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Irregular time periods
The NPV function as supplied in most spreadsheets will only work with equal intervals.
It assumes that each cash sum specified is received at the end of a regular time period.
If you want to calculate the PV of a cashflow at unequal intervals you can always use the basic formula on each payment - that is discount each sum and find the total.
For example, if you receive payments of:
- $500 after 6 months
- $1000 at the end of one year
- $1500 at the end of the second year
- $2000 at the end of the third year
then the NPV can be calculated as shown in below
An irregular cash flow
The cashflow and the time received are entered normally in column A with fractional years used to express 6 months i.e. 0.5 of a year.
The present value of each cashflow is calculated by entering:
into C3 and then copying this down the column. Notice the the discount formula uses the interest rate stored in B3.
Finally to get the NPV enter the formula:
This first principles method can always be used to calculate the NPV. As long as you record the time that each sum is received using fractions if necessary then the NPV calculated will be correct.
An alternative to the direct approach is to reduce all of the payments to the smallest interval of time and use the NPV function. For example, in the cashflow used earlier the smallest interval is 6 months so we could record all of the cashflows in terms of a basic 6 month period.
Of course some of these cashflows would be zero but this doesn’t alter the calculation of the NPV.
You can see the result of doing this below. You can see that the annual payments are simply the consequence of no payment half way through the year! The cashflow can now be reduced to its NPV using the NPV function in the usual way.
Converting irregular cash flows into regular.
The only question is what interest rate is appropriate.
You might think that as the rate is 6% per annum the appropriate rate for a half yearly cashflow is 6%/2.
This isn’t the case because it doesn’t take into account the different compounding periods.
The correct interest rate to use is the effective interest rate calculated from the annual rate. As explained in Chapter 3, the effective rate gives you an interest rate that can be applied over a different compounding period while still giving the same result as the original rate. The formula relating the two is simply
where m is the new compounding period measured in units of the original period. In this case m=.5 and the formula for the effective rate is:
which should be entered into B2. Now the NPV formula can be entered into B10:
and you can see that the result is the same as the first calculation.
Whenever you alter the period of an NPV calculation you must make sure that the interest rate that you are using is the appropriate one. As the NPV calculation is based on compound interest rates should be converted using the nominal to effective formula.
Net Future Value
In the same way that we can define the Net Present Value or an irregular cashflow we can define the Net Future Value
Each of the terms in the cashflow is multiplied by (1+I)^m where m is the number of periods from the end of the loan that the payment is received.
That is if $Si is received at the end of period i then:
You can see that the NFV is just the amount that you would accumulate if you invested each of the cash sums as soon as it was received at I% for the remainder of the term of the investment.
In more mathematical terms the NFV 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 Future Value is:
where I is the effective annual rate of interest.
If I is assumed to be 8% then the NFV is $832.64 which should be compared to the total income of $800 and the NPV of $660.98. You should be able to see from this example that the NFV really is just the amount you accumulate by investing each of the cashflows as they arrive.
The NPV-NFV relationship
For each cash sum in the cashflow the NPV discounts it back to what would have to be invested to produce the sum and the NFV compounds it forward to what it would produce if invested immediately it is received.
There is a very simple and general relationship between NPV and NFV:
In other words, if you invest the NPV for n years at I% the final sum is the NFV.
To see that this is true, skip this section if you are happy to take it all on trust, all we need to do is return to the general formula of the NPV and multiply it by (1+I)^n:
If you multiply though by (1+I)n you get:
which when you cancel the (1+I)^i factor on the bottom of each term from the (1+I)^n factor gives:
which is of course the definition of the NFV given earlier.
Hence the NFV really does always equal the NPV multiplied by (1+I)^n.
In more mathematical terms what we have just shown can be written as:
and cancelling out the (1+I)^i with the (1+I)^n gives:
and you should once again recognise the right-hand side as being the definition of the NFV.
Because of this simple relationship there is no need for an NFV financial function. In all cases you can use
where n is the number of cashflows.
Notice that all of this still assumes that the cashflows occur at the end of each period. If this is not the case then slight modifications have to be made but the general principles remain unchanged and the relationship between the two values still holds.