Simple and Compound Interest – Time is Money


Page 1 of 5

Chapter 2

We explore the idea of ​​borrowing money for a specified interest rate or earning interest on an investment.

Financial functions

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Spreadsheets make calculations simple, but you should always know how to do them. Financial functions with a spreadsheet are all about understanding and reasoning, using a spreadsheet to do the actual calculation.

  1. Understanding the percentages
    Percentages are familiar to all of us, but they present many pitfalls that should be avoided.
  2. Simple and compound interest
    We explore the idea of ​​borrowing money for a specified interest rate or earning interest on an investment. The ideas of current and future value PV and FV are introduced.
  3. Effective interest rates
    We explore the idea of ​​the “effective” annual interest rate, then the effective interest rate / annual percentage rate, the much quoted EIR or APR.
  4. Introduction to treasury – Savings plans
    In the first of three chapters covering how interest rates affect cash flow, we explore saving – but first introduce some general ideas that also apply to annuities and repayment loans.
  5. Cash flow continued – Annuities
    We turn to annuities in the second of three chapters devoted to exploring how the interest rate affects
  6. Explore repayment loans
    Repayable loans are the subject of the last of three chapters which examine the effects of regular cash flows.
  7. Present and future values
    The present and future value principles apply even if the cash flows are irregular. Calculations are simply breaking down the cash flow calculations into simple steps.

  8. Investment analysis
    How to evaluate investments that generate irregular cash flows? We explore how NPV can be used to make investment decisions.

  9. Advanced TRI and MIRR investment analysis
    IRR is perhaps the most complicated measure of the value of an investment with irregular cash flow. Understanding exactly what this means is a good step in putting it to good use.

The idea of ​​borrowing money for a specified interest rate or earning interest on an investment is something we are all familiar with.

Interest is a percentage, but it has a time component. Interest is calculated and paid at regular intervals, which makes its behavior a little 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 period.

Interest is a percentage – so many percent per month, so many percent per year, and so on. It’s a rate in the sense of something that involves the passage of time – miles per hour, kilometers per second, and 10% per month are all rates.

Before legislation hardened the way interest rates were quoted, it was not 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 of 10% per year.

Thus, there are two important components of any specification of interest:

  1. the percentage to pay
  2. the period of time governing how often it is paid

This conception of percentage as a rate highlights some of the difficulties that lie ahead.

For example, if you can make a return of 1% per month, 3% each quarter, or 11% per year, what is the best investment?

A small bank loan is offered at 20% per year, but a credit card charge only costs 2% per month which is better?

Clearly, converting between quoted interest for different time periods is something we’re going to have to look at. But first we need to look at how this interest is calculated.

Lenders and borrowers

Interest is paid on deposits and taken 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 the interest on the loan / investment.

It doesn’t matter if the borrower is actually called a bank, investment trust, or John Smith, the cash flows are the same.

Fig.  1

If the principal is $ M and the interest rate is 1%, the interest due in each payment period is simply:

=$M*I

Note that we do not contemplate repaying the loan or accumulating interest.

If the principal is a loan, it is assumed that the entire principal will be repaid as a lump sum in the future – that is, it is an interest-only loan. If the principal is an investment, the interest is paid to the investor and is 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 period is also constant, which results in a very manageable situation – simple interest.

Present and future value

It is common to use the terminology Present Value, or PV, for the amount of money involved at the start of a loan or investment and Future Value, or FV, for the ending balance.

In other words, FV is what results after interest has acted on PV.

This jargon applies to both investments and loans:

  • In the case of an investment, the amount of money deposited or invested is the PV and the final balance is the FV
  • In the case of a loan, the amount borrowed is the PV and the amount finally repaid is the FV

Other terms, such as principal, are used for PV but for the rest of this book PV and FV will be used to refer to the value before and after the action of interest, respectively.

Note that the relationship between PV and FV depends on the type of situation we are considering.

For example, in the case of a simple interest of I% over n interest-bearing periods, the FV is given by:

FV=PV+PV*I*n

Where:

FV=PV*(1+I*n)

You should be able to recognize this as a simple I * n% HP increase.

Simple interest comparison

In the situation where interest is paid on a PV that does not change over time, it is very easy to compare different interest rates.

For example, if a deposit earns 2% interest per month, then over a 12 month period the total amount paid in interest is simply:

=12*PV*0.02

Where:

=PV*0.24

This implies that receiving 2% per month is equivalent to receiving 24% per year.

This same reasoning applies to any interest rate over any period of time.

  • To compare rates, simply convert them to the equivalent rate per year

For example, 10% paid every six months, or two interest-bearing periods per year, equals a rate of 0.10 * 2, or 20% per year.

In other words, for simple interest rates, converting between different periods is actually multiplying by the ratio of the periods.

For example:

  • 0.5%, or half a percent, paid daily equals 0.5 * 365% or 185% per annum
  • 1% paid bi-monthly equals 0.5% paid monthly
  • a return of 50% over 10 years is equivalent to 5% per year

Note that for all of these examples to be correct, the situation must be simple interest, i.e. calculated interest is not added to PV.


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