Chapter 8 HW Solutions.doc

7/29/2019 Chapter 8 HW Solutions.doc

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CHAPTER 8

The Time Value of Money

THINKING BEYOND THE QUESTION

How much will it cost to borrow money?

A shorter borrowing period means that a creditor will receive repaymentof a loan sooner than if the borrowing period is longer. Consequently, themoney that is borrowed is at risk over a shorter period. The lender hasthe opportunity to lend the money out again and renegotiate the interestrate on the loan when it is paid back sooner.

Interest rates depend on the lenders evaluation of the ability of the bor-rower to repay the loan on a timely basis. A borrower with high operatingcash flows and net income and relatively low amounts of debt is less like-ly to have difficulty repaying a loan than a company with low cash flowsand profits and high amounts of debt. A good credit history also is impor-tant. A borrower who has a history of repaying loans and interest on atimely basis is likely to obtain lower interest loans than one who has hadpayment problems.

Lenders examine the income statement to determine a companys prof-

itability and whether profits are increasing or decreasing over time. Theyexamine the statement of cash flows to determine the amount of operat-ing cash flow a company is generating, how it is using that cash flow, andhow much debt and interest the borrower is paying. They examine thebalance sheet to determine how much debt and other liabilities the com-pany has outstanding.

Good financial statement numbers usually result in lower interest costs.

QUESTIONS

Q8-1 Future value and present value are based on the concept of interest. Bothrecognize that, when invested at a rate of interest, an amount of moneywill grow to a larger amount as time passes. Future value is the amountthat an investment will grow to over time. Higher rates of interest orlonger investment periods result in higher future value. Present value isthe amount that must be invested today to grow to a desired amount inthe future. Higher interest rates or longer investment periods result inlower present value.

201


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202 Chapter 8

Q8-2 The concept of future value assumes that money has a time value. Itassumes, for example, that a specific amount of money invested todaywill grow to a larger amount over time. Therefore, $1,000 invested todayat a rate of return greater than zero will grow to an amount larger than theamount originally deposited.

Q8-3 The concept of present value assumes that money has a time value. Itassumes, for example, that a specific amount of money to be received atsome future time has a lesser value (or benefit) than if that same amountwere available to be received today.

Q8-4 Any interest factor found on Table 1 reveals the amount that $1 will grow toif invested at the given interest rate for the number of periods indicated.Here, the number 5.55992 reveals that $1 invested at 10% for 18 periodswill grow to approximately $5.56.

Q8-5 Each year the interest earned will be larger than the interest earned in theprevious year. This is because the balance in the account gets largereach year as the interest from the previous year is added to it. Each year7% interest is earned on a larger and larger account balance.

Q8-6 Table 1 reveals what happens to $1 when it is invested at varying periodsand interest rates. As one moves from the upper left corner of the table tothe lower right corner of the table, the number of periods increases asdoes the interest rate. An increase in either one, while not reducing theother, results in additional return to the investor.

Q8-7 The problem can be solved in two steps using Table 1 only. First, find theanswer for a 25 period investment ($8,000 8.62308 = $68,984.64).Second, determine how much that amount will grow to over an additional4 periods ($68,984.64 1.41158 = $97,377.34).

Q8-8 $572,066.55 FVA = Amount Interest Factor = $5,000 114.41331= $572,066.55

Q8-9 Any interest factor found on Table 2 reveals the amount that a $1 ordinaryannuity will grow to if invested at the given interest rate for the number ofperiods indicated. Here, the number 18.88214 reveals that a 13-period, $1ordinary annuity, invested at 6%, will grow to approximately $18.88.

Q8-10 Table 3 reveals what happens to $1 when it is discounted for varyingperiods and rates. In general, the farther (in time) that money is awayfrom being collected, the less its present value. Similarly, the higher the


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The Time Value of Money 203

discount rate, the less is the present value. As one moves from the upperleft corner of Table 3 to the lower right corner of Table 3, the number ofperiods increases as does the discount rate. Therefore, such movementcauses the present value factors to decrease.

Q8-11 The problem can be solved in two steps using Table 3 only. First, discount

the future cash flow for 25 periods at 8% ($350,000 0.14602 = $51,107).Second, discount that result for another 7 periods at 8% ($51,107 0.58349 = $29,820.42).

Q8-12 Table 4 reveals what happens to an ordinary annuity of $1 when it isdiscounted for varying periods and rates. In general, the higher thediscount rate, the less is the present value. One would expect, therefore,that the interest factors would get smaller as one moves from left to rightin Table 4. This is correct. But, as the length of the annuity increases, theeffect of the higher discount rate is offset by the effect of the additionalannuity cash flows. As one moves down Table 4, there is an additional

cash flow each period adding to the value of the annuity. Overall,therefore, as one moves from left to right in Table 4, the interest factorsget smaller and smaller. As one moves from top to bottom of Table 4, theinterest factors get larger and larger.

Q8-13 If part of Jeraldos capital was returned to him at the end of each year($1,000 per year), the amount of his investment decreased each year. Ifhis investment got smaller each year, it is only reasonable that theinterest earned each year would also get smaller.

Q8-14 The rows in time value of money tables (whether in textbooks or

programmed into calculators or computers) represent periods rather thanyears. This is done so that the tables can also be used for compoundingperiods other than annual. For example, by denominating the rows inperiods, the tables can be used for monthly, quarterly, or semiannualcompounding. The table merely reflects what happens during a period. Itis up to the user to define the length of the period depending on thecircumstances of the event or transaction under consideration.

Q8-15 The size of the monthly payment is fixed at $288. When a payment is made,the interest incurred since the previous payment is deducted first and theremainder is subtracted from the balance of the loan. Therefore, as eachpayment is made, the amount she owes decreases. This causes theamount of interest owed for the following month to decrease also. Sinceeach months interest cost is always smaller than the prior monthsinterest cost, the amount of the payment left over to repay principal goesup each month. As the months go by, each payment contains lessinterest and more repayment of principal.


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204 Chapter 8

EXERCISES

E8-1 Definitions of terms are provided in the glossary.

E8-2 Owed on

March 31, 2008 $26,750 FV = $25,000 (1.07)1= $26,750

March 31, 2009 $28,622.50 FV = $25,000 (1.07)2= $28,622.50

March 31, 2009 $26,750 If the interest incurred during the firstyear is paid off before the second yearbegins, interest will accrue only on theprincipal balance of $25,000 during year2. The computation is $25,000 (1.07)1=$26,750.

E8-3 a. The future value $10,000 deposited for 25 years at 6% is computedas follows:

FVA = A IF= $10,000 4.29187= $42,918.70

b. The amount of interest earned is computed as follows:

Ending balance in the account $42,918.70Amount originally deposited 10,000.00Interest earned $32,918.70

E8-4 a. The future value of a seven-year, 5%, $2,000 annuity is computed asfollows:

FVA = A IF= $2,000 8.14201= $16,284.02

b. Amount resulting from deposits: Amount resulting from interest:

$2,000 7 deposits = $14,000 Ending balance $16,284.02Amount from deposits 14,000.00

$ 2,284.02

E8-5 a. $1,829,252 Future value of a four-year, 9%, $400,000 annuity:FVA = Amount Interest Factor

= $400,000 4.57313= $1,829,252

b. $3,680,172 Future value of a seven-year, 9%, $400,000 annuity:FVA = Amount Interest Factor

= $400,000 9.20043= $3,680,172


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c. The future value of an annuity grows from the contribution ofadditional deposits and by the compounding of interest on theexisting balance. As the length of the annuity increases, thecompounding of interest on the existing balance begins tocontribute more to the annuitys value than do the additionaldeposits. Therefore, the length of an annuity does not need to

double in order for the annuity balance to double.

E8-6 a. $226,566 =FV(0.08, 30, 2,000)b. $237,991 =FV(0.04, 60, 1,000)c. $244,129 =FV(0.02, 120, 500)d. $17,563 $244,129 $226,566 = $17,563e. $23,941 =FV(0.08/365,30365,2000/365) (using a 365-day year)

$250,507 $226,566 = $23,941f. $166,566 $226,566 ($2,000 30 deposits) = $166,566g. $271,547 $518,113 ($2,000 40 deposits) = $438,113

$438,113 $166,566 = $271,547 additional interest

E8-7 $925.93 $1,000 1.08 = $925.93 (or $1,000 0.92593)$917.43 $1,000 1.09 = $917.43 (or $1,000 0.91743)$909.09 $1,000 1.10 = $909.09 (or $1,000 0.90909)

As the rate of return increases, the present value of a future cash flow de-creases.

E8-8 $740.74 $800 1.08 = $740.74 (or $800 0.92593)$833.33 $900 1.08 = $833.33 (or $900 0.92593)$1,388.89 $1,500 1.08 = $1,388.89 (or $1,500 0.92593)

As the amount of the expected cash flow increases, the present value of afuture cash flow increases.

E8-9 $952.38 $1,000 1.05 = $952.38 (or $1,000 0.95238)$907.03 $1,000 (1.05)2= $907.03 (or $1,000 0.90703)$863.84 $1,000 (1.05)3= $863.84 (or $1,000 0.86384)

As the time until expected cash flow is received increases, the presentvalue of a future cash flow decreases.

E8-10 $600.00 The present value of $802.93 at 6% for five years is $802.93

0.74726 = $600.

E8-11 $547,953.91 PV of $50,000 received today $ 50,000.00PV of 19 period, $50,000 annuity @ 7% 497,953.91

Total $547,953.91

E8-12 $798.54 $200 3.99271 (from Table 4, 8%, 5 years) = $798.54$758.16 $200 3.79079 (from Table 4, 10%, 5 years) = $758.16


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206 Chapter 8

E8-13 a. $408.15 $500 0.81630 (from Table 3, 7%, 3 years) = $408.15b. $410.02 $100 4.10020 (from Table 4, 7%, 5 years) = $410.02

The annuity (b) has a slightly higher present value. The alternatives areapproximately the same, however. [Note: One consideration may be ex-pected investment opportunities at the end of three years. Since there is

little difference between (a) and (b), (a) may be the better choice since the$500 could be reinvested sooner than in (b).]

E8-14a. No. The present value of the net cash inflows to be received is less

than the present value of the investment made to obtain those cashinflows.

Proof:Expected additional cash inflow per year $50,500Expected additional cash outflow per year 30,200Net new cash inflow $20,300

The present value of an annuity of $20,300 for six years at 8%is:PVA = Amount IF (Table 4)PVA = $20,300 4.62288PVA = $93,844.46

Since the present value of the expected net cash inflows($93,844.46) is less than the investment required ($100,000),this was not a wise business decision.

b. $21,631.54

Proof:To earn exactly 8%, the present value of the future net cashflows must be equal to the $100,000 investment. Therefore, thepresent value of the annuity (future cash flows) = $100,000 at8% for six years.PVA = Amount IF (Table 4)$100,000 = Amount 4.62288Amount = $100,000 4.62288Amount = $21,631.54

c. $1,331.54Current expectationsExpected new cash inflows $50,500.00Expected new cash outflows 30,200.00Net expected cash flow $20,300.00Cash flow needed to earn 8% 21,631.54Amount of additional cash inflows needed $ 1,331.54


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E8-15a. $19,738.20 FV = Amount of single sum IF (Table 1)

FV = $10,000 1.97382FV = $19,738.20

$9,738.20 $19,738.20 $10,000 = $9,738.20

b. i. $2,464.51 FV = Amount of annuity IF (Table 2)$20,000 = Amount 8.11519Amount = $2,464.51

ii. Column E in the table that follows identifies the amount the in-vestment is worth at the end of each year.

iii. The total of column D is the amount invested over the six years.

iv. The total of column C is the amount of interest earned over thesix years.

A B C D E

Year

Valueat Beg.of Year

Int. Earned(Col. B Int. Rate)

AmountInvested atEnd of Year

Future Valueat End of Year(Cols. B + C +

D)

1 0.00 0.00 2,464.51 2,464.512 2,464.51 295.74 2,464.51 5,224.763 5,224.76 626.97 2,464.51 8,316.244 8,316.24 997.95 2,464.51 11,778.705 11,778.70 1,413.44 2,464.51 15,656.656 15,656.65 1,878.80 2,464.51 19,999.96

Totals 5,212.90 14,787.06

E8-16a. Present value of Option 1:

PV = Amount of single sum IF (Table 3)= $140,000 0.56447= $79,025.80

Present value of Option 2:PV = Amount of annuity IF (Table 4)

= $20,000 4.35526= $87,105.20

b. Option 2 is worth more (has a higher present value) even though hewill receive $20,000 more in the long run from Option 1 ($140,000versus 6 payments of $20,000 each).


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208 Chapter 8

E8-17 a.Compounding

FrequencyInterest Factor

(Table 1)FutureValue

AnnualSemiannualQuarterly

Monthly

1.254401.262481.26677

1.26973

$1,254.401,262.481,266.77

1,269.73

b. The more frequent the compounding, the greater is the increase inthe future value of a single sum. Compounding refers to the earningof interest on interest. When the compounding period is shorter, theearning of interest on interest begins sooner.

c. The same effect should take place. An annuity is merely a series ofseveral single sums. Whatever happens to one single sum wouldalso happen with a series of them.

d. CompoundingFrequency

Interest Factor(Table 3)

PresentValue

AnnualSemiannualQuarterlyMonthly

0.797190.792090.789410.78757

$797.19792.09789.41787.57

e. The more frequent the compounding period, the smaller is thepresent value of the sum involved. This is reasonable becausecomputing present value is the exact opposite of computing futurevalue. Because more frequent compounding causes future value toincrease, it should cause present value to decrease.

f. More frequent compounding reduces the interest earned eachperiod, but the annuity payments are received more frequently. Thetotal effect depends on frequency and interest rates.

E8-18 $410.02 (5 years) PVA = $100 4.10020 (from Table 4, 5 periods,7%) = $410.02

$702.36 (10 years) PVA = $100 7.02358 (from Table 4, 10 periods,7%) = $702.36

$1,059.40 (20 years) PVA = $100 10.59401 (from Table 4, 20 periods,7%) = $1,059.40

E8-19 $1,070.24 $80 7.02358 (from Table 4, 7%, 10 years) + $1,000 0.50835 (from Table 3, 7%, 10 years) = $561.89 + $508.35 =$1,070.24

$1,000.00 $80 6.71008 (from Table 4, 8%, 10 years) + $1,000 0.46319 (from Table 3, 8%, 10 years) = $536.81 + $463.19 =$1,000.00


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$935.82 $80 6.41766 (from Table 4, 9%, 10 years) + $1,000 0.42241 (from Table 3, 9%, 10 years) = $513.41 + $422.41 =$935.82

E8-20 $67.10 ($10 per year) PVA = $10 6.71008 (from Table 4, 10 years,8%) = $67.10

$671.00 ($100 per year) PVA = $100 6.71008 (from Table 4, 10 years,8%) = $671.00

E8-21 $1,081,434 The present value of an annuity of $300,000 per year forfive years at 12% is $300,000 3.60478 (Table 4) =$1,081,434. This is the maximum price a company shouldbe willing to pay. At any price above this amount, thecompany will earn less than a 12% return on its investment.If the company is able to pay less than this amount, its rateof return will be higher than 12%.

E8-22 a. $395,055 Because the three payments will pay off the amountborrowed, $1 million is the present value of the threepayments.PVA = A IF (3 periods, 9%)$1,000,000 = A 2.53129A = $1,000,000 2.53129A = $395,055

b. Year 1 = $90,000 $1,000,000 9% = $90,000Year 2 = $62,545 $694,945 9% = $62,545Year 3 = $32,619 $362,435 9% = $32,619

Proof:

Period

Present Valueat Beginning

of Period

InterestExpense

at 9% PaymentRepaymentof Principal

Valueof Debt at

End of Period

Year 1Year 2Year 3

$1,000,000$694,945$362,435

$90,000$62,545$32,619

$395,055$395,055$395,055

$305,055$332,510$362,436

$694,945$362,435

01

1Ignore $1 rounding difference.

(continued)

c.Journal Effect on Accounting Equation

Date Accounts Debits Credits A = L +OE

CC + RECash 1,000,0

001,000,000

Notes Payable 1,000,000 1,000,000


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210 Chapter 8

d.Journal Effect on Accounting Equation


CC + REInterest Expense 90,000 90,000Notes Payable 305,055 305,055

Cash 395,055 395,055

E8-23 a. The size of the payment is solved by the following equation:PVA = Payment IF (3 periods, 9%)$46,000 = Payment 2.53129Payment = $46,000 2.53129Payment = $18,173 ($18,172.55 before rounding to the nearest dollar)

b. The loan would be entered into the accounting system as follows:

Journal Effect on Accounting Equation

Date Accounts Debits Credits A = L +

OE

CC + RECash 46,000 46,000

Loan Payable 46,000 46,000

The three payments would be entered into the accounting system asfollows:



CC + RE

Interest Expense 4,140 4,140

Loan Payable 14,033 14,033

Cash 18,173 18,173



CC + REInterest Expense 2,877 2,877Loan Payable 15,296 15,296

Cash 18,173 18,173



OE


Cash 18,173 18,173

Presentation of an amortization table was not part of the require-ments. The following table, however, explains the entries to the ac-counting system above.


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YearBeginning of

Year BalanceInterestExpense

CashPayment

Reduction inLoan Balance

End of YearBalance

123

$46,00031,96716,671

$4,1402,8771,500

$18,17318,17318,173

$14,03315,29616,671

$31,967$16,671

0**Ignore $2 rounding difference.

E8-24a. PV = Amount IF

i. $251.89 $300 0.83962 (Table 3)ii. $237.63 $300 0.79209 (Table 3)iii. $246.81 $300 0.82270 (Table 3)iv. $1,039.53 $300 3.46511 (Table 4)v. $999.48 $100 0.94340 = $ 94.34 (Table 3)

$200 0.89000 = 178.00 (Table 3)$300 0.83962 = 251.89 (Table 3)$600 0.79209 = 475 .25 (Table 3)

Total $999.48b. Implications: 1. The present value of an investment decreases as

the time until the investment is received increases.2. The present value of an investment decreases as

the interest rate increases. A higher interest rateresults in a higher amount of interest being earnedfor investment ii ($62.37 = $300 $237.63) than forinvestment iii ($53.19 = $300 $246.81).

3. The present value of an investment increases as

the number of payments received increases. Thus,an annuity is more valuable than a single paymentwhen each annuity payment is as large as the sin-gle payment.

E8-25a. $3,851 PV(.11,18,500)

b. $114,545 FV(.05,25,2400)c. $80,610 FV(.064,30,950)d. $1,215 PMT(.09,8,13400)e. $273,555 PV(.02,40,10000)f. $6,637 FV(.07,16,238)g. $2,655 PV(.06375,3,1000)h. $9,214 PV(.05,22,700)i. $1,209 PMT(.065,12,9860)


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212 Chapter 8

j. $60,486,857 FV(.08,200,1)

PROBLEMS

P8-1

A. $8,289.87 FVA = Amount of annuity IF (Table 2)FVA = $600 13.81645FVA = $8,289.87

B. $44,992.69 FV = PV IF (Table 1)FV = $8,289.87 5.42743FV = $44,992.69

C. $1,790.47 FVA = Amount of annuity IF (Table 2)$44,992.69 = Amount of annuity 25.12902Amount = $44,992.69 25.12902Amount = $1,790.47

D. Total cash payments (parts A & B) = $6,000 [$600 10 years = $6,000]Total cash payments (part C) = $26,857.05

[$1,790.47 15 years = $26,857.05]

The difference relates to the length of time the money is invested.Time is a powerful component of the value of money. Even thoughan investor may be able to save only small amounts, it is importantto begin investing early rather than wait until later and have to investlarger amounts or look for higher paying, but riskier investments.

P8-2 Future value of plan #1:

Employee contribution per year $3,000Employer matching contribution @ 20% 600Total annual end-of-year contribution $3,600Future value annuity factor (25 periods, 8%) 73.10594Future value of plan #1 at age 65 $263,181.38

Future value of plan #2:

Employee contribution per year $2,500Employer matching contribution @ 85% 2,125Total annual end-of-year contribution $4,625Future value annuity factor (25 periods, 6%) 54.86451Future value of plan #2 at age 65 $253,748.35Choosing between plan #1 and plan #2:

Effective arguments can be made in favor of either alternative. Plan #1has the advantage of a future value that is $9,433.03 larger ($263,181.38$253,748.35). If the sole criterion is to choose the alternative with thehigher future value, the choice is to select plan #1.


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Plan #2 has an advantage too. Under plan #2, the employee contributes alesser amount each year and the employer a greater amount. In fact, underplan #2, the employee contributes $12,500 ($500 25 years) less than hewould under plan #1. At retirement, however, the fund balance is only$9,433.03 smaller. In other words, as compared to plan #2, plan #1 requires$12,500 more employee contribution but yields only $9,433.03 in additional

benefit. By this analysis, plan #2 is more favorable than plan #1.

P8-3A. Prudence = $16,000 $2,000 on each of 8 birthdays starting with her

16th and ending with her 23rd.

Margo = $80,000 $2,000 on 40 birthdays starting on her 26th andending with her 65th.

B. $539,053.35 FV single sum = Amount IF= $21,273.26 IF (42 periods @ 8%)= $21,273.26 25.33948

= $539,053.35

C. $518,113.04 FV annuity = Amount IF= $2,000 IF (40 periods @ 8%)= $2,000 259.05652= $518,113.04

D. $1,147,540.32 FV annuity = Amount IF= $2,000 IF (50 periods @ 8%)= $2,000 573.77016= $1,147,540.32

E. Compound interest is a powerful financial tool when savings arestarted early and continue over a long period of time. Here we seetwo examples. First, Prudence invested only $16,000 yet ended upwith a higher balance at age 65 than did her sister Margo whoinvested $80,000. The difference is the timing. Second, if Prudencehad continued making deposits to her IRA she would havecontributed only $84,0001 more, yet her ending balance would havebeen $608,486.972 higher. Time is your best friend when investing.

1$2,000 42 deposits (birthdays 24 through 65) = $84,0002$1,147,540.32 $539,053.35 = $608,486.97

P8-4

A. FV = PV IF = 44,400 2.25219 = 99,997. Almost. She will fall justshort of her goal.

B. No Starla would deposit $3,700 annually ($44,400 12). Then usingthis amount to calculate the future value of an annuity, therelationship would be:


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214 Chapter 8

FV = Amount of annuity IF (Table 2)FV = $3,700 17.88845FV = $66,187.27

This amount is considerably less than the desired $100,000 be-cause the initial investment of $44,400 is not on deposit for the

entire 12-year period. Therefore, it earns substantially less inter-est and does not grow to the desired $100,000 in the time al-lowed.

C. 16 years To determine the approximate number of equal annual deposits of $3,700 to equal$100,000 at the 7% rate determined above,the relationship for the future value of anannuity would be required.

FV = Amount of annuity IF (Table 2)$100,000 = $3,700 IFIF = $100,000 $3,700IF = 27.02703

In the 7% column of Table 2, this factor liesbetween 15 and 16 periods (closer to 16 peri-ods). Because an additional $3,700 paymentwould be made in the 16th year, a 15-year an-nuity would fall considerably short.


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P8-5 Investment A:PVA = $1,000 3.31213 (from Table 4, 8%, 4 years) = $3,312

Investment B:PV = $4,500 0.73503 (from Table 3, 8%, 4 years) = $3,308

Investment C:PVA = $600 2.57710 (from Table 4, 8%, 3 years) $1,546+ PV= $2,400 0.73503 (from Table 3, 8%, 4 years) 1,764

$ 3,310

The amount an investor should pay for each investment is the presentvalue of the cash flows expected from the investment. Investment A is anannuity. Investment B is a single amount, and investment C is a combina-tion of an annuity and a single amount. The present value provides a wayof equating the cash flows of each alternative. In this problem, the alter-native investments all have approximately the same present values.

Therefore, an investor would be relatively indifferent toward the choices,assuming the uncertainty of payments was the same for each investment.

P8-6

A. Account balance = $45,578.79 The future value of an annuity:FVA = Amount of annuity IF (Table 2)

Interest = $15,578.79 FVA = $3,000 15.19293FVA = $45,578.79Interest = $45,578.79 $30,000 =$15,578.79

B. Account balance = $49,680.84 see proof below

Interest = $19,680.84 $49,680.84 $30,000

Use the future value of the annuity formula in this problem. The pay-ments can be treated as 10 separate calculations of the future valueof a single amount. The relationship used is: FV = PV IF (Table 1)

Year

PVAmount

DepositedYears onDeposit

IFInterestFactor

FVFutureValue

1 $ 3,000 10 2.36736 $ 7,102.082 3,000 9 2.17189 6,515.67

3 3,000 8 1.99256 5,977.68

4 3,000 7 1.82804 5,484.12

5 3,000 6 1.67710 5,031.30

6 3,000 5 1.53862 4,615.86

7 3,000 4 1.41158 4,234.74

8 3,000 3 1.29503 3,885.09


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216 Chapter 8

9 3,000 2 1.18810 3,564.30

10 3,000 1 1.09000 3,270.00

Totals $30,000 $49,680.84

Laura earned $19,680.84 in interest.

C. The amounts are larger in part B because the amount of interest

earned is greater. The reason for the greater amount of interest isthat each $3,000 IRA contribution was on deposit one extra period.That is, in part B all 10 deposits were made at the beginning of theyear, while in part A the 10 deposits were made at the end of theyear. Each deposit, therefore, earned interest for one less period.Making deposits at the first of the year rather than at the end of theyear allows interest for each deposit to accumulate over a longerperiod of time.

P8-7A. $3,000,000 It is convenient to group the benefits by retirement

dates. The total cash payments required is obtainedby multiplying column (a) by column (b) by column (c).

(a) (b) (c)

Group

Numberof

employees

Annualpensionbenefit

Yearsto bepaid

Cashrequired

2011 10 $10,000 5 $ 500,0002016 20 10,000 5 1,000,0002021 30 10,000 5 1,500,000

$3,000,000

B. $1,235,890.49 It is convenient to group the benefits by retirementdates. The present value of each groups benefits,if payments started at year-end 2007, is obtained bymultiplying column (a) by column (b) by column (c).Because the benefits dont start until 2011, 2016, or2021, however, each groups amount must bediscounted as a single sum to the date theirpayments do start.

(a) (b) (c) (d) (e) (f)

Group #

Annualpensionbenefit

PV factor for5 period

annuity @ 7%

PV @year-end

2007

PV factorfor 4, 9 or14 years

Year-end2007 pension

liability2011 10 $10,000 4.10020 $ 410,020 0.76290 $ 312,804.262016 20 10,000 4.10020 820,040 0.54393 446,044.362021 30 10,000 4.10020 1,230,060 0.38782 477,041.87

$1,235,890.49


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C.Journal Effect on Accounting Equation


CC + REPension Expense 1,235,890 1,235,890

Pension Liability 1,235,890 1,235,890

P8-8 A. and B.Monthly rate(annual 12)

Numberof periods

Amountborrowed

Monthlypayment

Dealer financing 0.011667 72 $35,000 $721.21Bank financing 0.008333 60 29,750 632.09Credit union financing 0.006667 48 26,250 640.84

C. Responses will vary. Many students will automatically select theoption with the lowest monthly payment, bank financing. Some willnote, however, that in order to save $8.75 per month, as compared tothe credit union option, they will have to make 12 more payments.This leads some to select the slightly higher payment associatedwith the credit union option.

D. Dealer financing (72 payments of $721.21) $51,927.12Bank financing (60 payments of $632.09) 37,925.40Credit union financing (48 payments of $640.84) 30,760.32

E.

Total paymentsAmount

borrowed InterestDealer financing $51,927.12 $35,000 $16,927.12Bank financing 37,925.40 29,750 8,175.40

Credit union financing 30,760.32 26,250 4,510.32

F. Responses will vary. Many students will select credit union financingonce they see the comparative interest cost. Others may select thebank financing, even with the higher interest cost, because theywould prefer to have the extra money available for other uses.

P8-9

A. PMT(0.08/12,360,250000) = $1,834.41 monthly paymentPayments remaining = $1,834.41 324 = $594,348.84

B. $243,200 + $2,432 = $245,632 = new loan amountPMT(0.065/12,180,245632) = $2,139.72 = new payment

C. $2,139.72 180 = $385,149.60 payments remaining

D. If the Taylors can afford the additional $305.31 per month, they willpay a total of $209,200 less for the loan and pay it off 12 yearsearlier. If possible, they should do it.


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218 Chapter 8

P8-10 A. The amortization table is based on required monthly payments of$53,309.26 as determined below.PVA = Payment Interest factor$600,000 = Payment 11.25508 (Table 4, 1%, 12 months)

Payment = $600,000 11.25508Payment = $53,309.26

MonthPresent

Value (Beg.)InterestExpense

TotalPayment

PrincipalPayment Value (End)

May $600,000.00 $6,000.00 $53,309.26 $47,309.26 $552,690.74June 552,690.74 5,526.90 53,309.26 47,782.36 504,908.38July 504,908.38 5,049.08 53,309.26 48,260.18 456,648.20August 456,648.20 4,566.48 53,309.26 48,742.78 407,905.42September 407,905.42 4,079.05 53,309.26 49,230.21 358,675.21October 358,675.21 3,586.75 53,309.26 49,722.51 308,952.70

November 308,952.70 3,089.53 53,309.26 50,219.73 258,732.97December 258,732.97 2,587.33 53,309.26 50,721.93 208,011.04January 208,011.04 2,080.11 53,309.26 51,229.15 156,781.89February 156,781.89 1,567.82 53,309.26 51,741.44 105,040.45March 105,040.45 1,050.40 53,309.26 52,258.86 52,781.59April 52,781.59 527.82 53,309.26 52,781.44 01

1Note: Students may observe slight rounding errors between their solutionsand the solutions presented here. Rounding errors typically occur because ofthe number of significant digits Excel uses in its calculations.

B. Interest expense for the 2007 fiscal year would be the sum of interestexpense for MayDecember, $34,485.12.

C. Interest expense for the 2008 fiscal year would be the sum of interestexpense for JanuaryApril, $5,226.15.

D. The liability reported at the end of 2007 would be the present valueof the loan at the end of December, $208,011.04.

E. The company would have no liability at the end of 2008 for this loanbecause it would have been repaid.

P8-11 A. $2,577.10 The principal amount of the note was for the entire

purchase price of the machinery.B. 8% The first years interest was $206.17 on a principal

balance of $2,577.10. ($206.17 $2,577.10 = 8%). Thesame rate applies to the other years also.

C. $793.83 The difference between the beginning of year 1 balanceand the beginning of year 2 balance is amount ofreduction ($2,577.10 $1,783.27 = $793.83).


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Alternatively, end-of-year payment minus the year 1interest expense yields the same result ($1,000 $206.17 = $793.83).

D. $142.66 The year 2 interest expense will be reported onInterest Expense the income statement at $142.66.

E. $1,000 The year-end cash payment of $1,000 will bereported on the statement of cash flows.

F. $1,783.27 The year 1 ending balance will be reported on theLiabilities balance sheet under the category of liabilities.Note Payable The ending balance can be computed as follows:

Beginning balance + Year 1 interest expense End of year 1 payment ($2,577.10 + $206.17 $1,000 = $1,783.27).

P8-12 A. $147,763 PVA = Payment Interest factor$750,000 = Payment 5.07569 (from Table 4)Payment = $750,000 5.07569Payment = $147,763

B. The amortization table follows.

Period AmountInterest at

5% PaymentPayment of

Principal

Balanceat End of

PeriodYear 2007 $750,000 $37,500 $147,763 $110,263 $639,737Year 2008 639,737 31,987 147,763 115,776 523,961Year 2009 523,961 26,198 147,763 121,565 402,396Year 2010 402,396 20,120 147,763 127,643 274,753Year 2011 274,753 13,738 147,763 134,025 140,728Year 2012 140,728 7,036 147,763 140,727 1*

* Rounding difference because of using whole numbers in theamortization schedule.

C. 2007 interest = $37,5002008 interest = $31,987

P8-13 A. $13,529 PVA = Payment Interest factor$50,000 = Payment 3.69590 (from Table 4)Payment = $50,000 3.69590Payment = $13,529

B. First-year interest = $5,500 $50,000 11% = $5,500. Thedifference between the amount paidand the interest on the note is arepayment of principal: $13,529 $5,500 = $8,029. Thus, the balance of


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220 Chapter 8

the note at the end of the first yearwould be $41,971 ($50,000 $8,029).

(continued)

Second-year interest = $4,617 Beginning of second year principalbalance 11% interest ($41,971

11%)

C.Journal Effect on Accounting Equation



Cash 13,529 13,529



OE


Cash 13,529 13,529

P8-14 A. $305.98 The amount of monthly payments can be determined fromthe present value of an annuity equation:

PVA = Payment Interest factor (Table 4)$8,500 $2,000 = $6,500 = Payment 21.24339 (1%, 24 pe-riods)Payment = $6,500 21.24339Payment = $305.98 per month

B. $7,343.52 24 payments of $305.98 = $7,343.52

C. $843.52 Total amount paid ($7,343.52) Amount borrowed($6,500)


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D. $3,443.77 An amortization schedule is useful for determining theamount owed at the end of any period of a loan. The first12 months are shown here.

Period

AmountOwed at

Beginning1% Interest

Expense Payment

Amount ofPrincipal

PaidAmount

Owed at End1 $6,500.00 $65.00 $305.98 $240.98 $6,259.022 6,259.02 62.59 305.98 243.39 6,015.633 6,015.63 60.16 305.98 245.82 5,769.814 5,769.81 57.70 305.98 248.28 5,521.535 5,521.53 55.22 305.98 250.76 5,270.776 5,270.77 52.71 305.98 253.27 5,017.507 5,017.50 50.17 305.98 255.81 4,761.698 4,761.69 47.62 305.98 258.36 4,503.339 4,503.33 45.03 305.98 260.95 4,242.38

10 4,242.38 42.42 305.98 263.56 3,978.82

11 3,978.82 39.79 305.98 266.19 3,712.6312 3,712.63 37.13 305.98 268.85 3,443.78

Therefore, the amount owed at the end of the first year would be$3,443.78.

Alternatively, it could be pointed out that the amount owed at any point inthe life of the loan is equal to the present value of the remaining pay-ments. Here, there are 12 remaining payments of $305.98. The presentvalue is computed as follows:

PVA= $305.98 11.25508 (Table 4, 1%, 12 periods)= $3,443.83 (ignore $0.05 rounding difference)

Note: Students may observe slight rounding errors between their solu-tions and the solutions presented here. Rounding errors typically occurbecause of the number of significant digits Excel uses in its calculations.

P8-15 A. Column (i) refers to the year. This is customary. Column (iv) has tobe the annual cash payment because that is the only item that isconstant over time. If column (iv) is the annual cash payment, thereare two other columns (interest and reduction of principal) that mustadd up to the amount of the annual cash payment. These have to becolumn (iii), interest and column (v), reduction of principal. Since

column (vi) decreases by the amount of column (v) (reduction ofprincipal) each year, column (vi) must be the ending balance of thenote. This leaves only column (ii), which has to be the beginningbalance of the note each year.

(continued)


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222 Chapter 8

The complete amortization table is as follows:

(i)

Year

(ii)

Beginningof YearBalance

(iii)

Interestat 8%

(iv)

AnnualCash

Payment

(v)

Reductionof Principal

(vi)

End of YearBalance

1 130,000 10,400 50,444 40,044 89,9562 89,956 7,196 50,444 43,248 46,708

3 46,708 3,737 50,444 46,708 0

B. $130,000

C. 8% Any column (iii) entry divided by its correspondingcolumn (ii) entry. For example, $10,400 $130,000 = 8%.

D. Cannot tell This table fits either side of the transaction. A noteamortizes exactly the same way for the borrower (NotePayable) as for the lender (Note Receivable). The only

possible difference in the amortization table would be inthe manner in which column (iii) was labeled. It might belabeled interest expense if prepared by the borrower. Itmight be labeled interest revenue if prepared by thelender. Here, neither expense nor revenue wasspecified, making the amortization table equallyapplicable to either party.

E. $7,196 It will be reported as interest expense by the borrowerand as interest revenue by the lender.

F. $46,708 It will be reported as a short-term note payable by the

borrower and as a short-term note receivable by thelender.

G. $50,444 Cash outflows totaling $50,444 will be reported by theborrower ($7,196 interest expense under operatingactivities and $43,248 repayment of loan under financingactivities). Cash inflows of $50,444 will be reported bythe lender ($7,196 interest revenue under operatingactivities and $43,248 collection on loan under investingactivities).

P8-16 A. $80,603.62

This problem must be split into pieces. One approach is to assumetwo different annuities. The first annuity is $4,000 for 12 years. A sec-ond annuity is $3,000 for four years. Compute the future value ofeach annuity and add the totals together.

Annuity 1: FVA = $4,000 16.86994 (12 years @ 6%) = $67,479.76Annuity 2: FVA = $3,000 4.37462 (4 years @ 6%) = 13,123.86Balance in the account after 12 deposits $ 80,603.62


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Alternatively, one could treat the two annuities completely separate-ly. The first annuity grows for eight years (to $39,589.88) and thencontinues to grow as a single sum for four additional years (to a totalof $49,981.43). The second annuity grows to a balance of $30,622.34over its four-year life. Added together, the two annuities total to thesame amount as the first approach (subject to 15 cents of rounding

difference).

Annuity 1: FVA = $4,000 9.89747 (8 years, 6%) = $39,589.88

Annuity 1: FV = $39,589.88 1.26248 (4 years, 6%) = $49,981.43Annuity 2: FVA = $7,000 4.37462 (4 years, 6%) = 30,622.34Balance in the account after 12 deposits $ 80,603.77

B. $144,348.99 FV = $ 80,603.62 1.79085 (10 years, 6%) == $144,348.99

C. $20,603.62 Balance after 12th deposit $80,603.62Less: 8 deposits of $4,000 $32,000.00

4 deposits of $7,000 28,000.00 60,000.00Amount of interest earned $20,603.62

P8-17 A. $8,856.77 The four equal annual withdrawals constitute anannuity. The $30,000 gift is the present value of thatannuity.

PVA = Amount IF (Table 4)$30,000 = Amount 3.38724Amount = $30,000 3.38724Amount = $8,856.77

B. $5,427.08 Total withdrawals ($8,856.77 4) $35,427.08Less: Amount deposited 30,000.00Total interest earned $ 5,427.08

C. $7,659.13 Balance after 1 year, just beforefirst withdrawal ($30,000 1.07) $32,100

First withdrawal 12,000Remaining balance $20,100

$20,100 is the present value of the remaining with-drawals:

PVA = Amount IF (Table 4)$20,100 = Amount 2.62432Amount = $20,100 2.62432Amount = $7,659.13


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224 Chapter 8

P8-18 A. $13,490.05 To determine the maximum purchase price for theinvestment, it is necessary to combine the present valueof the annuity of $1,050 ($5,284.60) and the presentvalue of the lump sum to be received at the end of seven

years ($8,205.45), using a 9% interest rate.

PVA = Amount of annuity IF (Table 4)PVA = $1,050 5.03295PVA = $5,284.60

PV = FV IF (Table 3)PV = $15,000 0.54703PV = $8,205.45

B. $16,735.89 The calculation is the same as in part A, except that a5% interest rate is used. The cost of the investment toMilo would be $16,735.89 ($6,075.69 + $10,660.20)

PVA = Amount of annuity IF (Table 4)PVA = $1,050 5.78637PVA = $6,075.69

PV = FV IF (Table 3)PV = $15,000 0.71068PV = $10,660.20

C. The investment in part B has the higher cost. The cash flows to bereceived by Milo are generated from two sources. First, he willreceive the money he put in. Second, he will receive earnings on his

investment. When the interest rate is low, the amount of earnings onthe investment will be low. Therefore, to get back the same amountof money as in part A, he must put more in. That is, he must paymore for the investment.

P8-19 A. 0.007 8.4% 12 = 0.007 interest rate per month

B. 360 30 years 12 months per year = 360 monthsC. $188,000 $209,500 $21,500 down payment = $188,000

D. $1,432.25 (from the PMT function in Excel)

E. Interest = $1,316 Amount of payment $ 1,432.25Amount owed during first month $188,000.00

Principal = $116.25 Times: monthly interest rate 0.007First months interest cost $ 1,316.00Amount remaining forreduction of principal $116.25


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F. $1,796.63 Rate is 0.00666666 (8% 12). The number of periods is 180 (15 years 12 months). The amountof the loan is still $188,000.

G. $192,217 30-year mortgage:Total payments ($1,432.25 360) $515,610

Less: Mortgage amount 188,000Total interest paid $327,610

15 year mortgage:Total payments ($1,796.63 180) $323,393Less: Mortgage amount 188,000Total interest paid 135,393Difference $192,217

P8-20 A. Bob and Lisa can calculate the sales price of the vehicle as follows:

Loan amount = PV (interest rate, term, payment)$25,862.78 = PV (0.06/12,60,500)

$25,862.78 Loan amount2,000.00 (plus) Down payment

$ 27,862.78 Total sales price

B. The dealers behavior is not ethical because he is not telling Bob andLisa the whole story. The dealer reduces the amount of the monthlypayment, but does not tell Bob and Lisa that the present value of thepayments plus the down payment exceeds the agreed-upon price of$24,500.

C. Unscrupulous businesses can take advantage of customers who do

not understand the relationship among interest rate, term, andpayment amount. The present value of the payments (plus downpayment, if any) is the amount actually paid for a good or service. Byfocusing on the amount of the payment, Bob and Lisa are not gettingthe sales price that they negotiated.

P8-21 A. If the loan is paid off over 30 years at 8%, the monthly paymentwould be $1,174.02.

B. The amount owed on 12/31/08 after the monthly payment is$158,663.48.

The total interest incurred in the first year would be $12,751.72.Principal 160,000Period 360Interest Rate 0.00667Payment 1,174.02

(continued)


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226 Chapter 8

Month

Present Valueat Beginning of

MonthInterestIncurred

AmountPaid

PrincipalPaid

Value atEnd ofMonth

1 160,000.00 1,066.67 1,174.02 107.35 159,892.652 159,892.65 1,065.95 1,174.02 108.07 159,784.583 159,784.58 1,065.23 1,174.02 108.79 159,675.79

4 159,675.79 1,064.51 1,174.02 109.51 159,566.285 159,566.28 1,063.78 1,174.02 110.24 159,456.046 159,456.04 1,063.04 1,174.02 110.98 159,345.067 159,345.06 1,062.30 1,174.02 111.72 159,233.348 159,233.34 1,061.56 1,174.02 112.46 159,120.889 159,120.88 1,060.81 1,174.02 113.21 159,007.67

10 159,007.67 1,060.05 1,174.02 113.97 158,893.7011 158,893.70 1,059.29 1,174.02 114.73 158,778.9712 158,778.97 1,058.53 1,174.02 115.49 158,663.48

Total 12,751.72 14,088.24

Note: Students may observe slight rounding errors between their

solutions and the solutions presented here. Rounding errors typicallyoccur because of the number of significant digits Excel uses in itscalculations.

C. If the loan is paid off over 15 years at 8%, the monthly paymentwould be $1,529.04.The amount owed on 12/31/08 after the monthly payment is$154,243.48.The total interest incurred in the first year would be $12,591.96.

Principal 160,000Period 180

Interest Rate 0.00667Payment 1,529.04

Month


of MonthInterest In-

curredAmount

PaidPrincipal

PaidValue at End

of Month1 160,000.00 1,066.67 1,529.04 462.37 159,537.632 159,537.63 1,063.58 1,529.04 465.46 159,072.173 159,072.17 1,060.48 1,529.04 468.56 158,603.614 158,603.61 1,057.36 1,529.04 471.68 158,131.935 158,131.93 1,054.21 1,529.04 474.83 157,657.10

6 157,657.10 1,051.05 1,529.04 477.99 157,179.117 157,179.11 1,047.86 1,529.04 481.18 156,697.938 156,697.93 1,044.65 1,529.04 484.39 156,213.549 156,213.54 1,041.42 1,529.04 487.62 155,725.92

10 155,725.92 1,038.17 1,529.04 490.87 155,235.0511 155,235.05 1,034.90 1,529.04 494.14 154,740.9112 154,740.91 1,031.61 1,529.04 497.43 154,243.48

Total 12,591.96 18,348.48


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D. If the loan is paid off over 30 years at 9%, the monthly paymentwould be $1,287.40.The amount owed on 12/31/08 after the monthly payment is$158,906.83.The total interest incurred in the first year would be $14,355.63.

Principal 160,000Period 360Interest Rate 0.00750Payment 1,287.40

Month


of MonthInterest In-

curredAmount

PaidPrincipal

PaidValue at End

of Month1 160,000.00 1,200.00 1,287.40 87.40 159,912.602 159,912.60 1,199.34 1,287.40 88.06 159,824.543 159,824.54 1,198.68 1,287.40 88.72 159,735.824 159,735.82 1,198.02 1,287.40 89.38 159,646.445 159,646.44 1,197.35 1,287.40 90.05 159,556.396 159,556.39 1,196.67 1,287.40 90.73 159,465.667 159,465.66 1,195.99 1,287.40 91.41 159,374.258 159,374.25 1,195.31 1,287.40 92.09 159,282.169 159,282.16 1,194.62 1,287.40 92.78 159,189.38

10 159,189.38 1,193.92 1,287.40 93.48 159,095.9011 159,095.90 1,193.22 1,287.40 94.18 159,001.7212 159,001.72 1,192.51 1,287.40 94.89 158,906.83

Total 14,355.63 15,448.80

P8-22

1 2 3 4 5 6 7 8 9 10

b a b a d c b b a c


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228 Chapter 8

CASES

C8-1 The essential facts of the two alternatives can be summarized as follows:

Dealer 1 2Cost of car $20,500 $20,000

Rebate 1,400 1,000Trade-in 3,600 3,000Net price 15,500 16,000

Annual payments, dealer 1:$15,500 3.03735 (Table 4, 12%, 4 years) = $5,103.13Annual payments, dealer 2:$16,000 3.16987 (Table 4, 10%, 4 years) = $5,047.53

Darren would be better off trading with dealer 2. His payments would beabout $55 less each year.

C8-2 M E M O R A N D U M

DATE: (todays date)TO: HaroldFROM: (students name)SUBJECT: Evaluation of loan options

Your two financing options will result in different payments.

25-year loan: The payments will be $8,144.50 as determined by the follow-ing calculations:

PVA = Payment Interest factor$80,000 = Payment 9.82258 (Table 4, 9%, 25 years)Payment = $80,000 9.82258Payment = $8,144.50

15-year loan: The payments will be $9,346.36 as determined by the follow-ing calculations:

PVA = A IF$80,000 = A 8.55948 (Table 4, 8%, 15 years)

A = $80,000 8.55948A = $9,346.36

As a result of these payments differences, the total payments over the lifeof the loans and the total interest associated with them will be quite dif-ferent:


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25-year loan 15-year loanTotal payments $203,612.50 $140,195.40Less: Cost of house 80,000.00 80,000.00Interest paid $123,612.50 $ 60,195.40

It is apparent that you will pay more than twice as much interest over the

life of the loan if you finance your mortgage for 25 years rather than for 15years.

Therefore, if you can afford the higher payment of about $1,200 each year(about $100 a month), you should finance your loan for the shorter peri-od.

C8-3 To analyze the various contract proposals, we use present value concepts.We used Excel for our analysis; however, present value tables also canbe used. The following analysis assumes a 4% discount rate.

Proposal 1. The present value of Proposal 1 is $2,823,769. The contractguarantees $1 million per year to be paid quarterly. Thus, the quarterlycash flow is $250,000 ($1,000,000 4 = $250,000). The discount rate is 1%(4% 4 periods per year = 1%). The contract specifies 12 periods (3 years 4 quarters per year). Using Excel, the present value is determined asfollows:

=PV(0.01, 12, 250,000) = $2,813,769

Proposal 2. The present value of Proposal 2 is $3,629,895. The contractguarantees $1 million per year in each of four years. Using Excel, thepresent value is determined as follows:

=PV(0.04, 4, 1,000,000) = $3,629,895

Excel formula values were computed as follows:0.04 = discount rate4 = number of periods over which the payments are made1,000,000 = Annual payment

Proposal 3. The present value of Proposal 3 is $2,739,734. The contractguarantees a signing bonus of $900,000. Since the bonus is paid uponsigning the contract, the present value of the signing bonus is $900,000(the value today). In addition to a signing bonus, quarterly payments of$125,000 will be paid for four years. The present value of the proposal isthe sum of the present value of the signing bonus and the present value

of the quarterly payments. Using Excel, the present value is determinedas follows:

=PV(0.01, 16, 125,000) = $1,839,734

Excel formula values were computed as follows:1% (0.04/4 payments per year)16 periods (4 periods per year 4 years),125,000 = quarterly payment (continued)


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230 Chapter 8

Proposal 4. The present value of Proposal 4 is $10,487,956. The contractguarantees three annual payments of $400,000 plus a lump sum paymentof $25 million in 25 years. The present value of the contract is calculatedby adding the present value of the annual payments and the present val-ue of the lump sum payment. Excel formula values were computed as fol-lows:

=PV(0.04, 3, 400,000) = $1,110,036

Plus 25,000,000*(1/1.04^25) = $9,377,920 The present value of$25,000,000 received in 25years, discounted at 4%, asshown in Chapter 8.

Proposal 5. The present value of Proposal 5 is $2,500,000. No discount-ing is necessary because all cash is received upon signing. The presentvalue of $1 today is $1.

Proposal 6. The present value of Proposal 6 is $7,966,270. The contract

is not guaranteed. If Fleet continues to play, the contract will pay$1,500,000 a year ($375,000 per quarter) for six years. Using Excel, thepresent value is determined as follows:

=PV(0.01, 24, 375,000) = $7,966,270

Excel formula values were computed as follows:1% (0.04/4 payments per year)24 periods (4 periods per year 6 years),375,000 = quarterly payment.

Cash FlowEach Period

PresentValue

Proposal 13-year contract at $1 millionPayable quarterlyQuarterly payment $ 0,250,000 $ 2,813,769

Proposal 24-year contract at $1 millionPayable at the end of the year $ 1,000,000 $ 3,629,895

Proposal 34-year contract900,000 signing bonus $ 900,000 $ 900,000125,000 end-of-quarter payments 125,000 $1,839,734

$2,739,734Proposal 43-year contract of $400,000 at each year-end $ 400,000 $ 1,110,036$25 million to be paid 25 years after signing 25,000,000 $09,377,920

$10,487,956


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Cash FlowEach Period

PresentValue

Proposal 53-year contract$2.5 million signing bonus, no payments $ 2,500,000 $ 2,500,000

Proposal 66-year contract at $1,500,000 $ 375,000 $ 7,966,270Payable quarterlyCancelable if injured or cut

Key points:

The proposals with the highest present value (4 and 6) also have thegreatest risk.

Proposal 4 requires Fleet to wait 25 years to collect the majority of hiscontract. This contract is risky because Fleet (or the NFL) may not bearound in 25 years.

Proposal 6 is risky because the contract is cancelable.

If Fleet is risk averse, Proposal 2 is the most attractive.


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