From: Ronald Raygun 
Subject: Re: Bank base rate now 4.5%
Newsgroups: uk.finance
Date: Fri, 18 Jun 2004 15:57:02 GMT

Andy Pandy wrote:

> "Doug Ramage"  wrote
>>
>> An increase from 6% to 7.5% p.a. with 25 years remaining is about £189
>> per month (from £1288 to £1477) according to my Excel spreadsheet and
>> Loan Calculator. :)
> 
> I worked it out for a 4% to 5.5% increase.
> 
> Surprisingly it varies a lot depending on what the original rate was, and
> even more surprising, the lower the original rate the less impact any rate
> rise has.

This should not really be surprising, because if you plot the payment
as a function of the rate, it is (of course) an increasing function
(i.e. it has a positive first derivative), but also the rate of increase
increases, that is to say that the graph is not a straight line but it
curves upwards (i.e. it has a positive second derivative too).

Now, if it *were* a straight line, then if you superimposed the graph
for payment(rate) and payment(rate+0.005), the vertical separation
between the two payment graphs would be equal for all rate values.
But as both graphs curve upwards, you must expect the vertical gap
to increase with increasing rate.

> I tried coming up with a nice neat formula but couldn't really do any
> better than simply using the standard repayment formula LI/(1-(1+I)^-N))
> for the 2 different values of I.

It doesn't really get any simpler, I'm afraid.

One trick is to plug this formula into a plotting tool, keeping N as
a constant (e.g. 25) and forgetting L (treat it as 1), so put in a
function f(x) = x/(1-(1+x)^-25), and tell it to plot not only f(x)
between (say) x = 0.04 and x = 0.15, but also to plot f(x+0.005),
and, more to the point, f(x+0.005)-f(x).  You'll find that this
last expression increases asymptotically with increasing x to a ceiling
of 0.005, because, well, it should be obvious:  If the interest rate
increases more and more, to silly rates, such that the capital repayment
element of each payment is insignificant, then the difference in payments
is exactly the difference in the interest rates (per unit of capital).

Of course, if you're feeling ambitious, you could try differentiating
f(x) with respect to x, twice, looking at the answer, and reasoning why
it must be positive.