-- Mortgage
Han de Bruijn
http://www.eurofinancialservice.com/mortgage_request.html
Fixed instalment mortgage
-------------------------
The fixed instalments mortgages belong to the classical mortgages,
together with the annuity mortgages. The former are described at:
http://www.eurofinancialservice.com/lineair_mortgage.html
Mathematical idealization of the fixed instalments mortgage leads to
the following formula (i.e. _continuous_ instead of discrete model):
S(t) = B.(1 - t/T)
Here: B = principal ; T = term ; t = time ; S = debt .
Interest is calculated in this idealized model as:
r = S(t).v
Here: r = installment ; v = (interest) rate, assume fixed .
The total interest R is calculated as a definite integral:
dR = r.dt ==> R = integral(0..T) r.dt
= B.v . integral(0..T)(1 - t/T).dt = B.v.T/2 .
(Or much simpler as the area of a triangle times constants)
We are interested in the case where the total interest becomes greater
than the principal debt:
B.v.T/2 > B ==> v.T > 2 .
Thus the break even point for an idealized fixed instalments mortgage
is where (interest rate) times (term) is greater than two.
Did I make any mistakes (mathematics / terminology / whatever) so far?
Han de Bruijn
orang...@googlemail.com
dude, what you have done looks pretty good. but we generally use an
exponential function to work out compound interest. the best
explanation is in 'calculus made easy' by silvanus p thompson. its on
page 134 on google books -
its one of my favorite bit of math. you get the taylor series for e
from compound interest, which is cool.
Tonico
**************************************************************
Yes, you did a huge mistake...if you have a mortgage you can't pay
back and your house is about to be foreclosed.
If you're a banker and you gave money for mortgages in an
irresponsible and/or cheating way and now houses owners can't pay you
back don't worry: the government will pay you, and still will screw
owners and plain middle class tax payers in general.
Of course, the above is true in the USA...and in Europe...and in
Japan...and in Mexico, in Central America and in South America and in
Africa and in Australia and in the North Pole.
Wonders and marvels of this planet's monetary system: the less you
have the more you get screwed, and the more you have the best you can
get away with fraud, cheating, stupid management and lies and
exploitation.
Enjoy the Earth: the blue, screwed third planet from the star called
the Sun.
Regards
Tonio
Han de Bruijn
> dude, what you have done looks pretty good. but we generally use an
> exponential function to work out compound interest. the best
> explanation is in 'calculus made easy' by silvanus p thompson. its on
> page 134 on google books -
>
> http://books.google.co.uk/books?hl=en&id=5C-O8rRPnc4C&dq=silvanus+p+thompson+calculus+made+easy&printsec=frontcover&source=web&ots=DM80iN9Y4O&sig=vgCUW_cKRecy2-De_nsjyADz5Vc&sa=X&oi=book_result&resnum=7&ct=result#PPA134,M1
>
> its one of my favorite bit of math. you get the taylor series for e
> from compound interest, which is cool.
This is just a cautious start. Exponentials will be in the next posting.
Han de Bruijn
Han de Bruijn
> Main reference (w.r.t. English terminology):
>
> http://www.eurofinancialservice.com/mortgage_request.html
>
> Fixed instalment mortgage
> -------------------------
Annuity mortgage
----------------
The annuity mortgage belongs to the classical mortgages, together with
the fixed instalment mortgage. The former is described at:
http://www.eurofinancialservice.com/annuity_mortgage.html
Skip to "What does it mean ?" if you're only interested in conclusions.
Idealization of the annuity mortgage leads to still another continuous
instead of discrete mathematical model. In differential form it reads:
- dS + S.v.dt = m.dt
Here: t = time ; S = debt ; v = (interest) rate ; m = annuity .
This results in a first order Ordinary Differential Equation (ODE):
- dS/dt + v.S = m
The solution is found to be: S(t) = m/v + C . exp(v.t)
As can be verified by substituting it back into the ODE.
For t = 0 is S(0) = B = principal , so: B = m/v + C . Hence:
S(t) = m/v + (B - m/v) . exp(v.t)
At the end of the mortgage's life, the debt is zero:
S(T) = m/v + (B - m/v) . exp(v.T) = 0
Herewith the annuities can be solved:
m/v . ( 1 - exp(v.T) ) = - B . exp(v.T)
m = B . v . exp(v.T) / ( exp(v.T) - 1 )
How about the debt:
S(t) = m/v + (B - m/v) . exp(v.t) =
m/v . ( 1 - exp(v.t) ) + B . exp(v.t)
Where: m/v = B . exp(v.T) / ( exp(v.T) - 1 )
Giving: S = B . exp(v.T) ( 1 - exp(v.t) ) / ( exp(v.T) - 1 )
+ B . exp(v.t) . ( exp(v.T) - 1 ) / ( exp(v.T) - 1 )
Giving: S(t) = B . ( exp(v.T) - exp(v.t) ) / ( exp(v.T) - 1 )
Installment A as a function of time is the principal minus the debt:
A(t) = B - S(t) = B . ( exp(v.t) - 1 ) / ( exp(v.T) - 1 )
Of course A(t) is equal to the principal for t = T : A(T) = B .
The total interest paid is:
R(T) = m.T - B = B . v . T . exp(v.T) / ( exp(v.T) - 1 ) - B
We are INTERESTED in the case where the total interest becomes greater
than, or at least equal to the principal debt:
B . v . T . exp(v.T) / ( exp(v.T) - 1 ) - B = B
The outcome is independent of the principal:
v.T . exp(v.T) / ( exp(v.T) - 1 ) = 2
Substitute x = v.T , then we have the following equation:
x . exp(x) = 2 . ( exp(x) - 1 ) <=> (x - 2) . exp(x) + 2 = 0
The following computer program is used to solve, where Newton-Rhapson
iterations are employed:
program Mortgage;
type
funktie = function(x : double) : double;
function original(x : double) : double;
begin
original := (x-2)*exp(x)+2;
end;
function derivative(x : double) : double;
begin
derivative := exp(x)*(x-1);
end;
function Newton(P ,eps : double; zeroth, first : funktie) : double;
{
y - f(p_n) = f'(p_n).(x - p_n) met y = 0 en x = p_(n+1)
==> p_(n+1) = p_n - f(p_n)/(f'(p_n)
}
var
x : double;
begin
x := P;
while abs(zeroth(x)) > eps do
x := x - zeroth(x) / first(x);
Newton := x;
end;
function Equation : double;
const
eps : double = 1.e-9;
begin
Equation := Newton(2, eps, original,derivative);
end;
begin
Writeln('Break Even Point = ',Equation);
end.
And the outcome is:
Break Even Point = 1.59362426004024
What does it mean ?
Suppose you buy a house and you have an annuity mortgage. Then you pay
more interest to the bank than you've borrowed as a principal loan iff
v.T > 1.59362426
Thus the break even point for an idealized annuity mortgage is where
(interest rate) times (term) is greater than 1.59362426 .
If your mortgage has a lifetime of, say, 30 years, then we find:
v > 1.59362426 / 30 * 100 = 5.4 %
Now take a look at
And you will see that _any_ current mortgage rate takes care that you
shall pay back, at least, _TWICE_ the amount of money you've borrowed.
Did I make any mistakes (mathematics / terminology / whatever) so far?
If not, then I rest my case, so far.
Dutch version available, for quite some years, at:
http://hdebruijn.soo.dto.tudelft.nl/tomaatnet/theorie.htm#AH
Han de Bruijn
Jesse F. Hughes
> Now take a look at
>
> http://www.mortgage101.com/
>
> And you will see that _any_ current mortgage rate takes care that you
> shall pay back, at least, _TWICE_ the amount of money you've borrowed.
>
> Did I make any mistakes (mathematics / terminology / whatever) so far?
>
> If not, then I rest my case, so far.
What case would that be?
What is your point? That mortgages are profitable for banks (assuming
they're paid back)? Or that banks are bad for making lots of money
off of mortgages?
(Note: maybe this is a language issue. In English, "I rest my case"
is an idiom coming out of the courthouse, stated when one side has
presented its argument. But you don't seem to have an argument yet,
or at least I haven't figured out what it is.)
--
Jesse F. Hughes
Quincy (age 3 1/2, looking at a picture): Are these people Canadians?
Me: Uh, no, they're Australian Aborigines.
Quincy: Do they fight Canadians?
** Posted from http://www.teranews.com **
orang...@googlemail.com
> Han de Bruijn wrote:
> > Main reference (w.r.t. English terminology):
>
cool, we should all be taking out annuity mortgages. i got a couple of
questions, if you got time to explain
> >http://www.eurofinancialservice.com/mortgage_request.html
>
> > Fixed instalment mortgage
> > -------------------------
>
> Annuity mortgage
> ----------------
> The annuity mortgage belongs to the classical mortgages, together with
> the fixed instalment mortgage. The former is described at:
>
> http://www.eurofinancialservice.com/annuity_mortgage.html
>
> Skip to "What does it mean ?" if you're only interested in conclusions.
>
> Idealization of the annuity mortgage leads to still another continuous
> instead of discrete mathematical model. In differential form it reads:
>
> - dS + S.v.dt = m.dt
i'm reading this as (arbitrarily small)
- debt + (interest rate * debt) = payment
shouldn't the debt be positive, or am i getting this wrong?
>
> Here: t = time ; S = debt ; v = (interest) rate ; m = annuity .
>
> This results in a first order Ordinary Differential Equation (ODE):
>
> - dS/dt + v.S = m
>
or the change in the debt is the repayment minus the interest.
- ds /dt = m - vs
i'm ok up to here but i can't figure out what you did next.
Han de Bruijn
> On 15 Oct, 11:27, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
>>Han de Bruijn wrote:
>>
>>>Main reference (w.r.t. English terminology):
>
> cool, we should all be taking out annuity mortgages. i got a couple of
> questions, if you got time to explain
>
>>>http://www.eurofinancialservice.com/mortgage_request.html
>>
>>>Fixed instalment mortgage
>>>-------------------------
>>
>>Annuity mortgage
>>----------------
>>The annuity mortgage belongs to the classical mortgages, together with
>>the fixed instalment mortgage. The former is described at:
>>
>>http://www.eurofinancialservice.com/annuity_mortgage.html
>>
>>Skip to "What does it mean ?" if you're only interested in conclusions.
>>
>>Idealization of the annuity mortgage leads to still another continuous
>>instead of discrete mathematical model. In differential form it reads:
>>
>> - dS + S.v.dt = m.dt
>
> i'm reading this as (arbitrarily small)
>
> - debt + (interest rate * debt) = payment
>
> shouldn't the debt be positive, or am i getting this wrong?
Payment is (interest rate * debt) plus a _diminishing_ of the debt.
Hence the minus sign.
>>Here: t = time ; S = debt ; v = (interest) rate ; m = annuity .
>>
>>This results in a first order Ordinary Differential Equation (ODE):
>>
>> - dS/dt + v.S = m
>
> or the change in the debt is the repayment minus the interest.
>
> - ds /dt = m - vs
>
> i'm ok up to here but i can't figure out what you did next.
It's mathematics. If you're unable to figure out exactly what it's all
about, you could decide to jump to the conclusions and trust that I've
done my homework well (: at least hope so).
Han de Bruijn
Han de Bruijn
> Han de Bruijn wrote:
>
>> Main reference (w.r.t. English terminology):
>>
>> http://www.eurofinancialservice.com/mortgage_request.html
>>
>> Fixed instalment mortgage
>> -------------------------
>
> Annuity mortgage
> ----------------
How about _this_ (didn't say "agreed", but .. ):
http://www.fdrs.org/index.html
Han de Bruijn
Han de Bruijn
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
>>Now take a look at
>>
>>http://www.mortgage101.com/
>>
>>And you will see that _any_ current mortgage rate takes care that you
>>shall pay back, at least, _TWICE_ the amount of money you've borrowed.
>>
>>Did I make any mistakes (mathematics / terminology / whatever) so far?
>>
>>If not, then I rest my case, so far.
>
> What case would that be?
>
> What is your point? That mortgages are profitable for banks (assuming
> they're paid back)? Or that banks are bad for making lots of money
> off of mortgages?
Did I even _suggest_ a moral judgement ? For the simple minded among us:
- You have actually paid back TWO HOUSES, one for you, one for the bank.
> (Note: maybe this is a language issue. In English, "I rest my case"
> is an idiom coming out of the courthouse, stated when one side has
> presented its argument. But you don't seem to have an argument yet,
> or at least I haven't figured out what it is.)
Now, WHAT is the bank doing with that other house ? I mean, some people
find it strange that such a system _must_ explode.
Han de Bruijn
Jesse F. Hughes
> Jesse F. Hughes wrote:
>
>> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>>
>>>Now take a look at
>>>
>>>http://www.mortgage101.com/
>>>
>>>And you will see that _any_ current mortgage rate takes care that you
>>>shall pay back, at least, _TWICE_ the amount of money you've borrowed.
>>>
>>>Did I make any mistakes (mathematics / terminology / whatever) so far?
>>>
>>>If not, then I rest my case, so far.
>>
>> What case would that be?
>>
>> What is your point? That mortgages are profitable for banks (assuming
>> they're paid back)? Or that banks are bad for making lots of money
>> off of mortgages?
>
> Did I even _suggest_ a moral judgement ?
You didn't suggest any conclusion at all, hence the question. I just
took a couple of guesses at what your point might be.
> For the simple minded among us:
>
> - You have actually paid back TWO HOUSES, one for you, one for the
> bank.
Yes, when you take a loan, you pay back more than the principal.
Sometimes much more. And so?
>
>> (Note: maybe this is a language issue. In English, "I rest my case"
>> is an idiom coming out of the courthouse, stated when one side has
>> presented its argument. But you don't seem to have an argument yet,
>> or at least I haven't figured out what it is.)
>
> Now, WHAT is the bank doing with that other house ? I mean, some people
> find it strange that such a system _must_ explode.
Pardon?
Again, your point is very unclear.
--
Quincy (age 5): Baba, play some [computer games].
Mama: Quincy, if you want [Baba] to live, don't make those
suggestions.
Quincy: Make those suggestions. Got it.
amy666
you are one frustrated guy tonio !
Han de Bruijn
In number of houses (or equivalent money) as a function of time:
1 -> 2 -> 2.2 -> 2.2.2 -> 2^4 -> 2^5 -> .. -> 2^N
See the pattern ? Does that sequence diverge perhaps ?
Han de Bruijn
orang...@googlemail.com
> orangata...@googlemail.com wrote:
> > On 15 Oct, 11:27, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
> >>Han de Bruijn wrote:
>
> >>>Main reference (w.r.t. English terminology):
>
> > cool, we should all be taking out annuity mortgages. i got a couple of
> > questions, if you got time to explain
>
> >>>http://www.eurofinancialservice.com/mortgage_request.html
>
> >>>Fixed instalment mortgage
> >>>-------------------------
>
> >>Annuity mortgage
> >>----------------
> >>The annuity mortgage belongs to the classical mortgages, together with
> >>the fixed instalment mortgage. The former is described at:
>
> >>http://www.eurofinancialservice.com/annuity_mortgage.html
>
> >>Skip to "What does it mean ?" if you're only interested in conclusions.
>
> >>Idealization of the annuity mortgage leads to still another continuous
> >>instead of discrete mathematical model. In differential form it reads:
>
> >> - dS + S.v.dt = m.dt
>
> > i'm reading this as (arbitrarily small)
>
> > - debt + (interest rate * debt) = payment
>
> > shouldn't the debt be positive, or am i getting this wrong?
>
> Payment is (interest rate * debt) plus a _diminishing_ of the debt.
> Hence the minus sign.
>
isn't interest rate <1? if interest rate * debt < debt, m becomes
negative.
> >>Here: t = time ; S = debt ; v = (interest) rate ; m = annuity .
>
> >>This results in a first order Ordinary Differential Equation (ODE):
>
> >> - dS/dt + v.S = m
>
> > or the change in the debt is the repayment minus the interest.
>
> > - ds /dt = m - vs
>
> > i'm ok up to here but i can't figure out what you did next.
>
> It's mathematics. If you're unable to figure out exactly what it's all
> about, you could decide to jump to the conclusions and trust that I've
> done my homework well (: at least hope so).
>
took another look and i think you did the integration right. sorry!
the rest of the math looks good as well.
Han de Bruijn
These are your words. But - dS + S.v.dt = m.dt should be read as:
- (CHANGE in debt) + (debt * interest rate * time interval)
= (payment * time interval)
Han de Bruijn
orang...@googlemail.com
ah! yeah, it makes sense to me now.
Han de Bruijn
Title of the above: Debt Free Money.
How about _this_ (didn't say "agreed", but .. ):
http://userpage.fu-berlin.de/~roehrigw/kennedy/english/
Margrit Kennedy: Interest and Inflation Free Money.
How about Riegel's "Free Money" (didn't say "agreed", but ..):
http://kentennant.com/rm/riegel.php
So far so good. I may have missed a few subtle differences.
The money of a house market essentially consists of TWO components:
1. Debt
2. Interest
So, logically speaking, we ONLY have the following FOUR combinations:
Kind of money | debt | interest
--------------------------------------
Free money | no | no
Interest free money | yes | no
Debt free money | no | yes
Contemporary money | yes | yes
Which of the above four combinations would be the best ? And why ?
Han de Bruijn
orang...@googlemail.com
> How about _this_ (didn't say "agreed", but .. ):
>
> http://userpage.fu-berlin.de/~roehrigw/kennedy/english/
>
> Margrit Kennedy: Interest and Inflation Free Money.
>
this is a really good idea. artificially increasing the velocity of
money to increase gdp. in theory this would be more efficient than
commodity money, such as gold or silver. but the arguments against
other forms of fiat, such as the temptation for government to print
money, still apply.
on balance i think i prefer commodity money as it is more difficult to
manipulate. but this would defininitly be superior to the debt backed
money we use today.