## Effective Interest? Simple Interest? Compound Interest? Nominal Interest?

Simple interest is usually discussed and compared with compound interest. Simple interest is named as such because the interest calculated is not compounded.

In contrast, when compound interest is calculated, nominal interest rate and effective interest rate would be the relevant interest rates involved in the calculations or discussions.

Example

1. Simple Interest

ABC Co Ltd. placed \$100,000 deposit with Bank A for 1 year with interest of 3.5% per annum and calculated on simple interest method.

Interest earned at the end of the one year period is therefore calculated as follows: –

\$100,000 x 3.5%

= \$3,500

2. Compound Interest

In the context of compound interest calculation, you need to specify the following: –

1. The total length of the placement of deposit. In this example, one year.
2. The frequency of compound interest calculation. E.g. daily, monthly, quarterly & etc.
3. The nominal annual rate of interest used.

Assume monthly compounding of interest is adopted, with 3.5% nominal annual rate of interest, we usually say:

“ABC Co Ltd. placed \$100,000 deposit with Bank A for 1 year with 3.5% nominal annual rate of interest monthly compounding”.

The interest earned is calculated as follows:-

FV = PV x (1 + i)n

Where,

FV = future value of the deposit (the total value of the \$100,000 deposit at the end of the 1 year period.)

PV = the present value of the deposit (the value of the deposit at the beginning of the 1 year period, which is \$100,000)

n = the number of period in terms of compounding. For example, if the period of the deposit placement is 2 years with monthly compounding, n = 24 ( 2years x 12)

i = the interest rate in percent per period of compounding. In this example, i =0.29667% (3.5%/12). Word of caution here, when you do the calculation, i is 0.00296667. Many make mistake in the calculation because they did not realise that i is expressed in percentage!

Therefore, for this \$100,000 deposit placement,

FV = \$100,000 (1 + 3.5%/12)12

= \$103,556.70

The interest earned is therefore \$3,556,70 (\$103,556.70 – \$100,000)

Conclusion

On \$100,000 deposit placement, interest at 3.5% per annum monthly compounding for 1 year period will yield an extra \$56.70 compared to 3.5% per annum simple interest method.

There could be instances whereby ABC Co Ltd. is offered different options for deposit placement with Bank A. For example:-

Option 1

\$100,000 1 year at 3.5% per annum monthly compounding

Option 2

\$100,000 2 years at 4.75% per annum quarterly compounding

Option 3

\$100,000 5 years at 4.80% per annum yearly compounding

How does ABC Co Ltd. evaluate these options? Which is the best option? As the period of the investment i.e. the length of the placement is different (Option 1 = 1 year; Option 2 = 2 years; Option 3 = 5 years), a meaningful comparison on “level field” is desired – using Effective Interest Rate.

The formula for effective interest rate is:-

R = (1 + i)n – 1

Where,

R = the effective interest rate

i = the interest rate in percent per period of compounding

n
= the number of period in terms of compounding in a year

Option 1

R = (1 + 3.5%/12)12 – 1

= 3.56%

Option 2

R = (1 + 4.75%/4)4 – 1

= 4.84%

Option 3

R = (1 + 4.80%/1)1 – 1

= 4.80%

Based on the effective interest rates calculated for the three options, Option 2 gives the highest rate and appears to the best. However, in making the selection ABC Co Ltd. should also consider other factors including the future plan of ABC Co Ltd. in terms of when will the money is needed in future & etc. in one year? 3 years? 5 years?

How much will ABC Co Ltd earn for each option?

Option 1

FV = \$100,000 (1 + 3.5%/12)12

= \$103,556.70

Interest earned at the end of 1 year placement is \$3,556.70 (\$103,556.70 – \$100,000)

Option 2

FV = \$100,000 (1 + 4.75%/4)8

= \$109,904.36

Interest earned at the end of 2 years placement is \$9,904.36 (\$109,904.36 – \$100,000). But take note that this is over 2 years if you compare to that of Option 1. If you calculate using straight-line time proportion basis, Option 2 gives \$4,952.18 (\$9,904.36 divided by 2 years) and appears to yield much higher interest. Straight-line time proportion basis does not give a good picture for the comparison because the interest is compounded. Using the effective interest method, the FV under Option 2 at the end of Year 1 is: –

FV = \$100,000 (1 + 4.75%/4)4

= \$104,835.28

The interest earned for Year 1 under Option 2 is therefore \$4,835.28 (\$104,835.28 – \$100,000) and not \$4,952.18 calculated under the straight-line time proportion method.

The interest earned during Year 2 under Option 2 is: –

FV = \$104,835.28 (1 + 4.75%/4)4

= \$109,940.36

The interest earned for Year 2 is therefore 5,069.08 (\$109,940.36 – \$104,835.28).

Perform a summation proof by adding the interest earned in Year 1 and Year 2:-

Total interest (Year 1 + Year 2) = \$4,835.28 + \$5,069.08

= \$9,904.36!

Option 3

FV = \$100,000 (1 + 4.80%/1)5

= \$126,417.17

Interest earned at the end of 5 years placement is \$26,417.27 (\$126,417.17 – \$100,000). But take note that this is over 5 years if you compare to that of Option 1 or Option 2. If you calculate using straight-line time proportion basis, Option 5 gives \$5,283.54 (\$26,417.27 divided by 5 years) and appears to yield much higher interest. Straight-line time proportion basis does not give a good picture for the comparison because the interest is compounded. Using the effective interest method, the FV under Option 3 at the end of Year 1 is: –

FV = \$100,000 (1 + 4.80%/1)1

= \$104,800

The interest earned for Year 1 under Option 2 is therefore \$4,800 (\$104,800 – \$100,000) and not \$5,283.54 calculated under the straight-line time proportion method.

The interest earned during Year 2 under Option 3 is: –

FV = \$104,800 (1 + 4.80%/1)1

= \$109,830.40

The interest earned for Year 2 is therefore 5,030.40 (\$109.830.40 – \$104,800.00).

The interest earned during Year 3 under Option 3 is: –

FV = \$109,830.40 (1 + 4.80%/1)1

= \$115,102.26

The interest earned for Year 3 is therefore 5,271.86 (\$115,102.26 – \$109.830.40).

The interest earned during Year 4 under Option 3 is: –

FV = \$115,102.26 (1 + 4.80%/1)1

= \$120,627.17

The interest earned for Year 4 is therefore 5,524.91 (\$120,627.17 – \$115,102.26).

The interest earned during Year 5 under Option 3 is: –

FV = \$120,627.17 (1 + 4.80%/1)1

= \$126,417.27

The interest earned for Year 5 is therefore 5,790.10 (\$126,417.27 – \$120,627.17).

Perform a summation proof by adding the interest earned in Year 1,Year 2, Year 3, Year 4 and Year 5:-

Total interest (Year 1 + Year 2 + Year 3 + Year 4 + Year 5)

= \$4,800 + 5,030.40 + 5,271.86 + 5,524.91 + 5,790.10

= \$26,417.27!

Note: The excess of 10 cents (\$26,417.27 – \$26,417.17) is due to rounding error.