Unsecured loan formula

You won't get a calculation based on an APR which is accurate to two significant figures to to give you a result accurate to five significant figures. Your two results agree to three significant figures, which is as good as you can expect..

"Robert Morgan" wrote

Looks like A&L may be using a monthly rate of 0.482% (equates to an APR of

5.94% if there are no other charges), whereas you are effectively using an APR of about 6.06% (monthly rate a twelth of 5.9%).

Does that help?

This is incorrect. You must replace APR in what follows with NAR, the nominal annual rate. The periodic rate (what you'd call X-1) is the nominal annual rate divided by the number of periods per year, and in the absence of complicating factors such as fees, the APR is equal to (X^N-1)*100 rounded to two places.

Mistake: You mean payment per month (singular, not plural), since payments-per-month is 1 (assuming N). If you don't want to assume the period is the month, and you do seem to be trying to keep it general by using N and not 12, you shouldn't call it PPM, but PPP (payment per period).

Agreed.

Applying the formula in reverse, I calculate that £151.62 corresponds to a NAR of 5.784%, which in turn corresponds to an APR of 5.94%.

Does that help?

Hi all,

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?

Many thanks,

Rob

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.

Thanks Ronald - crystal clear.