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Repayment loans are the last of the three guises in which annuities appear (see Chapters Four and Five for the other two) and by now you should be growing accustomed to the type of reasoning involved in calculating the effects of regular cashflows.
A repayment loan is particularly easy to deal with because it is just an annuity seen from a different point of view. In the case of a repayment loan a sum of money - the principal or Present Value PV is borrowed at the start of the loan. At the end of each time period a regular sum S is paid back. This regular sum has to be large enough to pay back the interest due on the loan and some of the principal. As long as this is the case the proportion of interest to principal repaid each period changes so that more and more of the principal is repaid. Eventually the entire principal is repaid and the debt is cleared.
A repayment loan may be just an annuity seen from the other side of the table but the sort of questions you need to answer are subtly different. Partly this is due to a slight difference in psychology and the nature of the decisions you have to make - you are trying to minimise the interest rate and term rather than maximise them - but it is also the way that institutions and the law deal different with loans than annuities.
In a repayment loan what interests us most is the size of the repayment given the loan, interest and term but there are still lots of reasons to need to calculate any of the four quantities - repayment, interest rate, term and loan amount - given the remaining three.
The basic mathematics of repayment loans is the same as for an annuity.
At the start of the loan the amount PV is borrowed - this is a cashflow in and so positive, however the balance in the account at this point is negative to signify a debt.
At the end of the first period the debt has increased to:
but the payment of S has also reduced it. That is, at the end of the first period the balance stands at:
at the end of the second period the balance stands at:
[PV*(1+I)-S]*(1+I) - S =
and so on.
In general after n time periods the balance stands at
which is of course the same as the situation encountered in the case of the ordinary annuity.
Thus all of the equations and even the spreadsheets that we have constructed for the ordinary annuity apply to the repayment loan.
The only difference is that now the Present Value (PV) is the amount loaned and the payments are to repay the debt. After all one man’s annuity investment is another's repayment loan.
In more practical terms this means that in terms of the cash flows the present value is positive because it is what the borrower receives, i.e. cash in, and the regular payments are negative, i.e. cash out. One final possible confusion is that the financial functions return a negative value of the PV or FV because it represents a debt.
Cash flow in a loan