S ¼ a
1 À q
n
1 À q
(30)
These expressions are the basis for the mortgage formula and many other economic
calculations.
Appendix 10: Loan Repayments (Mortgage Formula)
Let us assume that we have borrowed a certain amount of money, C, that we plan to
return over a total of n periods such that at the end of each period, we repay the same
amount, A. We further assume that the rate of interest remains constant and is r for
each period.
Time Zero
Take the loan out. We owe C.
End of the First Period
We owe C(1 + r). We repay an amount A. Therefore, we now owe C(1 + r) – A.
End of the Second Period
We owe C 1 þ r
ð
ÞÀA
½
Š1 þ r
ð
Þ¼C 1 þ r
ð
Þ
2 À A 1 þ r
ð
Þ . Again we repay an
amount A. Therefore, we now owe C 1 þ r
ð
Þ
2 À A 1 þ r
ð
ÞÀA.
End of the Third Period
After repaying the amount A, we owe C 1 þ r
ð
Þ
2 À A 1 þ r
ð
ÞÀA
h
i
1 þ r
ð
ÞÀA, or
C 1 þ r
ð
Þ
3 À A 1 þ r
ð
Þ
2 þ 1 þ r
ð
Þþ1
h
i
.
Keep repeating the reasoning for all other intermediate periods.
End of the nth Period
After
making
the
last
payment,
A,
we
owe
C 1 þ r
ð
Þ
n À A 1 þ r
ð
Þ
nÀ1
ð
Þ þ 1 þ r
ð
Þ
nÀ2
ð
Þ þ Á Á Á þ 1 þ r
ð
Þþ1
h
i
. But since we are
repaying the loan in n periods, at this time the balance must be zero!
772
D.S.J. Jones and P.R. Pujado ´