Simple and Compound Interest – Time is Money

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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 simplify calculations, but you still need to 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 Percentages
    Percentages are familiar to us all, but they present many pitfalls to avoid.
  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 present and future value PV and FV are introduced.
  3. Effective interest rates
    We explore the idea of ​​the “effective” annual interest rate, then move on to the effective interest rate/annual effective rate, the oft-quoted EIR or APR.
  4. Introduction to Cash – Savings Plans
    In the first of three chapters covering how the interest rate affects cash flow, we explore savings – but first introduce some general ideas that apply equally 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
    Repayment loans are the subject of the last of three chapters that examine the effects of regular cash flows.
  7. Present and future values
    The principles of present and future value apply even if the cash flows are irregular. The math is just 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 analysis of IRR and MIRR investments
    The IRR is perhaps the most complicated measure of the value of an investment with irregular cash flow. Understanding exactly what it means is a good step towards using it correctly.

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 one that has a time-based component. Interest is calculated and paid at regular intervals, which makes their behavior a bit more varied than just a 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.

The 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.

In the days before the legislation tightened on how interest rates were quoted, it was not uncommon to find interest rates of 10%, but with no mention of the time period – and 10% per day is a very different amount of money than 10% per year.

There are therefore two important elements in any specification of interest:

  1. the percentage to pay
  2. the time period governing the frequency of payment

This view of percentage as rate highlights some of the difficulties ahead.

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

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

Clearly the conversion between quoted interest for different time periods is something we are going to have to consider. But first we need to look at how 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 the calculation of 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, an 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 each payment period is simply:

=$M*I

Note that we do not consider the repayment of the loan or the accumulation of interest.

If the principal is a loan, it is assumed that the entire principal will be repaid in a lump sum in the future – i.e. it is an interest-only loan. If the principal is an investment, the interest is paid to the investor and not reinvested.

The key factor is that the interest is paid in such a way that the value of the capital, ie 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 acts on PV.

This jargon applies to both investments and loans:

  • In the case of an investment, the amount of money that is 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 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 HP increase of I*n%.

Compare simple interest

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, 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.

  • All we have to do to compare the rates is 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, the conversion between different periods is simply a matter of multiplying by the ratio of the periods.

For instance:

  • 0.5%, i.e. half a percent, paid daily equals 0.5*365% or 185% per year
  • 1% paid every two months 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 correspond to simple interest, i.e. the calculated interest is not added to the PV.

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