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Chapter 6: The Time Value of Money

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• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 61

AACSB assurance of learning standards in accounting and business education require documentation of outcomes assessment. Although schools, departments, and faculty may approach assessment and its documentation differently, one approach is to provide specific questions on exams that become the basis for assessment. To aid faculty in this endeavor, we have labeled each question, exercise, and problem in Intermediate Accounting, 7e, with the following AACSB learning skills:

Questions AACSB Tags Exercises (cont.) AACSB Tags 61 Reflective thinking 69 Analytic 62 Reflective thinking 610 Analytic 63 Reflective thinking 611 Analytic 64 Reflective thinking 612 Analytic 65 Reflective thinking 613 Analytic 66 Reflective thinking 614 Analytic 67 Reflective thinking 615 Analytic 68 Reflective thinking 616 Analytic 69 Reflective thinking 617 Analytic

610 Analytic 618 Analytic 611 Analytic 619 Analytic 612 Reflective thinking 620 Analytic 613 Reflective thinking 621 Reflective thinking 614 Analytic CPA/CMA 615 Reflective thinking, Communications 1 Analytic

Brief Exercises 2 Analytic 61 Analytic 3 Analytic 62 Analytic 4 Analytic 63 Analytic 5 Analytic 64 Analytic 6 Analytic 65 Analytic 7 Analytic 66 Analytic 1 Analytic 67 Analytic 2 Reflective thinking 68 Analytic Problems 69 Analytic 61 Analytic

610 Analytic 62 Analytic 611 Analytic 63 Analytic 612 Analytic 64 Analytic 613 Analytic 65 Analytic

Exercises 66 Analytic 61 Analytic 67 Analytic 62 Analytic 68 Analytic 63 Analytic 69 Analytic 64 Analytic 610 Analytic 65 Analytic 611 Analytic 66 Analytic 612 Analytic 67 Analytic 613 Analytic 68 Analytic 614 Analytic

615 Analytic

Chapter 6 Time Value of Money Concepts

• The McGraw-Hill Companies, Inc., 2013 62 Intermediate Accounting, 7/e

Question 61 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 63 If interest is compounded more frequently than once a year, the effective rate or yield will be

higher than the annual stated rate.

Question 64 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 65 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 66 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 67 An annuity is a series of equal-sized cash flows occurring over equal intervals of time.

Question 68 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 69 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.

QUESTIONS FOR REVIEW OF KEY TOPICS

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 63

Answers to Questions (continued)

Question 610 Present Value ? 0 Year 1 Year 2 Year 3 Year 4

___________________________________________

\$200 \$200 \$200 \$200 n = 4, i = 10%

Question 611 Present Value ? 0 Year 1 Year 2 Year 3 Year 4

___________________________________________

\$200 \$200 \$200 \$200 n = 4, i = 10%

Question 612 A deferred annuity exists when the first cash flow occurs more than one period after the date

the agreement begins.

Question 613 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,

Annuity amount = PVA

Ordinary annuity factor

The annuity factor can be obtained from Table 4 at the intersection of the 8% column and 5 period row.

Question 614 Annuity amount =

\$5003.99271

Annuity amount = \$125.23

• The McGraw-Hill Companies, Inc., 2013 64 Intermediate Accounting, 7/e

Answers to Questions (concluded)

Question 615 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 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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 65

Brief Exercise 61 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 62 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 63

FV factor = \$26,600 = 1.33* \$20,000 * Future value of \$1: n = 3, i = ? (from Table 1, i = approximately 10%)

Brief Exercise 64 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 65

PV factor = \$13,200 = .825* \$16,000 * Present value of \$1: n = 4, i = ? (from Table 2, i = approximately 5%)

BRIEF EXERCISES

• The McGraw-Hill Companies, Inc., 2013 66 Intermediate Accounting, 7/e

Brief Exercise 66 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 67 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 68

PVA = \$10,000 (4.10020* ) = \$41,002 * Present value of an ordinary annuity of \$1: n =5, i = 7% (from Table 4)

Brief Exercise 69

PVAD = \$10,000 (4.38721*) = \$43,872 * Present value of an annuity due of \$1: n = 5, i = 7% (from Table 6)

Brief Exercise 610

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)

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 67

Brief Exercise 611

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 612

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 613

PVAD = \$55,000 (7.24689* ) = \$398,579 = Liability * Present value of an annuity due of \$1: n = 10, i = 8% (from Table 6)

• The McGraw-Hill Companies, Inc., 2013 68 Intermediate Accounting, 7/e

Exercise 61 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 62 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 63 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)

EXERCISES

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 69

Exercise 64 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 65 PV = \$85,000 (.82645* ) = \$70,248 = Note/revenue * Present value of \$1: n = 2, i = 10% (from Table 2)

Exercise 66 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)

• The McGraw-Hill Companies, Inc., 2013 610 Intermediate Accounting, 7/e

Exercise 67 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

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 611

Exercise 68 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

• The McGraw-Hill Companies, Inc., 2013 612 Intermediate Accounting, 7/e

Exercise 69 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)

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 613

Exercise 610 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

• The McGraw-Hill Companies, Inc., 2013 614 Intermediate Accounting, 7/e

Exercise 611 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 612 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

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 615

Exercise 613 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 614 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 615 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%

• The McGraw-Hill Companies, Inc., 2013 616 Intermediate Accounting, 7/e

Exercise 616 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 617 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)

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 617

Exercise 618 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 one-half years interest in 2013.

Exercise 619 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 620 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%)

• The McGraw-Hill Companies, Inc., 2013 618 Intermediate Accounting, 7/e

Exercise 621 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 amount of 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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 619

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

• The McGraw-Hill Companies, Inc., 2013 620 Intermediate Accounting, 7/e

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 three 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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 621

Problem 61 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 62 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)

PROBLEMS

• The McGraw-Hill Companies, Inc., 2013 622 Intermediate Accounting, 7/e

Problem 63 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 64 The restaurant should be purchased if the present value of the future cash

flows discounted at a 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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 623

Problem 65 The maximum amount that should be paid for the store is the present value of the

estimated cash flows. Years 15: PVA = \$70,000 x 3.99271* = \$279,490

* Present value of an ordinary annuity of \$1: n = 5, i = 8% (from Table 4)

Years 610: 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 1120:

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.

• The McGraw-Hill Companies, Inc., 2013 624 Intermediate Accounting, 7/e

Problem 66 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.

Annuity factor = PVA

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.

Annuity amount = PVA

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)

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 625

Problem 67 Requirement 1

Annuity amount = PVA

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

Annuity amount = PVA

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

Annuity factor = PVA

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

Annuity factor = PVA

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%)

• The McGraw-Hill Companies, Inc., 2013 626 Intermediate Accounting, 7/e

Problem 68 Requirement 1 Present value of payments 46:

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 13: \$25,000 x 2.48685* ) 74,736 (PV of payments 46 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 2013

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 627

Problem 69 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*

• The McGraw-Hill Companies, Inc., 2013 628 Intermediate Accounting, 7/e

Problem 610 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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 629

Problem 611 Requirement 1 PVAD = Annuity amount x Annuity factor

Annuity amount = PVAD

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

Annuity amount = PVAD PV of residual

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

• The McGraw-Hill Companies, Inc., 2013 630 Intermediate Accounting, 7/e

Problem 612 Requirement 1 PVA = Annuity amount x Annuity factor

Annuity amount = PVA

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

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 631

Problem 613 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.

• The McGraw-Hill Companies, Inc., 2013 632 Intermediate Accounting, 7/e

Problem 614 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

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 633

Problem 614 (concluded) Present value of pension obligations at 12/31/13: 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, 2016:

Employee PV as of 12/31/13 x FV of \$1 factor, = FV as of 12/31/16

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,

12/31/16

\$533,756

Amount of annual contribution: FVAD = Annuity amount x Annuity factor

Annuity amount = FVAD

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)

• The McGraw-Hill Companies, Inc., 2013 634 Intermediate Accounting, 7/e

Problem 615 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 companys 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).

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 635

Ethics Case 61 The ethical issue is that the 21% return implies an annual return of 21% on an

investment and misrepresents the funds 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.

CASES

• The McGraw-Hill Companies, Inc., 2013 636 Intermediate Accounting, 7/e

Analysis Case 62 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*

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 637

Communication Case 63

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

• The McGraw-Hill Companies, Inc., 2013 638 Intermediate Accounting, 7/e

Analysis Case 64

The settlement was determined by calculating the present value of lost future income (\$200,000 per year) discounted at a rate that 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 Johns retirement). Johns 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, Johns present salary was increased by 3% per year to reflect future salary increases.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 639

Judgment Case 65 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.

• The McGraw-Hill Companies, Inc., 2013 640 Intermediate Accounting, 7/e

Real World Case 66 Requirement 1

The effective interest rate can be determined by solving for the unknown present value of \$1 factor for 20 semiannual periods (20112020):

PV of \$1 factor = \$ 194 = .71193* \$272.5

* Present value of \$1: n = 20, i = ? (from Table 2, i = approximately 1.5%) So, 1.5% is the approximate effective semiannual interest rate. A financial

calculator or Excel will produce the same rate. The companys long-term debt disclosure note indicates that the annual rate is 3.0%

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.

• The McGraw-Hill Companies, Inc., 2013 Solutions Manual, Vol.1, Chapter 6 641

Real World Case 67 Requirement 1

The effective interest rate can be determined by solving for the unknown present value of an ordinary annuity of \$1 factor for seven periods:

PV of an ordinary annuity of \$1 factor = \$738 = 4.824* \$153

* Present value of an ordinary annuity \$1: n = 7, i = ? (from Table 4, i = approximately 10%) In row 7 of Table 4, the value of 4.86842 is in the 10% column. So, 10% 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 seven periods:

PV of an annuity due of \$1 factor = \$738 = 4.824 \$153

* Present value of an annuity due \$1: n = 7, i = ? (from Table 6, i = approximately 12%) In row 7 of Table 6, the value of 5.11141 is in the 12% column. So, the

approximate effective interest rate is slightly higher than 12%. A financial calculator or Excel will produce the same result.