Robert said:

Thanks for your help, think we're getting to the bottom of this.

Ronald, my maths skills are rusty at best, please could you explain how

you reverse the formula to get the NAR for a given PPP?

You can't do it algebraically. Didn't I explain this the other day?

There are two options:

You can do it graphically, if you have a suitable tool (I use gnuplot),

by plotting payment as a function of rate, treating all the other inputs

as constants, and reading the graph backwards, i.e. imagining a horizontal

line from the required output value (the payment) to the curve, then

vertically from there to the other axis where you can read off the input

value (rate) which would have produced the right output when fed into the

function you've plotted. If your tool lets you zoom in, you can get the

answer to suitable precision.

You can also do it numerically by an iterative process. This is in

effect the equivalent of graphic zooming in. You guess the rate and

work out the payment for that rate. This payment will be too high or

too low, and accordingly you revise your guess, refining it until you

get as near to the right answer as you like.

You can get there quite quickly (i.e. in fairly few guesses) by starting

with two very coarse guesses one of which is definitely too high and the

other definitely too low.

Call these the upper and lower bounds. In your example, where the

answer is in the vicinity of 5.8% you might start with the bounds

at 4% and 8%.

Let the next guess be halfway between the bounds (so the first guess

would be 6%), and depending on whether its answer is too high or low,

replace either the upper or the lower bound, as appropriate, with the

most recent guess before repeating the process. Here, the bounds

after the first step would become 4% and 6%. Next time 5% and 6%.

And so on. At each step the "ballpark" decreases in size by half.

The technical term for this is "binary search", sometimes called "How

to catch a lion in Africa". You simply erect a fence across the middle

of Africa. Your lion will be either in one half or the other. Pick

the half he's in, and erect a fence across the middle of that. It will

be either in one half or the other. Keep this up until the fenced-off

area is lion-sized, and you'll have caught your lion. It is left as an

exercise to the reader to work out how many fences, and how many miles

of fencing, are needed.