Re: Calculating mortgage payments

Joe Kelleher wrote:

I'm looking to take out a mortgage and I want to draw up a spread*** to
compare different mortgages and test different scenarios. In particular, I
want to be able to calculate an 'effective' interest rate that includes
repayment of any initial or redemption fees averaged over the life of the
mortgage, which looks like being only a few years before I switch. Can
anyone please explain or point me to an exact definition of the interest
rate that is quoted in a mortgage ad?

Sure

Is it the total proportion of the
loan value that would be paid in a year, if one were to continuously repay
only the interest on the loan so as to keep the debt exactly constant?

Yes and No

There are two methods in wide use. Both assume that you actually pay
a fixed amount per month.

One method assumes (incorrectly) that if, say, the amount outstanding at
the beginning of the year is £100k and the annual interest rate is 6%,
that you are borrowing the whole £100k for the whole year, and need
therefore to be charged £6k of interest for that year. Then also
calculate how much extra you should pay to reduce the balance in
order to pay it off over whatever term is agreed, and they divide this
amount by 12 and you pay that each month. In other words, the monthly
payments are a twelfth of £100k*0.06 / (1 - 1.06^-20) if the loan is
to be paid off over 20 years.

Another method assumes (correctly) that you are borrowing the whole
£100k only for a month, and are then maing an interest payment of £500
plus an amount to reduce the balance, and are then borrowing slightly
less than £100k for month 2, etc. The monthly interest rate is taken
to be a twelfth of the quoted annual rate. In other words, the monthly
payments are £100k*0.005 / (1 - 1.005^-240) if the loan is to be paid
off over the same period, i.e. 240 months.

I would say this second method is more popular these days. It means
you pay slightly less each month than with the first.

If repayments were made only once a month,

Only? It's extremely unusual to make repayments more often than monthly.

the interest paid would therefore
be slightly higher. Or is it something else? Also, I'd be interested to
learn about how the quoted APR figures are calculated, and whether it
saves me doing some maths.

The APR is as a rule a completely useless figure for comparison,
especially in the case of deals which start with a discount period,
because they all tend to assume a 25-year term, which in the case of
people switching lenders part way through an existing loan term
is inappropriate.

What they do is come up with a schedule of payments for the whole
300 months, add up all the payments, including any fees which apply
at the start and at the end or at any other time, having first scaled
them back, taking into account the time at which each payment is made,
using the *same* "discount rate" (a technical term for applying the
interest rate backwards in time to see what the amount would have been
worth at month 0, i.e. the day the loan was advanced) for each payment.
They use a guessed value for this discount rate, and re-run the
computation until this sum matches the amount lent. When this has been
achieved, the discount rate is the APR (or rather its inverse).

I trust this is now clear as mud. :-)

In the simplified case where no fees apply at all ever, and all monthly
payments are strictly equal, and the interest rate never changes, the APR
can be calculated by compounding the monthly rate. In other words, if
the monthly interest rate is 0.5%, the APR will be 1.005^12-1 or 6.168%.

In practice, life is never that simple. :-(

.

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