APR and Stuff - Confused

Scripsit "MadCrazyNewbie"

No.

Imagine that you get two loan offers with the same payout but different repayment schedules:

a) pay back £1200 each year for the next 3 years b) pay back £100 each month for the next 36 months

Each of these leads to paying back £3600, but (a) is more favourable because get to wait longer before paying back some of the £100's.

The purpose of APR is to take this kind of difference into account so that you can compare offers even when the schedules are so different that it is not immediate which one is objectively most attractive. In order to do this fairly for every kind of strange repayment option, APR needs to become a rather abstract number. It has no direct relation to the *sum* of the repayments you make.

Here is my attempt at explaining how APR is computed:

You borrow some money, which are typically all paid to you at once. In return you agree to give the lender certain amounts of money at certain times in the future. We do not care what the lender calls divides each of the future payments into "arrangement fees", "interest", "balance payment", "fee for your mandatory annual account statement" or whatever. To us they are all just "payments".

Now imagine that instead of spending the money you buy, in deposit them on an "ideal" bank account. It is "ideal" because no real-world bank offers such a thing, but we can imagine it nevertheless. It has the following properties:

1) You pay no fees at all, neither to have the account in the first place nor to deposit or withdraw money. 2) The balance is computed to the accuracy of a ridiculously small fraction of a penny. 3) Interest is paid nightly: Each night the bank increases the balance on the account by some small fixed fraction of the balance before the increase.

Now you have some money in your ideal bank account. Each time you have to make a payment to the lender, you withdraw it from the ideal account on exactly the day you have to pay. You do nothing else with the ideal bank account.

After some time the balance in the ideal account will obviously depend on what the small nightly interest fraction is. If it is too close to zero, you will run out of money before all your repayments are done, but if it is a bit higher you'll soon find yourself being fabulously rich. Somewhere in between there is a certain nightly interest fraction that is "just right" - i.e., when you have to make the very last payment it just happen to coincide with the balance on your ideal account, after all the various withdrawals and interest computations.

As far as I know, there is no easy way to calculate the "just right" rate directly - but instead we can let a computer try all of the computations with each a lot of possible rates and then select the one that comes closest. That gives a very good approximation. (And with a bit of higher math it is not very burdensome to *check* for oneself that the computer's answer is the right one, even if it would be hard to *find* it without a computer).

Now the fundamental assumption of APR is that the "just right" interest rate for the ideal account is a good measure of how expensive the loan offer is. If you have two offers to borrow the same amount of money, but one would require you to get a better ideal account in order to make the repayments, then all other things being equal you'd prefer the other one.

We could directly compare offers by the "nightly fractional rate" that I have alluded to above, but such nightly rates tend to be very small

-- on the order of a ten-thousandth of the balance -- so for ease of use we convert the nightly rate to an *annual* rate by computing how much total interest we would get if we paid £1 into the ideal account and left it there for a year. If we measure this interest in pence we get the rate as a *percentage*, and there is your _A_nnual _P_ercentage _R_ate.

When the interest rate is variable, banks compute APRs *as if* the rate were to stay forever at its current level. This makes the APR doubly fictional, but it is the best that one can do without knowing the future interest levels.

If the loan was repaid in a single installment one year later, then yes.

Yes.

Every month, they charge 2.1% interest on the outstanding balance. There must be some very high arrangement fees or similar to push the APR up to

43.9%.

More likely the 2.1% is an introductory rate valid for only a short discount period, after which it rises, perhaps to 4% per month.

Bitstring , from the wonderful person Jonathan Bryce said

Alternatively they charge 2.1% of the original sum every month, regardless of the fact you have paid some of it off (so called 'flat rate' interest, iirc). This is quite likely to work out at 43.9% APR (although I must confess I didn't do the maths, but it's in that ballpark).

That's why the government made people quote APR .. 2.1% per month sounds quite rational, doesn't it.