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| Guest |
 Posted: Sat Jul 15, 2006 9:18 am |
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Suppose I take out a loan of 1600 and repay the capital at 100 p/a for
16 years paid at the end of each year.
I repay the interest at 80 p/a for the first 4 years, and 10 p/a for
the remaining 12 years.
How do I calculate the overall annual interest rate for the full 16
years?
Many thanks
Tony
PS
I've got as far as
average loan over first 4 years = (1600+1300)/2 =1450
interest rate for first 4 years is therefore = 80/1450=5.5%
average loan over the remaining 12 yrs is (1200-0)/2=600
the interest rate for the 12 years is therefore =10/600= 1.6%
But I don't think that helps!
(The figures I've used aren't important if its easier to explain with
others - I'm just trying to understand the principles invovled)
Thanks to anyone who can advise |
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| Arthur Dent |
| Posted: Sat Jul 15, 2006 11:31 am |
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Guest
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Correct me if I'm wrong, not done this for a while. I think if it's a
continuous function you could use logs. For discrete functions, eg
interest calculated anually you could use a geometric series.
in the case of 100% pa repayments would be 1600^(n-1) +1600 where n is
the number of years.
at 80% repayments would be 1600^(4n/5 - 1) +1600
etc
tonyjeffs@tonyjeffs.com wrote:
[quote:080f618295]Suppose I take out a loan of 1600 and repay the capital at 100 p/a for
16 years paid at the end of each year.
I repay the interest at 80 p/a for the first 4 years, and 10 p/a for
the remaining 12 years.
How do I calculate the overall annual interest rate for the full 16
years?
Many thanks
Tony
PS
I've got as far as
average loan over first 4 years = (1600+1300)/2 =1450
interest rate for first 4 years is therefore = 80/1450=5.5%
average loan over the remaining 12 yrs is (1200-0)/2=600
the interest rate for the 12 years is therefore =10/600= 1.6%
But I don't think that helps!
(The figures I've used aren't important if its easier to explain with
others - I'm just trying to understand the principles invovled)
Thanks to anyone who can advise[/quote:080f618295] |
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| Barry Schwarz |
| Posted: Sat Jul 15, 2006 2:01 pm |
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Guest
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On 15 Jul 2006 05:18:30 -0700, tonyjeffs@tonyjeffs.com wrote:
[quote:556f8139b4]Suppose I take out a loan of 1600 and repay the capital at 100 p/a for
16 years paid at the end of each year.
I repay the interest at 80 p/a for the first 4 years, and 10 p/a for
the remaining 12 years.
How do I calculate the overall annual interest rate for the full 16
years?
It depends on what you mean by overall annual interest rate. I assume[/quote:556f8139b4]
you mean the constant interest rate at which 1600 would be repaid at
the same total cost to you using conventional payment calculations.
Note that this assumption is not consistent with your description. In
a conventional loan, the first payment reduces the principal (you
called it capital) very little. The first payment is almost all
interest. On the other hand, the last payment is almost all
principal.
Your total payment was 2040 which averages to 127.5 annually. Your
payment rate is 79.688/thousand. You should have no trouble
determining the interest rate which requires that payment rate for a
period of 16 years.
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| Guest |
| Posted: Sat Jul 15, 2006 2:40 pm |
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tonyjeffs@tonyjeffs.com wrote:
[quote:2c583f3da4]Suppose I take out a loan of 1600 and repay the capital at 100 p/a for
16 years paid at the end of each year.
I repay the interest at 80 p/a for the first 4 years, and 10 p/a for
the remaining 12 years.
How do I calculate the overall annual interest rate for the full 16
years?
[/quote:2c583f3da4]
The "obvious" calculation is to work out the total interest paid
according to the above schedule, and calculate an annual rate that
would entail paying that same amount of interest. However, I don't
think that this method holds water because you are equating amounts
paid at different times without taking into account the time value of
money.
Maybe the equivalent annual rate should be the rate that makes the
present value of all your repayments equal to 1600. Let this annual
rate be r, and define d, the discount factor, as d = 1/(1+r). The
present value of all the repayments is then
180*d + 180*d^2 + 180*d^3 + 180*d^4 + 110*d^5 + 110*d^6 + ... +
110*d^16
Set this equal to 1600, solve for d, and you get d = 0.9673 approx., so
r = 0.0338 approx, or about 3.38%. |
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| Arthur Dent |
| Posted: Sat Jul 15, 2006 5:49 pm |
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Guest
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matt271829-news@yahoo.co.uk wrote:
[quote:cb03340908]tonyjeffs@tonyjeffs.com wrote:
Suppose I take out a loan of 1600 and repay the capital at 100 p/a for
16 years paid at the end of each year.
I repay the interest at 80 p/a for the first 4 years, and 10 p/a for
the remaining 12 years.
How do I calculate the overall annual interest rate for the full 16
years?
The "obvious" calculation is to work out the total interest paid
according to the above schedule, and calculate an annual rate that
would entail paying that same amount of interest. However, I don't
think that this method holds water because you are equating amounts
paid at different times without taking into account the time value of
money.
Maybe the equivalent annual rate should be the rate that makes the
present value of all your repayments equal to 1600. Let this annual
rate be r, and define d, the discount factor, as d = 1/(1+r). The
present value of all the repayments is then
180*d + 180*d^2 + 180*d^3 + 180*d^4 + 110*d^5 + 110*d^6 + ... +
110*d^16
Set this equal to 1600, solve for d, and you get d = 0.9673 approx., so
r = 0.0338 approx, or about 3.38%.
[/quote:cb03340908]
yeah, that sounds right. don't know what i was thinking... |
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| Guest |
| Posted: Sun Jul 16, 2006 6:19 am |
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Definitely true that time of payment is important
And I also note that typically, loan Capital repayments do not remain
the same year by year.
..........................................
I eventually went for: the follwing, but will try again, incorporating
what you've said.
Here's where I got to...
I turned it around to an investment because it's easier for me to
visualise,
I invest:
year 1 80 for 16 years
year 2 80 for 15 years
year 3 80 for 14 years
year 4, 80 for 13 years
year 5, 10 for 12 years
....and so on until....
year 16, 10 for 1 year
.....................................................................................
So in total I've invested 80, 4 times for an average of 14.5 years
-the equivalent of making a single investment of 80 for 58 years
-and the equivalent of making a single investment of (80 x 5 =4640
for 1 year
plus
10, invested 12 times for an average of 6.5 years
-the equivalent of making a single investment of 10 for 78 years.
-and the equivalent of making a single investment of 10x 7=780 for
1 year
In total that is the equivalent of making a single investment of
(4640+780)=5420 for 1 year
We are looking for the equivalent constant unchanging investment over
16 years,
the unknown 'N'
which is invested for (16 yrs+15 yrs + ....+ 2yrs + 1 yr), a total of
136 yrs.
so
N=5420/136 =38.85
So investing 39.85 every year for 16 years is equivalent to investing
80 for 4yrs then 10 for 12 years.
Since that is interest on 1600, the apr is 39.5/1600 = 2.5%
................................
That looks like a realistic answer, but it's different to Matt's which
is also a realistic answer.
.......................................................................................
Maybe something to do with the way I'm dealing with the capital
payment.
Thanks Tony
and to get my answer, I'm saying that's the same as investing £N for
16 x (an average of 8.5 years)
i.e I invest 'N' each year, firstly for 16 years, then 15, then
14.....then 2 then 1..
so I make sixteen investments of average duration of
(16+15+...2+...1)/16= 8.5 years
(.80 x 4 x 14.5) + (10 x 12 x 6.5) = N x 16 x 8.5
£N is the annual interest equivalent
100 x N/1600 is the APR |
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| Guest |
| Posted: Sun Jul 16, 2006 6:21 am |
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Definitely true that time of payment is important
And I also note that typically, loan Capital repayments do not remain
the same year by year.
..........................................
I eventually went for: the follwing, but will try again, incorporating
what you've said.
Here's where I got to...
I turned it around to an investment because it's easier for me to
visualise,
I invest:
year 1 80 for 16 years
year 2 80 for 15 years
year 3 80 for 14 years
year 4, 80 for 13 years
year 5, 10 for 12 years
....and so on until....
year 16, 10 for 1 year
.....................................................................................
So in total I've invested 80, 4 times for an average of 14.5 years
-the equivalent of making a single investment of 80 for 58 years
-and the equivalent of making a single investment of (80 x 5=4640
for 1 year
plus
10, invested 12 times for an average of 6.5 years
-the equivalent of making a single investment of 10 for 78 years.
-and the equivalent of making a single investment of (10 x 7=780
for 1 year
In total that is the equivalent of making a single investment of
(4640+780)=5420 for 1 year
We are looking for the equivalent constant unchanging investment over
16 years,
the unknown 'N'
which is invested for (16 yrs+15 yrs + ....+ 2yrs + 1 yr), a total of
136 yrs.
so
N=5420/136 =38.85
So investing 39.85 every year for 16 years is equivalent to investing
80 for 4yrs then 10 for 12 years.
Since that is interest on 1600, the apr is 39.5/1600 = 2.5%
................................
That looks like a realistic answer, but it's different to Matt's which
is also a realistic answer.
.......................................................................................
Maybe something to do with the way I'm dealing with the capital
payment.
Thanks Tony
and to get my answer, I'm saying that's the same as investing £N for
16 x (an average of 8.5 years)
i.e I invest 'N' each year, firstly for 16 years, then 15, then
14.....then 2 then 1..
so I make sixteen investments of average duration of
(16+15+...2+...1)/16= 8.5 years
(.80 x 4 x 14.5) + (10 x 12 x 6.5) = N x 16 x 8.5
£N is the annual interest equivalent
100 x N/1600 is the APR |
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| Guest |
| Posted: Sun Jul 16, 2006 7:31 am |
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Sorry I sent that twice, and included some half-thought-out rubbish
after my signature
Now this problem with the capital.....!
tony |
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| Guest |
| Posted: Sun Jul 16, 2006 8:43 am |
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tonyjeffs@tonyjeffs.com wrote:
[quote:b06fc7a2c2]Definitely true that time of payment is important
And I also note that typically, loan Capital repayments do not remain
the same year by year.
.........................................
I eventually went for: the follwing, but will try again, incorporating
what you've said.
Here's where I got to...
I turned it around to an investment because it's easier for me to
visualise,
I invest:
year 1 80 for 16 years
year 2 80 for 15 years
year 3 80 for 14 years
year 4, 80 for 13 years
year 5, 10 for 12 years
...and so on until....
year 16, 10 for 1 year
...................................................................................
So in total I've invested 80, 4 times for an average of 14.5 years
-the equivalent of making a single investment of 80 for 58 years
-and the equivalent of making a single investment of (80 x 5=4640
for 1 year
plus
10, invested 12 times for an average of 6.5 years
-the equivalent of making a single investment of 10 for 78 years.
-and the equivalent of making a single investment of (10 x 7=780
for 1 year
In total that is the equivalent of making a single investment of
(4640+780)=5420 for 1 year
We are looking for the equivalent constant unchanging investment over
16 years,
the unknown 'N'
which is invested for (16 yrs+15 yrs + ....+ 2yrs + 1 yr), a total of
136 yrs.
so
N=5420/136 =38.85
So investing 39.85 every year for 16 years is equivalent to investing
80 for 4yrs then 10 for 12 years.
Since that is interest on 1600, the apr is 39.5/1600 = 2.5%
...............................
That looks like a realistic answer, but it's different to Matt's which
is also a realistic answer.
.....................................................................................
Maybe something to do with the way I'm dealing with the capital
payment.
Thanks Tony
[/quote:b06fc7a2c2]
Hmmm. I'm no financial expert, but I'm not convinced by your method for
a variety of reasons (including, as you mentioned, the fact that you
don't seem to have considered the capital repayment schedule, which
must have a bearing on the equivalent annual rate).
Perhaps it might help to think of it this way. On day one you have 1600
(units of currency) that you've borrowed. How can you invest that 1600
at a fixed annual rate so that the return on your investment exactly
matches your repayments on the loan? If you can find such a rate then
that must be the effective fixed rate you are paying on the loan. Let
that fixed rate be r.
First of all, you need to find 180 at the end of year 1 for the first
repayment. To get back 180 in one year you need to invest 180/(1+r)
now. Next you need 180 at the end of year 2. To get back 180 in two
years (with compounding) you need to invest 180/(1+r)^2 now. Then you
need 180 at the end of year 3, which entails investing 180/(1+r)^3 now.
And so on.
The sum of all these initial investments must equal 1600, because
that's the total you have to invest. With the definition d = 1/(1+r)
this leads to the formula
180*d + 180*d^2 + 180*d^3 + 180*d^4 + 110*d^5 + 110*d^6 + ... +
110*d^16 = 1600
that I posted earlier. |
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| Guest |
| Posted: Sun Jul 16, 2006 8:52 am |
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matt271829-n...@yahoo.co.uk wrote:
[quote:901ea8b9e9]
Hmmm. I'm no financial expert, but I'm not convinced by your method for
a variety of reasons (including, as you mentioned, the fact that you
don't seem to have considered the capital repayment schedule, which
must have a bearing on the equivalent annual rate).
Perhaps it might help to think of it this way. On day one you have 1600
(units of currency) that you've borrowed. How can you invest that 1600
at a fixed annual rate so that the return on your investment exactly
matches your repayments on the loan? If you can find such a rate then
that must be the effective fixed rate you are paying on the loan. Let
that fixed rate be r.
First of all, you need to find 180 at the end of year 1 for the first
repayment. To get back 180 in one year you need to invest 180/(1+r)
now. Next you need 180 at the end of year 2. To get back 180 in two
years (with compounding) you need to invest 180/(1+r)^2 now. Then you
need 180 at the end of year 3, which entails investing 180/(1+r)^3 now.
And so on.
The sum of all these initial investments must equal 1600, because
that's the total you have to invest. With the definition d = 1/(1+r)
this leads to the formula
180*d + 180*d^2 + 180*d^3 + 180*d^4 + 110*d^5 + 110*d^6 + ... +
110*d^16 = 1600
that I posted earlier.
[/quote:901ea8b9e9]
Or, if you like, you can look at it from the point of view of the
lender. The lender has lent you 1600 and in return gets an annual
stream of payments 180, 180, 180, 180, 110, 110, etc.
How can the lender invest that 1600 at a fixed annual rate so as to
achieve an identical payment stream, and what would that rate be? The
method and answer are the same as above. |
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| Guest |
| Posted: Sun Jul 16, 2006 11:20 am |
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Matt,
I've got a feeling you're right. I like to think I'm ok at maths, but
I'm rusty as a car of the same age.
I'll get some sunglasses and study your method in the garden.
Back later
Tony |
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| Guest |
| Posted: Sun Jul 16, 2006 12:27 pm |
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Matt
I've done it as a spreadsheet, as it'd be described on a bank
statement..
Proves you're right. Congrats & thanks.
KEY
Outstanding = remaining debt
Payment= end of year payment
interest = Outstanding * 0.0338 = interest component paid
Capital = payment - interest = capital component paid
interest + capital = 180
***interest=0.0338***
outs- pay-
year tanding ment intrst Capitl
1 1600.00 180.00 54.12 125.88
2 1474.12 180.00 49.86 130.14
3 1343.98 180.00 45.46 134.54
4 1209.44 180.00 40.91 139.09
5 1070.35 110.00 36.20 73.80
6 996.56 110.00 33.71 76.29
7 920.27 110.00 31.13 78.87
8 841.40 110.00 28.46 81.54
9 759.86 110.00 25.70 84.30
10 675.56 110.00 22.85 87.15
11 588.41 110.00 19.90 90.10
12 498.31 110.00 16.86 93.14
13 405.17 110.00 13.70 96.30
14 308.87 110.00 10.45 99.55
15 209.32 110.00 7.08 102.92
16 106.40 110.00 3.60 106.40
17 0.00 110.00 0.00 110.00
.. |
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| Guest |
| Posted: Sun Jul 16, 2006 12:30 pm |
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| Maybe that payment is made at the beginning of each year for it to work! |
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| Guest |
| Posted: Sun Jul 16, 2006 2:09 pm |
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tonyjeffs@tonyjeffs.com wrote:
[quote:44a3402a4c]Maybe that payment is made at the beginning of each year for it to work!
[/quote:44a3402a4c]
Though I've not checked the individual figures, your spreadsheet looks
to work out OK to me, with payments made at the end of each year -
except that you've just gone one payment too far, if that's what you
were querying.
Over year 16 you had capital outstanding of 106.40. One year's interest
at 3.38% is 3.60, so at the end of year 16 you owe 106.40 + 3.60 = 110.
You pay 110 at the end of that year, your debt is exactly paid off on
time, and the loan is done.
Actually I'm rather surprised it comes out exact to the nearest penny,
since the 3.38% figure is approximate. To more decimal places it should
be 3.382528546867... |
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| Guest |
| Posted: Sun Jul 16, 2006 3:45 pm |
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Matt, I got the following figure, but set the display to 2 decimal
places so it was readable.
The difference may be due to Excel itself, but they're identical to
within 4dp which is very good..
3.38252678930608 mine
3.382528546867... yours
You're right again about the one payment too many shown on the last
line, but it didn't affect the calculation since there's no next line.
I made a mistake by copying the entire formula from the line above.
Thanks again for the interesting discussion. I asked a couple of maths
teachers who couldn't do it, so I'll send them your solution, and my
spreadsheet.
Cheers
tony |
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