1 Interest Rate Conversions Definitions:  Length of Compounding period (n) How often (in years) is interest computed (calculated)? e.g. if interest is compounded…...annually -> n = 1 …semi-annually -> n = 1/2 …every decade -> n = 10  Effective Annual Interest Rate (EAR): Actual percentage interest (simple & compounded) charged during a year  Stated (nominal) Annual Interest Rate (r n ) Only simple percentage interest charged during a year (What the bank usually quotes) Supplement to Chapter 4

Jacoby, Stangeland and Wajeeh, Interest Rate Conversions

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Jacoby, Stangeland and Wajeeh, First, clear previous data, and check that your calculator is set to 1 P/YR: The display should show: 1 P_Yr Input data (based on example 2 (ii)): Interest Rate Conversions in your HP 10B Calculator Key in EAR Key in number of compounding periods per year (1/n) 15 Display should show: PV EFF% Yellow 2 Compute stated (nominal) annual rate compounded s.a. I/YR NOM% Yellow When finished - don’t forget to set your calculator to 1P_Yr Yellow C C ALL PMT P/YR

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Jacoby, Stangeland and Wajeeh, Canadian Mortgages What is special about Canadian Mortgages: uCanadian banks quote the annual interest compounded semi-annually for mortgages, although interest is calculated (compounded) every month uThe terms of the mortgage are usually renegotiated during the term of the mortgage. For example, the interest of my 25-year mortgage will be negotiated 5 years after the initiation of my mortgage.

12 Canadian Mortgages Q. You have negotiated a 25-year, $100,000 mortgage at a rate of 7.4% per year compounded semi-annually with the Toronto-Dominion Bank. To answer most mortgage questions, we first have to convert the quoted (stated) annual interest compounded semi-annually, to the actual interest rate charged each month. That is, we need to calculate the Effective monthly Period rate, EPR 1/12. We have: r 1/2 = 7.4%. By equation (3): EPR 1/2 =0.5 % 0.074= Thus, using equation (1), we get: Equation (4) gives us EPR 1/12 :

13 ÷ Calculating the Effective Monthly Mortgage Rate Key in stated (nominal) annual rate compounded semiannually Key in 2 compounding periods per year Display should show: % Yellow 2 Compute EAR Yellow 7.4 Key in 12 compounding periods per year 12 Yellow Display should show: % Yellow Compute stated (nominal) annual rate compounded monthly Display should show: % Divide by 12 to get the Effective Monthly Rate 12 = RCL STO Yellow Store this result for future calculations 1 I/YR NOM% PMT P/YR PV EFF% PMT P/YR I/YR NOM%

14 We are now ready to solve mortgage problems: Q-a What is the monthly payment on the above mortgage? A-a We have: PV 0 = $100,000, EPR 1/12 = %, and T=25 % 12=300 months. Since a mortgage is an annuity with equal monthly payments, we use the present value of annuity formula: Solving for PMT, we get the monthly payment on the mortgage: PMT = $725.28

Jacoby, Stangeland and Wajeeh, Set your calculator to 1 P/YR: Input data (based on above PV example) Mortgage Payments in your HP 10B Calculator I/YR N PV Key in this rate Key in number of monthly periods Key in PV PMT Compute monthly PMT Display should show: Recall Effective Monthly Rate RCL 1 Display should show: % 300 +/- 100,000 Key in 1 compounding periods per year 1 Yellow PMT P/YR

16 Q-b How much of the first three mortgage payment, goes toward principal and interest? A-b In general, to calculate the interest portion of each monthly payment, use: EPR 1/12 % (Balance of Principal at the Beginning of Month) The principal portion of each monthly payment is given by: PMT - Interest Payment For the first three payments:

Jacoby, Stangeland and Wajeeh, Q-c Assuming that the mortgage rate remains at 7.4% for the remaining time of the mortgage, what is the total amount of interest paid during the 25-year period? A-c The total amount of interest to be paid is given by: Total Payments - Total Principal Payments Since after 25 years the entire principal will be paid, the value total principal payments is $100,000. With 300 monthly payments of $ each, we get a total amount of interest to be paid: Total Payments - Total Principal Payments = (300 % ) - 100,000 = 217, ,000 = $117,584

18 Q-d Assuming that the mortgage rate remains at 7.4% for the remaining time of the mortgage, after you have paid two-thirds of your monthly payments, what is the amount still remaining to be paid on the mortgage? A-d Two-thirds of your monthly payments will be paid right after the 200 th payment. The remaining value of the mortgage at that time is given by the present value of the remaining 100 monthly payments. We have: EPR 1/12 = %, PMT=$ and T=100 months. We use the present value of annuity formula to get:

Jacoby, Stangeland and Wajeeh, Continuous Compounding uEAR of continuous compounding = e r - 1 uThe TVM Relationship for continuous compounding: FV t+T = PV t % e (r % T) where r is the stated annual continuously compounded interest rate Note: r = ln(1+EAR)

20 Q.Your bank offers you a certificate which pays you $100 at the end of each of the following three years, carrying a stated annual continuously compounded interests of 8%. What is the present value of this security? A.A Time Line: To find the PV, we use the present value of annuity formula with an EAR of: EAR = e r - 1 = e = = % Time: Cashflow: 100 r = 8% c.c.