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In the case of compound interest the value of the principal changes because of the interest added to it. This is just the simplest case of a regular alteration of the principal at the end of each compounding period. The next step is to include a regular payment or withdrawl. This is a remarkably common situation encounted in three major financial transactions - repayment loans, savings plans and annuities.
Loans and annuities
In the case of a repayment loan or a mortgage (i.e. a repayment loan secured on property) the principal is reduced by the regular payment of the interest due plus an additional amount to pay off the loan. The repayments continue until the loan is paid off. Repayment loans are commonly encountered as part of Hire Purchase agreements for all types of goods and as repayment mortgages.
Although repayment loans are the most common example of this type of transaction every loan situation has its corresponding incarnation as an investment. Seen from the other side of the table, as it were, a repayment loan is called an annuity.
In the case of an annuity a sum of money is placed on deposit and regular withdrawals composed of interest earned and capital are made. As in the case of a repayment loan the withdrawals continue until the deposit is reduced to zero.
Annuities are most commonly encountered as part of a pension provision but they are applicable in any situation where an asset has to be converted into a regular cashflow.
As well as the lender/borrower view point there is also another variation on the repayment loan/annuity situation. In both the repayment loan and annuity the flow of cash acts to reduce the principal. In a regular savings plan the flow of cash acts to increase the principal. Apart from this change in direction of the cashflow the mathematics and many other consideration are identical.
Clearly the sorts of questions that interest us about repayment loans and annuities concern the size of the regular payment needed to reduce the principal to zero in a given number of years at the interest stated. As you might guess the formulas are the same for repayment loans and for annuities.
PV and FV
So to summarise before moving on to consider each case in turn.
The basic idea is that the is a sum of money at the start of the process - the Present value PV and this is increased by the action of interest over time either increased of decreased by a regular payment to give a sum of money in the future - the Future value FV.
We use the convention that the calculation is done from your point of view and money that you pay out is negative, i.e. it decreases your wealth.
In the case of a savings plan you initially deposit the present value (which is often zero) and the regular payment each time period. As both of these are cashflows away from you, both are negative. Over time the action of interest and the regular payment increase the size of the deposit and hence its future value grows. Notice that the FV is positive because it is money you own.
Figure 1: A savings plan - the deposit gets bigger due to the payments and the action of interest.
The situation with an annuity is very similar. In this case the deposit i.e the PV starts out big and gets smaller as you withdraw money from it. Interest still acts to increase the deposit but usually the effect of the withdrawl slowly reduces the deposit to its future value which is usually taken to be zero. In this case the cashflow in the first period is -PV representing the flow out of the present value and +S representing the payment flowing in. In most cases the FV, which is also positive in that it represents wealth you still hold, usually slowly decreases - however this isn't always the case.
Figure 2: An annuity - the principle value is decreased by payments but increased by interest.
Finally a repayment loan once again involves an initial sum only in this case the principle value is paid to you so it is positive and a regular payment from you, i.e. another negative cashflow slowly reduces the debt. Notice that the future value is negative as it represents a debt i.e. a negative asset.
Figure 3: A repayment loan - the principle is slowly eroded by a cashflow.
It has to be admitted that it is difficult to keep track of the signs of the cashflows but if you apply the principle that a cashflow out is negative and a cashflow in is positive it should all work out.