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Chapter 06 - Time Value of Money Concepts Chapter 6 th Time Value of Money Concepts (Spiceland 6 ed) QUESTIONS FOR REVIEW OF KEY TOPICS Question 6-1 Interest is the amount of money paid or received in excess of the amount borrowed or lent. Question 6-2 Compound interest includes interest not only on the original invested amount but also on the accumulated interest from previous periods. Question 6-3 If interest is compounded more frequently than once a year, the effective rate or yield will be higher than the annual stated rate. Question 6-4 The three items of information necessary to compute the future value of a single amount are the original invested amount, the interest rate (i) and the number of compounding periods (n). Question 6-5 The present value of a single amount is the amount of money today that is equivalent to a given amount to be received or paid in the future. Question 6-6 Monetary assets and monetary liabilities represent cash or fixed claims/commitments to receive/pay cash in the future and are valued at the present value of these fixed cash flows. All other assets and liabilities are nonmonetary. Question 6-7 An annuity is a series of equal-sized cash flows occurring over equal intervals of time. Question 6-8 An ordinary annuity exists when the cash flows occur at the end of each period. In an annuity due the cash flows occur at the beginning of each period. Question 6-9 Table 2 lists the present value of $1 factors for various time periods and interest rates. The factors in Table 4 are simply the summation of the individual PV of $1 factors from Table 2. 6-1 Chapter 06 - Time Value of Money Concepts Answers to Questions (continued) Question 6-10 Present Value ? 0 Year 1 Year 2 Year 3 Year 4 ___________________________________________ $200 $200 $200 $200 n = 4, i = 10% Question 6-11 Present Value ? 0 Year 1 Year 2 Year 3 Year 4 ___________________________________________ $200 $200 $200 $200 n = 4, i = 10% Question 6-12 A deferred annuity exists when the first cash flow occurs more than one period after the date the agreement begins. Question 6-13 The formula for computing present value of an ordinary annuity incorporating the ordinary annuity factors from Table 4 is: PVA = Annuity amount x Ordinary annuity factor Solving for the annuity amount, PVA Annuity amount = Ordinary annuity factor The annuity factor can be obtained from Table 4 at the intersection of the 8% column and 5 period row. Question 6-14 $500 Annuity amount = 3.99271 Annuity amount = $125.23 6-2 Chapter 06 - Time Value of Money Concepts Answers to Questions (concluded) Question 6-15 Companies frequently acquire the use of assets by leasing rather than purchasing them. Leases usually require the payment of fixed amounts at regular intervals over the life of the lease. Certain long-term, noncancelable leases are treated in a manner similar to an installment sale by the lessor and an installment purchase by the lessee. In other words, the lessor records a receivable and the lessee records a liability for the several installment payments. For the lessee, this requires that the leased asset and corresponding lease liability be valued at the present value of the lease payments. 6-3 Chapter 06 - Time Value of Money Concepts BRIEF EXERCISES Brief Exercise 6-1 Fran should choose the second investment opportunity. More rapid compounding has the effect of increasing the actual rate, which is called the effective rate, at which money grows per year. For the second opportunity, there are four, three-month periods paying interest at 2% (one-quarter of the annual rate). $10,000 invested will grow to $10,824 ($10,000 x 1.0824*). The effective annual interest rate, often referred to as the annual yield, is 8.24% ($824 ÷ $10,000), compared to just 8% for the first opportunity. * Future value of $1: n=4, i=2% (from Table 1) Brief Exercise 6-2 Bill will not have enough accumulated to take the trip. The future value of his investment of $23,153 is $347 short of $23,500. FV = $20,000 (1.15763* ) = $23,153 * Future value of $1: n=3, i=5% (from Table 1) Brief Exercise 6-3 FV factor = $26,600 = 1.33* $20,000 * Future value of $1: n=3, i=? (from Table 1, i = approximately 10%) Brief Exercise 6-4 John would be willing to invest no more than $12,673 in this opportunity. PV = $16,000 (.79209* ) = $12,673 * Present value of $1: n=4, i=6% (from Table 2) Brief Exercise 6-5 PV factor = $13,200 = .825* $16,000 * Present value of $1: n=4, i=? (from Table 2, i = approximately 5%) 6-4 Chapter 06 - Time Value of Money Concepts Brief Exercise 6-6 Interest is paid for 12 periods at 1% (one-quarter of the annual rate). FVA = $500 (12.6825* ) = $6,341 * Future value of an ordinary annuity of $1: n=12, i=1% (from Table 3) Brief Exercise 6-7 Interest is paid for 12 periods at 1% (one-quarter of the annual rate). FVAD = $500 (12.8093* ) = $6,405 * Future value of an annuity due of $1: n=12, i=1% (from Table 5) Brief Exercise 6-8 PVA = $10,000 (4.10020* ) = $41,002 * Present value of an ordinary annuity of $1: n=5, i=7% (from Table 4) Brief Exercise 6-9 PVAD = $10,000 (4.38721*) = $43,872 * Present value of an annuity due of $1: n=5, i=7% (from Table 6) Brief Exercise 6-10 PVA = $10,000 x 4.10020* = $41,002 * Present value of an ordinary annuity of $1: n=5, i=7% (from Table 4) PV = $41,002 x .87344* = $35,813 * Present value of $1: n=2, i=7% (from Table 2) Or alternatively: From Table 4, PVA factor, n=7, i=7% = 5.38929 – PVA factor, n=2, i=7% = 1.80802 = PV factor for deferred annuity = 3.58127 PV = $10,000 x 3.58127 = $35,813 (rounded) 6-5 Chapter 06 - Time Value of Money Concepts Brief Exercise 6-11 Annuity = $100,000 = $14,903 = Payment 6.71008* * Present value of an ordinary annuity of $1: n=10, i=8% (from Table 4) Brief Exercise 6-12 PV = $6,000,0001 (12.40904* ) + 100,000,000 (.13137** ) PV = $74,454,240 + 13,137,000 = $87,591,240 = price of the bonds 1 $100,000,000 x 6% = $6,000,000 * Present value of an ordinary annuity of $1: n=30, i=7% (from Table 4) ** Present value of $1: n=30, i=7% (from Table 2) Brief Exercise 6-13 PVAD = $55,000 (7.24689* ) = $398,579 = Liability * Present value of an annuity due of $1: n=10, i=8% (from Table 6) 6-6 Chapter 06 - Time Value of Money Concepts EXERCISES Exercise 6-1 1. FV = $15,000 (2.01220* ) = $30,183 * Future value of $1: n=12, i=6% (from Table 1) 2. FV = $20,000 (2.15892* ) = $43,178 * Future value of $1: n=10, i=8% (from Table 1) 3. FV = $30,000 (9.64629* ) = $289,389 * Future value of $1: n=20, i=12% (from Table 1) 4. FV = $50,000 (1.60103* ) = $80,052 * Future value of $1: n=12, i=4% (from Table 1) Exercise 6-2 1. FV = $10,000 (2.65330* ) = $26,533 * Future value of $1: n=20, i=5% (from Table 1) 2. FV = $10,000 (1.80611* ) = $18,061 * Future value of $1: n=20, i=3% (from Table 1) 3. FV = $10,000 (1.81136* ) = $18,114 * Future value of $1: n=30, i=2% (from Table 1) Exercise 6-3 1. PV = $20,000 (.50835* ) = $10,167 * Present value of $1: n=10, i=7% (from Table 2) 2. PV = $14,000 (.39711* ) = $5,560 * Present value of $1: n=12, i=8% (from Table 2) 3. PV = $25,000 (.10367* ) = $2,592 * Present value of $1: n=20, i=12% (from Table 2) 4. PV = $40,000 (.46651* ) = $18,660 * Present value of $1: n=8, i=10% (from Table 2) 6-7 Chapter 06 - Time Value of Money Concepts Exercise 6-4 PV of $1 Payment i=8% PV n First payment: $5,000 x .92593 = $ 4,630 1 Second payment 6,000 x .85734 = 5,144 2 Third payment 8,000 x .73503 = 5,880 4 Fourth payment 9,000 x .63017 = 5,672 6 Total $21,326 Exercise 6-5 PV = $85,000 (.82645* ) = $70,248 = Note/revenue * Present value of $1: n=2, i=10% (from Table 2) Exercise 6-6 1. PV = $40,000 (.62092* ) = $24,837 * Present value of $1: n=5, i=10% (from Table 2) 2. $36,289 = .55829* $65,000 * Present value of $1: n=10, i=? (from Table 2, i = approximately 6%) 3. $15,884 = .3971* $40,000 * Present value of $1: n=?, i=8% (from Table 2, n = approximately 12 years) 4. $46,651 = .46651* $100,000 * Present value of $1: n=8, i=? (from Table 2, i = approximately 10%) 5. FV = $15,376 (3.86968* ) = $59,500 * Future value of $1: n=20, i=7% (from Table 1) 6-8 Chapter 06 - Time Value of Money Concepts Exercise 6-7 1. FVA = $2,000 (4.7793* ) = $9,559 * Future value of an ordinary annuity of $1: n=4, i=12% (from Table 3) 2. FVAD = $2,000 (5.3528* ) = $10,706 * Future value of an annuity due of $1: n=4, i=12% (from Table 5) 3. FV of $1 Deposit i=3% FV n First deposit: $2,000 x 1.60471 = $ 3,209 16 Second deposit 2,000 x 1.42576 = 2,852 12 Third deposit 2,000 x 1.26677 = 2,534 8 Fourth deposit 2,000 x 1.12551 = 2,251 4 Total $10,846 4. $2,000 x 4 = $8,000 6-9 Chapter 06 - Time Value of Money Concepts Exercise 6-8 1. PVA = $5,000 (3.60478* ) = $18,024 * Present value of an ordinary annuity of $1: n=5, i=12% (from Table 4) 2. PVAD = $5,000 (4.03735* ) = $20,187 * Present value of an annuity due of $1: n=5, i=12% (from Table 6) 3. PV of $1 Payment i = 3% PV n First payment: $5,000 x .88849 = $ 4,442 4 Second payment 5,000 x .78941 = 3,947 8 Third payment 5,000 x .70138 = 3,507 12 Fourth payment 5,000 x .62317 = 3,116 16 Fifth payment 5,000 x .55368 = 2,768 20 Total $17,780 6-10 Chapter 06 - Time Value of Money Concepts Exercise 6-9 1. PVA = $3,000 (3.99271* ) = $11,978 * Present value of an ordinary annuity of $1: n=5, i=8% (from Table 4) 2. $242,980 = 3.23973* $75,000 * Present value of an ordinary annuity of $1: n=4, i=? (from Table 4, i = approximately 9%) 3. $161,214 = 8.0607* $20,000 * Present value of an ordinary annuity of $1: n=?, i= 9% (from Table 4, n = approximately 15 years) 4. $500,000 = 6.20979* $80,518 * Present value of an ordinary annuity of $1: n=8, i=? (from Table 4, i = approximately 6%) 5. $250,000 = $78,868 3.16987* * Present value of an ordinary annuity of $1: n=4, i=10% (from Table 4) 6-11 Chapter 06 - Time Value of Money Concepts Exercise 6-10 Requirement 1 PV = $100,000 (.68058* ) = $68,058 * Present value of $1: n=5, i=8% (from Table 2) Requirement 2 Annuity amount = $100,000 5.8666* * Future value of an ordinary annuity of $1: n=5, i=8% (from Table 3) Annuity amount = $17,046 Requirement 3 Annuity amount = $100,000 6.3359* * Future value of an annuity due of $1: n=5, i=8% (from Table 5) Annuity amount = $15,783 6-12 Chapter 06 - Time Value of Money Concepts Exercise 6-11 1. Choose the option with the highest present value. (1) PV = $64,000 (2) PV = $20,000 + $8,000 (4.91732* ) * Present value of an ordinary annuity of $1: n=6, i=6% (from Table 4) PV = $20,000 + $39,339 = $59,339 (3) PV = $13,000 (4.91732* ) = $63,925 Alex should choose option (1). 2. FVA = $100,000 (13.8164* ) = $1,381,640 * Future value of an ordinary annuity of $1: n=10, i=7% (from Table 3) Exercise 6-12 PVA = $5,000 x 4.35526* = $21,776 * Present value of an ordinary annuity of $1: n=6, i=10% (from Table 4) PV = $21,776 x .82645* = $17,997 * Present value of $1: n=2, i=10% (from Table 2) Or alternatively: From Table 4, PVA factor, n=8, i=10% = 5.33493 – PVA factor, n=2, i=10% = 1.73554 = PV factor for deferred annuity = 3.59939 PV = $5,000 x 3.59939 = $17,997 6-13 Chapter 06 - Time Value of Money Concepts Exercise 6-13 Annuity = $20,000 – 5,000 = $670 = Payment 22.39646* * Present value of an ordinary annuity of $1: n=30, i=2% (from Table 4) Exercise 6-14 PVA factor = $100,000 = 7.46938* $13,388 * Present value of an ordinary annuity of $1: n=20, i=? (from Table 4, i = approximately 12%) Exercise 6-15 Annuity = $12,000 = $734 = Payment 16.35143* * Present value of an ordinary annuity of $1: n=20, i=2% (from Table 4) 5 years x 4 quarters = 20 periods 8% ÷ 4 quarters = 2% 6-14 Chapter 06 - Time Value of Money Concepts Exercise 6-16 PV = ? x .90573* = 1,200 PV = $1,200 = $1,325 .90573* * Present value of $1: n=5, i=2% (from Table 2) PVA = ? x 14.99203* = $1,325 annuity amount PVA = $1,325 = $88 = Payment 14.99203* * Present value of an ordinary annuity of $1: n=18, i=2% (from Table 4) Exercise 6-17 To determine the price of the bonds, we calculate the present value of the 40- period annuity (40 semiannual interest payments of $12 million) and the lump-sum payment of $300 million paid at maturity using the semiannual market rate of interest of 5%. In equation form, PV = $12,000,0001 (17.15909* ) + 300,000,000 (.14205** ) PV = $205,909,080 + 42,615,000 = $248,524,080 = price of the bonds 1 $300,000,000 x 4 % = $12,000,000 * Present value of an ordinary annuity of $1: n=40, i=5% (from Table 4) ** Present value of $1: n=40, i=5% (from Table 2) 6-15 Chapter 06 - Time Value of Money Concepts Exercise 6-18 Requirement 1 To determine the price of the bonds, we calculate the present value of the 30- period annuity (30 semiannual interest payments of $6 million) and the lump-sum payment of $200 million paid at maturity using the semiannual market rate of interest of 2.5%. In equation form, PV = $6,000,0001 (20.93029* ) + 200,000,000 (.47674) PV = $125,581,740 + 95,348,000 = $220,929,740 = price of the bonds 1 $200,000,000 x 3 % = $6,000,000 * Present value of an ordinary annuity of $1: n=30, i=2.5% (from Table 4) ** Present value of $1: n=30, i=2.5% (from Table 2) Requirement 2 $220,929,740 x 2.5% = $5,523,244 Because the bonds were outstanding only for six months of the year, Singleton reports only ½ year’s interest in 2011. Exercise 6-19 Requirement 1 PVA = $400,000 (10.59401* ) = $4,237,604 = Liability * Present value of an ordinary annuity of $1: n=20, i=7% (from Table 4) Requirement 2 PVAD = $400,000 (11.33560* ) = $4,534,240 = Liability * Present value of an annuity due of $1: n=20, i=7% (from Table 6) Exercise 6-20 PVA factor = $2,293,984 = 11.46992* $200,000 * Present value of an ordinary annuity of $1: n=20, i=? (from Table 4, i = 6%) 6-16 Chapter 06 - Time Value of Money Concepts Exercise 6-21 List A List B e 1. Interest a. First cash flow occurs one period after agreement begins. m 2. Monetary asset b. The rate at which money will actually grow during a year. j 3. Compound interest c. First cash flow occurs on the first day of the agreement. i 4. Simple interest d. The amount of money that a dollar will grow to. k 5. Annuity e. Amount of money paid/received in excess of amount borrowed/lent. l 6. Present value of a single f. Obligation to pay a sum of cash, the amount of amount which is fixed. c 7. Annuity due g. Money can be invested today and grow to a larger amount. d 8. Future value of a single h. No fixed dollar amount attached. amount a 9. Ordinary annuity i. Computed by multiplying an invested amount by the interest rate. b 10. Effective rate or yield j. Interest calculated on invested amount plus accumulated interest. h 11. Nonmonetary asset k. A series of equal-sized cash flows. g 12. Time value of money l. Amount of money required today that is equivalent to a given future amount. f 13. Monetary liability m. Claim to receive a fixed amount of money. 6-17 Chapter 06 - Time Value of Money Concepts CPA / CMA REVIEW QUESTIONS CPA Exam Questions 1. b. PV = FV x PV factor, PV=$25,458 x 0.3075 = $7,828 2. d. The sales price is equal to the present value of the note payments: Present value of first payment $ 60,000 Present value of last six payments: $60,000 x 4.36 261,600 Sales price $321,600 3. a. PVA = $100 x 4.96764 = $497 4. b. First solve for present value of a four-year ordinary annuity: PVA = $100 x 3.03735 = $304 Then discount back two years: PV = $304 x 0.79719 = $242 5. d. PVAD = $100,000 x 9.24424 = $924,424 6. a. PVA = $100 x 5.65022 = $565 (present value of the interest payments) PV = $1,000 x 0.32197 = $322 (present value of the face amount) Total present value = $887 = current market value of the bond 7. a. PVA = PMT x PVA factor $15,000 = PMT x 44.955 PMT = $334 6-18 Chapter 06 - Time Value of Money Concepts CMA Exam Questions 1. d. Both future value tables will be used because the $75,000 already in the account will be multiplied times the future value factor of 1.26 to determine the amount 3 years hence, or $94,500. The three payments of $4,000 represent an ordinary annuity. Multiplying the three-period annuity factor (3.25) by the payment amount ($4,000) results in a future value of the annuity of $13,000. Adding the two elements together produces a total account balance of $107,500. 2. a. An annuity is a series of cash flows or other economic benefits occurring at fixed intervals, ordinarily as a result of an investment. Present value is the value at a specified time of an amount or amounts to be paid or received later, discounted at some interest rate. In an annuity due, the payments occur at the beginning, rather than at the end, of the periods. Thus, the present value of an annuity due includes the initial payment at its undiscounted amount. This lease should be evaluated using the present value of an annuity due. 6-19 Chapter 06 - Time Value of Money Concepts PROBLEMS Problem 6-1 Choose the option with the lowest present value of cash outflows, net of the present value of any cash inflows (Cash outflows are shown as negative amounts; cash inflows as positive amounts). Machine A: PV = – $48,000 – 1,000 (6.71008* ) + 5,000 (.46319** ) * Present value of an ordinary annuity of $1: n=10, i=8% (from Table 4) ** Present value of $1: n=10, i=8% (from Table 2) PV = – $48,000 – 6,710 + 2,316 PV = - $52,394 Machine B: PV = – $40,000 – 4,000 (.79383) – 5,000 (.63017) – 6,000 (.54027) PV of $1: i=8% n=3 n=6 n=8 (from Table 2) PV = - $40,000 - 3,175 - 3,151 - 3,242 PV = - $49,568 Esquire should purchase machine B. Problem 6-2 1. PV = $10,000 + 8,000 (3.79079* ) = $40,326 = Equipment * Present value of an ordinary annuity of $1: n=5, i=10% (from Table 4) 2. $400,000 = Annuity amount x 5.9753* * Future value of an annuity due of $1: n=5, i=6% (from Table 5) Annuity amount = $400,000 5.9753 Annuity amount = $66,942 = Required annual deposit 3. PVAD = $120,000 (9.36492* ) = $1,123,790 = Lease liability * Present value of an annuity due of $1: n=20, i=10% (from Table 6) 6-20 Chapter 06 - Time Value of Money Concepts Problem 6-3 Choose the option with the lowest present value of cash payments. 1. PV = $1,000,000 2. PV = $420,000 + 80,000 (6.71008* ) = $956,806 * Present value of an ordinary annuity of $1: n=10, i=8% (from Table 4) 3. PV = PVAD = $135,000 (7.24689* ) = $978,330 * Present value of an annuity due of $1: n=10, i=8% (from Table 6) 4. PV = $1,500,000 (.68058* ) = $1,020,870 * Present value of $1: n=5, i=8% (from Table 2) Harding should choose option 2. Problem 6-4 The restaurant should be purchased if the present value of the future cash flows discounted at 10% rate is greater than $800,000. PV = $80,000 (4.35526* ) + 70,000 (.51316** ) + 60,000 (.46651**) n=7 n=8 + $50,000 (.42410**) + 40,000 (.38554**) + 700,000 (.38554**) n=9 n=10 n=10 * Present value of an ordinary annuity of $1: n=6, i=10% (from Table 4) ** Present value of $1:, i=10% (from Table 2) PV = $718,838 < $800,000 Since the PV is less than $800,000, the restaurant should not be purchased. 6-21 Chapter 06 - Time Value of Money Concepts Problem 6-5 The maximum amount that should be paid for the store is the present value of the estimated cash flows. Years 1-5: PVA = $70,000 x 3.99271* = $279,490 * Present value of an ordinary annuity of $1: n=5, i=8% (from Table 4) Years 6-10: PVA = $70,000 x 3.79079* = $265,355 * Present value of an ordinary annuity of $1: n=5, i=10% (from Table 4) PV = $265,355 x .68058* = $180,595 * Present value of $1: n=5, i=8% (from Table 2) Years 11-20: PVA = $70,000 x 5.65022* = $395,515 * Present value of an ordinary annuity of $1: n=10, i=12% (from Table 4) PV = $395,515 x .62092* = $245,583 * Present value of $1: n=5, i=10% (from Table 2) PV = $245,583 x .68058* = $167,139 * Present value of $1: n=5, i=8% (from Table 2) End of Year 20: PV = $400,000 x .32197* x .62092 x .68058 = $54,424 * Present value of $1: n=10, i=12% (from Table 2) Total PV = $279,490 + 180,595 + 167,139 + 54,424 = $681,648 The maximum purchase price is $681,648. 6-22 Chapter 06 - Time Value of Money Concepts Problem 6-6 1. PV of $1 factor = $30,000 = .5000* $60,000 * Present value of $1: n=? , i=8% (from Table 2, n = approximately 9 years) 2. PVA Annuity factor = Annuity amount Annuity factor = $28,700 = 4.1000* $7,000 * Present value of an ordinary annuity of $1: n= 5, i=? (from Table 4, i = approximately 7%) 3. PVA Annuity amount = Annuity factor Annuity amount = $10,000 = $1,558 = Payment 6.41766* * Present value of an ordinary annuity of $1: n=10, i=9% (from Table 4) 6-23 Chapter 06 - Time Value of Money Concepts Problem 6-7 Requirement 1 PVA Annuity amount = Annuity factor Annuity amount = $250,000 = $78,868 = Payment 3.16987* * Present value of an ordinary annuity of $1: n=4, i=10% (from Table 4) Requirement 2 PVA Annuity amount = Annuity factor Annuity amount = $250,000 = $62,614 = Payment 3.99271* * Present value of an ordinary annuity of $1: n=5, i=8% (from Table 4) Requirement 3 PVA Annuity factor = Annuity amount Annuity factor = $250,000 = 4.86845* $51,351 * Present value of an ordinary annuity of $1: n=? , i= 10% (from Table 4, n = approximately 7 payments) Requirement 4 PVA Annuity factor = Annuity amount Annuity factor = $250,000 = 2.40184* $104,087 * Present value of an ordinary annuity of $1: n= 3, i= ? (from Table 4, i = approximately 12%) 6-24 Chapter 06 - Time Value of Money Concepts Problem 6-8 Requirement 1 Present value of payments 4-6: PVA = $40,000 x 2.48685* = $99,474 * Present value of an ordinary annuity of $1: n= 3, i= 10% (from Table 4) PV = $99,474 x .75131* = $74,736 * Present value $1: n= 3, i= 10% (from Table 2) Present value of all payments: $ 62,171 (PV of payments 1-3: $25,000 x 2.48685* ) 74,736 (PV of payments 4-6 calculated above) $136,907 The note payable and corresponding building should be recorded at $136,907. Or alternatively: PV = $25,000 (2.48685* ) + 40,000 (1.86841** ) = $136,907 * Present value of an ordinary annuity of $1: n=3, i=10% (from Table 4) From Table 4, PVA factor, n=6, i=10% = 4.35526 – PVA factor, n=3 i=10% = 2.48685 = PV factor for deferred annuity = 1.86841** Requirement 2 $136,907 x 10% = $13,691 = Interest in the year 2011 6-25 Chapter 06 - Time Value of Money Concepts Problem 6-9 Choose the alternative with the highest present value. Alternative 1: PV = $180,000 Alternative 2: PV = PVAD = $16,000 (11.33560* ) = $181,370 * Present value of an annuity due of $1: n=20, i=7% (from Table 6) Alternative 3: PVA = $50,000 x 7.02358* = $351,179 * Present value of an ordinary annuity of $1: n=10, i=7% (from Table 4) PV = $351,179 x .54393* = $191,017 * Present value of $1: n=9, i=7% (from Table 2) John should choose alternative 3. Or, alternatively (for 3): PV = $50,000 (3.82037* ) = $191,019 (difference due to rounding) From Table 4, PVA factor, n=19, i=7% = 10.33560 – PVA factor, n=9, i=7% = 6.51523 = PV factor for deferred annuity = 3.82037* or, From Table 6, PVAD factor, n=20, i=7% = 11.33560 — PVAD factor, n=10, i=7% = 7.51523 = PV factor for deferred annuity = 3.82037* 6-26 Chapter 06 - Time Value of Money Concepts Problem 6-10 PV = $20,000 (3.79079* ) + 100,000 (.62092** ) = $137,908 * Present value of an ordinary annuity of $1: n=5, i=10% (from Table 4) ** Present value of $1: n=5, i=10% (from Table 2) The note payable and corresponding merchandise should be recorded at $137,908. 6-27 Chapter 06 - Time Value of Money Concepts Problem 6-11 Requirement 1 PVAD = Annuity amount x Annuity factor PVAD Annuity amount = Annuity factor Annuity amount = $800,000 7.24689* * Present value of an annuity due of $1: n=10, i=8% (from Table 6) Annuity amount = $110,392 = Lease payment Requirement 2 Annuity amount = $800,000 6.71008* * Present value of an ordinary annuity of $1: n=10, i=8% (from Table 4) Annuity amount = $119,224 = Lease payment Requirement 3 PVAD = (Annuity amount x Annuity factor) + PV of residual PVAD – PV of residual Annuity amount = Annuity factor PV of residual = $50,000 x .46319* = $23,160 * Present value of $1: n=10, i=8% (from Table 2) Annuity amount = $800,000 – 23,160 7.24689* * Present value of an annuity due of $1: n=10, i=8% (from Table 6) Annuity amount = $107,196 = Lease payment 6-28 Chapter 06 - Time Value of Money Concepts Problem 6-12 Requirement 1 PVA = Annuity amount x Annuity factor PVA Annuity amount = Annuity factor Annuity amount = $800,000 7.36009* * Present value of an ordinary annuity of $1: n=10, i=6% (from Table 4) Annuity amount = $108,694 = Lease payment Requirement 2 Annuity amount = $800,000 15.32380* * Present value of an annuity due of $1: n=20, i=3% (from Table 6) Annuity amount = $52,206 = Lease payment Requirement 3 Annuity amount = $800,000 44.9550* * Present value of an ordinary annuity of $1: n=60, i=1% (given) Annuity amount = $17,796 = Lease payment 6-29 Chapter 06 - Time Value of Money Concepts Problem 6-13 Choose the option with the lowest present value of cash outflows, net of the present value of any cash inflows. (Cash outflows are shown as negative amounts; cash inflows as positive amounts) 1. Buy option: PV = - $160,000 - 5,000 (5.65022* ) + 10,000 (.32197** ) * Present value of an ordinary annuity of $1: n=10, i=12% (from Table 4) ** Present value of $1: n=10, i=12% (from Table 2) PV = - $160,000 - 28,251 + 3,220 PV = - $185,031 2. Lease option: PVAD = - $25,000 (6.32825* ) = - $158,206 * Present value of an annuity due of $1: n=10, i=12% (from Table 6) Kiddy Toy should lease the machine. 6-30 Chapter 06 - Time Value of Money Concepts Problem 6-14 Requirement 1 Tinkers: PVA = $20,000 x 7.19087* = $143,817 * Present value of an ordinary annuity of $1: n=15, i=11% (from Table 4) PV = $143,817 x .81162* = $116,725 * Present value of $1: n=2, i=11% (from Table 2) Evers: PVA = $25,000 x 7.19087* = $179,772 * Present value of an ordinary annuity of $1: n=15, i=11% (from Table 4) PV = $179,772 x .73119* = $131,447 * Present value of $1: n=3, i=11% (from Table 2) Chance: PVA = $30,000 x 7.19087* = $215,726 * Present value of an ordinary annuity of $1: n=15, i=11% (from Table 4) PV = $215,726 x .65873* = $142,105 * Present value of $1: n=4, i=11% (from Table 2) Or, alternatively: Deferred annuity factors: Deferred annuity Employee PVA factor, i=11% - PVA factor, i=11% = factor Tinkers 7.54879 (n=17) - 1.71252 (n=2) = 5.83627 Evers 7.70162 (n=18) - 2.44371 (n=3) = 5.25791 Chance 7.83929 (n=19) - 3.10245 (n=4) = 4.73684 6-31 Chapter 06 - Time Value of Money Concepts Problem 6-14 (concluded) Present value of pension obligations at 12/31/11: Tinkers: $20,000 x 5.83627 = $116,725 Evers: $25,000 x 5.25791 = $131,448* Chance: $30,000 x 4.73684 = $142,105 *rounding difference Requirement 2 Present value of pension obligations as of December 31, 2014: Employee PV as of 12/31/11 x FV of $1 factor, = FV as of 12/31/14 n=3, i=11% Tinkers $116,725 x 1.36763 = $159,637 Evers 131,448 x 1.36763 = 179,772 Chance 142,105 x 1.36763 = 194,347 Total present value, 12/31/14 $533,756 Amount of annual contribution: FVAD = Annuity amount x Annuity factor FVAD Annuity amount = Annuity factor Annuity amount = $533,756 = $143,881 3.7097* * Future value of an annuity due of $1: n=3, i=11% (from Table 5) 6-32 Chapter 06 - Time Value of Money Concepts Problem 6-15 Bond liability: PV = $4,000,0001 (18.40158* ) + 100,000,000 (.17193** ) PV = $73,606,320 + 17,193,000 = $90,799,320 = initial bond liability 1 $100,000,000 x 4 % = $4,000,000 * Present value of an ordinary annuity of $1: n=40, i=4.5% (from Table 4) ** Present value of $1: n=40, i=4.5% (from Table 2) Lease liability: Lease A: PVAD = $200,000 (9.36492* ) = $1,872,984 = Liability * Present value of an annuity due of $1: n=20, i=10% (from Table 6) Lease B: PVAD = $220,000 x 8.82371* = $1,941,216 * Present value of an annuity due of $1: n=17, i=10% (from Table 6) PV = $1,941,216 x .75131* = $1,458,455 * Present value of $1: n=3, i=10% (from Table 2) Or, alternatively for Lease B: PVA = $220,000 x 8.02155* = $1,764,741 * Present value of an ordinary annuity of $1: n=17, i=10% (from Table 4) PV = $1,764,741 x .82645** = $1,458,470 (difference due to rounding) **Present value of $1: n=2, i=10% (from Table 2) Or, alternatively for Lease B: PV = $220,000 (6.62938* ) = $1,458,464 (difference due to rounding) From Table 4, PVA factor, n=19, i=10% = 8.36492 – PVA factor, n=2, i=10% = 1.73554 = PV factor for deferred annuity = 6.62938* The company’s balance sheet would include a liability for bonds of $90,799,320 and a liability for leases of $3,331,439 ($1,872,984 + 1,458,455). 6-33 Chapter 06 - Time Value of Money Concepts CASES Ethics Case 6-1 The ethical issue is that the 21% return implies an annual return of 21% on an investment and misrepresents the fund’s performance to all current and future stakeholders. Interest rates are usually assumed to represent an annual rate, unless otherwise stated. Interested investors may assume that the return for $100 would be $21 per year, not $21 over two years. The Damon Investment Company ad should explain that the 21% rate represented appreciation over two years. 6-34 Chapter 06 - Time Value of Money Concepts Analysis Case 6-2 Sally should choose the alternative with the highest present value. Alternative 1: PV = $50,000 Alternative 2: PV = PVAD = $10,000 (5.21236* ) = $52,124 * Present value of an annuity due of $1: n=6, i=6% (from Table 6) Alternative 3: PVA = $22,000 x 2.67301* = $58,806 * Present value of an ordinary annuity of $1: n=3, i=6% (from Table 4) PV = $58,806 x .89000* = $52,337 * Present value of $1: n=2, i=6% (from Table 2) Sally should choose alternative 3. Or, alternatively (for 3): PV = $22,000 (2.37897* ) = $52,337 From Table 4, PVA factor, n=5, i=6% = 4.21236 – PVA factor, n=2, i=6% = 1.83339 = PV factor for deferred annuity = 2.37897* or, From Table 6, PVAD factor, n=6, i=6% = 5.21236 – PVAD factor, n=3, i=6% = 2.83339 = PV factor for deferred annuity due = 2.37897* 6-35 Chapter 06 - Time Value of Money Concepts Communication Case 6-3 Suggested Grading Concepts and Grading Scheme: Content (65%) _____ 25 Explanation of the method used (present value) to compare the two contracts. _____ 30 Presentation of the calculations. 49ers PV = $6,989,065 Cowboys PV = $6,492,710 _____ 10 Correct conclusion. ____ _____ 65 points Writing (35%) _____ 5 Proper letter format. _____ 6 Terminology and tone appropriate to the audience of a player's agent. _____ 12 Organization permits ease of understanding. ____ Introduction that states purpose. ____ Paragraphs that separate main points. _____ 12 English ____ Sentences grammatically clear and well organized, concise. ____ Word selection. ____ Spelling. ____ Grammar and punctuation. ___ _____ 35 points 6-36 Chapter 06 - Time Value of Money Concepts Analysis Case 6-4 The settlement was determined by calculating the present value of lost future income ($200,000 per year)1 discounted at a rate which is expected to approximate the time value of money. In this case, the discount rate, i, apparently is 7% and the number of periods, n, is 25 (the number of years to John’s retirement). John’s settlement was calculated as follows: $200,000 x 11.65358* = $2,330,716 annuity amount * Present value of an ordinary annuity of $1: n=25, i=7% (from Table 4) Note: In the actual case, John’s present salary was increased by 3% per year to reflect future salary increases. 1In the actual case, John’s present salary was increased by 3% per year to reflect future salary increases. 6-37 Chapter 06 - Time Value of Money Concepts Judgment Case 6-5 Purchase price of new machine $150,000 Sales price of old machine (100,000) Incremental cash outflow required $ 50,000 The new machine should be purchased if the present value of the savings in operating costs of $8,000 ($18,000 - 10,000) plus the present value of the salvage value of the new machine exceeds $50,000. PV = ($8,000 x 3.99271* ) + ($25,000 x .68058** ) PV = $31,942 + 17,015 PV = $48,957 * Present value of an ordinary annuity of $1: n=5, i=8% (from Table 4) ** Present value of $1: n=5, i=8% (from Table 2) The new machine should not be purchased. 6-38 Chapter 06 - Time Value of Money Concepts Real World Case 6-6 Requirement 1 The effective interest rate can be determined by solving for the unknown present value of $1 factor for 22 semiannual periods (2010-2020): PV of $1 factor = $ 189 = .693578* $272.5 * Present value of $1: n= 22, i= ? (from Table 2, i = approximately 1.5%) There is no row 22 in Table 2. The 24-period factor in the 1.5% column is .69954. So, 1.5% is the approximate effective semiannual interest rate. A financial calculator or Excel will produce the same rate. Requirement 2 Using a 1.5% effective semiannual rate and 40 periods: PV = $1,000 (.55126* ) = $551.26 * Present value of $1: n=40, i=1.5% (from Table 2) The issue price of one, $1,000 maturity-value bond was $551.26. 6-39 Chapter 06 - Time Value of Money Concepts Real World Case 6-7 Requirement 1 The effective interest rate can be determined by solving for the unknown present value of an ordinary annuity of $1 factor for 3 periods: PV of an ordinary annuity of $1 factor = $39 = 2.7857* $14 * Present value of an ordinary annuity $1: n= 3, i= ? (from Table 4, i = approximately 4%) In row 3 of Table 4, the value of 2.77509 is in the 4% column. So, 4% is the approximate effective interest rate. A financial calculator or Excel will produce the same result. Requirement 2 The effective interest rate can be determined by solving for the unknown present value of an annuity due $1 factor for 4 periods: PV of an annuity due of $1 factor = $39 = 2.7857* $14 * Present value of an annuity due $1: n= 3, i= ? (from Table 6, i = approximately 8%) In row 3 of Table 6, the value of 2.78326 is in the 8% column. So, 8% is the approximate effective interest rate. A financial calculator or Excel will produce the same result. 6-40

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