     Home Simple & Compound Interest Present Value Ordinary Annuities Present Value Annuities General Annuities & Equivalent Rates Mortgages Review&Test UNIT 9  :  MATHEMATICS OF INVESTMENT

LESSON 1: SIMPLE AND COMPOUND INTEREST

Simple Interest:

When you borrow or invest money, interest is paid or earned.  If the interest is calculated only on the money originally invested, it is called simple interest.  Example 1:

Jenna borrows \$5000 from her parents to start a landscaping business.  She agrees to repay the loan in 3 years and to pay simple interest at a rate of 5%/a [5% per annum].  Calculate the interest due in 3 years and the total amount repayable.

Solution:

 p = 5000 r = 0.05 t = 3 I = ? A = ? Compound Interest:

If interest is calculated at the end of each year (or interest period) and added on at this point, then this is called compound interest.  Example 2:

a)  Suppose Jenna had borrowed the money from a bank at 5%/a compounded yearly.  This means interest is calculated at the end of each year and added on.  Hence at the end of the second year you would be paying interest on the original amount and interest on the first year’s interest as well.  This is compound interest.

Solution:

 p = 5000 n = 3 i = 0.05 A = ? b)  How much interest did Jenna pay?

Solution:

Subtract the amount repayable from the principle.

Interest = \$5788.13 - \$5000 = \$788.13

Time Value of Money:

The investment above can be illustrated on a time line.  Time = 0 means today.  The arrow moving to the right shows the investment increasing over the 3 year period.    0                              1                                  2                                      3   \$5000                                                                                                              \$5000(1.05)3

Note:  We can evaluate a give sum of money at any point using a time line simply using the compound interest formula, the correct interest rate and the correct number of time periods.

Example 3: Using different compounding periods.

a)  Shelby invested \$8 000 in a 5-year term deposit which pays interest at a rate of 5 %/a [per annum], compounded semi-annually.  What will the investment be worth at the end of the 5 year period?

Solution:

Because the interest is compounded every 6 months, we must adjust both the number of interest periods n and the interest rate i.

If interest is paid twice a year, then the number of interest periods [compounding periods] will be 5 x 2 = 10.  Hence n = 10.

The appropriate time line is shown below.  Since there are 10 interest periods, we will put a mark every 6 months over the 5-year period.

0                1                2                       .   .   .  .   .                                        4              5          \$8000                                                                                                                                        \$8000(1.025)10 b)  What will the investment be worth if interest is 6.5%/a, compounded quarterly.

Solution:

Because the interest is compounded every 3 months, we must adjust both the number of interest periods n and the interest rate i.

If interest is paid 4 times a year, then the number of interest periods [compounding periods] will be 5 x 4 = 20.  Hence n = 20.

The appropriate time line is shown below.  Since there are 20 interest periods, we will put a mark every 3 months over the 5-year period.

0                                  1                       .   .   .  .   .                      4                                5            \$8000                                                                                                                                        \$8000(1.01625)20 c)  What will the investment be worth if interest is 5.5%/a, compounded monthly.

Solution:

Because the interest is compounded every  month, we must adjust both the number of interest periods n and the interest rate i.

If interest is paid 12 times a year, then the number of interest periods [compounding periods] will be 5 x 12 = 60.  Hence n = 60.

The appropriate time line is shown below.  Since there are 60 interest periods, we will number the line using months.

0       1       2       3       4                                   .   .   .  .   .                          58     59      60 [5 years]          \$8000                                                                                                                                        \$8000(1.004583333)60 