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

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 nwhere

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) + Pwhere

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) + PA = ($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


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


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


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


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


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


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

• 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

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

?

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

1st. 1,000 1,574

2nd. 1,0001,405

3rd. 1,000 1,254

4th. 1,0001,120

5th. 1,000

Total 6,353First payment earns 4 years of interest. Last earns none.

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

The faster approach is to use Table A-3

• 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)

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

Another ordinary annuity situation


• We will receive five annual payments of $1,000 each starting on December 31, 2001


• What are the payments worth to us today ?

Present value of an ordinary annuity

• 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

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000?

We could discount each payment

893 1,000

797 1,000

712 1,000

636 1,000

5671,000

3,605First payment discounted for one year, last for five years

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000?

But using Table A-5 is much faster

• 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)

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000?


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


• We will receive five annual payments of $1,000 each starting today


• What will the payments be worth on December 31, 2005 ?

Future value of an annuity due

• The five payments come at the beginning of the periods, so this is an annuity due

• We are looking for the future value

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000 ?

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

1,000 (1st.) 1,762

2nd. 1,000 1,574

3rd. 1,0001,405

4th. 1,000 1,254

5th. 1,0001,120

Total 7,115Even the last payment earns interest for one year.

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000 ?

Table A-4 solves the problem fast

• 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)

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000 ?

Another annuity due situation


• We will receive five annual payments of $1,000 each starting today


• What is the series of payments worth to us today ?

Present value of an annuity due

• The five payments come at the beginning of the periods, so this is an annuity due

• We are looking for the present value

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

?

Once again, we could discount each payment

1,000 (First payment needs no discounting)

893 1,000

797 1,000

712 1,000

635 1,000

4,037

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

?

Table A-6 is the fast way

• 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)

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

?

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


• 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

2001 2002 2003 2004 2005

We arehere

1stpayment

2ndpayment

3rdpayment

We could discount the payments individually:

712 1,000

636 1,000

5671,000

1,915

This is OK if there are only a few payments

2001 2002 2003 2004 2005

We arehere

1stpayment

2ndpayment

3rdpayment

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

The Texas Two-Step MethodRequires use of two tables

• 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

We arehere

1stpayment

2ndpayment

3rdpayment

• 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

We arehere

1stpayment

2ndpayment

3rdpayment


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


• How much needs to be invested today to provide for Bubba’s education ?

Bubba goes to college

• 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

We arehere

$100,000needed

here

Bubba goes to college

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

We arehere

$100,000needed

here

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 ?

• 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

1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 11

• 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

1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 11

0

• 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

1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10 11

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 ?

• $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

2001 2002 2003 2004 20051 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

2001 2002 2003 2004 20051 2 3 4 5

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 ?

• Two steps are necessary in this problem

• Finding the present value of the $10,000 par value to be received in five years

2001 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

• 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

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

$10,000

?

• 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

2001 2002 2003 2004 2005

$1,000 $1,000 $1,000 $1,000 $1,000

$10,000

?

Documents

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