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A regular cashflow
Now that we have a definition of both the NPV and NFV it it time to use them to look at some situations that we have already analysed in earlier chapters.
In particular what is the NPV or NFV of a perfectly regular cashflow as might be found as part of a repayment loan, annuity or savings plan.
If you receive $S at the end of each period for n periods the NPV is given by:
which should remind you of the calculation to find the future value of a cashflow. In that case the sum was the same but each value of S was multiplied by (1+I) raised to a power. By comparing the two expressions it isn’t difficult to work out that:
and comparing this to the formulas given in Chapter 4 also reveals that:
NPV = PV(I,n,S)
In other words the NPV of a regular cashflow is just its present value as calculated in Chapter 4.
This is very reasonable as the present value of a cashflow is simply what it would be fair to pay if you were purchasing an annuity.
Put another way, it is the amount you need to invest now to generate the cashflow at I% per annum.
If you regard the cashflow as the repayments of a loan then the present value is equal to the amount of the loan. Again this is perfectly reasonable.
In the case of a savings plan the reasoning is slightly different because there is no sum of money changing hands now - only the cashflow being invested. However there is a well defined future value - the final value of the investment.
In this case the present value is the amount that you would have to invest now to produce the same future value as the cashflow.
For example,if you invest $1000 per annum at 10% (effective) for 10 years the future value is:
and the present value is
If you invest $6144.56 at 10% for 10 years you will receive:
So investing $1000 per year for 10 years is the same as a making a one-off investment of approximately $6,144. Again the concept and terminology of ‘present value’ of a cashflow makes perfect sense.
You can choose to look at either the present value or the future value of a cashflow. In each case the future value is what the cashflow is finally worth and the present value is what it is worth now allowing for interest at I%.
No matter what the source or the purpose of the cashflow is the present value is calculated using:
or the corresponding PV function and the future value is calculated using:
or the corresponding FV function.
In each case the PV and FV are equal to the NPV and NFV and they are related in by the standard formula
For simplicity we are assuming that all cashflows are at the end of each time period.
This idea of present and future values of a cashflow has a pleasing symmetry but the interpretation of PV and FV is different in each of the standard situations of a repayment loan, annuity and savings plan.
These are summarised in the table below.
If we assume that a transaction generates a series of equal cash payments then the present and future values can be interpreted as follows:
|Cash flow resulting from
|| Future value
||The amount of loan
||The final value of the repayments if reinvested at I%. Equivalent to investing the amount of the loan at I%
||The amount that has to be invested to generate the cash flow i.e. the price of the annuity.
||The final value of the cash flow if reinvested at I%. Also equivalent to investing the price of the annuity at I%
| Savings plan
||The amount that has to be invested to give the same future value as the cash flow
||The final value of the plan
You can see that in the case of a repayment loan and the annuity the present value is more meaningful than the future value and in the case of the savings plan both are meaningful.
Including all of the payments
There seems to be a contradiction contained in the previous section.
In Chapter Six we worked out the formula for a repayment loan based on the fact that its future value was zero and yet in the last section the idea that a repayment loan has a large positive future value was introduced.
The difference is that in in the previous chapter we included the amount of the loan as the first amount in the cashflow.
That is, the cashflow was assumed to start with a negative amount equal to the loan. If you include the initial negative amount in either the repayment loan or the annuity then we no longer have the simple regular cashflow that was analysed in the previous section.
Fortunately it isn’t too difficult to arrive at similar results but they are different.
Now that we have a general definition of the NFV and a relationship between the NPV and NFV we can re-examine the repayment loan and the annuity including their initial payments.
Notice that in both cases the first cashflow occurs right at the very start of the investment, that is at the start of the first period. This means that in the calculation of the NPV you simply include the first cash amount without the need to discount it. This is true of any sort of investment that involves an initial deposit which then generates a series of cashflows.
NPV of investment= -initial sum +
PV of cashflow
If you include the amount of the loan or the purchase price of an annuity as the first item in the cashflow the NPV is zero.
The reason is quite simply that the NPV of the cashflow is equal to the loan or purchase price as discussed earlier.
To see that this is so let’s look at an example.
If you borrow $1000 for 5 years at 10% (effective) per annum the repayments are $263.80 annually for 5 years.
Taking the cashflow as five yearly payments (at the end of each year) of $263.80, we can ask what the present value of this cashflow is:
i.e. the amount of the loan.
If we now include the loan in the cashflow as an initial -1000 outflow of cash then the Net Present Value is -1000+1000 which is of course $0.
This is also the result computed by the NPV function as long as you remember to include the initial sum separately.
If we include the initial amount of the loan as a negative sum then it is clear that the Net Future Value of a repayment loan is also zero.
Once again an example will make this clear.
If you borrow $1000 at 10% per annum to be repaid over 10 years then repayments are $162.75 per annum and the Net Future Value is:
which works out to zero.
You can arrive at the same conclusion by recalling the fact that the NFV is also given by the NPV invested at I% for the same period.
Of course this implies that if the NPV is zero then then the NFV also has to be zero.
The same sort of reasoning applies to an annuity.
If you include the initial deposit needed to fund the annuity then its present value and its future value are both zero. In the case of a savings plan there is no initial deposit and so the earlier interpretation of the present and future values hold.
What is important here is that we get different answers depending on whether or not we include all of the cash sums involved in a transaction.
It is perfectly reasonable to analyse the cashflow that results from a transaction like a repayment loan in isolation from the value of the loan.
It is also perfectly reasonable to analyse the cashflow including the loan but not only are the results different but they have to be interpreted in different ways.