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
$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
• 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
• 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
• 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?
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 ?
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
• 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 ?
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
• 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
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
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
• 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
?