# CHAPTER 6. Accounting and the Time Value of Money. 2. Use of tables. 13, a. Unknown future amount. 7, 19 1, 5, 13 2, 4, 6, 7, 11

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1 CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, Use of tables. 13, Present and future value problems: a. Unknown future amount. 7, 19 1, 5, 13 2, 4, 6, 7, 11 b. Unknown payments. 10, 11, 12 6, 12, 17 7, 16, 17 2, 6 c. Unknown number of periods. 4, 9 8, 15 2 d. Unknown interest rate. 15, 18 3, 11, 16 7, 9, 14 2, 7 e. Unknown present value. 8, 19 2, 7, 8, 10, 14 2, 3, 4, 5, 6, 8, 12, 17, 18, Value of a series of irregular deposits; changing interest rates. 1, 4, 7, 13, 14 3, 5, 8 5. Valuation of leases, pensions, bonds; choice between projects , 12, 13, 14, 15 3, 5, 6, 8, 9, 10, 13, Deferred annuity Expected Cash Flows. 20, 21 13, 14 *8. Uses of a calculator. 22, 23, 24 15, 16,

2 ASSIGNMENT CLASSIFICATION TABLE (BY LEARNING OBJECTIVE) Learning Objectives Brief Exercises Exercises Problems 1. Identify accounting topics where the time value of money is relevant. 2. Distinguish between simple and compound interest Use appropriate compound interest tables Identify variables fundamental to solving interest problems. 5. Solve future and present value of problems. 1, 2, 3, 4, 7, 8 2, 3, 6, 9, 10, 12, 15, 18, 19 1, 2, 3, 5, 7, 9, Solve future value of ordinary and annuity due problems. 5, 6, 9, 13 3, 4, 5, 6, 12, 15, 16 1, 2, 3, 4, 5, 7, 8, 9, Solve present value of ordinary and annuity due problems. 10, 11, 12, 14, 16, 17 3, 4, 5, 6, 11, 12, 16, 17, 18, 19 1, 2, 3, 4, 5, 7, 8, 9, 10, 13, Solve present value problems related to deferred annuities and bonds. *9. Apply expected cash flows to present value measurement. *10. Use a financial calculator to solve time value of money problems. 15 7, 8, 13, 14 5, 6, 11, 12 20, 21 13, 14 22, 23, 24 15, 16,

3 ASSIGNMENT CHARACTERISTICS TABLE Item Description Level of Difficulty Time (minutes) E6-1 Using interest tables. Simple 5 10 E6-2 Simple and compound interest computations. Simple 5 10 E6-3 Computation of future values and present values. Simple E6-4 Computation of future values and present values. Moderate E6-5 Computation of present value. Simple E6-6 Future value and present value problems. Moderate E6-7 Computation of bond prices. Moderate E6-8 Computations for a retirement fund. Simple E6-9 Unknown rate. Moderate 5 10 E6-10 Unknown periods and unknown interest rate. Simple E6-11 Evaluation of purchase options. Moderate E6-12 Analysis of alternatives. Simple E6-13 Computation bond liability. Moderate E6-14 Computation of pension liability. Moderate E6-15 Investment decision. Moderate E6-16 Retirement of debt. Simple E6-17 Computation of amount of rentals. Simple E6-18 Least costly payoff ordinary annuity. Simple E6-19 Least costly payoff annuity due. Simple E6-20 Expected cash flows. Simple 5 10 E6-21 Expected cash flows and present value. Moderate *E6-22 Determine the interest rate (with a calculator). Simple 3 5 *E6-23 Determine the interest rate (with a calculator). Simple 3 5 *E6-24 Determine the interest rate (with a calculator). Simple 3 5 P6-1 Various time value situations. Moderate P6-2 Various time value situations. Moderate P6-3 Analysis of alternatives. Moderate P6-4 Evaluating payment alternatives. Moderate P6-5 Analysis of alternatives. Moderate P6-6 Purchase price of a business (deferred annuities). Moderate P6-7 Time value concepts applied to solve business problems. Complex P6-8 Analysis of alternatives. Moderate P6-9 Analysis of business problems. Complex P6-10 Analysis of lease versus purchase. Complex P6-11 Pension funding, deferred annuity. Complex P6-12 Pension funding ethics Moderate P6-13 Expected cash flows and present value. Moderate P6-14 Expected cash flows and present value. Moderate *P6-15 Various time value of money situations. Moderate *P6-16 Various time value of money situations. Moderate *P6-17 Various time value of money situations. Moderate

4 ANSWERS TO QUESTIONS 1. Money has value because with it one can acquire assets and services and discharge obligations. The holding, borrowing or lending of money can result in costs or earnings. And the longer the time period involved, the greater the costs or the earnings. The cost or earning of money as a function of time is the time value of money. Accountants must have a working knowledge of compound interest, annuities, and present value concepts because of their application to numerous types of business events and transactions which require proper valuation and presentation. These concepts are applied in the following areas: (1) sinking funds, (2) installment contracts, (3) pensions, (4) long-term assets, (5) leases, (6) notes receivable and payable, (7) business combinations, and (8) amortization of premiums and discounts. 2. Some situations in which present value measures are used in accounting include: (a) Notes receivable and payable these involve single sums (the face amounts) and may involve annuities, if there are periodic interest payments. (b) Leases involve measurement of assets and obligations, which are based on the present value of annuities (lease payments) and single sums (if there are residual values to be paid at the conclusion of the lease). (c) Pensions and other deferred compensation arrangements involve discounted future annuity payments that are estimated to be paid to employees upon retirement. (d) Bond pricing the price of bonds payable is comprised of the present value of the principal or face value of the bond plus the present value of the annuity of interest payments. (e) Long-term assets evaluating various long-term investments or assessing whether an asset is impaired requires determining the present value of the estimated cash flows (may be single sums and/or an annuity). 3. Interest is the payment for the use of money. It may represent a cost or earnings depending upon whether the money is being borrowed or loaned. The earning or incurring of interest is a function of the time, the amount of money, and the risk involved (reflected in the interest rate). Simple interest is computed on the amount of the principal only, while compound interest is computed on the amount of the principal plus any accumulated interest. Compound interest involves interest on interest while simple interest does not. 4. The interest rate generally has three components: (1) Pure rate of interest This would be the amount a lender would charge if there were no possibilities of default and no expectation of inflation. (2) Expected inflation rate of interest Lenders recognize that in an inflationary economy, they are being paid back with less valuable dollars. As a result, they increase their interest rate to compensate for this loss in purchasing power. When inflationary expectations are high, interest rates are high. (3) Credit risk rate of interest The government has little or no credit risk (i.e., risk of nonpayment) when it issues bonds. A business enterprise, however, depending upon its financial stability, profitability, etc. can have a low or a high credit risk. Accountants must have knowledge about these components because these components are essential in identifying an appropriate interest rate for a given company or investor at any given moment. 5. (a) Present value of an ordinary annuity at 8% for 10 periods (Table 6-4). (b) Future value of 1 at 8% for 10 periods (Table 6-1). (c) Present value of 1 at 8% for 10 periods (Table 6-2). (d) Future value of an ordinary annuity at 8% for 10 periods (Table 6-3). 6-4

5 Questions Chapter 6 (Continued) 6. He should choose quarterly compounding, because the balance in the account on which interest will be earned will be increased more frequently, thereby resulting in more interest earned on the investment. As shown in the following calculation: Semiannual compounding, assuming the amount is invested for 2 years: n = 4 \$1,000 X = \$1, i = 4 Quarterly compounding, assuming the amount is invested for 2 years: n = 8 \$1,000 X = \$1, i = 2 Thus, with quarterly compounding, Bill could earn \$1.80 more. 7. \$24, = \$18,000 X (future value of 1 at 2 1 / 2 for 12 periods). 8. \$27, = \$50,000 X (present value of 1 at 6% for 10 periods). 9. An annuity involves (1) periodic payments or receipts, called rents, (2) of the same amount, (3) spread over equal intervals, (4) with interest compounded once each interval. Rents occur at the end of the intervals for ordinary annuities while the rents occur at the beginning of the intervals for annuities due. \$30, Amount paid each year = (present value of an ordinary annuity at 12% for 4 years). Amount paid each year = \$9, Amount deposited each year = \$160, (future value of an ordinary annuity at 10% for 4 years). Amount deposited each year = \$34, Amount deposited each year = \$160, [future value of an annuity due at 10% for 4 years ( X 1.10)]. Amount deposited each year = \$31, The process for computing the future value of an annuity due using the future value of an ordinary annuity interest table is to multiply the corresponding future value of the ordinary annuity by one plus the interest rate. For example, the factor for the future value of an annuity due for 4 years at 12% is equal to the factor for the future value of an ordinary annuity times The basis for converting the present value of an ordinary annuity table to the present value of an annuity due table involves multiplying the present value of an ordinary annuity factor by one plus the interest rate. 6-5

6 Questions Chapter 6 (Continued) 15. Present value = present value of an ordinary annuity of \$25,000 for 20 periods at? percent. \$210,000 = present value of an ordinary annuity of \$25,000 for 20 periods at? percent. Present value of an ordinary annuity for 20 periods at? percent = The factor 8.4 is closest to in the 10% column (Table 6-4). \$210,000 \$25,000 = Present value of ordinary annuity at 12% for eight periods Present value of ordinary annuity at 12% for three periods Present value of ordinary annuity at 12% for eight periods, deferred three periods. The present value of the five rents is computed as follows: X \$10,000 = \$25, (a) Present value of an annuity due. (b) Present value of 1. (c) Future value of an annuity due. (d) Future value of \$27,000 = PV of an ordinary annuity of \$6,900 for five periods at? percent. \$27,000 \$6,900 = PV of an ordinary annuity for five periods at? percent = PV of an ordinary annuity for five periods at? = approximately 9%. 19. The IRS argues that the future reserves should be discounted to present value. The result would be smaller reserves and therefore less of a charge to income. As a result, income would be higher and income taxes may therefore be higher as well. 6-6

7 SOLUTIONS TO BRIEF EXERCISES BRIEF EXERCISE 6-1 8% annual interest i = 8% PV = \$10,000 FV =? n = 3 FV = \$10,000 (FVF 3, 8% ) FV = \$10,000 ( ) FV = \$12, % annual interest, compounded semiannually i = 4% PV = \$10,000 FV =? n = 6 FV = \$10,000 (FVF 6, 4% ) FV = \$10,000 ( ) FV = \$12,

8 BRIEF EXERCISE % annual interest i = 12% PV =? FV = \$20, n = 4 PV = \$20,000 (PVF 4, 12% ) PV = \$20,000 (.63552) PV = \$12, % annual interest, compounded quarterly i = 3% PV =? FV = \$20, n = 16 PV = \$20,000 (PVF 16, 3% ) PV = \$20,000 (.62317) PV = \$12,

9 BRIEF EXERCISE 6-3 i =? PV = \$30,000 FV = \$222, n = 21 FV = PV (FVF 21, i ) PV = FV (PVF 21, i ) OR \$222,000 = \$30,000 (FVF 21, i ) \$30,000 = \$222,000 (PVF 21, i ) FVF 21, i = PVF 21, i = i = 10% i = 10% BRIEF EXERCISE 6-4 i = 5% PV = \$10,000 FV = \$13,400 0? n =? FV = PV (FVF n, 5% ) PV = FV (PVF n, 5% ) OR \$13,400 = \$10,000 (FVF n, 5% ) \$10,000 = \$13,400 (PVF n, 5% ) FVF n, 5% = PVF n, 5% = n = 6 years n = 6 years 6-9

10 BRIEF EXERCISE 6-5 First payment today i = 12% R = FV AD = \$5,000 \$5,000 \$5,000 \$5,000 \$5,000? n = 20 FV AD = \$5,000 (FVF OA 20, 12% ) 1.12 FV AD = \$5,000 ( ) 1.12 FV AD = \$403,494 First payment at year-end i = 12% FV OA =? \$5,000 \$5,000 \$5,000 \$5,000 \$5, n = 20 FV OA = \$5,000 (FVF OA 20, 12% ) FV OA = \$5,000 ( ) FV OA = \$360,

11 BRIEF EXERCISE 6-6 i = 11% FV OA = R =???? \$200, n = 10 \$200,000 = R (FVF OA 10, 11% ) \$200,000 = R ( ) \$200, = R R = \$11,960 BRIEF EXERCISE % annual interest i = 12% PV =? FV = \$350, n = 5 PV = \$350,000 (PVF 5, 12% ) PV = \$350,000 (.56743) PV = \$198,

12 BRIEF EXERCISE 6-8 With quarterly compounding, there will be 20 quarterly compounding periods, at 1/4 the interest rate: PV = \$350,000 (PVF 20, 3% ) PV = \$350,000 (.55368) PV = \$193,788 BRIEF EXERCISE 6-9 i = 10% FV OA = R = \$100,000 \$12,961 \$12,961 \$12, n n =? \$100,000 = \$12,961 (FVF OA n, 10% ) \$100,000 FVF OA n, 10% = 12,961 = Therefore, n = 6 years 6-12

13 BRIEF EXERCISE 6-10 First withdrawal at year-end i = 8% PV OA = R =? \$20,000 \$20,000 \$20,000 \$20,000 \$20, n = 10 PV OA = \$20,000 (PVF OA 10, 8% ) PV OA = \$20,000 ( ) PV OA = \$134,202 First withdrawal immediately i = 8% PV AD =? R = \$20,000 \$20,000 \$20,000 \$20,000 \$20, n = 10 PV AD = \$20,000 (PVF AD 10, 8% ) PV AD = \$20,000 ( ) PV AD = \$144,

14 BRIEF EXERCISE 6-11 i =? PV = R = \$1, \$75 \$75 \$75 \$75 \$ n = 18 \$1, = \$75 (PVF OA 18, i ) \$1, PVF 18, i = = Therefore, i = 2% per month BRIEF EXERCISE 6-12 i = 8% PV = \$200,000 R =????? n = 20 \$200,000 = R (PVF OA 20, 8% ) \$200,000 = R ( ) R = \$20,

15 BRIEF EXERCISE 6-13 i = 12% R = \$20,000 \$20,000 \$20,000 \$20,000 \$20,000 12/31/06 12/31/07 12/31/08 12/31/12 12/31/13 12/31/14 n = 8 FV OA = \$20,000 (FVF OA 8, 12% ) FV OA = \$20,000 ( ) FV OA = \$245,994 BRIEF EXERCISE 6-14 i = 8% PV OA = R =? \$20,000 \$20,000 \$20,000 \$20, n = 4 n = 8 PV OA = \$20,000 (PVF OA 12 4, 8% ) PV OA = \$20,000 (PVF OA 8, 8% )(PVF 4, 8% ) OR PV OA = \$20,000 ( ) PV OA = \$20,000 ( )(.73503) PV OA = \$84,479 PV OA = \$84,

16 BRIEF EXERCISE 6-15 i = 8% PV =? PV OA = R = \$1,000,000? \$70,000 \$70,000 \$70,000 \$70,000 \$70, n = 10 \$1,000,000 (PVF 10, 8% ) = \$1,000,000 (.46319) = \$463,190 70,000 (PVF OA 10, 8% ) = \$70,000 ( ) 469,706 \$932,896 BRIEF EXERCISE 6-16 PV OA = \$20,000 \$4, \$4, \$4, \$4, \$20,000 = \$4, (PV OA 6, i% ) (PV OA 6, i% ) = \$20,000 \$4, (PV OA 6, i% ) = Therefore, i% =

17 BRIEF EXERCISE 6-17 PV AD = \$20,000 \$? \$? \$? \$? \$20,000 = Payment (PV AD 6, 12% ) \$20,000 (PV AD 6, 12% ) = Payment \$20, = \$4,

18 SOLUTIONS TO EXERCISES EXERCISE 6-1 (5 10 minutes) (a) (b) Rate of Interest Number of Periods 1. a. 9% 9 b. 3% 20 c. 5% a. 9% 25 b. 5% 30 c. 3% 28 EXERCISE 6-2 (5 10 minutes) (a) Simple interest of \$1,600 per year X 8 \$12,800 Principal 20,000 Total withdrawn \$32,800 (b) (c) Interest compounded annually Future value of 8% for 8 periods X \$20,000 Total withdrawn \$37, Interest compounded semiannually Future value of 4% for 16 periods X \$20,000 Total withdrawn \$37, EXERCISE 6-3 (10 15 minutes) (a) \$7,000 X = \$10, (b) \$7,000 X = \$3, (c) \$7,000 X = \$222, (d) \$7,000 X = \$87,

19 EXERCISE 6-4 (15 20 minutes) (a) (b) (c) (d) Future value of an ordinary annuity of \$4,000 a period for 20 periods at 8% \$183, (\$4,000 X ) Factor (1 +.08) X 1.08 Future value of an annuity due of \$4,000 a period at 8% \$197, Present value of an ordinary annuity of \$2,500 for 30 periods at 10% \$23, (\$2,500 X ) Factor (1 +.10) X 1.10 Present value of annuity due of \$2,500 for 30 periods at 10% \$25, (Or see Table 6-5 which gives \$25,924.03) Future value of an ordinary annuity of \$2,000 a period for 15 periods at 10% \$63, (\$2,000 X ) Factor (1 + 10) X 1.10 Future value of an annuity due of \$2,000 a period for 15 periods at 10% \$69, Present value of an ordinary annuity of \$1,000 for 6 periods at 9% \$4, (\$1,000 X ) Factor (1 +.09) X 1.09 Present value of an annuity date of \$1,000 for 6 periods at 9% \$4, (Or see Table 6-5) EXERCISE 6-5 (10 15 minutes) (a) \$30,000 X = \$149, (b) \$30,000 X = \$249, (c) (\$30,000 X X = \$46, or ( ) X \$30,000 = \$46, (difference of \$.08 due to rounding). 6-19

20 EXERCISE 6-6 (15 20 minutes) (a) Future value of 10% for 10 years (\$12,000 X ) = \$31, (b) Future value of an ordinary annuity of \$600,000 at 10% for 15 years (\$600,000 X ) \$19,063, Deficiency (\$20,000,000 \$19,063,488) \$936, (c) \$70,000 discounted at 8% for 10 years: \$70,000 X = \$32, Accept the bonus of \$40,000 now. (Also, consider whether the 8% is an appropriate discount rate since the president can probably earn compound interest at a higher rate without too much additional risk.) EXERCISE 6-7 (12 17 minutes) (a) \$50,000 X = \$15, \$5,000 X = 42, \$58, (b) \$50,000 X = \$11, \$5,000 X = 38, \$49, The answer should be \$50,000; the above computation is off by 10 due to rounding. (c) \$50,000 X = \$ 9, \$5,000 X = 34,054,30 \$43,

21 EXERCISE 6-8 (10 15 minutes) (a) Present value of an ordinary annuity of 1 for 4 8% Annual withdrawal X \$20,000 Required fund balance on June 30, 2013 \$66, (b) Fund balance at June 30, 2013 \$66, Future value of an ordinary annuity at 8% for 4 years = \$14, Amount of each of four contributions is \$14, EXERCISE 6-9 (10 minutes) The rate of interest is determined by dividing the future value by the present value and then finding the factor in the FVF table with n = 2 that approximates that number: \$123,210 = \$100,000 (FVF 2, i% ) \$123,210 \$100,000 = (FVF 2, i% ) = (FVF 2, i% ) reading across the n = 2 row reveals that i = 11%. EXERCISE 6-10 (10 15 minutes) (a) (b) The number of interest periods is calculated by first dividing the future value of \$1,000,000 by \$92,296, which is the value \$1.00 would accumulate to at 10% for the unknown number of interest periods. The factor or its approximate is then located in the Future Value of 1 Table by reading down the 10% column to the 25-period line; thus, 25 is the unknown number of years Mike must wait to become a millionaire. The unknown interest rate is calculated by first dividing the future value of \$1,000,000 by the present investment of \$182,696, which is the amount \$1.00 would accumulate to in 15 years at an unknown interest rate. The factor or its approximate is then located in the Future Value of 1 Table by reading across the 15-period line to the 12% column; thus, 12% is the interest rate Venus must earn on her investment to become a millionaire. 6-21

22 EXERCISE 6-11 (10 15 minutes) (a) (b) Total interest = Total payments Amount owed today \$162, (10 X \$16,274.53) \$100,000 = \$62, Sosa should borrow from the bank, since the 9% rate is lower than the manufacturer s 10% rate determined below. PV OA 10, i% = \$100,000 \$16, = Inspection of the 10 period row reveals a rate of 10%. EXERCISE 6-12 (10 15 minutes) Building A PV = \$600,000. Building B Rent X (PV of annuity due of 25 periods at 12%) = PV \$69,000 X = PV \$606, = PV Building C Rent X (PV of ordinary annuity of 25 periods at 12%) = PV \$7,000 X = PV \$54, = PV Cash purchase price \$650, PV of rental income 54, Net present value \$595, Answer: Lease Building C since the present value of its net cost is the smallest. 6-22

23 EXERCISE 6-13 (15 20 minutes) Time diagram: Lance Armstrong, Inc. PV =? i = 5% PV OA =? Principal \$2,000,000 interest \$110,000 \$110,000 \$110,000 \$110,000 \$110,000 \$110, n = 30 Formula for the interest payments: PV OA = R (PVF OA n, i ) PV OA = \$110,000 (PVF OA 30, 5% ) PV OA = \$110,000 ( ) PV OA = \$1,690,970 Formula for the principal: PV = FV (PVF n, i ) PV = \$2,000,000 (PVF 30, 5% ) PV = \$2,000,000 ( ) PV = \$462,760 The selling price of the bonds = \$1,690,970 + \$462,760 = \$2,153,

24 EXERCISE 6-14 (15 20 minutes) Time diagram: i = 8% R = PV OA =? \$700,000 \$700,000 \$700, n = 15 n = 10 Formula: PV OA = R (PVF OA n, i ) PV OA = \$700,000 (PVF OA 25 15, 8% ) PV OA = \$700,000 ( ) PV OA = \$700,000 ( ) PV OA = \$1,480,710 OR Time diagram: i = 8% R = PV OA =? \$700,000 \$700,000 \$700, FV(PV n, i ) (PV OA n, i ) 6-24

25 EXERCISE 6-14 (Continued) (i) Present value of the expected annual pension payments at the end of the 10 th year: PV OA = R (PVF OA n, i ) PV OA = \$700,000 (PVF OA 10, 8% ) PV OA = \$700,000 ( ) PV OA = \$4,697,056 (ii) Present value of the expected annual pension payments at the beginning of the current year: PV = FV (PVF n, i ) PV = \$4,697,056 (PVF 15,8% ) PV = \$4,697,056 ( ) PV = \$1,480,700* *\$10 difference due to rounding. The company s pension obligation (liability) is \$1,480,

26 EXERCISE 6-15 (15 20 minutes) (a) i = 8% PV = \$1,000,000 FV = \$1,999, n =? FVF( n, 8% ) = \$1,999,000 \$1,000,000 = reading down the 8% column, corresponds to 9 periods. (b) By setting aside \$300,000 now, Andrew can gradually build the fund to an amount to establish the foundation. PV = \$300,000 FV =? FV = \$300,000 (FVF 9, 8% ) = \$300,000 (1.999) = \$599,700 Thus, the amount needed from the annuity: \$1,999,000 \$599,700 = \$1,399,300. \$? \$? \$? FV = \$1,399, Payments = FV (FV OA 9, 8% ) = \$1,399, = \$112,

27 EXERCISE 6-16 (10 15 minutes) Amount to be repaid on March 1, Time diagram: i = 6% per six months PV = \$70,000 FV =? 3/1/05 3/1/06 3/1/07 3/1/13 3/1/14 3/1/15 n = 20 six-month periods Formula: FV = PV (FVF n, i ) FV = \$70,000 (FVF 20, 6% ) FV = \$70,000 ( ) FV = \$224,500 Amount of annual contribution to retirement fund. Time diagram: i = 10% R R R R R FV AD = R =????? \$224,500 3/1/10 3/1/11 3/1/12 3/1/13 3/1/14 3/1/

28 EXERCISE 6-16 (Continued) 1. Future value of ordinary annuity of 1 for 5 periods at 10% Factor (1 +.10) X Future value of an annuity due of 1 for 5 periods at 10% Periodic rent (\$224, ) \$33,430 EXERCISE 6-17 (10 15 minutes) Time diagram: i = 11% R R R PV OA = \$365,755??? n = 25 Formula: PV OA = R (PV OA n, i ) \$365,755 = R (PVF OA 25, 11% ) \$365,755 = R ( ) R = \$365, R = \$43,

29 EXERCISE 6-18 (10 15 minutes) Time diagram: i = 8% PV OA =? \$300,000 \$300,000 \$300,000 \$300,000 \$300, n = 15 Formula: PV OA = R (PVF OA n, i ) PV OA = \$300,000 (PVF OA 15, 8% ) PV OA = \$300,000 ( ) R = \$2,567,844 The recommended method of payment would be the 15 annual payments of \$300,000, since the present value of those payments (\$2,567,844) is less than the alternative immediate cash payment of \$2,600,

31 EXERCISE 6-20 (5 10 minutes) Expected Cash Flow Probability Cash Estimate X Assessment = Flow (a) \$ 3,800 20% \$ 760 6,300 50% 3,150 7,500 30% 2,250 Total Expected Value \$ 6,160 (b) \$ 5,400 30% \$ 1,620 7,200 50% 3,600 8,400 20% 1,680 Total Expected Value \$ 6,900 (c) \$(1,000) 10% \$ 100 2,000 80% 1,600 5,000 10% 500 Total Expected Value \$ 2,000 EXERCISE 6-21 (10 15 minutes) Estimated Cash Probability Outflow X Assessment = Expected Cash Flow \$ % \$ % % % 75 X PV Factor, n = 2, I = 6% Present Value \$ 505 X 0.89 \$

32 *EXERCISE ? 19, ,000 N I/YR. PV PMT FV 9.94% *EXERCISE ? 42,000 6,500 0 N I/YR. PV PMT FV 8.85% *EXERCISE ? 178,000 14,000 0 N I/YR. PV PMT FV 7.49% (semiannual) 6-32

33 TIME AND PURPOSE OF PROBLEMS Problem 6-1 (Time minutes) Purpose to present an opportunity for the student to determine how to use the present value tables in various situations. Each of the situations presented emphasizes either a present value of 1 or a present value of an ordinary annuity situation. Two of the situations will be more difficult for the student because a noninterest-bearing note and bonds are involved. Problem 6-2 (Time minutes) Purpose to present an opportunity for the student to determine solutions to four present and future value situations. The student is required to determine the number of years over which certain amounts will accumulate, the rate of interest required to accumulate a given amount, and the unknown amount of periodic payments. The problem develops the student s ability to set up present and future value equations and solve for unknown quantities. Problem 6-3 (Time minutes) Purpose to present the student with an opportunity to determine the present value of the costs of competing contracts. The student is required to decide which contract to accept. Problem 6-4 (Time minutes) Purpose to present the student with an opportunity to determine the present value of two lottery payout alternatives. The student is required to decide which payout option to choose. Problem 6-5 (Time minutes) Purpose to provide the student with an opportunity to determine which of four insurance options results in the largest present value. The student is required to determine the present value of options which include the immediate receipt of cash, an ordinary annuity, an annuity due, and an annuity of changing amount. The student must also deal with interest compounded quarterly. This problem is a good summary of the application of present value techniques. Problem 6-6 (Time minutes) Purpose to present an opportunity for the student to determine the present value of a series of deferred annuities. The student must deal with both cash inflows and outflows to arrive at a present value of net cash inflows. A good problem to develop the student s ability to manipulate the present value table factors to efficiently solve the problem. Problem 6-7 (Time minutes) Purpose to present the student an opportunity to use time value concepts in business situations. Some of the situations are fairly complex and will require the student to think a great deal before answering the question. For example, in one situation a student must discount a note and in another must find the proper interest rate to use in a purchase transaction. Problem 6-8 (Time minutes) Purpose to present the student with an opportunity to determine the present value of an ordinary annuity and annuity due for three different cash payment situations. The student must then decide which cash payment plan should be undertaken. 6-33

34 Time and Purpose of Problems (Continued) Problem 6-9 (Time minutes) Purpose to present the student with the opportunity to work three different problems related to time value concepts: purchase versus lease, determination of fair value of a note, and appropriateness of taking a cash discount. Problem 6-10 (Time minutes) Purpose to present the student with the opportunity to assess whether a company should purchase or lease. The computations for this problem are relatively complicated. Problem 6-11 (Time minutes) Purpose to present the student an opportunity to apply present value to retirement funding problems, including deferred annuities. Problem 6-12 (Time minutes) Purpose to provide the student an opportunity to explore the ethical issues inherent in applying time value of money concepts to retirement plan decisions. Problem 6-13 (Time minutes) Purpose to present the student an opportunity to compute expected cash flows and then apply present value techniques to determine a warranty liability. Problem 6-14 (Time minutes) Purpose to present the student an opportunity to compute expected cash flows and then apply present value techniques to determine the fair value of an asset. *Problems 6-15, 6-16, 6-17 (Time minutes each) Purpose to present the student an opportunity to use a financial calculator to solve time value of money problems. 6-34

35 SOLUTIONS TO PROBLEMS PROBLEM 6-1 (a) Given no established value for the building, the fair market value of the note would be estimated to value the building. Time diagram: i = 9% PV =? FV = \$275,000 1/1/07 1/1/08 1/1/09 1/1/10 n = 3 Formula: PV = FV (PVF n, i ) PV = \$275,000 (PVF 3, 9% ) PV = \$275,000 (.77218) PV = \$212, Cash equivalent price of building \$212, Less: Book value (\$250,000 \$100,000) 150, Gain on disposal of the building \$ 62,

36 PROBLEM 6-1 (Continued) (b) Time diagram: i = 11% Principal \$200,000 Interest PV OA =? \$18,000 \$18,000 \$18,000 \$18,000 1/1/07 1/1/08 1/1/09 1/1/2016 1/1/2017 n = 10 Present value of the principal FV (PVF 10, 11% ) = \$200,000 (.35218) = \$ 70, Present value of the interest payments R (PVF OA 10, 11% ) = \$18,000 ( ) = 106, Combined present value (purchase price) \$176, (c) Time diagram: i = 8% PV OA =? \$4,000 \$4,000 \$4,000 \$4,000 \$4, n = 10 Formula: PV OA = R (PVF OA n,i ) PV OA = \$4,000 (PVF OA 10, 8% ) PV OA = \$4,000 ( ) PV OA = \$26, (cost of machine) 6-36

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