General Mathematics q2

1. Compounding More than
One a Year

2. Lesson Outline
1. Compounding more than once a year
2. Maturity value, interest, and present value
when compound interest is computed more
than once a year.

3. Sometimes interest may be computed more
than once a year. Consider the following
example.

4. Given a principal of 10,000, which of the
following options will yield greater interest after
5 years:
A. Earn an annual interest rate of 2% at the
end of the year, or
B. Earn an annual interest rate of 2% in two
portions – 1% after 6 months, and 1% after
another 6 months?

5. Solution
Option A: interest is compounded annually
Time (t) in
years
Principal = 10,000
Annual interest rate = 2%, compounded annually
Amount at the end of the year
1 10,000 x 1.02 = 10,200
2 10,200 x 1.02 = 10, 404
3 10, 404 x 1.02 = 10,612.08
4 10,612.08 x 1.02 = 10,824.32
5 10,824.32 x 1.02 = 11, 040.81

6. Solution
Option B: interest is compounded semi-annually, or every 6
months Time (t)
in years
Principal = 10,000, Annual Interest rate = 2%, compounded
semi-annually
Amount at the end of the year
½ 10,000 x 1.01 = 10, 100
1 10, 100 x 1.01 = 10, 201
1 ½ 10, 201 x 1.01 = 10, 303.01
2 10, 303.01 x 1.01 = 10,406.04
2 ½ 10,406.04 x 1.01 = 10,510.10
3 10,510.10 x 1.01 = 10,615.20
3 ½ 10,615.20 x 1.01 = 10,721.35
4 10,721.35 x 1.01 = 10828.56
4 ½ 10,828.56 x 1.01 = 10,936.85
5 10,936.85 x 1.01 = 10,046.22

7. What option will you choose? A or B?

8. Option B will give the higher interest after 5 years.
If all else is equal, a more frequent compounding
will result in a higher interest, which is why Option
B gives higher interest than Option A.

9. The investment scheme in Option B introduces
new concepts because interest is compounded
twice a year, the conversion period is 6 months,
and the frequency of conversion is 2. As the
investment runs for 5 years, the total number of
conversion periods is 10. the nominal rate is 2%
and the rate of interest for each conversion period
is 1%. These terms are generally define in the next
slide.

10. Frequency of conversion (m) – number of conversion periods in
one year
Conversion or interest period – time between successive
conversions of interest
Total number of conversion periods n
n = mt = (frequency of conversion) x (time in years)
Nominal rate (ⅈ 𝒎 ) – annual rate of interest
Rate (j) of interest for each conversion period
𝑗 =
ⅈ 𝑚
𝑚
=
𝑎𝑛𝑛𝑢𝑎𝑙 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛
Definition of Terms