Lesson 26: Compounding
More than Once a Year
Lesson Outcome(s):
• At the end of the lesson, the learner is able to compute
maturity value, interest, and present value, and solve
problems involving compound interest when compound
interest is computed more than once a year.
Lesson Outline:
1. Compounding more than a year
2. Maturity value, interest, and present value when
compound interest is computed more than once a year.
Example 1. Given a proncipal of Php
10,000, which of the following options will
yield greater interest after 5 years.
Option A: Earn an annual interest rate of
2% at the end of the year, or
Option B: Earn an annual interest rate of
2% in two portions-1% after 6 months, and
1% after another 6 months?
SOLUTION
Option A: Interest is compounded annually
Time (t) in years
Principal=Php 10,000
Annual int. 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
Option B: Interest is compunded semi-annually, or every 6
months.
Under this option, the interest rate every six months is 1% (2% divided
by 2) .
Time (t) in years
Principal= Php 10,000
Annual int. rate= 2%, compounded semi-annually
Amount at the end of the year
1/2
10,000 x 1.01= 10,100
1
10,100 x 1.01= 10,201
1 1/2
10,201 x 1.01= 10,303.01
2
10,303.01 x 1.01= 10,406.04
2 1/2
10,406.04 x 1.01= 10,510.10
3
10,510.10 x 1.01= 10,615.20
3 1/2
10,615.20 x 1.01= 10,721.35
4
10,721.35 x 1.01= 10,826.56
4 1/2
10,826.56 x 1.01= 10,936.85
5
10,936.85 x 1.01= 11,046.22
Answer:
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
a higher interest than Option A.
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%.
DEFINITION OF TERMS:
• Frequency of conversion(m) - number of conversion
periods in 1 year.
• Conversion of 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
Note on rate notation: r, i(m),j
In earlier lessons, r was used to denote the
interest rate. Now that an interest rate can refer
to two rates (either nominal or rate per
conversion period), the symbols i(m) and j will be
used instead.
Examples of nominal rates and the corresponding frequencies of
conversion and interest rate for each period:
i(m)= Nominal Rate
(Annual Interest Rate)
m= Frequency
of conversions
j= Interest Rate per
conversion period
One
conversion
period
2% compounded
annually;
i(1)=0.02
1
0.02/1=0.02=2%
1 year
2% compounded semiannually;
i(2)=0.02
2
0.02/2=0.01=1%
6 months
2% compounded
quarterly;
i(3)=0.02
4
0.02/4=0.005=0.5%
3 months
2% compounded
monthly;
i(12)= 0.02
12
0.02/12=0.0016=0.16%
1 month
2% compounded daily;
i(365)= 0.02
365
0.02/365
1 day
From Lesson 25, the formula for the maturity value F when
principal P is invested at an annual interest rate j compounded
annually is F=P(1＋j)t.
Because the rate for each conversion period is j=i(m)/m, then in t
years, interest is compunded mt times.
Maturity Value, Compounding m times a year
F=P(1＋i(m)/m)mt
where:
F= maturity (future) value
P= principal
i(m)= nominal rate of interest (annual rate)
m= frequency of conversion
t= term/time in years