Chap006.doc

Chapter 06 - Time Value of Money Concepts

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 6 Time Value of Money Concepts

QUESTIONS FOR REVIEW OF KEY TOPICS


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 factorSolving for the annuity amount,Annuity amount = The annuity factor can be obtained from Table 4 at the intersection of the 8% column and 5

period row.

Question 6-14 Annuity amount =Annuity amount = $125.23

6-2


Answers to Questions (concluded)

Question 6-15Companies 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


BRIEF EXERCISESFran 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.

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

FV factor = $26,600 = 1.33 $20,000

John would be willing to invest no more than $12,673 in this opportunity.

PV = $16,000 (.79209) = $12,673Present value of $1: n=4, i=6% (from Table 2)

PV factor = $13,200 = .825 $16,000

Present value of $1: n=4, i=? (from Table 2, i = approximately 5%)

6-4

Brief Exercise 6-1

Brief Exercise 6-2

Brief Exercise 6-3

Brief Exercise 6-4

Brief Exercise 6-5


Brief Exercise 6-6

Interest is paid for 12 periods at 1% (one-quarter of the annual rate).

FVA = $500 (12.6825) = $6,341Future value of an ordinary annuity of $1: n=12, i=1% (from Table 3)

Interest is paid for 12 periods at 1% (one-quarter of the annual rate).

FVAD = $500 (12.8093) = $6,405Future value of an annuity due of $1: n=12, i=1% (from Table 5)

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

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

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

PV = $41,002 x .87344 = $35,813Present 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

Brief Exercise 6-7

Brief Exercise 6-8

Brief Exercise 6-9

Brief Exercise 6-10


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)

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% = $6,000,000Present 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)

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

6-6

Brief Exercise 6-12

Brief Exercise 6-13


EXERCISES1. FV = $15,000 (2.01220) = $30,183

Future value of $1: n=12, i=6% (from Table 1)

2. FV = $20,000 (2.15892) = $43,178Future value of $1: n=10, i=8% (from Table 1)



1. FV = $10,000 (2.65330) = $26,533

Future value of $1: n=20, i=5% (from Table 1)



1. PV = $20,000 (.50835) = $10,167

Present value of $1: n=10, i=7% (from Table 2)

2. PV = $14,000 (.39711) = $5,560Present value of $1: n=12, i=8% (from Table 2)



6-7

Exercise 6-1

Exercise 6-2

Exercise 6-3


PV of $1

Payment i=8% PV nFirst payment: $5,000 x .92593 = $ 4,630 1Second payment 6,000 x .85734 = 5,144 2Third payment 8,000 x .73503 = 5,880 4Fourth payment 9,000 x .63017 = 5,672 6

Total $21,326

PV = $85,000 (.82645) = $70,248 = Note/revenue

Present value of $1: n=2, i=10% (from Table 2)1. PV = $40,000 (.62092) = $24,837


2. $36,289 = .55829$65,000


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



6-8

Exercise 6-4

Exercise 6-5

Exercise 6-6


Exercise 6-71. FVA = $2,000 (4.7793) = $9,559Future value of an ordinary annuity of $1: n=4, i=12% (from Table 3)

2. FVAD = $2,000 (5.3528) = $10,706Future value of an annuity due of $1: n=4, i=12% (from Table 5)

3. FV of $1Deposit i=3% FV n

First deposit: $2,000 x 1.60471 = $ 3,209 16Second deposit 2,000 x 1.42576 = 2,852 12Third deposit 2,000 x 1.26677 = 2,534 8Fourth deposit 2,000 x 1.12551 = 2,251 4

Total $10,846

4. $2,000 x 4 = $8,000

6-9


Exercise 6-81. PVA = $5,000 (3.60478) = $18,024Present value of an ordinary annuity of $1: n=5, i=12% (from Table 4)

2. PVAD = $5,000 (4.03735) = $20,187Present value of an annuity due of $1: n=5, i=12% (from Table 6)

3. PV of $1Payment i = 3% PV n

First payment: $5,000 x .88849 = $ 4,442 4Second payment 5,000 x .78941 = 3,947 8Third payment 5,000 x .70138 = 3,507 12Fourth payment 5,000 x .62317 = 3,116 16Fifth payment 5,000 x .55368 = 2,768 20

Total $17,780

6-10


Exercise 6-91. PVA = $3,000 (3.99271) = $11,978Present 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


6-11


Exercise 6-10Requirement 1

PV = $100,000 (.68058) = $68,058Present value of $1: n=5, i=8% (from Table 2)

Requirement 2Annuity amount = $100,000

5.8666Future value of an ordinary annuity of $1: n=5, i=8% (from Table 3)

Annuity amount = $17,046


6.3359Future value of an annuity due of $1: n=5, i=8% (from Table 5)

Annuity amount = $15,783

6-12


Exercise 6-111. 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) = $1,381,640Future value of an ordinary annuity of $1: n=10, i=7% (from Table 3)

PVA = $5,000 x 4.35526= $21,776


PV = $21,776 x .82645= $17,997Present 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 = $5,000 x 3.59939 = $17,997

6-13

Exercise 6-12


Exercise 6-13Annuity = $20,000 – 5,000 = $670 = Payment

22.39646Present value of an ordinary annuity of $1: n=30, i=2% (from Table 4)

PVA factor = $100,000 = 7.46938

$13,388Present value of an ordinary annuity of $1: n=20, i=? (from Table 4, i = approximately 12%)

Annuity = $12,000 = $734 = Payment


5 years x 4 quarters = 20 periods8% ÷ 4 quarters = 2%

6-14

Exercise 6-14

Exercise 6-15


Exercise 6-16PV = ? x .90573= 1,200

PV = $1,200 = $1,325 .90573


PVA = ? x 14.99203= $1,325annuity amount

PVA = $1,325 = $88 = Payment 14.99203


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 % = $12,000,000Present 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

Exercise 6-17


Exercise 6-18

Requirement 1To 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 % = $6,000,000Present 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.

Requirement 1

PVA = $400,000 (10.59401) = $4,237,604 = LiabilityPresent value of an ordinary annuity of $1: n=20, i=7% (from Table 4)

Requirement 2PVAD = $400,000 (11.33560) = $4,534,240 = LiabilityPresent value of an annuity due of $1: n=20, i=7% (from Table 6)

PVA factor = $2,293,984 = 11.46992

$200,000Present value of an ordinary annuity of $1: n=20, i=? (from Table 4, i = 6%)

6-16

Exercise 6-19

Exercise 6-20


Exercise 6-21

List A List B

e 1. Interest a. First cash flow occurs one period afteragreement begins.

m 2. Monetary asset b. The rate at which money will actually growduring 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 ofamount borrowed/lent.

l 6. Present value of a single f. Obligation to pay a sum of cash, the amountof amount which is fixed.

c 7. Annuity due g. Money can be invested today and grow to alarger 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


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 $ 60,000Present value of last six payments: $60,000 x 4.36 261,600Sales 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

CPA / CMA REVIEW QUESTIONS


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


PROBLEMSChoose 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.1. PV = $10,000 + 8,000 (3.79079) = $40,326 =

EquipmentPresent value of an ordinary annuity of $1: n=5, i=10% (from Table 4)

2. $400,000 = Annuity amount x 5.9753Future 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 liabilityPresent value of an annuity due of $1: n=20, i=10% (from Table 6)

6-20

Problem 6-1

Problem 6-2


Problem 6-3Choose the option with the lowest present value of cash payments.

1. PV = $1,000,000

2. PV = $420,000 + 80,000 (6.71008) = $956,806Present value of an ordinary annuity of $1: n=10, i=8% (from Table 4)

3. PV = PVAD = $135,000 (7.24689) = $978,330Present value of an annuity due of $1: n=10, i=8% (from Table 6)

4. PV = $1,500,000 (.68058) = $1,020,870Present value of $1: n=5, i=8% (from Table 2)

Harding should choose option 2.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

Problem 6-4


Problem 6-5The 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= $279,490Present value of an ordinary annuity of $1: n=5, i=8% (from Table 4)

Years 6-10:

PVA = $70,000 x 3.79079= $265,355Present value of an ordinary annuity of $1: n=5, i=10% (from Table 4)


Years 11-20:

PVA = $70,000 x 5.65022 = $395,515Present value of an ordinary annuity of $1: n=10, i=12% (from Table 4)



End of Year 20:

PV = $400,000 x .32197x .62092 x .68058 = $54,424Present 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


Problem 6-61.PV of $1 factor = $30,000 = .5000

$60,000Present value of $1: n=? , i=8% (from Table 2, n = approximately 9 years)

2.Annuity factor =

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 =

Annuity amount = $10,000 = $1,558 = Payment 6.41766


6-23


Problem 6-7

Requirement 1Annuity amount =



Requirement 2Annuity amount =



Requirement 3Annuity factor =

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 4Annuity factor =

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


Problem 6-8

Requirement 1Present value of payments 4-6:

PVA = $40,000 x 2.48685 = $99,474Present value of an ordinary annuity of $1: n= 3, i= 10% (from Table 4)

PV = $99,474 x .75131 = $74,736Present 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) = $136,907Present 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


Requirement 2

$136,907 x 10% = $13,691 = Interest in the year 2011

6-25


Problem 6-9Choose the alternative with the highest present value.

Alternative 1:

PV = $180,000

Alternative 2:

PV = PVAD = $16,000 (11.33560) = $181,370Present value of an annuity due of $1: n=20, i=7% (from Table 6)

Alternative 3:



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


or, From Table 6,

PVAD factor, n=20, i=7% = 11.33560— PVAD factor, n=10, i=7% = 7.51523


6-26


Problem 6-10PV = $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


Requirement 1

PVAD = Annuity amount x Annuity factor

Annuity amount =

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 3PVAD = (Annuity amount x Annuity factor) + PV of residual

Annuity amount =

PV of residual = $50,000 x .46319= $23,160Present value of $1: n=10, i=8% (from Table 2)

Annuity amount = $800,000 – 23,160 7.24689



6-28

Problem 6-11


Requirement 1

PVA = Annuity amount x Annuity factor

Annuity amount =

Annuity amount = $800,000 7.36009




15.32380Present value of an annuity due of $1: n=20, i=3% (from Table 6)



44.9550Present value of an ordinary annuity of $1: n=60, i=1% (given)


6-29

Problem 6-12


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) = - $158,206Present value of an annuity due of $1: n=10, i=12% (from Table 6)

Kiddy Toy should lease the machine.

6-30

Problem 6-13


Problem 6-14

Requirement 1Tinkers:



Evers:



Chance:



Or, alternatively:

Deferred annuity factors:

Deferred annuityEmployee PVA factor, i=11% - PVA factor, i=11% = factor

Tinkers 7.54879 (n=17) - 1.71252 (n=2) = 5.83627Evers 7.70162 (n=18) - 2.44371 (n=3) = 5.25791Chance 7.83929 (n=19) - 3.10245 (n=4) = 4.73684

6-31


Problem 6-14 (concluded)

Present value of pension obligations at 12/31/11:Tinkers: $20,000 x 5.83627 = $116,725Evers: $25,000 x 5.25791 = $131,448*Chance: $30,000 x 4.73684 = $142,105

*rounding difference

Requirement 2Present value of pension obligations as of December 31, 2014:

Employee PV as of 12/31/11 x FV of $1 factor, = FV as of 12/31/14n=3, i=11%

Tinkers $116,725 x 1.36763 = $159,637Evers 131,448 x 1.36763 = 179,772Chance 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

Annuity amount =

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


Problem 6-15Bond 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 % = $4,000,000Present 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


Lease B: PVAD = $220,000 x 8.82371 = $1,941,216


PV = $1,941,216 x .75131 = $1,458,455Present value of $1: n=3, i=10% (from Table 2)

Or, alternatively for Lease B:

PVA = $220,000 x 8.02155 = $1,764,741Present 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)


= 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


CASESThe 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

Ethics Case 6-1


Analysis Case 6-2Sally should choose the alternative with the highest present value.

Alternative 1:

PV = $50,000

Alternative 2:

PV = PVAD = $10,000 (5.21236) = $52,124Present value of an annuity due of $1: n=6, i=6% (from Table 6)

Alternative 3:PVA = $22,000 x 2.67301 = $58,806Present value of an ordinary annuity of $1: n=3, i=6% (from Table 4)


Sally should choose alternative 3.

Or, alternatively (for 3):

PV = $22,000 (2.37897) = $52,337



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


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 = $6,989,065Cowboys PV = $6,492,710

_______ 10 Correct conclusion.______

_______ 65 points

Writing (35%)_______ 5 Proper letter format.

_______ 6 Terminology and tone appropriate to the audience ofa 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


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= $2,330,716annuity amount


Note: In the actual case, John’s present salary was increased by 3% per year to reflect future salary increases.

1 In the actual case, John’s present salary was increased by 3% per year to reflect future salary increases.

6-37


Judgment Case 6-5Purchase price of new machine $150,000Sales 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


Real World Case 6-6

Requirement 1The 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 2Using 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


Real World Case 6-7


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.


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

Chap006.doc

Documents

Transcript of Chap006.doc