Re: The actual interest rate on a loan

Rich wrote:

What it is, is that I'm wanting a loan for a business I want to start.
I have a (dormant) business account with my bank. I have
no security by way of mortage etc, but manage to pay off credit card
obligations. Anyway, despite my relatively weak position, my bank
would give me a loan, but it would bepersonal loan wrapped up as a
business loan.

Okay, a loan of £7.500 was chosen over a 66 month term and the following
figures were returned:

Condition: No PPI.
Monthly Repayment: £160.16
Premium: 0.00
Amount Repayable: £10.570.56
APR (%): 13.95

I can pay back the loan early.

I was hoping that if the business was successful I could arrange to pay
off the loan after one year. And I was wanting to know what the loan
interst rate would be if I paid off the loan in a year.

But you already know that. It's 13.95% per year.
That's what APR means, that *is* the *effective* interest rate.

I did a rough calculation, I worked out that if there was no interest on
£7,500 monthly payment would be £113.63. Since I'm paying £160.16 I
figured interest paid on every payment was £46.53. £46.53 * 12 =
£558.36. So, after one year the interest rate would have been £558.36 /
£7,500 * 100 = 7.44%.

Okay, that wrong because the calculation is too simple, because the
amount loaned goes down in time. I think what happens is that as
time goes on the monthly payments remain the same, but an
increasing portion is paid as interest and a decreasing amount
is paying off the loan.

It's the other way round. The interest proportion decreases with
time, and the "paying off" (capital or principal) proportion increases.

The chart I have says APR 13.95%. Anyway, what is inportant is how much
money will I have been considered to have loaned after 1 year. It's less
than £7.500. It's going to be something like (£7500 (amount of money
owed at start) + £6504.06 (amount of money owed at end of year) / 2 =
£7002.03.

Yes, that's approximately right. It's not exactly right because the
amount owed doesn't go down by exactly the same amount each month (as
a result of the capital payoff rate accelerating). The actual average
debt during the first 12 months is nearer £7050.

Now if there was no interest at all I would pay back £995.64
according to your calculation. I've paid out £160.16 * 12 = £1921.92.
So, due to interest, Ive paid out £926.28.

So, it looks to me that the effective loan is £7002.03 and I've paid out
£926.28 for the privaledge, which is an effective interest rate of
13.22%.

That's approximately right. More exactly it's about £926.26/£7054.42,
which works out at 13.13%. Divide this by 12 to get the monthly rate
of 1.0942%, of which the compound annual equivalent is 13.95%. You get
this by raising 1.01942 to the power 12 and getting 1.13950.

Not sure how you got your £995.64. Probably it was debt remaining after
N payments = original debt * f(66-N)/f(66), where f(x) = 1 -
1.010942^-x.

IOW f (66 - N)
------------
f (66)

where f(x) = 1 - 1.010942^-x.

Not sure how to use this formula.

Basically, if m is the monthly interest factor (m=1.010942), then the
amount A(k) owing after k out of N payments on a loan of amount A have
been made, is:

A(k) = A * (1 - m^(k-N)) / (1 - m^-N)

[This is a consequence of A(k) being equal to A(k-1)*m-P, where
P is the monthly payment, itself equal to A*(m-1)/(1-m^-N).]

I tried to "simplify" this a bit above by rewriting "1-m^-x" as "f(x)".

Notice how A(0)=A and A(n)=0, as you'd expect.

So for k=12 this is £7500 * (1 - 1.010942^-54) / (1 - 1.010942^-66)
which is £6504.36. Hence if after one year you still owe that amount,
it means that during that year you have paid off £7500-£6504.36,
which is £995.64.

This rate I'm sure is quite high in the present climate. I think folks
are going to say, there are better deals around. But are there. I must
Google I think.

For an unsecured personal loan, 13.95% is not really too bad, but you
may well find a better deal. Remember, the APR is the basis on which
you want to compare whatever deals you find.

.

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