Compound Interest Part Three – Guide to Enhancing Your Finances

Question

In this third part of our compound interest series, we explore essential methods and shortcuts that can improve your financial decision-making. This guide covers a range of important concepts related to compound interest, including managing varying interest rates, comparing simple and compound interest, and calculating instalments for loans.

Here&rsquo;s what you&rsquo;ll learn in this guide:

Calculating Compound Interest with Different Rates Across Multiple Years: 
	Calculating Compound Interest When Time Is a Fraction:
	Difference Between Compound Interest and Simple Interest Over 2 or 3 Years: 
	How to Calculate the Time It Takes for Money to Double or Multiply: 
	Deriving Interest Rates from Growth
	Calculating Interest Rates Based on Growth Over Different Periods:
	Calculating Instalments for Loans:

Compound Interest with Different Rates for Different Years

When the rates of interest are different for different years (say, R1, R2, and R3 percent for the first, second, and third years, respectively), the amount after the third year is calculated as:

Amount=     P  1+   R 1    100     1+   R 2    100     1+   R 3    100

Example:

Dupe invests $5,000 in a bond which gives interest at 4% per annum during the first year, 5% during the second year, and 10% during the third year. How much does she get at the end of the third year?

Given:
P = 5000 
R1 = 4%, R2 = 5%, R3 = 10%

The amount at the end of the third year:

Amount=50001+41001+51001+10100=6,006

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Fractional Years

When time is given as a fraction, say   t=3 1 2  &#160;  years, calculate the interest separately for the full years and the fraction of the year.

Difference Between Compound and Simple Interest

a. The difference between compound interest and simple interest on a certain sum of money for 2 years at R% per annum is given by:

Difference=  P&#215; R 2      100 2

Where:
P = Principal
R = Rate of interest

Example

What will be the difference between simple interest and compound interest on a sum of \$4,500 for 2 years at 5% per annum?

Given:
P = 4500
R = 5%
The difference:

Difference=  4500&#215; 5 2      100 2        =  4500&#215;25  10000  =$11.25

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b For 3 years at R% per annum, the difference between compound and simple interest is given by:

Difference=  P&#215; R 2  &#215;(3+R)    100 3

Time for a Sum to Become 'n' Times at Compound Interest

If a sum becomes n times in t years at compound interest, it will become nm times in m * t years.

Example

A sum of money placed at compound interest doubles in 3 years. In how many years will it amount to four times itself?

Solution:
Given:
n = 2 (double)
t = 3 years
m = 2 (four times)

It will become four times in 2 * 3 = 6 years.

Rate of Interest When a Sum Becomes 'n' Times

If a sum becomes n times in t years, the rate of interest R% per annum is given by:

R=100&#215;       n 1      1 t    &#8722;1

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Example

At what rate of compound interest does a sum of money become four times in 2 years?

Given:
n = 4
t = 2

The required rate:

R=100&#215;   4   1 2    &#8722;1      =100&#215;(2&#8722;1)=100%

Rate of Interest Given Two Amounts Over Two Different Years

If a certain sum of money at compound interest amounts to Nx in A years and Ny in B years, then the rate of interest per annum is given by solving the equation derived from the formula for compound interest.

Equal Yearly Instalments to Repay a Loan

If a loan of P at R% compound interest per annum is to be repaid in n equal yearly instalments, then the value of each instalment is given by:

Each&#160;instalment=  P&#215;R  1&#8722;    1+ R  100     &#8722;n

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Example

If a sum of $13,040 is to be paid back in two equal annual instalments at 3.75% per annum, what is the amount of each instalment?