In a recent post Professor Gowers writes on the new proposal for the A-level exam on the “Use of Mathematics”. In it he discusses the following discrete problem which he approaches in a way that uses continuous techniques.
” Suppose for simplicity that the interest rate for an interest-only mortgage would be 5% and that this rate never changes. If I take out a repayment mortgage of £50,000 and pay £500 a month, then roughly how long will it take me to pay off the mortgage? “
Here are two comments of mine concerning this problem.
1. “For your discrete problem about repayment mortgages I present a proposal for a direct solution. This solution, even if correct, is of course far less instructive than (*) your reasoning!
In general we have to find the value of constant payments given the principal during monthly periods. The value of in the period is equivalent to the present value monetary units, where is the interest rate in each capitalization period. Summing in , from 1 to , we get the sum
Now we have to sum a geometric progression with ratio (**) and first term
For the given problem, the payments will be made during months, with and . Thus
Solving for , we get
months ( years)
and, as you proved
2. “Please let me only add one further interpretation of mine: the continuous effective interest rate can be derived from the 5% nominal interest rate, compounded (***) times per year as follows:
approximates your formula is an approximation for (****)), and in the general case of an annual nominal interest rate as
From a point of view of a purely mathematical problem the model you are discussing
in the context of the “use-of-maths A’ level “ is much more interesting and worthy.”
Corrections to the original comments:
(*) “that” in original
(**) “rate” in original
(***) “compound” in original
(****) [Edited on July 28, 2009,. In original: "which approximates your formula for "]
[Edited on July 14, 2009: First paragraph corrected and title changed]
[Edited on July 15, 2009: Last paragraph. Remark: in a comment Professor Gowers wrote "I wasn’t suggesting that the mortgage question would be a suitable one for a use-of-maths A’level" ]