Interest Simple and Compound

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 Interest Simple and Compound
Article Index
Interest Simple and Compound
Present and future value
Compound interest
Financial functions
Inflation
Inflation and interest
Summary and key points

## Inflation

Inflation is the gradual erosion of the purchasing power of money. As this is usually reported as a percentage rate per annum it has many of the properties of compound interest. However compound interest normally works to increase the size of the principal - inflation reduces the value of the principal.

For example, if inflation is running at 10% per annum, \$100 held at the start of the year will be worth only \$90 at the start of the next year. At this point it is tempting to use the simplistic argument that the \$90 will be further reduced by \$10 in the following year and so on until after 10 years there will be no value left at all! This is clearly incorrect but it is surprising how often the argument is encountered.

The correct reasoning is. of course, that in the second year the value is reduced by 10% of \$90, which is only \$9 not a further \$10 and so on..

You can see how inflation acts like compound interest in that the reduction in value affects the next time period’s calculation. To be more exact, after one year the value of \$M is:

`    =M*(1-I)`

after two years it is:

`    =M*(1-I)*(1-I)`

and after n years it is:

`    =M*(1-I)^n`

This is of course just the general formula for the FV under compound interest but with -I instead of +I. If we regard inflation as a negative interest rate then we can use all of the formulas listed earlier without change to calculate the effect of inflation.

For example, what will be the value of \$100 in ten years time in today’s terms if inflation is running at 3% per annum? The answer is simply

`    =100*(1-0.03)^10`

or

`    =FV(-3%,10,0,-100)`

which both work out to \$73.74. That is, with 3% inflation in 10 years time \$100 will only buy what \$73.74 will buy today.

Notice that in the case of inflation there is a potential for confusion over the use of the terms Future Value and Present Value!

There is a subtle point in the previous calculation that has been ignored so far. The annual rate of inflation may be quoted as 8% per annum but what is the compounding period? The answer is that it depends on how the quoted rate has been calculated. Inflation is a process that happens continuously and so there is no sensible or reasonable compounding period to choose for calculations. (This point is discussed in more detail in Chapter 3). In most cases the quoted rate gives the correct results for an annual compounding period, hence the calculations above. This is often referred to as the “year on year inflation rate”.

To give you an example of how the other compound interest formulas can be used how would you calculate the number of years it takes for the value of money to exactly halve, given any particular inflation rate? This is just a matter of using the formulas that gives n in terms of PV, FV and I.

That is, time to half value is:

`    =LN(PV/2PV)/LN(1+I)`

which simplifies to

`    =ln(1/2)/ln(1+I)`

or using the NPER function:

`    =NPER(-I%,0,-PV,PV/2)`

For example, if inflation is 6% then the number of years to halve the value is:

`    =LN(0.5)/LN(1-0.06)`

or using the NPER function and a nomial present value of \$100:

`    =NPER(-6%,0,-100,100/2)`

both of which work out to 11.2 years. Notice that it is important to remember that the inflation rate is treated as a negative rate in all of the compound interest formulas - hence the -0.06 or -6% in the above.

The time it takes inflation to reduce the value of currency to 50% of its original value could be called the `half life’ because it is exactly analogus to the definition of the half life of radioactive elements. How the inflation rate determines monetary half life is quite an useful way of trying to gain an understanding of the effects of inflation. It is quite easy to construct a spreadsheet that tabulates half life, see Figure 6.

Figure 6 - the half life of money

Enter the labels and data as shown in column A and B1 and the formula

`=LN(0.5)/LN(1-A2)`

into B2 and then copy it from B2 into B3..B20.

It also helps to examine the half life using a graph. This makes it quite clear that the largest changes in half life occur when the inflation rate is small. With inflation running at 10% or more money loses half its value in less than 7 years and if the inflation rate hits 30% it halves the value of money in less than 2 years.

Figure 7: The half-life of money during inflation

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