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Savings plans
The workings of a repayment loan or annuity are actually easier to understand if we first consider the closely related situation of a regular saving plan.
In this case a regular sum S (a payment denoted PMT in Excel's terminolgy) is deposited each period and the total sum accumulated attracts interest at I% (Rate in Excel) at the end of each period. It is a convention that the money is deposited at the end of each period and money that you pay out is negative. Thus after the first period the balance is simply the first regular payment:
= S
After the second period the amount is the first regular payment plus the interest it has earned:
=S*(1+I) + S
After the third period the amount is:
=[S*(1+I)+S]*(1+I) + S
and so on. You can see that as each period passes the amount already on deposit is increased by being multiplied by (1+I) and has S added to it.
To work out the Future Value of the savings plan after n periods you could simply construct a spreadsheet that calculates the repeated multiplication by (1+I) and adding S but there is a simpler way of working out the same thing.
After a little math it can be shown that the Future Value is given by:
(1+I)^{n}  1 FV= S*  I
This is such a complicated looking formula that nearly all spreadsheets provide a single financial function, usually called FV, to calculate the same result.
This is just the function we met in Chapter Two in the calculation of compound interest only now we can specify an initial deposit  the Present Value PV  and a regular payment S.
FV function
The Excel/Open Office function:
=FV(I,nper,S,PV)
will calculate the Future Value FV of a regular saving of S, at I% for nper periods with a starting blance of PV which is often zero. If you don't specify the PV then it is assumed to be zero by default. Notice that as with nearly all financial functions it is best to regard the sums specified in the function, i.e. S and PV, as flowing from the depositor. So a deposit of $100 is $100 and a payment of $10 per month into the account is $10 per month. However the result of the function is the status of the account  postive when in credit and negative when in debt.
Figure 4: The cashflows are PV and S.
As with all financial calculations the interest rate has to be expressed in terms of the regular payment period and for a savings plan the money paid into the acount is negative.
For example, if you start a savings plan that calls for $10 per month at 6% per annum paid monthly then after 5 years the Future Value will be
=10*((1+0.06/12)^(5*12)1)/(0.06/12) =697.70
This is a horrible looking formulae and one that you are very likely to make mistakes in when trying to enter it into a spreadsheet. How much simpler the equivalent financial functions appears:
=FV(0.06/12,5*12,10)
and this of course gives exactly the same result, $697.70. Notice that there is a minus sign in front of the payment  it is a convention that cash paid in (e.g. S) is negative while cash paid out (e.g. FV) is positive.
If you make an initial deposit of PV at the start of the savings plan to get it going then the formula have to be altered to:
(1+I)^{n}  1 FV= S*  + PV*(1+I)^{n} I
That is, the compounded interest earned by the initial deposit has to be added to the Future Value.
In Excel/Open Office you simply include the initial deposit as the optional Present Value (PV) parameter at the end of the previous formula.
=FV(I,nper,S,PV)
For example, if deposit $100 and save $10 per month at 6% per annum paid monthly then after 5 years the Future Value will be:
=FV(0.06/12,5*12,10,100)
which works out to $832.59 which is more than the previous result by $134.89  the amount of interest earned by the initial $100 at 6% for 5 years.
<ASIN:0814472966>
