Car finance, APR rates and per week in adverts; help understanding them

Question

Looking at a car advert [for a car I was considering buying in April 2008] but in the end I didn't - this was what the advert stated:

2005 TOYOTA COROLLA 1.4 VVTi 5 door hatchback £7195 From £38 per week

The advert stated that it was 16.1% APR typical, a 60 month payment, 260 weekly payments in the small print.

According to the Bank of England the interest rate was 5.00% at the time

Would this be the correct type of calculation:

Total price x interest rate ÷ number of weeks

The advert stated APR was 16.1% APR typical.

Another advert from the same issue of the magazine, and an example:

2004 HONDA CIVIC 1.6 i-VTEC SE 5 door Hatchback £6,999 £113.15 per month

APR 9.9% [as quoted in advert], 58 monthly payments

There was also this in an advert from another dealer:

2003 BMW 325i £7477 TYPICAL APR 12.9% 60 monthly payments £167.05

What are the calculations that i would I need to do to work out how the advert comes to a £ per week figure or the monthly payment figures?

This is not homework help - it is simply a request to try and understand this complex situation.

I would appreciate it if anyone could help me and give me a basic understanding of the calculations for this as a sort of ready reckoner.

Answer 1

Easier to copy paste than type this out. Credit: www.financeformulas.net

Note that the present value would be the initial loan amount, which is likely the sale price you noted minus a down payment.

The loan payment formula is used to calculate the payments on a loan. The formula used to calculate loan payments is exactly the same as the formula used to calculate payments on an ordinary annuity. A loan, by definition, is an annuity, in that it consists of a series of future periodic payments.

The PV, or present value, portion of the loan payment formula uses the original loan amount. The original loan amount is essentially the present value of the future payments on the loan, much like the present value of an annuity.

It is important to keep the rate per period and number of periods consistent with one another in the formula. If the loan payments are made monthly, then the rate per period needs to be adjusted to the monthly rate and the number of periods would be the number of months on the loan. If payments are quarterly, the terms of the loan payment formula would be adjusted accordingly.

I like to let loan calculators do the heavy lifting for me. This particular calculator lets you choose a weekly pay back scheme.

http://www.calculator.net/loan-calculator.html

Answer 2

Taking the last case first, this works out exactly.

(Note the Bank of England interest rate has nothing to do with the calculation.)

The standard loan formula for an ordinary annuity can be used (as described by BobbyScon), but the periodic interest rate has to be calculated from an effective APR, not a nominal rate. For details, see APR in the EU and UK, where the definition is only valid for effective APR, as shown below.

2003 BMW 325i £7477 TYPICAL APR 12.9% 60 monthly payments £167.05

effective annual interest rate, i = 12.9% = 0.129
∴ monthly interest rate, r = (1 + i)^(1/12) - 1 = 0.01016

present value of loan, pv = £7477
number of months, n = 60
∴ monthly payment, p = r pv/(1 - (1 + r)^-n) = £167.05

How does this work? See the section Calculating the Present Value of an Ordinary Annuity. The payment formula is derived from the sum of the payments, each discounted to present value. I.e.

pv = p/(1 + r)^1 + p/(1 + r)^2 + ... + p/(1 + r)^59 + p/(1 + r)^60

∴ pv = Σ p (1 + r)^-k for k = 1 to n

∴ by induction, pv = (p - p (1 + r)^-n)/r

∴ p = r pv/(1 - (1 + r)^-n)

The example relates to the EU APR definition like so.

A and R = 0

∴ S = £7477 = Σ £167.05 (1 + 12.9/100)^-(k/12) for k = 1 to 60

Next, the second case doesn't make much sense (unless there is a downpayment).

2004 HONDA CIVIC 1.6 i-VTEC SE 5 door Hatchback £6,999 £113.15 per month

"At APR 9.9% [as quoted in advert], 58 monthly payments"

effective annual interest rate, i = 9.9% = 0.099
∴ monthly interest rate, r = (1 + i)^(1/12) - 1 = 0.0079

monthly payment = £113.15
number of months, n = 58
∴ present value of loan, pv = (p - p (1 + r)^-n)/r = £5248.75

58 monthly payments at 9.9% only amount to £5248.75 which is £1750.25 less than the price of the car.

Finally, the first case is approximate.

2005 TOYOTA COROLLA 1.4 VVTi 5 door hatchback £7195 From £38 per week

"16.1% APR typical, a 60 month payment, 260 weekly payments"

effective annual interest rate, i = 16.1% = 0.161
∴ weekly interest rate, r = (1 + i)^(1/52) - 1 = 0.002875

present value of loan, pv = £7195
number of weeks, n = 260
∴ weekly payment, p = r pv/(1 - (1 + r)^-n) = £39.33

A weekly payment of £38 would imply an APR of 14.3%.