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- anonymous

You have just grduated from college and landed youyr first job, you purchase a house cost 250,000 and find a bank interest rate 6% that compounds monthly for 30 years.
what is the monthly payment for this loan?
what is the unpaid balance at the end of 5 years?
what is the unpaid balance at the end of the 10th year?
I think I should use the formula A=p(1+I)n but I am not sure please help.

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- anonymous

can someone please help me please

- anonymous

- anonymous

This is actually very interesting problem and requires some tricks

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- anonymous

If you think about it, mortgages have inverse relationship with saving accounts.
With mortgage, you start with lump sum and end with zero while in saving account we start with 0 and end with a lump sum. With this relationship in mind, we can solve this problem.

- anonymous

We will start by working saving account problem.
1 st - D
2 nd - D (1 + R) + D = D (1 + (1 + R))
3 rd - D (1 + (1 + R) + (1 + R)^2)
Nth - D (1 + (1 + R) + (1 + R)^2 + ... + (1 + R)^N)
D (Sum of Geometric Series)
\[=D\frac{1-(1+R)^N}{-R}\]
Simplified further:
Lump Sum after N month =\[\frac{D \left(-1+(1+R)^N\right)}{R}\]
Now the trick is going from saving account formula to mortgage formula.
Recall that in saving account the lump sum in at the end while in mortgage, the lump sum exists in beginning and diminishes.
Formula:
A_N=Sum
A_0=Balance
\[A_N=A_0(1+R)^N\]
Sum=Balance * (1+R)^N
\[\text{Sum*}(1+R)^{-N}=\text{Balance}\]
\[\frac{D -1+(1+R)^N}{R} *(1+R)^{-N}\]
Simplified:
\[\frac{D \left(1-(1+R)^{-N}\right)}{R}=Balance\]
By setting Balance to 0 and solving,n=30year *12, we can solve for D ,monthly payment.

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