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We explore the idea of borrowing money for a specified rate of interest or earning interest on an investment.
The idea of borrowing money for a specified rate of interest or earning interest on an investment.is something that we are all familiar with. Interest is a percentage but one that has a time based component. Interest is calculated and paid at a regular intervals and this makes its behaviour rather more varied than a simple static percentage.
Interest - a percentage rate
Many financial arrangements are specified in terms of interest which is a percentage of the total per time period. Interest is a percentage rate - so many percent per month, so many percent per year and so on. It is a rate in the sense of something that involves the passage of time - miles per hour, kilometres per second and 10% per month are all rates.
In the days before legislation tightened up on how interest rates were quoted it wasn’t uncommon to find quotes of 10% interest but without any mention of the time period involved - and 10% per day is a very different amount of money from 10% per annum.
Thus there are two important components of any interest specification -
- the percentage to be paid
- the time period governing how often it is paid
This view of percentage as a rate makes clear some of the difficulties in store for us.
For example, if you can make a return of 1% per month, 3% every quarter or 11% per annum which is the better investment?
A small bank loan is offered at 20% per annum but a credit card load costs only 2% per month which is better?
Clearly converting between interest quoted for different time periods is something that we are going to have to examine. But first we need to look at the way that interest is calculated.
Lenders and borrowers
Interest is paid on deposits and charged on loans. These two situations are in fact identical from the point of view of calculating interest.
In each case there is an investor/lender who provides the lump sum - the principal and a borrower who pays interest on the loan/investment, see Figure 1. It doesn’t really matter if the borrower is in fact called a bank, an investment trust or John Smith, the cash flows are the same.
If the principal is $M and the interest rate is I% the interest due each payment period is simply:
Notice that we are not considering the repayment of the loan nor the accumulation of interest.
If the principal is a loan then it is assumed that the whole principal will be paid back as a lump sum sometime in the future - i.e. it is an `interest only’ loan. If the principal is an investment then the interest is paid out to the investor and not reinvested.
The key factor is that the interest is paid in such a way that the value of the principal, i.e. $M, remains constant over time. In this case the amount of interest paid in each time period is also constant and this results in a very easy to manage situation - simple interest.