# The actual interest rate on a loan

R

#### Rich

I can borrow £7,500 for 66 months (5.5 years)

Monthly permium is £160.16.

If I pay my premiums for a year then I'd pay £1921.92, of which £558.36
would be interest, £1363.56 paying off the loan.

So, if I then were able to pay off the loan after a year, the actual
percentage interest on my loan is £558.36 / £7,500 = 7.44 %.

Is this all correct? I cannot see why not, but just checking.

Would typically, a bank make any adjustments when I came to pay off my
loan after a year?

If I paid back the loan in 5.5 years, the interest would be £3,070.56.
That's an affective interst on my loan of 40.94%.

Quite some difference.

N

#### Norman Wells

Rich said:
I can borrow £7,500 for 66 months (5.5 years)

Monthly permium is £160.16.

If I pay my premiums for a year then I'd pay £1921.92, of which
£558.36 would be interest, £1363.56 paying off the loan.

So, if I then were able to pay off the loan after a year, the actual
percentage interest on my loan is £558.36 / £7,500 = 7.44 %.

Is this all correct? I cannot see why not, but just checking.

Would typically, a bank make any adjustments when I came to pay off my
loan after a year?

If I paid back the loan in 5.5 years, the interest would be £3,070.56.
That's an affective interst on my loan of 40.94%.

Quite some difference.
There are more accurate ways of looking at it and doing the maths, but you
seem to be comparing apples with oranges anyway. The rate of 7.44% you
calculated is not directly comparable with 40.94% because, in the first case
you would have borrowed the money for only 1 year, whereas in the second
case you'd have been borrowing it for 5.5 years and you'd surely expect to
pay about 5.5 times as much interest. 5.5 x 7.44 comes to 40.92%, which is
very similar to the overall interest you've calculated.

In fact, the calculation is somewhat wrong because you're actually paying
off the loan in part every month, but the same principle applies.

I do not know if your loan agreement allows you to pay it off early, but
obviously if you can pay it off early, you're borrowing the money for less
time and would expect to pay proportionately less in interest. So, the
earlier you can pay it off the better because 7.44% is far higher than you
can hope to get as a return on your savings.

R

#### Rich

Norman Wells said:
There are more accurate ways of looking at it and doing the maths, but
you seem to be comparing apples with oranges anyway. The rate of
7.44% you calculated is not directly comparable with 40.94% because,
in the first case you would have borrowed the money for only 1 year,
whereas in the second case you'd have been borrowing it for 5.5 years
and you'd surely expect to pay about 5.5 times as much interest. 5.5
x 7.44 comes to 40.92%, which is very similar to the overall interest
you've calculated.

In fact, the calculation is somewhat wrong because you're actually
paying off the loan in part every month, but the same principle
applies.

I do not know if your loan agreement allows you to pay it off early,
but obviously if you can pay it off early, you're borrowing the money
for less time and would expect to pay proportionately less in
interest. So, the earlier you can pay it off the better because 7.44%
is far higher than you can hope to get as a return on your savings.
If I spend pretty much the whole £7,500 for a purpose, which I would
want to otherwise why am I borrowing that figure, then it seems to me
the interest on that loan is 7.44%

If I pay the loan back in 5.5 years time then the interest in 40.94%.

From my point of view it's simply a question of what is the percentage
interest on a loan of £7,500. It's either 7.44% or 40.94%. I'd spend the
£7,500 in a year. And if I can pay the loan back in a year, it's cost me
only 7.44% interest. If I dally, or cannot afford to pay off the loan
for 5.5 years it's costing me an arm and a leg. Of course, I've had
the priviledge of taking my time over getting the money back. At a
price.

R

#### Ronald Raygun

Rich said:
I can borrow £7,500 for 66 months (5.5 years)

Monthly permium is £160.16.

If I pay my premiums for a year then I'd pay £1921.92,
Yes, 12 x £160.16 = £1921.92, so that's correct.
of which £558.36 would be interest, £1363.56 paying off the loan.
But this is incorrect. The calculation you have probably done
goes like this:

66 x £160.16 = £10570.56 is the total payable including
£7500 repayment of capital, and therefore the total interest
over the 66 month term is £3070.56.

This means you would be paying £3070.56/66 (which is £46.52)
interest per month *on average*.

But that does *not* mean you would be paying £46.52 interest
in *every actual* month.

The thing is that at the end of each month you actually pay an
amount of interest proportional to the amount of money you owed
during that month. Since the debt gradually reduces during the
term, you are paying less interest (and more capital) each month
than in the previous month. This means you are paying more
interest (and are therefore less capital) in the early months
than in the late months.
So, if I then were able to pay off the loan after a year, the actual
percentage interest on my loan is £558.36 / £7,500 = 7.44 %.

Is this all correct? I cannot see why not, but just checking.
No.

Would typically, a bank make any adjustments when I came to pay off my
loan after a year?
Yes, because after a year you will not have paid off as much as
the £1363.56 you think. According to my calculations, you would
have paid off only £995.64.
If I paid back the loan in 5.5 years, the interest would be £3,070.56.
That's an affective interst on my loan of 40.94%.

Quite some difference.
Well, yes, but interest is generally specified as a rate *per year*,
so you need to divide 40.94% by 5.5 years, which gives 7.44% per year.

However, that's not the true interest rate. Consider that as you are
paying the money off gradually, the average debt throughout the term
is rather less than £7500. If the balance were to shrink linearly (it
doesn't, but it's a reasonable first approximation), then the average
debt is roughly half that, so the interest rate is roughly twice your
7.4%pa.

In your case the actual interest rate is about 1.0942% per month,
which corresponds to an APR of 13.95%.

At the rate of 1.0942% per month, once you have made 12 payments of
£160.16, your debt would have reduced from £7500 to £6504.36. The
relevant formula is:

Debt remaining after N payments = original debt * f(66-N)/f(66),
where f(x) = 1 - 1.010942^-x.

N

#### Norman Wells

Rich said:
From my point of view it's simply a question of what is the percentage
interest on a loan of £7,500. It's either 7.44% or 40.94%. I'd spend
the £7,500 in a year. And if I can pay the loan back in a year, it's
cost me only 7.44% interest. If I dally, or cannot afford to pay off
the loan for 5.5 years it's costing me an arm and a leg.
Loans cost. If you're paying 7.44% a year, you'd expect to end up paying
over 40% in interest over 5.5 years.

If you had that amount to invest, you'd expect to receive 5.5 times the
annual interest if you invested for 5.5 years, wouldn't you?

It's how it works both ways.

J

From my point of view it's simply a question of what is the percentage
interest on a loan of £7,500. It's either 7.44% or 40.94%.
It depends how long you borrow it (pay interest) for

Obviously!

..

J

Norman Wells said:
Loans cost. If you're paying 7.44% a year, you'd expect to end up paying
over 40% in interest over 5.5 years.

If you had that amount to invest, you'd expect to receive 5.5 times the
annual interest if you invested for 5.5 years, wouldn't you?
Whilst ever there are people like this - there will always be money to be

R

#### Rich

It depends how long you borrow it (pay interest) for

Obviously!
Yes, obviously.

When someone is quoted some figures for a 66 month loan and one knows
one can pay back the loan early, one might want to know what interest is
paid after say one year or two years.

Obviously you cannot forsee that.

R

#### Ronald Raygun

Rich said:
Yes, obviously.

When someone is quoted some figures for a 66 month loan and one knows
one can pay back the loan early, one might want to know what interest is
paid after say one year or two years.

Obviously you cannot forsee that.
Eh? You may not be able to foresee exactly when you would be in a
position to pack back early, but one certainly can foresee how much
interest and capital has been paid at any given point in time during
the term, and how much remains owing, assuming the regular payments
have gone to plan, and (as is usually the case with this type of
loan) the interest rate does not change.

R

#### Ronald Raygun

Rich said:
A great many things are forseeable if one spends the time on it.
Things are either foreseeable (in this context that means precalculable)
or they are not. Spending time on it has nothing to do with whether it
is possible to precalculate a thing, only with whether it's possible
or practicable for any particular person to do so.
are trying to save time by using others people's knowledge, it's not
that they have no ability or means to calculate something correctly,
eventually thinking through the problem or reading books.
I understand all that.
I did a quick calculation, without spending much time on it admittedly,
and it turns out that the figure for interest is about twice 7.44%. You
pointed out that actually it's more nearer twice that amount. Fair
enough you put me right. You could have said that the figure is
foreseeable, so why post the question - but that would be a bit daft.
You misunderstand where I'm coming from. I'm not saying that because
you ought to be capable of answering a question yourself (given sufficient
training) you shouldn't ask the question. All I did was take issue with
your bizarre claim that "obviously you cannot foresee that" (the "that" in
question presumably being how much interest and capital will have been paid
at a particular stage during the loan term).

Naturally, you personally may not be able to work it out unassisted, but
when you say "you cannot" this means that it's impossible, that no-one can.
*That* is what I was objecting to, because it clearly *is* possible.

R

#### Rich

Ronald Raygun said:
Rich wrote:

You misunderstand where I'm coming from. I'm not saying that because
you ought to be capable of answering a question yourself (given
sufficient
training) you shouldn't ask the question. All I did was take issue
with
your bizarre claim that "obviously you cannot foresee that" (the
"that" in
question presumably being how much interest and capital will have been
paid
at a particular stage during the loan term).

Naturally, you personally may not be able to work it out unassisted,
but
when you say "you cannot" this means that it's impossible, that no-one
can.
*That* is what I was objecting to, because it clearly *is* possible.
Missunderstanding. It was somewthing else I was refering to about being
foreseeable. There isn't a problem.

R

#### Rich

Ronald Raygun said:
Yes, 12 x £160.16 = £1921.92, so that's correct.

But this is incorrect. The calculation you have probably done
goes like this:

66 x £160.16 = £10570.56 is the total payable including
£7500 repayment of capital, and therefore the total interest
over the 66 month term is £3070.56.

This means you would be paying £3070.56/66 (which is £46.52)
interest per month *on average*.

But that does *not* mean you would be paying £46.52 interest
in *every actual* month.

The thing is that at the end of each month you actually pay an
amount of interest proportional to the amount of money you owed
during that month. Since the debt gradually reduces during the
term, you are paying less interest (and more capital) each month
than in the previous month. This means you are paying more
interest (and are therefore less capital) in the early months
than in the late months.

Yes, because after a year you will not have paid off as much as
the £1363.56 you think. According to my calculations, you would
have paid off only £995.64.

Well, yes, but interest is generally specified as a rate *per year*,
so you need to divide 40.94% by 5.5 years, which gives 7.44% per year.

However, that's not the true interest rate. Consider that as you are
paying the money off gradually, the average debt throughout the term
is rather less than £7500. If the balance were to shrink linearly (it
doesn't, but it's a reasonable first approximation), then the average
debt is roughly half that, so the interest rate is roughly twice your
7.4%pa.

In your case the actual interest rate is about 1.0942% per month,
which corresponds to an APR of 13.95%.

At the rate of 1.0942% per month, once you have made 12 payments of
£160.16, your debt would have reduced from £7500 to £6504.36. The
relevant formula is:

Debt remaining after N payments = original debt * f(66-N)/f(66),
where f(x) = 1 - 1.010942^-x.
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

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
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.

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.

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. 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%.

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.

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

R

#### Rich

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
Actually I'm looking at ways to finance my business. No security and
out of work! Like I say, a weak position/ :c)

But, my bank would loan me money all said and done on my credit rating.

I've more investigating to do.

R

#### Ronald Raygun

Rich said:
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

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
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

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
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.

J

Missunderstanding. It was somewthing else I was refering to about being
foreseeable. There isn't a problem.
Your problem is that during the whole thread you don't seem to want to
acknowledge that *time* is a factor at all

From my point of view it's simply a question of what is the percentage
interest on a loan of £7,500. It's either 7.44% or 40.94%.
ergo you are puzzled by what is essentially a matter of fact

J