1 2 3 $112,825 $112,825 4 9 10 Discount rate from Table 4 is 7.72173. Present value of 10 payments of $112,825 each discounted at 5% is therefore $871,204.18 ($112,825 X 7.72173). Hung-Chao Yu Company should receive $871,204.18 from the issuance of the note. BRIEF EXERCISE C-19 i = 15% ? $35,000 $45,000 $55,000 0 1 2 3 To determine the present value of the future cash flows, discount the future cash flows at 15%, using Table 3. Year 1 ($35,000 X .86957) = Year 2 ($45,000 X .75614) = Year 3 ($55,000 X .65752) = Present value of future cash flows $ 30,434.95 34,026.30 36,163.60 $100,624.85 To achieve a minimum rate of return of 15%, Ramos Company should pay no more than $100,624.85. If Ramos pays less than $100,624.85, its rate of return will be greater than 15%. Appendix C-8 BRIEF EXERCISE C-20 i=? $2,090 0 1 2 3 $10,000 4 14 15 Present value = Future value X Present value of 1 factor $2,090 = $10,000 X Present value of 1 factor Present value of 1 factor = $2,090 ÷ $10,000 = .20900 The .20900 for 15 periods is found in the 11% column. Kerry Rodriguez will receive an 11% return. BRIEF EXERCISE C-21 $43,233 i = 15% $100,000 n=? Present value = Future value X Present value of 1 factor $43,233 = $100,000 X Present value of 1 factor Present value of 1 factor = $43,233 ÷ $100,000 = .43233 The .43233 at 15% is found in the 6 years column. Maloney Cork therefore must wait 6 years to receive $100,000. Appendix C-9 BRIEF EXERCISE C-22 i=? ? 0 $1,000 $1,000 $1,000 $1,000 $1,000 $1,000 $9,818.15 1 2 3 4 5 6 $1,000 $1,000 19 20 Present value = Future value X Present value of an annuity factor $9,818.15 = $1,000 X Present value of an annuity factor Present value of an annuity factor = $9,818.15 ÷ $1,000 = 9.81815 The 9.81815 for 20 periods is found in the 8% column. Annie Dublin will therefore earn a rate of return of 8%. BRIEF EXERCISE C-23 i = 9% $1,000 $1,000 $1,000 $1,000 $1,000 $1,000 $7,786.15 n=? Present value = Future value X Present value of an annuity factor $7,786.15 = $1,000 X Present value of an annuity factor Present value of an annuity factor = $7,786.15 ÷ $1,000 = 7.78615 The 7.78615 at an interest rate of 9% is shown in the 14-year column. Andy Sanchez therefore will receive 14 payments. Appendix C-10