# How do I calculate the monthly PCP repayment figure for a car purchase in the UK?

How would I calculate the PCP monthly repayment figure from the following information please?

Car offer price: £30,000
APR: 3.9%
Number of repayments: 48
Balloon payment: £15,000

I've found similar PCP questions on here but most seem to require the repayment figure to calculate the loan total. e.g.

``````L = (B + (M ((1 + R)^N - 1)/R))/(1 + R)^N

where

L = present value of loan
M = periodic repayment
R = periodic rate
B = balloon payment
N = number of periods
``````

I suspect the formula just needs reworking to have M = something, but my maths is no where near good enough to work this out, so hoping someone would be able to help please? Thanks.

A balloon loan - where the balloon payment is paid at the same time as the final periodic payment - can be written in terms of net present values (NPVs) like so ``````∴ L = (B + (M ((1 + R)^N - 1)/R))/(1 + R)^N

∴ M = (R (B - L (1 + R)^N))/(1 - (1 + R)^N)
``````

where

``````L = present value of loan
M = periodic repayment
R = periodic rate
B = balloon payment
N = number of periods
``````

``````L = £30000
B = £15000
R = (1 + 3.9/100)^(1/12) - 1 = 0.00319331 per month
N = 48 months

∴ M = £385.46
``````

So on the 48th month the final periodic payment of £385.46 is made along with the balloon payment of £15000.

There can be variations on this. Some would expect the payment on the 48th month to be £15000 and no more, (in which case there are 47 periodic payments). ``````∴ L = ((1 + R)^-N (B R + M (-1 - R + (1 + R)^N)))/R

∴ M = (B R - L R (1 + R)^N)/(1 + R - (1 + R)^N)

L = £30000
B = £15000
R = (1 + 3.9/100)^(1/12) - 1 = 0.00319331 per month
N = 48 months

∴ M = £393.05
``````

47 monthly payments of £393.05 with a final payment of £15000 on the 48th month.

Another variation is to make 48 monthly payments and the balloon on the 49th month, so this uses the previous formula but with N changed to 49.

``````M = (B R - L R (1 + R)^N)/(1 + R - (1 + R)^N)

L = £30000
B = £15000
R = (1 + 3.9/100)^(1/12) - 1 = 0.00319331 per month
N = 49 months

∴ M = £386.38
``````

You will have to find out from the lender exactly when the balloon payment is expected and whether it is to be added to one of the regular payments.

• Hi Chris, I saw some of your excellent replies on other posts so thank you very much for your help here. Something doesn't quite seem right though as the repayment value of £1285 is much higher than I expected. Calculating this comes out as: 48 * 1285.08 = 61,683.84 which is much higher than any of the initial figures. Also, I suspect L is a number that needs to be calculated first based on the car offer price of £30,000, so the loan amount will be higher based on the 3.9% APR? My apologies for not being clearer with the question. Jun 21 '19 at 9:46
• @allemtura You are quite right. I forgot to convert the interest to a monthly rate. Note that in the UK APR is quoted as an effective annual rate, so it is not simply divided by 12 to obtain the monthly rate, as would be the case for a nominal annual rate (as used in the US). Jun 21 '19 at 12:06
• As you can see `R` compounds to give the expected APR: `(1 + 0.00319331)^12 - 1 = 3.9%` Jun 21 '19 at 12:26
• Thanks @chris-degnen! I have some real quotes to test this against and can confirm that the first formula works. e.g. L=26,218, R=0.004788517(5.9%), B=15,057.50, N=43 (quoted as 42 repayments, but your suggested variation to make it include an extra balloon payment works). Equals 367.6256. Quoted repayment is 367.69 (finance charges 4282.48 if it helps). I have another set of figures where it varies a bit more but nothing too bad - L=38911.80, R=0.003193314(3.9%), B=18026.25, N=49. Equals: 528.67877. Quoted: 527.53 (finance charges 4435.89) Jun 21 '19 at 18:16
• Just to confirm, it was this second formula that worked for me: M = (B * R - L * R * (1 + R)^N)/(1 + R - (1 + R)^N), along with the extra repayment number. Jun 21 '19 at 20:14