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Calculation of mortgage interest from the amount of the mortgage, annuity monthly installments and number of years

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Dan Richter
Dan Richter 2023-12-9
评论： Walter Roberson 2023-12-9
I am having troubles to find a equation for interst (x) from the mortgage equation:
a=M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
x=? (Online equation solvers can't calculate it)
For Mortage 100,000 (M), interest x=10 and 1 year (r) monthly annuity a = 8,791.59

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回答（1 个）

Walter Roberson
Walter Roberson 2023-12-9
Ran in:
M = 100000;
a = 8791.59;
r = 1;
syms x
eqn = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn = 
sol = solve(eqn, x);
vsol = vpa(sol, 16)
vsol = 
vsol = vsol(imag(vsol)==0)
vsol = 
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Dan Richter
Dan Richter 2023-12-9
编辑：Dan Richter 2023-12-9
OK, Thank you for detail explanation :-). Actually, I am interested to express "x" i.e. interest rate per year.
Walter Roberson
Walter Roberson 2023-12-9
Ran in:
As you can see from the below, if you do not know r, there just isn't much you can do to get a useful expression of a solution.
You can also see that the value of r can substantially influence the number of solutions -- and that for some values of r, you can get fully explicit solutions.
%M = 100000;
%a = 8791.59;
%r = 1;
syms x
syms M a r positive
eqn = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn = 
X = solve(eqn, x);
Warning: Unable to find explicit solution. For options, see help.
r = 1;
eqn2 = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn2 = 
X2 = solve(eqn2, x)
Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.
X2 = 
r = 2;
eqn3 = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn3 = 
X3 = solve(eqn3, x)
Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.
X3 = 
r = sym(1)/3
r = 
eqn4 = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn4 = 
X4 = solve(eqn4, x, 'maxdegree', 4)
Warning: Solutions are only valid under certain conditions. To include parameters and conditions in the solution, specify the 'ReturnConditions' value as 'true'.
X4 = 
M = 100000;
a = sym(879159)/sym(100);
%r = 1;
syms x
syms r positive
eqn5 = a == M*(x/1200*(1+x/1200)^(r*12))/((x/1200+1)^(r*12)-1)
eqn5 = 
X5 = solve(eqn5, x)
Warning: Unable to find explicit solution. For options, see help.
X5 = Empty sym: 0-by-1

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