# formula for calculating compound interest

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Simple and Compound Interest

Objectives

On completion of this unit you should understand:
1.    Simple Interest
2.    Compound Interest

Written by Gareth Lotwick, March 2004
Education Drop-in Centre

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Simple and Compound Interest

What is Interest?

Money invested in a financial institution, e.g. a bank or building society, will normally
earn interest (that is, it will ‘earn’ extra money). The amount invested is called the
principal and is denoted by p. The interest rate is a percentage figure which enables us
to calculate the interest earned on the principal. We usually denote the interest rate by r,
where r is a decimal.

For example, if the interest rate is 4% per annum, then

4
r=       = 0.04 p.a.
100

Interest rates can be for any period, although by far the most usual is annually (that is, per
year or per annum or p.a.). Less commonly, you may be given a monthly interest rate.

Example 1

₤200 was invested in an account offering 4% interest per annum. Find the interest earned
after 1 year.

Solution
r = 0.04
p = £200

Interest earned = 0.04 × 200 = £8

Simple or Compound Interest?

When you invest money you usually have the option of earning ‘Simple’ or ‘Compound’
interest.

Simple interest means that you earn the same amount of money every year on the
principal you have invested. In the example above, you would earn ₤8 every year on your
₤200 principal at 4% p.a. So after 3 years you will have earned 3 × 8 = £24 .

The formula for calculating simple interest is

I = prt

where I is the interest earned, p is the principal invested, r is the interest rate per unit of
time (e.g. p.a.), expressed as a decimal, and t is the number of units of time (e.g. years).

With compound interest, you reinvest the interest you have earned each year, thus
increasing your principal every year, and so increasing the interest earned each year. In
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Simple and Compound Interest

the example above, if you add the ₤8 interest you earned at the end of year 1 to your
original principal of ₤200, you start the second year with a principal of ₤208.

So in the second year,

r = 0.04, p = 208, t = 1
I = prt = 0.04 × 208 × 1 = 8.32

At the beginning of the 3rd year you add the second year’s interest of ₤8.32 to the second
year’s principal of ₤208, to make the 3rd year’s principal of ₤216.32.

So at the end of the third year,

p = 216.32, r = 0.04, t = 1
I = prt = 216.32 × 0.04 × 1 = 8.65 (to 2 dp).

The total compound interest earned in 3 years on ₤200 at 4% p.a. is
₤ (8 + 8.32 + 8.65) = £24.97 , which is quite a lot more than the ₤24 earned in simple
interest over the same period.

Is there a formula for calculating Compound Interest?

As you can see from the calculations above, it is quite a lengthy business calculating
compound interest year by year.

We will now develop a formula to do it, using p for the principal, r for the interest rate
per time period, and n for the number of time periods.

Interest earned in the first time period = pr
Amount in account after 1 time period = p + pr = p (1 + r )

Interest earned in second time period = p (1 + r ) r
Amount in account after second time period = p (1 + r ) + p (1 + r ) r
= p (1 + r )(1 + r )
= p (1 + r ) 2

… and so on …

The amount in the account, A, after n time periods is given by

A = p (1 + r ) n

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Simple and Compound Interest

Example 2

₤500 is invested in an account paying compound interest at 3.5% per annum for 4 years.
Find (i) the total amount in the account at the end of the 4 years
(ii) the total amount of interest earned.

Solution
(i)     Compound interest formula: A = p (1 + r ) n .
In this case, p = 500, r = 0.035 , n = 4 ,
So A = 500(1 + 0.035) 4
A = 500(1.035) 4
A = 573.76
Amount after 4 years is ₤573.76

(ii) Total interest earned in 4 years is ₤(573.76 – 500) = ₤73.76

Example 3

An amount of ₤200 is invested in an account paying monthly interest of      1
4   % per month.
What is the amount of interest earned after one year?

0.25
Using A = p (1 + r ) n with p = 200, r =        = 0.0025, n = 12 :
100

Amount after 1 year: A = 200(1 + 0.0025)12
A = 200(1.0025)12
A = £206.08

So, interest earned = 206.08 − 200 = £6.08

Comparing compound interest rates for different time periods

Example 4

Is a rate of 6% p.a. compound interest equivalent to 3% per half-year (i.e. 6% ÷ 2) or
0.5% per month (i.e. 6% ÷ 12)?

Solution
Choose any amount as the initial principal, and find how much interest is earned in 1 year
in each of the 3 cases: (i) 6% p.a (ii) 3% per half-year (iii) 0.5% per month. Compare
these amounts.

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Simple and Compound Interest

We will choose the principal to be £100 (nice easy number to work with, especially when
working with percentages). Use the formula A = p (1 + r ) n each time.

Case 1: p = 100, r = 0.06, n = 1       A = 100(1 + 0.06)1 = £106 .
Interest earned = 106 − 100 = £6 .

Case 2: p = 100, r = 0.03, n = 2      A = 100(1 + 0.03) 2 = £106.09 .
Interest earned = 106.09 − 100 = £6.09 .

Case 3: p = 100, r = 0.005, n = 12    A = 100(1 + 0.005)12 = £106.17 .
Interest earned = 106.17 − 100 = £6.17 .

A compound interest rate of 6% p.a. is less than both the other rates. On a principal of
£100, in 1 year: 6% p.a. generates interest of £6, 3% per half-year generates interest of
£6.09, 0.5% per month generates interest of £6.17.

Exercises

1. Calculate the simple interest on:
(a) £150 for 2 years at 4% p.a.
(b) £300 for 3 years at 3.5% p.a.

2. James invests £10,000 in an account which pays monthly interest, and he spends
this interest each month. The monthly interest rate is 0.6%. How much interest
does he have to spend in each of the first 6 months? What about each of the
second 6 months?

3. How much should be invested for 2 years to obtain interest amounting to £20 in
total if the interest rate is 4% per annum? Assume the interest is simple interest.

Assuming Compound Interest in each of the following cases:

4. A person invests £1500 in a two-year bond paying 4.5% interest per year. Money
is left in the account for the whole of the two-year period. What amount will be in
the account at the end of the two-year period?

5. Calculate the interest earned by investing
a. £500 for 3 years at 4% per annum
b. £400 for 2 years at 0.5% per month
c. £300 for 1 year at 2% per half-year.

6. Jane has £1,000 to invest for one year and is considering three different accounts:
a) A one-year bond offering 4% per annum
b) And account offering 0.35% per month

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Simple and Compound Interest

c) An account offering 2.1% per half-year.
She does not need the interest until the end of the year. Into which account should
she invest her money to maximize the interest?

Solutions

1. Using the formula I = prt in both cases, to find I the interest earned:
a)     p = 150        r = 0.04       t=2
So I = 150 × 0.04 × 2 = 12
Simple Interest earned = £12

b)      p = 300        r = 0.035       t=3
So I = 300 × 0.035 × 3 = 31.50
Simple Interest earned = £31.50

2. Amount invested = £10,000
Interest rate = r = 0.006 per month (0.6%)
Time invested is one month because interest is received and used each month, i.e.
t =1

Using formula I = prt , for I the monthly interest:
I = 10000 × 0.006 × 1 = 60 . So I = £60 per month.

James has £60 per month to spend in the first 6 months, and also £60 per month to
spend in the second 6 months.

3. Interest rate = r = 0.04 per annum (4%)
Time invested is 2 years, i.e. t = 2
Interest earned = £20
I
Rearranging formula I = prt , to make p the subject gives p =
rt
p = 20/(0.04 × 2) = £250

4. Amount invested = p = £1,500
Interest rate = r = 0.045 p.a. (4.5%)
Time invested = n = 2

Using formula A = p (1 + r ) n to find A, the value of the investment after 2 years:
A = 1500(1 + 0.045) 2
A = 1638.04
Amount paid out after two years is £1,638.04

5. a)   p = 500          r = 0.04       n=3
Amount after 3 years is obtained using A = p (1 + r ) n

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Simple and Compound Interest

So      A = 500(1 + 0.04) 3
A = 500(1.04)3 = 562.43

Interest earned = £(562.43 – 500) = £62.43

b)      p = 400       r = 0.005       n = 24 (2 years = 24 months)
Amount after 2 years is obtained using A = p (1 + r ) n

So      A = 400(1 + 0.005) 24
A = 400(1.005)24 = 450.86

Interest earned = £(450.86 – 400) = £50.86

c)      p = 300       r = 0.02        n = 2 (1 year = 2 half-years)
Amount after 1 years is obtained using A = p (1 + r ) n

So      A = 300(1 + 0.02) 2
A = 300(1.02)2 = 312.12

Interest earned = £(312.12 – 300) = £12.12

6. Using A = p (1 + r ) n in each case:

a)      p = 1000       r = 0.04         n=1

So      A = 1000(1 + 0.04)1
A = 1000(1.04) = 1040

b)      p = 1000       r = 0.0035       n = 12

So      A = 1000(1 + 0.0035)12
A = 1000(1.00354)12 = 1042.82

c)      p = 1000       r = 0.021        n=2

So      A = 1000(1 + 0.021) 2
A = 1000(1.021)2 = 1042.44

Now subtract 1000 from each of your answers to a), b), and c), to find the interest
earned in each case. b) gives the highest interest (£42.82), so Jane should invest in
the account offering 0.35% per month.

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