Compound Interest, Future and Present Values

MM255-07

Professor Nicolle Brooks

Calculate the future value and the compound interest amount by compounding manually

Calculate the future value and the compound interest amount using a $1.00 future value table

Calculate the future value and the compound interest amount using a formula

Find the effective interest rate ( APY) Calculate the interest compounded daily

using a table

Some loans may use compounding formula to calculate the interest amount. In this case, the interest is calculated more than one time during the term of the loan or investment and the interest of a period is added to the principal before the interest of the next period is calculated. So, the interest of one period also earns some interest in the next period.

The process of adding interest to the principal before calculating the interest amount for the next period is called compounding interest.

There are two or more interest periods when the interest is compounded. The interest period can be one day, one week, one month, one quarter, one year, or any portion of the year.

The more the number of interest periods, the higher the amount of total interest will be because the interest of more interest periods contribute to the total interest amount.

Future value or maturity value or compound amount is the sum of compound interest and the original principal.

The simple interest formula can be used to calculate the future value of principal when annual number of interest periods and the annual interest rate are known.

First, we need to find the period interest rate or interest rate per period. This can be done by dividing the annual interest rate by the total number of annual interest periods.

Period interest rate = annual interest rate / number of interest periods per year

Calculate the principal at the end of the first period:

Principal at the end of first period = original principal x (1 + period interest rate)

  If original principal = P; principal at the end of the first period = A; period

interest rate = R then

  A = P(1 + R)  For each remaining period: 

End-of-period principal = previous end-of-period principal x (1 + period interest rate)

  The last end-of-period principal is the future value.

Example 1:Susan secured a loan of $8000 for three years compounded annually. If

the interest rate was9%,find:

1. Future value2. Total compound interest paid3. Compare the total compound interest with total simple interest over

the same loan period, original principle, and annual interest rate.

Solution: Principal at the end of first period = $8,000 (1 + 0.09) = $8,000 (1.09) = $8,720   Second end-of-period principal = $8,720 (1 + 0.09) = $8,720 (1.09) = $9,504.80

Third end-of-period principal = $9,504.80 (1 + 0.09) = $9,504.80 (1.09) = $10,360.23

So, the future value of the loan is $10,360.23   Total compound interest = A – P

= $10,360.23 - $8,000 = $2,360.23

The compound interest is $2,360.23   To find the simple interest: I = PRT I = $8,000 * 0.09 * 3 = $2,160 Total Simple Interest = $2,160

Different between total compound and total simple interests= 2,360.23 - $2,160 = $200.23

Example 2:Find the future value of a $10,000 investment at 8% annual interest

compoundedsemiannually for three years.

Solution:

Interest per period = annual interest rate / number of periods = 0.08 / 2 = 0.04 Total number of periods = 3 years * 2 periods per year= 6

o Principal at the end of the first period = $10,000 (1+ 0.04) = $10,000 (1.04) = $10,400   Principal at the end of the second period = $10,400 (1+ 0.04) = $10,400 (1.04) = $10,816

o Principal at the end of the third period = $10,816 (1+ 0.04) = $10,816 (1.04) = $11,248.64

 

o Principal at the end of the fourth period = $11,248.64 (1+ 0.04) = $11,248.64 (1.04) =

$11,698.59

 o Principal at the end of the fifth period = $11,698.59 (1+ 0.04)

= $11,698.59 (1.04) = $12,166.53

 o Principal at the end of the sixth period = $12,166.53 (1+ 0.04)

= $12,166.53 (1.04) = $12,653.19

Calculate the future value and the compound interest amount using a $1.00 future value table

We can also find the amount of compound interest using the Table. Here is the steps involve in this process.

Find total number of interest periods Total number of interest periods = number of years x number of interest periods

per year  Find interest rate per period Interest rate per period = annual interest rate / number of interest periods per

year

Select the row that corresponds to the number of periods and the column that corresponds to the interest rate per period from the Table.

Locate the value in the table at the intersection of the row and column mentioned in step 3.

Multiply the original principal by the value from step 4 to find the future value .  Future value = principal x table value  Total compound interest = future value – original principal

Example 1:

Use table to find the compound interest on a $5000 loan for six years at 8%, compounded annually.

o Solution:o 8% = 0.08o Total interest periods = number of years x number of interest

periods per yearo = 6 x 1 = 6o Interest rate per period = 0.08 / 1 = 0.08

o The table value for row = 6 and column = 8% is 1.58687

o Future value = principal x table value= $5,000 x 1.58687 = $7,934.35

 o Total compound interest = $7,934.35 - $5,000 = $2,934.35

Example 2:

A loan of $3,000 at 8% annual interest is compounded quarterly (4 times a year) for three years. Find the future value and the total compound interest.

o Solution:o 8% = 0.08o Total interest periods = number of years x number of interest

periods per yearo = 3 x 4 = 12

o Interest rate per period = 0.08 / 4 = 0.02

o The table value for row = 12 and column = 2% is 1.26824

o Future value = principal x table value= $3,000 x 1.26824= $3,804.72

 o Total compound interest = $3,804.72 - $3,000 = $804.72

 The following formula can be used to find the future value (FV).

FV = P(1 + R)N Where FV is the future value; R is interest rate per period ; n is the total number of

periods. Example 1:Find the future value and total interest amount of a $5,000 investment that earns 6%

compounded monthly after 3 years.Solution:6% = 0.06 ; R = Interest rate per period

Total interest periods = number of years x number of interest periods per year= 3 x 12 = 36

 R = 0.06 / 12 = 0.005

FV = $5,000 (1 + 0.005)36 = $5,000 (1.005)36

= $5,000 x 1.19668 = $5,983.40

Total interest amount = FV - Original principal = $5,983.40 - $5,000 = $983.40

o How can we compare two loans with different interest rates and number of periods compounded a year? To make such comparison we need to convert both to a rate of interest called, effective rate.

o The effective rate equates the compound interest rates to equivalent simple interest rates so that a comparison can be made. The effective rate is also referred to as the annual percentage yield (APY).

o Here is how to find the effective interest rate of a compound interest rate

o Effective annual interest rate = compound interest for first year / principal

o We can also find the effective interest rate by using a table by using the formula:Effective annual interest rate = [ (future value of $1.00 after 1 year - $1.00) / $1.00 ] * 100%

Example 1:John borrowed $6,000 at 10% compounded semiannually. What is the effective

rate?  Solution: 10% = 0.10 Interest rate per period = 0.10 / 2 = 0.05 First end-of-period principal = $6,000 (1 + 0.05)

= $6,000 (1.05) = $6,300  Second end-of- period principal = $6,000 (1 + 0.05)

= $6,300 (1.05) = $6,615

Compound interest after first year = $6,615 - $6,000 = $615  Effective interest after first year = ($615 / $6,000) * 100%

= 0.1025 * 100% = 10.25%

o On Table 6.1, we go to the row of 2 (for two periods) and the column of 5% and read the value at the intersection of the row and column. This value is 1.10250.

o Effective annual interest rate =[ (future value of $1.00 after 1 year - $1.00) /

$1.00 ] * 100%

o Effective annual interest rate = [ (1.10250 – 1) / 1 ] * 100%= [ 0.10250 /

1 ] * 100%= 0.10250 * 100% = 10.25%

Calculate the interest compounded daily using a table

o To find the compounded daily interest using a table we follow the steps below:

o We first determine the amount of money the table uses as the principal ($1, $100, or $1000).

o Then, divide the loan principal by table principal. o Then, using the table 6-2 find the days row and interest rate column

corresponding to the interest rate of the loan and find the value at the intersection of this row and column.

o Finally, multiply the value from the result of step 2 by step 3.

o Example:o Find the interest on $800 at 7.5% annually, compounded daily, for 28

days.o  o Solution:o First we find the number of $100 units in the principal. $800 / $100

= 8o The Table 6.2 value is 0.576941. So,o $8 ( 0.576941) = $4.615528 rounded to two digits after the decimal

= $.62

Learning outcomeso Calculate the present value bases on annual compounding for one yearo Calculate the present value using a $1.00 present value tableo Calculate the present value using a formula

o Present Value tells us how much to invest at present value in order to yield a certain amount at some specified future date.

 o Calculate the present value bases on annual compounding for

one yearo Calculating present value manually is time consuming. When there are

many interest periods out there. We find the Present Value formula by adjusting the Future value formula.

o Future Value = principal (1 + annual interest rate) so to find for Principal we need to divide the equation by (1 + annual interest rate). So,

 o Present value = future value / (1 + annual interest rate)

Example :Find the amount of money that Mary needs to

set aside today to ensure that $10,000 will be available in her account to buy a new TV in one year if the annual interest rate is 4% compounded annually.

 Solution:1 +0.04 = 1.04$10,000 / 1.04 = $9,615.38

To calculate the present value based on using the $1.00 present value table we use the following formula:

o Interest per period = annual interest rate / number of interest periods per year

o We can also follow the Table 6-3 to find the days row and interest rate column corresponding to the interest rate of the loan and find the value at the intersection of this row and column.

o Finally, multiply the value from the result of step 1 by the loan amount .

o Example: o AZZ company needs $20,000 in five years to buy a new machine. How much

must the firm invest at the present if it is received 5% interest compounded annually?

o  o Solution:o Going to table 6-3 row of 5 and rate of 5% column, the table value would be

0.78353.o  o Present value = $20,000 * 0.78353 = $15,670.60

The following is the present value formula where P is present value, R is the annual interest rate, and FV is the future value.

PV = FV / (1 + R)N  Example:Hotel Victoria would like to put away some of the holiday profits to save for a planned

expansion. A total of $8,000 is needed in three years. How much money in a 5.2% three-year certificate of deposit that is compounded monthly must be invested now to have the $8,000 in three years?

  Solution: 5.2% = 0.052 Interest rate per period = 0.052 / 12 = 0.00433 Number of periods = 3 * 12 = 36 PV = FV / (1 + R)N PV = $8,000 / (1 + 0.00433)36 PV = $8,000 / (1.00433)36 PV = $8,000 / 1.16829 PV = $6,847.61

Discussion?