Applications - A Guide to MATLAB for Beginners and Experienced Users - page 146

[0,A]
[

1, -R+A*m]

[ 2, -R-m*R+A*mˆ2]

[ 3, -R-m*R-mˆ2*R+A*mˆ3]

[ 4, -R-m*R-mˆ2*R-mˆ3*R+A*mˆ4]

[ 5, -R-m*R-mˆ2*R-mˆ3*R-

mˆ4*R+A*mˆ5]
[ 6, -R-m*R-mˆ2*R-mˆ3*R-

mˆ4*R-mˆ5*R+A*mˆ6]

We can write this in a simpler way by noticing that P = A ∗ m n + (terms

divisible by R ). For example, with n = 7wehave

factor(p - A*mˆ7)

ans =
-R*(1+m+mˆ2+mˆ3+mˆ4+mˆ5+mˆ6)

But the quantity inside the parentheses is the sum of a geometric series

n − 1

m n

− 1
m − 1 .

m k

=

k = 1

So we see that the principal after n payments can be written as
P = A ∗ m n − R ∗ ( m n − 1) / ( m − 1) .

Now we can solve for the monthly payment amount R under the assumption
that the loan is paid off in N installments, that is, P is reduced to 0 after N
payments:

syms N; solve(A*mˆN - R*(mˆN - 1)/(m - 1), R)

ans =
A*mˆN*(m-1)/(mˆN-1)

R = subs(ans, m,J+1)

R=
A*(J+1)ˆN*J/((J+1)ˆN-1)

For example, withan initial loan amount A = $150,000 and a loan lifetime
of 30 years (360 payments), we get the following table of payment amounts
as a function of annual interest rate: