1. Appendix A--Learning Objectives
1. Differentiate between simple and compound interest 1

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. $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

9. $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

10. $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

11. $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

12. $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

13. 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

14. 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

15. $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

16. 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

17. 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

18. 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

19. 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

20. FANCY CALCULATORS ARE LIKE FOUR-WHEEL DRIVE
THEY WILL NOT KEEP YOU FROM GETTING STUCK 20

21. BUT THEY WILL LET YOU GET STUCK
IN MORE REMOTE PLACES 21

22. 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

23. 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

24. 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

25. 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

26. Appendix A--Learning Objectives
2. Distinguish a single sum from an annuity 26

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. Appendix A--Learning Objectives
3. Differentiate between an ordinary annuity and an annuity due 37

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. ? 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

47. 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

48. 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

49. 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

50. 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

51. 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

52. The Texas Two-Step Method Requires use of two tables52

53. 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

54. 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

55. Appendix A--Learning Objectives
4. Solve representative problems based on the time value of money 60

56. 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

57. 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

58. 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

59. 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

60. 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

61. This is the amount of each payment1 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

62. This is the amount of each payment1 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

63. 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

64. $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

65. making the total cost of the car $29,9751 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

66. 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

67. ? 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

68. ? 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

69. $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