# Formula for calculating the monthly payment for a Personal Contract Purchase

How can I calculate the monthly payment for a PCP (Personal Contract Purchase) when given the variables below?

1. 'Amount to be financed' - finance amount
2. APR (annual percentage rate - %)
3. Length of finance (duration)
4. Final Payment (GMFV / balloon payment)

Need to get a formula to calculate this.

• Is there any way to calculate from APR? @ChrisDegnen – Agent 47 Oct 12 '20 at 8:33
• Could you please advise how to calculate monthly payment if balloon is not paid in monthly? carwow.co.uk/pcp-calculator I'm using this site to calculate. @ChrisDegnen – Agent 47 Oct 12 '20 at 8:42
• M = (R (B - L (1 + R)^N))/(1 - (1 + R)^N) This one, right? @ChrisDegnen Also, Here R=APR/12, Right? – Agent 47 Oct 12 '20 at 8:53
• For United Kingdom? – Agent 47 Oct 12 '20 at 9:00
• I've edited my question and added an image reference(Contains the values). @ChrisDegnen – Agent 47 Oct 12 '20 at 9:07

``````With

s = principal
n = no. periods
m = periodic payment
r = periodic rate
b = balloon
``````

where the balloon is paid at the same time as the final payment in month n

$s=\sum_{k=1}^{n}\frac{m}{(1+r)^{k}}+\frac{b}{(1+r)^{n}}$

The present value of the principal is equated to the net present values of the payments; then the summation is converted to a closed-form expression by induction.

``````∴ s = (m - m (1 + r)^-n)/r + b/(1 + r)^n

∴ m = (r ((1 + r)^n s - b))/((1 + r)^n - 1)
``````

Assuming the dealer contribution is deducted from the initial amount.

``````s = 20000 - 2000 - 1000
b = 1000
r = (1 + 7/100)^(1/12) - 1
n = 36

∴ m = (r ((1 + r)^n s - b))/((1 + r)^n - 1) = 498.12
``````

or calculating with APR as a nominal rate compounded monthly

``````s = 20000 - 2000 - 1000
b = 1000
r = 7/100/12
n = 36

∴ m = (r ((1 + r)^n s - b))/((1 + r)^n - 1) = 499.87
``````

Looks like the website is using nominal rates. However, UK uses effective rates.

• I think the following method is correct for my case. s = 20000 - 2000 - 1000 b = 1000 r = (1 + 7/100)^(1/12) - 1 n = 36 ∴ m = (r ((1 + r)^n s - b))/((1 + r)^n - 1) = 498.12 – Agent 47 Oct 12 '20 at 9:36
• r = (1 + 7/100)^(1/12) - 1 – Agent 47 Oct 12 '20 at 9:36
• That is correct for the UK where APR is advertised as the effective rate. – Chris Degnen Oct 12 '20 at 9:37