# Unsecured loan formula

R

#### Robert Morgan

Hi all,

I need a formula for calculating the monthly payment on an unsecured
loan, given the APR, loan amount and term.

All the formulas I can find seem to be a variation on the following:

X = 1 + ((APR x 0.01) / PPY)
Y = N x PPY
Z = X^Y

PPM = (Amount x Z x ((APR x 0.01) / PPY)) / (Z - 1)

Where:

Amount = Principle amount of the loan
APR = Annual Percentage Rate (%)
PPY = Number of payments per year
N = Term of the loan in years
PPM = Payments per month

Taking a £5000 loan as an example, borrowed at 5.9% over 36 months,
this formula calculates the monthly payment to be £151.88.

However, if I head over to Alliance and Leicester's web-site and use
their loan calculator, it reckons the monthly payment to be £151.62
for the same criteria.

Only a few pennies I know, but I really need to get it spot on. Any
idea what's causing the innaccuracy?

Rob

A

#### Alec McKenzie

I need a formula for calculating the monthly payment on an unsecured
loan, given the APR, loan amount and term.

All the formulas I can find seem to be a variation on the following:

X = 1 + ((APR x 0.01) / PPY)
Y = N x PPY
Z = X^Y

PPM = (Amount x Z x ((APR x 0.01) / PPY)) / (Z - 1)

Where:

Amount = Principle amount of the loan
APR = Annual Percentage Rate (%)
PPY = Number of payments per year
N = Term of the loan in years
PPM = Payments per month

Taking a £5000 loan as an example, borrowed at 5.9% over 36 months,
this formula calculates the monthly payment to be £151.88.

However, if I head over to Alliance and Leicester's web-site and use
their loan calculator, it reckons the monthly payment to be £151.62
for the same criteria.

Only a few pennies I know, but I really need to get it spot on. Any
idea what's causing the innaccuracy?
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..

T

#### Tim

I need a formula for calculating the monthly payment on
an unsecured loan, given the APR, loan amount and term.

All the formulas I can find seem to be a variation on the following:

X = 1 + ((APR x 0.01) / PPY)
Y = N x PPY
Z = X^Y

PPM = (Amount x Z x ((APR x 0.01) / PPY)) / (Z - 1)

Where:

Amount = Principle amount of the loan
APR = Annual Percentage Rate (%)
PPY = Number of payments per year
N = Term of the loan in years
PPM = Payments per month

Taking a £5000 loan as an example, borrowed at 5.9% over 36
months, this formula calculates the monthly payment to be £151.88.

However, if I head over to Alliance and Leicester's
web-site and use their loan calculator, it reckons the
monthly payment to be £151.62 for the same criteria.

Only a few pennies I know, but I really need to get
it spot on. Any idea what's causing the innaccuracy?
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?

R

#### Ronald Raygun

Robert said:
Hi all,

I need a formula for calculating the monthly payment on an unsecured
loan, given the APR, loan amount and term.
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.
All the formulas I can find seem to be a variation on the following:

X = 1 + ((APR x 0.01) / PPY)
Y = N x PPY
Z = X^Y

PPM = (Amount x Z x ((APR x 0.01) / PPY)) / (Z - 1)

Where:

Amount = Principle amount of the loan
APR = Annual Percentage Rate (%)
PPY = Number of payments per year
N = Term of the loan in years
PPM = Payments per month
Mistake: You mean payment per month (singular, not plural), since
payments-per-month is 1 (assuming N=12). 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).
Taking a £5000 loan as an example, borrowed at 5.9% over 36 months,
this formula calculates the monthly payment to be £151.88.
Agreed.

However, if I head over to Alliance and Leicester's web-site and use
their loan calculator, it reckons the monthly payment to be £151.62
for the same criteria.

Only a few pennies I know, but I really need to get it spot on. Any
idea what's causing the innaccuracy?
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?

R

#### Robert Morgan

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

R

#### Ronald Raygun

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

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
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.