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Chapter 4

The Time Value

of Money(Part 2)

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2013 Pearson Education, Inc. All rights reserved.4-2

1. Compute the future value of multiple cash flows.2. Determine the future value of an annuity.

3. Determine the present value of an annuity.

4. Adjust the annuity equation for present value and future valuefor an annuity due and understand the concept of a perpetuity.

5. Distinguish between the different types of loan repayments:discount loans, interest-only loans and amortized loans.

6. Build and analyze amortization schedules.

7. Calculate waiting time and interest rates for an annuity.

8. Apply the time value of money concepts to evaluate the lotterycash flow choice.

9. Summarize the ten essential points about the time value ofmoney.

Learning Objectives

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4.1 Future Value of Multiple PaymentStreams

With unequal periodic cash flows, treat eachof the cash flows as a lump sum andcalculate its future value over the relevantnumber of periods.

Sum up the individual future values to getthe future value of the multiple paymentstreams.

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Figure 4.1 The time line of anest egg

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4.1 Future Value of MultiplePayment Streams (continued)

Example 1: Future Value of an Uneven CashFlow Stream:

Jim deposits $3,000 today into an account thatpays 10% per year, and follows it up with 3 more

deposits at the end of each of the next three years.Each subsequent deposit is $2,000 higher than theprevious one. How much money will Jim haveaccumulated in his account by the end of threeyears?

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4.1 Future Value of Multiple PaymentStreams (Example 1 Answer)

FV= PVx (1+r)n

FV of Cash Flow at T0 = $3,000 x (1.10)3 = $3,000 x 1.331= $3,993.00

FV of Cash Flow at T1

= $5,000 x (1.10)2 = $5,000 x 1.210 = $6,050.00

FV of Cash Flow at T2 = $7,000 x (1.10)1 = $7,000 x 1.100 = $7,700.00

FV of Cash Flow at T3 = $9,000 x (1.10)0 = $9,000 x 1.000 = $9,000.00

Total = $26,743.00

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7/57 2013 Pearson Education, Inc. All rights reserved.4-7

4.1 Future Value of Multiple PaymentStreams (Example 1 Answer)

ALTERNATIVE METHOD:

Using the Cash Flow (CF) keyof the calculator, enter the

respective cash flows.

CF0=-$3000;CF1=-$5000;CF2=-$7000;CF3=-$9000;

Next calculate the NPV using I=10%; NPV=$20,092.41;

Finally, using PV=-$20,092.41; n=3; i=10%;PMT=0;CPT FV=$26,743.00

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4.2 Future Value of an AnnuityStream

Annuities are equal, periodic outflows/inflows., e.g. rent,lease, mortgage, car loan, and retirement annuity payments.

An annuity stream can begin at the start of each period

(annuity due) as is true of rent and insurance payments or

at the end of each period, (ordinary annuity) as in the case

of mortgage and loan payments.

The formula for calculating the future value of an annuity

stream is as follows:

FV = PMT * (1+r)n -1

r

where PMTis the term used for the equal periodic cash flow, ris the rate of interest, and n is the number of periodsinvolved.

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4.2 Future Value of an AnnuityStream (continued)

Example 2: Future Value of an OrdinaryAnnuity Stream

Jill has been faithfully depositing $2,000 at the endof each year since the past 10 years into anaccount that pays 8% per year. How much moneywill she have accumulated in the account?

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Example 2 Answer

Future Value of Payment One = $2,000 x 1.089 = $3,998.01

Future Value of Payment Two = $2,000 x 1.088 = $3,701.86

Future Value of Payment Three = $2,000 x 1.087 = $3,427.65

Future Value of Payment Four = $2,000 x 1.086 = $3,173.75

Future Value of Payment Five = $2,000 x 1.085 = $2,938.66

Future Value of Payment Six = $2,000 x 1.084 = $2,720.98

Future Value of Payment Seven = $2,000 x 1.083 = $2,519.42

Future Value of Payment Eight = $2,000 x 1.082 = $2,332.80

Future Value of Payment Nine = $2,000 x 1.081 = $2,160.00

Future Value of Payment Ten = $2,000 x 1.080 = $2,000.00

Total Value of Account at the end of 10 years $28,973.13


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Example 2 (Answer)FORMULA METHOD

FV = PMT * (1+r)n -1

r

where, PMT = $2,000; r = 8%; and n=10.FVIFA [((1.08)10 - 1)/.08] = 14.486562,

FV = $2000*14.486562 $28,973.13

USING A FINANCIAL CALCULATOR

N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13

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USING AN EXCEL SPREADSHEET

Enter =FV(8%, 10, -2000, 0, 0); Output = $28,973.13

Rate, Nper, Pmt, PV,TypeType is 0 for ordinary annuities and 1 for annuitiesdue

USING FVIFA TABLE (A-3)

Find the FVIFA in the 8% column and the 10 periodrow; FVIFA = 14.486

FV = 2000*14.4865 = $28.973.13

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FIGURE 4.3 Interest and principalgrowth with different interest ratesfor $100-annual payments.

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4.3 Present Value of an Annuity

r

rPMTPV

n

111

To calculate the value of a series of equalperiodic cash flows at the current point in time,we can use the following simplified formula:

The last portion of the equation, is the

Present Value Interest Factor of an Annuity (PVIFA).

Practical applications include figuring out the nest egg needed

prior to retirement or lump sum needed for college expenses.

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FIGURE 4.4 Time line of presentvalue of annuity stream.

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4.3 Present Value of an Annuity(continued)

Example 3: Present Value of anAnnuity.

John wants to make sure that he has saved upenough money prior to the year in which his

daughter begins college. Based on currentestimates, he figures that college expenses willamount to $40,000 per year for 4 years(ignoring any inflation or tuition increasesduring the 4 years of college). How muchmoney will John need to have accumulated inan account that earns 7% per year, just prior tothe year that his daughter starts college?

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rr1

11

PMTPV

n

Example 3 AnswerUsing the following equation:

1. Calculate the PVIFA value for n=4 and r=7%3.387211.

2. Then, multiply the annuity payment by this factor to getthe PV,PV = $40,000 x 3.387211 = $135,488.45

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Example 3 Answercontinued

FINANCIAL CALCULATOR METHOD:Set the calculator for an ordinary annuity (END mode) and

then enter:N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45

SPREADSHEET METHOD:Enter =PV(7%, 4, 40,000, 0, 0); Output = $135,488.45

Rate, Nper, Pmt, FV, Type

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Example 3 Answercontinued

PVIFA TABLE (APPENDIX A-4) METHOD

For r =7% and n = 4; PVIFA =3.3872PVA = PMT*PVIFA = 40,000*3.3872

= $135,488 (Notice the slight rounding error!)

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4.4 Annuity Due and Perpetuity

A cash flow stream such as rent, lease, andinsurance payments, which involves equal periodiccash flows that begin right away or at the beginningof each time interval is known as an annuity due.

Figure 4.5 An ordinary annuity versus an annuitydue.

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4.4 Annuity Due and Perpetuity

PV annuity due = PV ordinary annuity x (1+r)FV annuity due = FV ordinary annuity x (1+r)PV annuity due > PV ordinary annuityFV annuity due > FV ordinary annuityCan you see why?

Financial calculatorMode BGN for annuity dueMode END for an ordinary annuity

SpreadsheetType =0 or omitted for an ordinary annuityType = 1 for an annuity due.

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4.4 Annuity Due and Perpetuity(continued)

Example 4: Annuity Due versus OrdinaryAnnuity

Lets say that you are saving up for retirementand decide to deposit $3,000 each year for the

next 20 years into an account which pays a rateof interest of 8% per year. By how much willyour accumulated nest egg vary if you makeeach of the 20 deposits at the beginning of the

year, starting right away, rather than at theend of each of the next twenty years?

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Example 4 Answer

Given information: PMT = -$3,000; n=20; i= 8%; PV=0;

r

1r1PMTFV

n

FV ordinary annuity = $3,000 * [((1.08)20 - 1)/.08]= $3,000 * 45.76196

= $137,285.89FV of annuity due = FV of ordinary annuity * (1+r)FV of annuity due = $137,285.89*(1.08) = $148,268.76

i d i

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PerpetuityA Perpetuity is an equal periodic cash flowstream that will never cease.

The PV of a perpetuity is calculated by usingthe following equation:

r

PMT

PV

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Example 5: PV of a perpetuityIf you are considering the purchase of a consol thatpays $60 per year forever, and the rate of interestyou want to earn is 10% per year, how much

money should you pay for the consol?Answer:

r=10%, PMT = $60; and PV = ($60/.1) = $600

$600 is the most you should pay for the consol.

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4.5 Three Loan PaymentMethods

Loan payments can be structured in one of 3ways:

1)Discount loan

Principal and interest is paid in lump sum at end

2)Interest-only loan

Periodic interest-only payments, principal due at end.

3)Amortized loan

Equal periodic payments of principal and interest

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4.5 Three Loan PaymentMethods (continued)

Example 6: Discount versus Interest-only versusAmortized loans

Roseanne wants to borrow $40,000 for a period of 5 years.

The lenders offers her a choice of three payment structures:

1) Pay all of the interest (10% per year) and principal in one lump sumat the end of 5 years;

2) Pay interest at the rate of 10% per year for 4 years and then a finalpayment of interest and principal at the end of the 5th year;

3) Pay 5 equal payments at the end of each year inclusive of interestand part of the principal.

Under which of the three options will Roseanne pay the least interestand why? Calculate the total amount of the payments and the amountof interest paid under each alternative.

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Method 1: Discount Loan.Since all the interest and the principal is paid at theend of 5 years we can use the FV of a lump sumequation to calculate the payment required, i.e.

FV = PV x (1 + r)nFV5 = $40,000 x (1+0.10)

5

= $40,000 x 1.61051

= $64, 420.40

Interest paid = Total payment - Loan amount

Interest paid = $64,420.40 - $40,000 = $24,420.40

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Method 2: Interest-Only Loan.

Annual Interest Payment (Years 1-4)

= $40,000 x 0.10 = $4,000

Year 5 payment

= Annual interest payment + Principal payment

= $4,000 + $40,000 = $44,000

Total payment = $16,000 + $44,000 = $60,000

Interest paid = $20,000

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Method 3: Amortized Loan.n = 5; I = 10%; PV=$40,000; FV = 0; CPT PMT=$10,551.86

Total payments = 5*$10,551.8 = $52,759.31

Interest paid = Total Payments - Loan Amount

= $52,759.31-$40,000Interest paid = $12,759.31

Loan Type Total Payment Interest Paid

Discount Loan $64,420.40 $24,420.40Interest-only Loan $60,000.00 $20,000.00

Amortized Loan $52,759.31 $12,759.31

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4.6 Amortization Schedules

Tabular listing of the allocation of each loanpayment towards interest and principal reduction

Helps borrowers and lenders figure out the payoffbalance on an outstanding loan.

Procedure:1) Compute the amount of each equal periodic

payment (PMT).

2) Calculate interest on unpaid balance at the end of

each period, minus it from the PMT, reduce theloan balance by the remaining amount,

3) Continue the process for each payment period,until we get a zero loan balance.

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4.6 Amortization Schedules(continued)

Example 7: Loan amortization schedule.Prepare a loan amortization schedule for theamortized loan option given in Example 6above. What is the loan payoff amount at

the end of 2 years?

PV = $40,000; n=5; i=10%; FV=0;

CPT PMT = $10,551.89

4 6 Amo ti ation Sched les

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4.6 Amortization Schedules(continued)

The loan payoff amount at the end of 2years is $26,241.03

Year Beg. Bal Payment Interest Prin. Red End. Bal

1 40,000.00 10,551.89 4,000.00 6,551.89 33,448.11

2 33,448.11 10,551.89 3,344.81 7,207.08 26,241.03

3 26,241.03 10,551.89 2,264.10 7,927.79 18,313.24

4 18,313.24 10,551.89 1,831.32 8,720.57 9,592.67

5 9,592.67 10,551.89 959.27 9,592.67 0

4 7 Waiting Time and Interest

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4.7 Waiting Time and InterestRates for Annuities

Problems involving annuities typically have 4variables, i.e. PVorFV, PMT, r, n

If any 3 of the 4 variables are given, we can easilysolve for the fourth one.

This section deals with the procedure of solvingproblems where either n or ris not given.

For example:

Finding out how many deposits (n) it would take to reach a

retirement or investment goal; Figuring out the rate of return (r) required to reach a

retirement goal given fixed monthly deposits,


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4.7 Waiting Time and InterestRates for Annuities (continued)

Example 8: Solving for the number ofannuities involved

Martha wants to save up $100,000 as soonas possible so that she can use it as a downpayment on her dream house. She figuresthat she can easily set aside $8,000 peryear and earn 8% annually on her deposits.

How many years will Martha have to waitbefore she can buy that dream house?


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4.7 Waiting Time and InterestRates for Annuities (continued)

Example 8 AnswerMethod 1: Using a financial calculator

INPUT ? 8.0 0 -8000 100000

TVM KEYS N I/Y PV PMT FV

Compute N 9.00647Method 2: Using an Excel spreadsheet

Using the =NPER function we enter the following:

Rate = 8%; Pmt = -8000; PV = 0;

FV = 100000; Type = 0 or omitted;i.e. =NPER(8%,-8000,0,100000,0)

The cell displays 9.006467.

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4.8 Solving a Lottery Problem

In the case of lottery winnings, 2 choices1) Annual lottery payment for fixed number of

years, OR

2) Lump sum payout.

How do we make an informed judgment?

Need to figure out the implied rate of returnof both options using TVM functions.

4 8 Solving a Lottery Problem

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4.8 Solving a Lottery Problem(continued)

Example 9: Calculating an implied rateof return given an annuity

Lets say that you have just won the statelottery. The authorities have given you achoice of either taking a lump sum of$26,000,000 or a 30-year annuity of$1,625,000. Both payments are assumed to

be after-tax. What will you do?

4 8 Solving a Lottery Problem

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4.8 Solving a Lottery Problem(continued)

Example 9 AnswerUsing the TVM keys of a financial calculator, enter:

PV=26,000,000; FV=0; N=30; PMT = -1,625,000;

CPT I = 4.65283%

4.65283% = rate of interest used to determine the 30-year annuity of $1,625,000 versus the $26,000,000 lumpsum pay out.

Choice: If you can earn an annual after-tax rate of

return higher than 4.65% over the next 30 years,go with the lump sum.

Otherwise, take the annuity option.

4 9 Ten Important Points about

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4.9 Ten Important Points aboutthe TVM Equation

1. Amounts of money can be added or subtractedonly if they are at the same point in time.

2. The timing and the amount of the cash flow arewhat matters.

3. It is very helpful to lay out the timing and amountof the cash flow with a timeline.

4. Present value calculations discount all future cashflow back to current time.

5. Future value calculations value cash flows at asingle point in time in the future


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4.9 Ten Important Points aboutthe TVM Equation (continued)

6. An annuity is a series of equal cash payments atregular intervals across time.

7. The time value of money equation has four variablesbut only one basic equation, and so you must knowthree of the four variables before you can solve forthe missing or unknown variable.

8. There are three basic methods to solve for anunknown time value of money variable:

(1) Using equations and calculating the answer;(2) Using the TVM keys on a calculator;(3) Using financial functions from a spreadsheet.


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4.9 Ten Important Points aboutthe TVM Equation (continued)

9. There are 3 basic ways to repay a loan:(1) Discount loans,

(2) Interest-only loans, and

(3) Amortized loans.

10. Despite the seemingly accurate answers fromthe time value of money equation, in manysituations not all the important data can beclassified into the variables of present value,

i.e., time, interest rate, payment, or futurevalue.

Additional Problems with Answers

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Additional Problems with AnswersProblem 1

Present Value of an Annuity Due. Julie has justbeen accepted into Harvard and her father isdebating whether he should make monthly leasepayments of $5,000 at the beginning of eachmonth, on her flashy apartment or to prepay therent with a one-time payment of $56, 662.If Juliesfather earns1% per month on his savings should hepay by month or take the discount by making thesingle annual payment?


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Additional Problems with AnswersProblem 1 (Answer)

P/Y = 12; C/Y = 12; MODE = BGNINPUT 12 -56,662 5,000 0


OUTPUT 12.70%

Monthly rate = 12.7%/12 = 1.0583%

If he can get 1% interest per month...then his annual rate is 12% andhe can generate $4,984.51 per month with the $56,662 it would taketo pay off the rent. He is ahead $15.49 per month by making the onetime payment.

INPUT 12 12 -56,662 0


OUTPUT 4,984.51


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Future Value ofUneven cash flows. If Marydeposits $4000 a year for three years, starting ayear from today, followed by 3 annual deposits of$5000, into an account that earns 8% per year,how much money will she have accumulated in heraccount at the end of 10 years?


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Future Value in Year 10 =$4000*(1.08)9(she deposits 4K a year from today)

+ $4000*(1.08)8+ $4000*(1.08)7+ $5000*(1.08)6

+ $5000*(1.08)5

+ $5000*(1.08)4

= $4000*1.999 + $4000*1.8509 + $4000*1.7138 +$5000*1.5868 + $5000*1.4693 + $5000*1.3605

=$7,996 + $7,403.6+ $6,855.2+ $7,934 + $7,346.5 +6,802.5

=$44,337.8


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Additional Problems with AnswersProblem 2 (Answer) (continued)

ALTERNATIVE METHOD:Using the Cash Flow (CF) key of the calculator, enterthe respective cash flows.

CF0=0;CF1=-$4000;CF2=-$4000;CF3=-$4000;

CF4=-$5000; CF5=-$5000; CF6=-$5000Next calculate the NPV using I=8%;NPV=$20,537.30;Finally, using PV=-$20,537.30; n=10; i=8%; PMT=0;CPT FV$44,338


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Present Value of Uneven Cash Flows:Jane Bryant has just purchased someequipment for her beauty salon. She plansto pay the following amounts at the end of

the next five years: $8,250, $8,500,$8,750, $9,000, and $10,500. If she uses adiscount rate of 10 percent, what is the costof the equipment that she purchased today?


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$33,765.58

$6,519.67$6,147.12$6,574$7,024.79$7,500

(1.10)

$10,500

(1.10)

$9,000

(1.10)

$8,750

(1.10)

$8,500

(1.10)

$8,250PV

5432


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Computing Annuity Payment: The Corner Bar & Grillis in the process of taking a five-year loan of $50,000with First Community Bank. The bank offers therestaurant owner his choice of three payment options:

1) Pay all of the interest (8% per year) and principal

in one lump sum at the end of 5 years;2) Pay interest at the rate of 8% per year for 4 years

and then a final payment of interest and principalat the end of the 5th year;

3) Pay 5 equal payments at the end of each year

inclusive of interest and part of the principal.Under which of the three options will the owner pay theleast interest and why?


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Option 1

Discount Loan:Principal and Interest Due at end

Payment at the end of year 5 = FVn = PV x (1 + r)n

FV5= $50,000 x (1+0.08)5

= $50,000 x 1.46933

= $73,466.5

Interest paid = Total payment - Loan amount

Interest paid = $73,466.5 - $50,000 = $23,466.50

5 N-50000 PV

8 I

0 PMT

CPT FV 73,466.5


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Option 2: Interest-only LoanAnnual Interest Payment (Years 1-4)

= $50,000 x 0.08 = $4,000

Year 5 payment = Annual interest payment +

Principal payment= $4,000 + $50,000 = $54,000

Total payment = $16,000 + $54,000= $70,000

Interest paid = Total Payment LoanAmount = 70000 50000 = $20,000


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Option 3: Amortized Loan.To calculate the annual payment of principal andinterest we can use the PV of an ordinary annuityequation and solve for the PMTvalue using n = 5; I =8%; PV = $50,000, and FV = 0.

CPTPMT $12,522.82Total payments = 5*$12,522.82 = $62,614.11

Total Interest = Total Payments - Loan Amount

= $62,614.11-$50,000= $12,614.11


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Comparison of total payments and interest paidunder each method

Loan Type Total Payment InterestPaidDiscount Loan $73,466.5 $23,466.50Interest-only Loan $70,000.00 $20,000.00Amortized Loan $62,614.11 $12,614.11

So, the amortized loan is the one with the lowestinterest expense, since it requires a higher annualpayment, part of which reduces the unpaidbalance on the loan and thus results in lessinterest being charged over the 5-year term.


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Loan amortization. Lets say that the restaurantowner in Problem 4 above decides to go with theamortized loan option and after having paid 2payments decides to pay off the balance. Using an

amortization schedule calculate his payoff amount.Amount of loan = $50,000;Interest rate = 8%;Term = 5 years;Annual payment (PMT) = $12,522.82


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AMORTIZATION SCHEDULEYear

Beg. Bal. PMT Interest Prin. Red. End Bal.

1 50,000.00 12,522.82 4,000.00 8,522.82 41,477.18

2 41,477.18 12,522.82 3,318.17 9,204.65 32,272.53

3 2,272.53 12,522.82 2,581.80 9,941.02 22,331.51

4 22,331.51 12,522.82 1,786.52 10,736.30 11,595.21

5 11,595.21 12,522.82 927.62 11,595.21 0

The loan payoff amount at the end of 2years is $32,272.53

Figure 4.2 The time line of a $1,000-

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Figure 4.2 The time line of a $1,000per year nest egg.

LN04Brooks671956_02_LN04

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