Interest rates and inflation

The effect of inflation on interest rates and investments is a very general concern and it could be discussed in almost any chapter of this book. However it is particularly relevant to a discussion of the time value of money.

If we are discounting future cash amounts to allow for the ‘natural’ growth in money due to the availability of a safe interest rate, then why not also discount to adjust for inflation?

This is a perfectly reasonable procedure and something that can be done quite easily if you have an estimate of the inflation rate. If you receive $F n years in the future then its deflated value now is simply:

P=F/(1-R)^n

where R is the effective annual inflation rate.

Notice that this calculation has nothing at all to do with the mechanism by which the future sum of money is acquired. It could be the promised cash lump sum received as part of a pension, or the accumulated growth of an investment.

The calculation of the deflated value is identical in form to the calculation of the present value. This raises the question of how the two are related and whether or not we should also deflate as well as discount the future value?

The deflated value takes account of the loss of purchasing power of money. That is having $F in the future will allow you to buy the same goods that having $P today would do. The present value on the other hand does not take into account the purchasing power of money, only the lost investment revenue due to not receiving money for n years.

The point is that inflation does not directly alter the amount of money that you need to invest now to receive a another sum in n years time. In this sense inflation does not affect the concept or method of calculation of the present value nor the relationship between present and future value.

That is

• the present value of a future cash sum is not affected by inflation even though the purchasing power of the future value is.

The true rate

It is worth while pointing out that there is a generally accepted connection between inflation and the prevailing safe interest rate.

Many investors are familiar with the notion of the ‘true’ interest rate - which roughly means an interest rate adjusted for inflation.

For example, if you are offered an interest rate of 8% and inflation is running at 3% then the ‘true’ interest rate is only 8-3% i.e. 5%. What this means is that the purchasing power of this investment is growing at a rate of 5% and not 8% per annum.

Simply subtracting the inflation rate from the interest to give the true interest rate is a useful rough and ready approximation but it isn’t quite accurate.

If inflation is running at R% per annum and the investment is made at I% per annum then each year the purchasing power of the deposit $M decreases by (1-R) and the amount on deposit increases by (1+I). Thus each year the overall change in purchasing power of the investment is:

 =M*(1+I)(1-R)

If i is the ‘true’ interest rate then the purchasing power of the deposit is given by

 =M*(1+i)

You can see from these two formulas that

 (1+i)=(1+I)(1-R)

If you multiply out the right-hand side you get

1+i=1+I-R-IR

or

 i=I-R-IR

In other words, the true interest rate is worked out by subtracting the inflation rate and subtracting IR. For example, in the case of 8% interest and 3% inflation the true interest rate is (0.08-0.03 -0.08*0.03)=0.0476 or 4.8%.

As you can see the additional -IR usually makes only a small difference and you can safely carry on using the ‘interest minus inflation’ rule to estimate the true interest rate.

Of course the need to subtract IR to get the accurate true rate is also the reason why you need more than R% interest to compensate for inflation. For example, if the interest rate is 6% and inflation is also 6% the true rate of interest is -0.06*0.06 or -0.36%.

You can make use of the true interest rate to work out the increase in the purchasing power of an investment by simply using the true rate in place of the interest rate in any of the calculations described earlier.

For example, if you make a single investment of $1000 at an annual interest rate of 8% for 10 years then the final balance will be:

=1000*(1+0.08)^10
=$2158.92

but if inflation is running at 5% per annum the true interest rate is only 2.6% making the final balance only worth:

=1000*(1+0.026)^10
=$1292.63

in today's terms.

You can arrive at similar inflation adjusted estimates by using the true interest rate in place of the quoted interest rate.

In the case of a loan the estimate of the true interest is usually sufficient in itself. For example, if a mortgage is offered at 8% per annum and inflation is running at 4% then the true interest rate is only 3.68%. Clearly in times of high inflation it is possible for the true rate to be negative.

Notice that the calculation of the true interest rate assumes no underlying theory of how interest rates are set. It is a purely arithmetic relationship between the action of inflation and interest.

However, it seems a reasonable assumption that the quoted interest rate should reflect the estimated inflation rate. That is, the true rate of interest is determined by the real growth in wealth and then the quoted rate is ‘adjusted’ by inflation. Even if this theory does not apply and the true interest rate is not a fundamental economic quantity this makes no difference to the fact that it does at least measure the real increase in wealth.