Mortgage

A mortgage (literal translation: "death pledge") is a device developed in the common law world, whereby the ownership of property is passed from one person -- the mortgagor -- to another -- the mortgagee -- in return for the loan of money. The mortgagee is prevented from exercising his rights of ownership by the rules of equity so long as the interest on the loan is paid.

Fixed rate mortgage calculations

First the nomenclature:

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• 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:

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• 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:

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:B_0 ,

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Now calculate the balance after one period or payment:

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:B_1 = B_0 (1 + i) - P ,

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During the first period the initial balance has grown by the period interest and has been decreased by the first payment. Similarly:

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:B_2 = B_1 (1 + i) - P = B_0 (1 + i)^2 - P (1 + i) - P,

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Again:

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:B_3 = B_2 (1 + i) - P = B_0 (1 + i)^3 - P (1 + i)^2 - P (1 + i) - P,

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After n periods or payments we have:

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:B_n = B_0 (1 + i)^n - P (1 + i)^{n-1} ..... - P (1 + i)^2 - P (1 + i) - P,

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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:

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:B_0 (1 + i)^n = P ,

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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.

Related Topics:
Geometric series - Geometric sequence

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Solving for P gives:

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:P = B_0 /,

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The payment can be readily calculated to the penny with a spread sheet or scientific calculator.

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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.

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Now let's do some calculations. Mortgages are usually for 10, 15, 20 or 30 years. Interest rates used to be around 9%/year and today around 6%/year. For all calculations B0 = 1

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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.

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Alternatively, for a given payment P, it is possible to solve for the number of periods needed n to repay the loan:

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: n = leftlceil - log_{1+i}left ight ceil,

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where the n^ extrm{th} (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>iB_0. If P

Related Topics:
Reverse mortgage - Interest-only loan

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