The major disadvantages or the 15-year fixed rate mortgage are the higher monthly payments. But if saving on total interest costs and cutting the to free and clear ownership are important to you, the 15-year fixed rate mortgage is a good option. The biweekly mortgage shortens the loan term to 18 to 19 years by requiring a payment for half the monthly amount every two weeks. The biweekly payments increase the annual amount paid by about 8 percent and in effect pay 13 monthly payments(26 biweekly payments) per year. The shortened loan term decreases the total interest costs substantially. The interest costs for the biweekly mortgage are decreased even farther, however, by the application of each payment to the principal upon which the interest is calculated every 14 days. By nibbling away at the principal faster, the homeowner saves additional interest. Remember, however, that you trade lower total interest costs for fewer mortgage interest deductions on your federal income tax. Your ability to qualify for this type of loan is based on a 30-year term, and most lenders who offer this mortgage will allow the homebuyer to convert to a more traditional 30-year loan without penalty. Availability is limited on this mortgage, but it can be worth looking for.
Fixed rate mortgage calculations
First the nomenclature:
* I - The stated interest rate, for example, 5%/year. This is not the APR (annualized percentage rate).
* m - The number of periods in the time frame of I. I is usually based on a year but it could be based on any amount of time.
* i - The interest rate for the compounding period which is needed for the calculation. For example, a real property mortgage is usually based on a monthly period. In this case i=I/12 where I is based on the normal yearly period. In general i=I/m. Also I needs to be a decimal not a percent thus it also needs to be divided by 100.
* n - The total number of periods or payments. Things like mortgages usually cover multiple years.
* B - The balance, for example, the balance remaining on the mortgage at any point in time.
Mortgage Calculations:
* Let B0 be the original mortgage.
* Let B1, B2, B3 etc. be the balance after the first, second, third period respectively.
Obviously, one can think of B0 as the balance after the zeroth period namely the beginning balance.
* P - The mortgage payment.
Now let's write down the balances. First the initial balance, the amount of the mortgage:
B_0 \,
Now calculate the balance after one period or payment:
B_1 = B_0 (1 + i) - P \,
During the first period the initial balance has grown by the period interest and has been decreased by the first payment. Similarly:
B_2 = B_1 (1 + i) - P = B_0 (1 + i)^2 - P (1 + i) - P\,
Again:
B_3 = B_2 (1 + i) - P = B_0 (1 + i)^3 - P (1 + i)^2 - P (1 + i) - P\,
After n periods or payments we have:
B_n = B_0 (1 + i)^n - P (1 + i)^{n-1} ..... - P (1 + i)^2 - P (1 + i) - P\,
Bn is set equal to zero. When the mortgage is paid off the balance is zero. Now one can solve for P the payment. Rearranging gives:
B_0 (1 + i)^n = P [1 + (1 + i) + (1 + i)^2 + .... + (1 + i)^{n-1}]\,
The righthand side is a geometric series where each term is equal to the preceding term multiplied by (1 + i) which is known as the common ratio. See geometric sequence for additional details.
Solving for P gives:
P = B_0 [i(1 + i)^n]/[(1 + i)^n - 1]\,
The payment can be readily calculated to the penny with a spread sheet or scientific calculator.
Note: B0 is just a simple multiplier. Therefore one can do the calculation for a unit of currency such as a dollar and then multiply the result by the amount of the loan. In essence B0 is just a scale factor. For example think of the loan amount as my dollar where my dollar is just a currency whose exchange rate is just the loan amount difference.
First calculate (1 + i)n since it occurs in both the numerator and the denominator. Then complete the calculation for the payment P. In the first case, for each dollar of loan the payment is a little over a penny per month. Multiplying the amount of the payment P by the number of payments n gives the total amount paid. In the case with 9% interest over 15 years, for each dollar of loan the repayment is a little over a dollar and 82 cents. The 1.82 is also the ratio of the repayment amount to the amount of the loan.
Alternatively, for a given payment P, it is possible to solve for the number of periods needed n to repay the loan:
n = \left\lceil - \log_{1+i}\left[1-\frac{iB_0}{P}\right]\right\rceil\,
where the nth (final) payment will be less than or equal to the others. This is useful for loans with no prepayment penalty, and a mortgagor who wishes to repay the loan as quickly as possible (in order to minimze the interest paid). Of course, for this formula to apply, P > iB0. If P < iB0, this corresponds to a Reverse mortgage; P = iB0 corresponds to an Interest-only loan.
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