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SS.02.6 - Mortgages

MCR3U – Mr. Santowski

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(A) Terms Related to Mortgages

• a mortgage is special loan that is repaid over a longer period of time

• the amortization period refers to the length of time that you have to repay the mortgage

• the stated interest rates for mortgages (say 6%) means that the compounding period is semi-annual

•  so if you make monthly payments, then our payment schedule is not the same as the compounding period ==> hence we get into a concept called equivalent rates

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(B) Equivalent Rates

• two interest rates are equivalent if they yield the same amounts at the end of one year (or at the end of any number of years)

• ex. You invest $1 in an account that earns 10% compounded semi-annually. Likewise, you invest $1 in an account that compounds the interest monthly. If the two future values are to be the same in half a year, determine the interest rate in the second investment.

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(B) Equivalent Rates

• Investment #1• A = P(1 + i)n

• A = 1(1 + 0.05)1

• A = $1.05

• Investment #2• A = P(1 + i)n

• 1.05 = 1(1 + i)6

• (1.05)1/6 = (1 + i)• 1.008164846 = 1 + i• 0.008164846 = i

• So when compounding, a monthly interest rate of 0.8164846% is equivalent to a semi-annual rate of 5%

• Or a rate of 10% compounded semi-annually is equivalent to a rate of 12(0.8164846) = 9.798%

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(C) Applying Equivalent Rates to Mortgages

• since mortgages are loans wherein the money is loaned “in the present”, we have to use the present value formula of an annuity

• ex 1. Determine the monthly payments on a $150,000 mortgage amortized over 25 years if your terms are 6.5%

• (i) Step 1 is to determine the equivalent monthly interest rate (since by implication, the 6.5% is compounded semi-annually)

• 1.0325 = (1 + i)6

• i = (1.0325)1/6 - 1• i = 0.00534474 (Or an annual rate of 6.414%)

• (ii) Step 2 is to simply use the formula to find R• 150,000 = R x [(1 - (1.00534474)-300)/0.00534474]• R = 1004.74

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(D) Using the TVM Solver on the GC

• We can analyze the same mortgage using the GDC

• (1) Hit the APPS key• (2) Select 1:Finance• (3) Select 1:TVM Solver• (4) set N = 300 (why?)• (5) set I%= 6.5• (6) set PV = 150000• (7) set PMT to 0• (8) set FV = 0 (why?)• (9) set P/Y = 12 (why?)• (10) set C/Y = 2 (why?)• (11) move cursor to PMT• (12) hit ALPHA and then the ENTER key• (13) you should see the value -1004.7356...

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(E) Examples

• ex 2. If you have $750 per month to spend on a mortgage payment, what would be the total amount of your mortgage if mortgage rates were 7%.

• i = (1.035)1/6 - 1• i = 0.00575 (or 6.9% annually) • so now make some assumptions as per the amortization

period (let=s say 15, 20, 25 years) • PV = 750[(1 - (1.00575)-180,-240,-300)/0.00575]• PV = 83,963.11• PV = 97,489.88• PV = 107,079.26

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(E) Examples • ex 3. If your $125,000 mortgage is amortized over 22 years at

6.75%, how much interest have you paid when you finally have paid off your mortgage?

 • i = (1.03375)1/6 - 1• i = 0.0055475 • Now find R: • 125,000 = R x [(1 - (1.0055475)-264)/0.0055475]• R = 903.06 • so if you make 264 monthly payments of $903.06, you pay

$238,406.98 on the principal of $125,000, meaning you have paid $113,406.98 in interest over the 22 years of your mortgage.

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(F) Internet Links

• To help you out with the mortgage calculations, many banks have websites with on-line calculators, that will allow you to quickly enter the relevant numbers and immediately calculate mortgage amounts, etc...

• Some of these "calculators" are found at the following websites:

• From the Bank of Montreal ==> go to the Calculator option (in the column on the bottom right where is says "Payment Calculator“

• From the Bank of Nova Scotia ==> go to the "Mortgage Payment Calculator", halfway down the column on the right side of the page ==> you will find this calculator gives you some great additional information about your mortgage!!

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(F) Homework

• Nelson Text, p178, Q3,6,8 is work with equivalent rates• Nelson Text, p192, Q6-12 is work with mortgages

• page 170, Q1ace,2ac,4-7,12,19