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Appendix A--Learning Objectives

- 1. Differentiate between simple and compound interest

Interest

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

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

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

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 ?

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

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

$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

$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

$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

$10,000 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

$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

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

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

$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 1.7623
- A = $17,623
- This bears an awesome resemblance
- to what we got a minute ago

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%
- 1 1.1100 1.1200
- 2 1.2321 1.2544
- 3 1.3676 1.4049
- 4 1.5181 1.5735
- 5 1;6851 1.7623

The table is faster !

- Multiply the number from the table
- 1.7623
- times the principal
- $10,000
- and we have the answer
- $17,623

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

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

FANCY CALCULATORSARE LIKE FOUR-WHEEL DRIVE

- THEY WILL NOT KEEP YOU FROM GETTING STUCK

BUT THEY WILL LET YOU GET STUCK

- IN MORE REMOTE PLACES

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 ?

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 ?

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 1.7623
- and the reciprocal is
- 1 / 1.7623 or .5674

Factors for the present value of 1are 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

Appendix A--Learning Objectives

- 2. Distinguish a single sum from an annuity

Annuity

- A series of equal payments
- at equal intervals
- at a constant interest rate

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

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 ?

Future value of an ordinary annuity

2001

2002

2003

2004

2005

- 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

$1,000

$1,000

$1,000

$1,000

$1,000

?

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

2001

2002

2003

2004

2005

- 1st. 1,000 1,574
- 2nd. 1,000 1,405
- 3rd. 1,000 1,254
- 4th. 1,000 1,120
- 5th. 1,000
- Total 6,353
- First payment earns 4 years of interest. Last earns none.

$1,000

$1,000

$1,000

$1,000

$1,000

The faster approach is to use Table A-3

2001

2002

2003

2004

2005

- Table A-3 gives us a factor of 6.3528 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)

$1,000

$1,000

$1,000

$1,000

$1,000

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 ?

Present value of an ordinary annuity

2001

2002

2003

2004

2005

- 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

$1,000

$1,000

$1,000

$1,000

$1,000

?

We could discount each payment

2001

2002

2003

2004

2005

- 893 1,000
- 797 1,000
- 712 1,000
- 636 1,000
- 567 1,000
- 3,605
- First payment discounted for one year, last for five years

?

$1,000

$1,000

$1,000

$1,000

$1,000

But using Table A-5 is much faster

2001

2002

2003

2004

2005

- Table A-5 gives us a factor of 3.6048 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)

?

$1,000

$1,000

$1,000

$1,000

$1,000

Appendix A--Learning Objectives

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

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

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 ?

$1,000

$1,000

$1,000

$1,000

Future value of an annuity due2001

2002

2003

2004

2005

- The five payments come at the beginning of the periods, so this is an annuity due
- We are looking for the future value

?

$1,000

$1,000

$1,000

$1,000

A slow solution approach--finding the FV of each payment2001

2002

2003

2004

2005

- 1,000 (1st.) 1,762
- 2nd. 1,000 1,574
- 3rd. 1,000 1,405
- 4th. 1,000 1,254
- 5th. 1,000 1,120
- Total 7,115
- Even the last payment earns interest for one year.

?

$1,000

$1,000

$1,000

$1,000

Table A-4 solves the problem fast2001

2002

2003

2004

2005

- The table factor is 7.1152
- 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)

?

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 ?

$1,000

$1,000

$1,000

$1,000

Present value of an annuity due2001

2002

2003

2004

2005

- The five payments come at the beginning of the periods, so this is an annuity due
- We are looking for the present value

?

$1,000

$1,000

$1,000

$1,000

Once again, we could discount each payment?

2001

2002

2003

2004

2005

- 1,000 ( First payment needs no discounting)
- 893 1,000
- 797 1,000
- 712 1,000
- 635 1,000
- 4,037

$1,000

$1,000

$1,000

$1,000

Table A-6 is the fast way?

2001

2002

2003

2004

2005

- The table factor is 4.0373
- 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)

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

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 ?

Here is the fact situation:

- 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

We are

here

1st

payment

2nd

payment

3rd

payment

2001

2002

2003

2004

2005

here

1st

payment

2nd

payment

3rd

payment

- We could discount the payments individually:
- 712 1,000
- 636 1,000
- 567 1,000
- 1,915
- This is OK if there are only a few payments

2001

2002

2003

2004

2005

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

here

1st

payment

2nd

payment

3rd

payment

- 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 2.4018
- Making the present value $2,401.80

2001

2002

2003

2004

2005

here

1st

payment

2nd

payment

3rd

payment

- Now we know that the payments would be worth $2,401.80 at the end of year 2
- We need to know what they are worth at the start of year 1
- We discount the $2,401.80 as a single sum for two years
- The factor from Table A-2 is .7972
- And the result is $1,915 (nearest dollar)

2001

2002

2003

2004

2005

Appendix A--Learning Objectives

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

Representative problem # 1Bubba 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 ?

Bubba goes to college

We are

here

$100,000

needed

here

- 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 .3152 from Table A-2

15 years

Bubba goes to college

We are

here

$100,000

needed

here

- 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

15 years

Representative problem # 2Ima 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 ?

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

9

10

- 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

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

9

10

0

- The future value factor for an ordinary annuity of eleven payments at 7% is 15.7836
- Now we solve for the amount of each payment:
- $150,000 = X x 15.7836
- 15.7836 X = $150,000
- X = $9,504
- This is the amount of each payment
- that Ima needs to make

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

9

10

- The future value factor for an annuity due of tenpayments at 7% is 14.7836
- Now we solve for the amount of each payment plus the payment at the end:
- $150,000 = 14.7836 X +X
- 15.7836 X = $150,000
- X = $9,504
- This is the amount of each payment
- that Ima needs to make

Representative problem # 3Can 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 ?

2002

2003

2004

2005

1

2

3

4

5

- $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 4.1699

2002

2003

2004

2005

1

2

3

4

5

- Solving for the amount of each payment:
- $25,000 = X x 4.1699
- 4.1699 X = $25,000
- X = $5,995
- Each payment is $5,995
- making the total cost of the car $29,975

Representative problem # 4The 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 ?

2002

2003

2004

2005

$10,000

?

- Two steps are necessary in this problem
- Finding the present value of the $10,000 par value to be received in five years

2002

2003

2004

2005

$10,000

?

$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

2002

2003

2004

2005

$10,000

?

$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 3.6048
- 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

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