Math Is Fun Forum

Nestea

Member

Registered: 2005-11-12

Posts: 9

Help Me!!

If you went to the bank for a 15 year, $150,000 mortgage at an interest rate of 6.75% pa, compounding quarterly and you wanted to give payments semi-annually how much would each of your payments have to be over the 15 years?

Answer:$8,057.37

Last edited by Nestea (2005-12-09 16:44:54)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Help Me!!

C = Principal Balance
P = Payment every six months
Q = 100% + 6.75%/4
Work backwards in time, perhaps to get an equation.
At 15.0 years, principal = $0
At 14.99 years, C = P, just before sent out last check.
At 14.98 years, C = P/Q
At 14.5 years, C = P/Q/Q
At 14.49 years C = P+(P/Q/Q)
continued on next post...

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Help Me!!

Then a long equation can be written with thirty P's and sixty Q's in it and set it equal to $150000.
I don't know how to solve it, though.
C=P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+
   (P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P+(P/Q/Q)/Q/Q
     )/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q
      )/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q
       )/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q)/Q/Q at the
beginning of the loan, so C = $150000 in above equation.
I don't know how to solve this though, sorry

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igloo myrtilles fourmis

dublet

Member

Registered: 2005-12-16

Posts: 16

Re: Help Me!!

If you went to the bank for a 15 year, $150,000 mortgage at an interest rate of 6.75% pa, compounding quarterly and you wanted to give payments semi-annually how much would each of your payments have to be over the 15 years?

Answer:$8,057.37

You can solve this by a recurrence relation.

Out of this recurrence, you can formulate the following formula

Now for f(15) this is 399585.399585.31053
Following that, the total payments formula can easily be described as

Which leads to p(15) = f(15) - 150000 = 399585.399585.31053 - 150000 = 249585.31053.

Dividing that over 15 * 2 is 30 payments would be 8319.51035 per payment.

The difference is in the fact that my formula doesn't account for the decreasing of the overall amount by your payment. Left as exercise for the reader.

Last edited by dublet (2005-12-16 03:26:25)

Ms. Bitters

Member

Registered: 2008-07-31

Posts: 19

Re: Help Me!!

If you went to the bank for a 15 year, $150,000 mortgage at an interest rate of 6.75% pa, compounding quarterly and you wanted to give payments semi-annually how much would each of your payments have to be over the 15 years?

Answer:$8,057.37

The general annuity formula for the present value of an ordinary annuity (i.e. end of interval payment) is given by

where
A = $150,000 (amount of mortgage loan)
R = semi-annual payments (to be determined)
j = .0675 (nominal interest rate)
m = 4 (interest period, as in quarterly)
t = 15 years (term of loan)
p = 2 (payment interval, as in semi-annually)

Solving for R gives us: R ≈ 8,057.36905 or $8,057.37.
Thus, your answer checks out.

Last edited by Ms. Bitters (2008-08-25 07:25:26)

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