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**Appendix A--Learning Objectives**

1. Differentiate between simple and compound interest 1

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**The charge for the use of money for a specified period of time**

Interest The charge for the use of money for a specified period of time 2

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**The basic interest formula is**

I = P x r x n where I = the amount of interest P = the principal r = the rate n = the number of periods or time 3

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**Another useful formula is**

A = (P x r x n) + P where A = is the final amount or maturity value P = the principal r = the rate n = the number of periods or time 4

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**Interest accrues on the principal only How much money will we have**

Simple interest Interest accrues on the principal only Suppose we have $10,000 We can earn 12 percent and we can wait 5 years: How much money will we have at the end of that time ? 5

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**At the end of the five years,**

Simple interest A = (P x r x n) + P A = ($10,000 x .12 x 5) + $10,000 A = $6,000 + $10,000 A = $16,000 At the end of the five years, we will have $16,000 6

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**Compound interest Is nothing more than simple interest**

over and over again with interest on the interest as well as the principal Let’s check it out 7

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**$10,000 in 5 years at 12 % compounded annually**

The first year A = ( P x r x n ) + P A = ($10,000 x .12 x 1) + $10,000 A = $1,200 + $10,000 A = $11,200 8

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**$10,000 in 5 years at 12 % compounded annually**

It gets better in the second year (because we have more money) A = ( P x r x n ) + P A = ($11,200 x .12 x 1) + $11,200 A = $1,344 + $11,200 A = $12,544 9

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**$10,000 in 5 years at 12 % compounded annually**

The third year is even better A = ( P x r x n ) + P A = ($12,544 x .12 x 1) + $12,544 A = $1,505 + $12,544 A = $14,049 10

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**$10,000 in 5 years at 12 % compounded annually**

The fourth year is better yet A = ( P x r x n ) + P A = ($14,049 x .12 x 1) + $14,049 A = $1,686 + $14,049 A = $15,735 11

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**$10,000 in 5 years at 12 % compounded annually**

And the fifth year is best A = ( P x r x n ) + P A = ($15,735 x .12 x 1) + $15,735 A = $1,888 + $15,735 A = $17,623 12

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**Note the difference With compound interest we got $17,623**

With simple interest we got $16,000 The difference of $1,623 is not bad compensation for getting the words “compounded annually” into the agreement 13

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**The “over and over” method worked, but it was a lot of trouble**

Another approach is to use the formula A = P x ( 1 + r ) n where A = Amount P = Principal 1 = The loneliest number r = Rate n = number of periods 14

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**$10,000 in 5 years at 12 % compounded annually**

A = P x ( 1 + r ) n A = $10,000 x ( 1.12 ) 5 A = $10,000 x A = $17,623 This bears an awesome resemblance to what we got a minute ago 15

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**Another way is with the table ( Table A-1 in our book )**

Interest rates are across the top And number of periods down the side Just find the intersection n/r 11% 12% 5 1; 16

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**Multiply the number from the table**

The table is faster ! Multiply the number from the table 1.7623 times the principal $10,000 and we have the answer $17,623 17

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**Future value $17,623 could be referred to as the future value**

of $10,000 at 12 percent for 5 years compounded annually That is what we will usually call it 18

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**A note about financial calculators**

A number of calculators have built-in financial functions and can solve problems of this type very quickly Your instructor will advise you as to what the calculator policies are for your course and your school But remember, a fancy calculator will not solve all of your problems for you 20

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**FANCY CALCULATORS ARE LIKE FOUR-WHEEL DRIVE**

THEY WILL NOT KEEP YOU FROM GETTING STUCK 20

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**BUT THEY WILL LET YOU GET STUCK**

IN MORE REMOTE PLACES 21

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**Now for a change Instead of having $10,000 now**

let’s say we have to wait 5 years to get the $10,000 the interest rate is still 12% compounded annually What is that worth to us now ? 22

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**In other words What is the present value**

of $10,000 to be received in 5 years if the interest rate is 12 percent compounded annually ? 23

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**A reciprocal The future value interest formula was ( 1 + r ) n**

and the basic present value formula is 1 / [ ( 1 + r ) n ] the future value example was ( 1.12 ) 5 or and the reciprocal is 1 / or 24

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**Factors for the present value of 1 are found in Table A-2**

The present value factor for $1 to be received in five years at 12 percent compounded annually is .5674 We are looking for the present value of $10,000 All we need to do is multiply the factor by the amount to obtain the answer of $5,674 In other words, the present value of $10,000 to be received five years from now is $5,674 if the interest rate is 12% compounded annually 25

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**Appendix A--Learning Objectives**

2. Distinguish a single sum from an annuity 26

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**A series of equal payments at a constant interest rate**

Annuity A series of equal payments at equal intervals at a constant interest rate 27

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Types of annuities Ordinary annuity--payments at the ends of the periods Annuity due--payments at the beginnings of the periods Deferred annuity--one or more periods pass before payments start 28

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**Ordinary annuity assumptions**

Today is January 1, 2001 We will receive five annual payments of $1,000 each starting on December 31, 2001 Money is worth 12 percent per year compounded annually What will the payments be worth on December 31, 2005 ? 29

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**Future value of an ordinary annuity**

2001 2002 2003 2004 2005 $1,000 $1,000 $1,000 $1,000 $1,000 ? The five payments are equal amounts at equal intervals at a constant interest rate They come at the ends of the periods, so this is an ordinary annuity We are looking for the future value 30

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**A slow solution approach-- finding the FV of each payment**

2001 2002 2003 2004 2005 $1,000 $1,000 $1,000 $1,000 $1,000 1st. 1, ,574 2nd. 1, ,405 3rd. 1,000 1,254 4th. 1,000 1,120 5th ,000 Total 6,353 First payment earns 4 years of interest. Last earns none. 31

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**The faster approach is to use Table A-3**

2001 2002 2003 2004 2005 $1,000 $1,000 $1,000 $1,000 $1,000 Table A-3 gives us a factor of for 12% interest and five payments (periods) For annuities, we multiply the factor by the amount of each payment--$1,000 in this case The result is the same answer--$6,353 (rounded to the nearest dollar) 32

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**Another ordinary annuity situation**

Today is January 1, 2001 We will receive five annual payments of $1,000 each starting on December 31, 2001 Money is worth 12 percent per year compounded annually What are the payments worth to us today ? 33

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**Present value of an ordinary annuity**

2001 2002 2003 2004 2005 ? $1,000 $1,000 $1,000 $1,000 $1,000 This is an ordinary annuity with the payments at the ends of the periods We want to know what the 5 payments are worth to us NOW 34

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**We could discount each payment**

2001 2002 2003 2004 2005 ? $1,000 $1,000 $1,000 $1,000 $1,000 893 1,000 797 1,000 ,000 ,000 ,000 3,605 First payment discounted for one year, last for five years 35

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**But using Table A-5 is much faster**

2001 2002 2003 2004 2005 ? $1,000 $1,000 $1,000 $1,000 $1,000 Table A-5 gives us a factor of for 12% interest and five payments (periods) Multiply by the payment amount--$1,000 The result is the same answer--$3,605 (rounded to the nearest dollar) 36

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**Appendix A--Learning Objectives**

3. Differentiate between an ordinary annuity and an annuity due 37

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**Another type of annuity is the annuity due**

The ordinary annuity has the payments at the ends of the periods But the annuity due has the payments at the beginnings of the periods 38

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**An annuity due situation**

Today is January 1, 2001 We will receive five annual payments of $1,000 each starting today Money is worth 12 percent per year compounded annually What will the payments be worth on December 31, 2005 ? 39

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**Future value of an annuity due**

2001 2002 2003 2004 2005 $1,000 ? The five payments come at the beginning of the periods, so this is an annuity due We are looking for the future value 40

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**A slow solution approach-- finding the FV of each payment**

2001 2002 2003 2004 2005 $1,000 ? 1,000 (1st.) 1,762 2nd. 1, ,574 3rd. 1, ,405 4th. 1,000 1,254 5th. 1,000 1,120 Total 7,115 Even the last payment earns interest for one year. 41

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**Table A-4 solves the problem fast**

2001 2002 2003 2004 2005 $1,000 ? The table factor is Once again, we multiply by the amount of each payment--$1,000 in this example The result is the same number--$7,115 (rounded to the nearest dollar) 42

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**Another annuity due situation**

Today is January 1, 2001 We will receive five annual payments of $1,000 each starting today Money is worth 12 percent per year compounded annually What is the series of payments worth to us today ? 43

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**Present value of an annuity due**

2001 2002 2003 2004 2005 $1,000 ? The five payments come at the beginning of the periods, so this is an annuity due We are looking for the present value 44

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**Once again, we could discount each payment**

? 2001 2002 2003 2004 2005 $1,000 1, ( First payment needs no discounting) 893 1,000 797 1,000 ,000 ,000 4,037 45

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**? Table A-6 is the fast way The table factor is 4.0373**

2001 2002 2003 2004 2005 $1,000 The table factor is Once again, we multiply by the amount of each payment--$1,000 in this example The result is the same number--$4,037 (rounded to the nearest dollar) 46

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**The last type of annuity we will look at is the deferred annuity**

A deferred annuity is also a series of equal payments at equal intervals at a constant interest rate but two or more periods elapse before the first payment is made 47

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**Deferred annuity example**

Today is January 1, 2001 We are going to receive three annual payments of $1,000 each We get the first payment on December 31, 2003, the second on December 31, 2004, and the third on December 31, 2005 The interest rate is 12% compounded annually What is the series of payments worth to us today ? 48

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**Here is the fact situation:**

We are here 1st payment 2nd payment 3rd payment 2001 2002 2003 2004 2005 Each of the three payments is $1,000 We want to know the value as of January 1, 2001 The first payment does not occur until the end of the third year 49

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**This is OK if there are only a few payments**

We are here 1st payment 2nd payment 3rd payment 2001 2002 2003 2004 2005 We could discount the payments individually: 712 1,000 636 1,000 ,000 1,915 This is OK if there are only a few payments 50

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**Let’s look at two other approaches**

There is no “instant” solution to a deferred annuity problem Both approaches require at least two steps One involves use of two tables, the other requires only one One could be called The Texas Two-Step Method The other could be called The Ghost Payment Method 51

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**The Texas Two-Step Method Requires use of two tables**

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**In this case, the start of year 3 (end of year 2) **

We are here 1st payment 2nd payment 3rd payment 2001 2002 2003 2004 2005 First we pick a point that will make the series of payments an ordinary annuity In this case, the start of year 3 (end of year 2) Then we find the present value of the ordinary annuity at that time The factor from Table A-5 is Making the present value $2,401.80 53

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**We need to know what they are worth at the start of year 1 **

We are here 1st payment 2nd payment 3rd payment 2001 2002 2003 2004 2005 Now we know that the payments would be worth $2, at the end of year 2 We need to know what they are worth at the start of year 1 We discount the $2, as a single sum for two years The factor from Table A-2 is .7972 And the result is $1,915 (nearest dollar) 54

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**Appendix A--Learning Objectives**

4. Solve representative problems based on the time value of money 60

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**Representative problem # 1 Bubba Goes to College**

Bubba will start college in 15 years He will need $100,000 Money is worth 8 percent per year compounded annually How much needs to be invested today to provide for Bubba’s education ? 61

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Bubba goes to college We are here $100,000 needed here 15 years In this case, we know the future value, the time and the interest rate We are looking for the present value The PV factor for 8% for 15 years is from Table A-2 62

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**Bubba goes to college PV = 100,000 x .3152 PV = $31,520**

We are here $100,000 needed here 15 years PV = 100,000 x .3152 PV = $31,520 $31,520 must be invested today at 8 % compounded annually in order for Bubba to have $100,000 in 15 years 63

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**Representative problem # 2 Ima Geezer plans his Retirement**

Ima wants to retire in 10 years He wants to save $150,000 for his retirement He wants to start making annual deposits today and will make the last one on the day he retires If money is worth 7 percent compounded annually, how much must each deposit be ? 64

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**He wants a total of $150,000 at the end of the tenth year **

2 3 4 5 6 7 8 9 10 11 Since Ima plans to make his first payment immediately, and his last when he retires, there will be a total of eleven payments He wants a total of $150,000 at the end of the tenth year We can consider $150,000 as the known future value of an ordinary annuity of eleven payments It is an ordinary annuity because the last payment comes at the end of the process 65

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**This is the amount of each payment**

1 2 3 4 5 6 7 8 9 10 11 The future value factor for an ordinary annuity of eleven payments at 7% is Now we solve for the amount of each payment: $150,000 = X x X = $150,000 X = $9,504 This is the amount of each payment that Ima needs to make 66

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**This is the amount of each payment**

1 2 3 4 5 6 7 8 9 10 11 The future value factor for an annuity due of tenpayments at 7% is Now we solve for the amount of each payment plus the payment at the end: $150,000 = X +X X = $150,000 X = $9,504 This is the amount of each payment that Ima needs to make

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**Representative problem # 3 Can we afford those new wheels ?**

Our dream car costs $25,000 We can buy it with five annual payments at 10 percent compounded annually The first payment we make today How much are the payments ? 67

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**$25,000 is the known present value of an annuity due of five payments **

1 2 3 4 5 2001 2002 2003 2004 2005 $25,000 is the known present value of an annuity due of five payments It is an annuity due because the first payment is made immediately and we are concerned with the present value The present value factor for an annuity due of 5 payments at 10 % is 68

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**making the total cost of the car $29,975**

1 2 3 4 5 2001 2002 2003 2004 2005 Solving for the amount of each payment: $25,000 = X x X = $25,000 X = $5,995 Each payment is $5,995 making the total cost of the car $29,975 69

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**Representative problem # 4 The sale price of the James Bonds**

James Company is selling bonds with a par value of $10,000 on January 1, 2001 The bonds pay interest at 10 percent annually on December 31 and mature in five years (real bonds would take longer) The market interest rate for investments of comparable quality and risk on the sale date is 12 percent What will the bonds sell for ? 70

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**? Two steps are necessary in this problem**

$10,000 ? 2001 2002 2003 2004 2005 Two steps are necessary in this problem Finding the present value of the $10,000 par value to be received in five years 71

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**? Two steps are necessary in this problem**

$10,000 ? 2001 2002 2003 2004 2005 $1,000 $1,000 $1,000 $1,000 $1,000 Two steps are necessary in this problem Finding the present value of the $10,000 par value to be received in five years And finding the present value of the five $1,000 annual interest payments, the first of which will be received on December 31, 2001 We use the effective or market interest rate--12 percent in this case 72

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$10,000 ? 2001 2002 2003 2004 2005 $1,000 $1,000 $1,000 $1,000 $1,000 The PV factor for a single sum in 5 years at 12 % from Table A-2 is .5674 So the present value of the $10,000 par value is $5,674 The PV factor for an ordinary annuity of 5 payments at 12 % from Table A- 5 is So the present value of the interest payments is $3,605 The sum of the two is $9,279 which will be the selling price of the bonds 73

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NPV and the Time Value of Money

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