|Present and Future Values|
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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:
If i is the ‘true’ interest rate then the purchasing power of the deposit is given by
You can see from these two formulas that
If you multiply out the right-hand side you get
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:
but if inflation is running at 5% per annum the true interest rate is only 2.6% making the final balance only worth:
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.
Net present value - NPV
The idea of using the present value to estimate the worth of a sum of money received in the future can be extended to an arbitrary cashflow.
If there are a series of cash amounts that become available on different dates then the best way to gauge their value is to reduce each one to its present value and then sum them to form the ‘Net Present Value’.
That is if $Si is recieved at the end of time period i then the NPV is given by:
That is, the NPV is sum of all of the cash sums each one discounted by the appropriate factor.
In more mathematical terms the NPV is
For example, if an investment promises to generate a cashflow of $100 at the end of the first year, $200 at the end of the second and $500 as a closing payment then the Net Present Value is:
where I is the effective annual rate of interest.
If I is assumed to be 8% then the NPV is $660.98 which should be compared to the total income of $800.
Most spreadsheets have an NPV function which will calculate the present value of a cashflow:
where range is the part of the row or column that holds the cashflow values.
An alternative form is often provided:
where the values are entered directly into the formula.
The cashflows are assumed to arrive at the end of equal periods and the interest rate specified has to be appropriate for this period.
For example, in the case of the cashflows described earlier, the spreadsheet shown below demonstrates the two methods of calculation - calculating the PV for each individual value and then adding the results up or more directly in one step using the NPV function.
The cashflows are entered into B3..B5 and the present values of amount is calculate in column C. The NPV in C7 is obtained by summing the present values listed above. That is in C3 the formula is:
which is copied down the column and summed in C7.
Alternatively the formula:
entered into D7 calculates the NPV of all of the values in one step.