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Interest Rate Conversions

Supplement to Chapter 4. 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

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Interest Rate Conversions

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  1. Supplement to Chapter 4 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 (rn) Only simple percentage interest charged during a year (What the bank usually quotes)

  2. Interest Rate Conversions Jacoby, Stangeland and Wajeeh, 2000

  3. Interest Rate Conversions in your HP 10B Calculator 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)): Yellow C C ALL When finished - don’t forget to set your calculator to 1P_Yr 15 Key in EAR Yellow PV EFF% Key in number of compounding periods per year (1/n) 2 Yellow PMT P/YR Compute stated (nominal) annual rate compounded s.a. Display should show: 14.47610590 Yellow I/YR NOM% Jacoby, Stangeland and Wajeeh, 2000

  4. Jacoby, Stangeland and Wajeeh, 2000

  5. Jacoby, Stangeland and Wajeeh, 2000

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  8. Canadian Mortgages What is special about Canadian Mortgages: • Canadian banks quote the annual interest compounded semi-annually for mortgages, although interest is calculated (compounded) every month • The 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. Jacoby, Stangeland and Wajeeh, 2000

  9. 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, EPR1/12. We have: r1/2= 7.4%. By equation (3): EPR1/2=0.5%0.074=0.037. Thus, using equation (1), we get: Equation (4) gives us EPR1/12:

  10. Calculating the Effective Monthly Mortgage Rate Key in stated (nominal) annual rate compounded semiannually 7.4 Yellow I/YR NOM% Key in 2 compounding periods per year 2 Yellow PMT P/YR Display should show: 7.5369% Compute EAR Yellow PV EFF% Key in 12 compounding periods per year 12 Yellow PMT P/YR Compute stated (nominal) annual rate compounded monthly Display should show: 7.28843047% Yellow I/YR NOM% Display should show: 0.607369% Divide by 12 to get the Effective Monthly Rate ÷ 12 = Store this result for future calculations Yellow RCL STO 1

  11. We are now ready to solve mortgage problems: Q-a What is the monthly payment on the above mortgage? A-a We have: PV0 = $100,000, EPR1/12=0.607369%, 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

  12. Mortgage Payments in your HP 10B Calculator Set your calculator to 1 P/YR: Input data (based on above PV example) Key in 1 compounding periods per year 1 Yellow PMT P/YR Recall Effective Monthly Rate Display should show: 0.607369% RCL 1 Key in this rate 0.607369 I/YR Key in number of monthly periods 300 N +/- 100,000 Key in PV PV Compute monthly PMT Display should show: 725.28464244 PMT Jacoby, Stangeland and Wajeeh, 2000

  13. 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: EPR1/12%(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:

  14. 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 $725.28 each, we get a total amount of interest to be paid: Total Payments - Total Principal Payments = (300%725.28) - 100,000 = 217,584 - 100,000 = $117,584 Jacoby, Stangeland and Wajeeh, 2000

  15. 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 200th payment. The remaining value of the mortgage at that time is given by the present value of the remaining 100 monthly payments. We have: EPR1/12=0.607369%, PMT=$725.28 and T=100 months. We use the present value of annuity formula to get:

  16. Continuous Compounding • EAR of continuous compounding = er - 1 • The TVM Relationship for continuous compounding: FVt+T = PVt%e(r%T) where r is the stated annual continuously compounded interest rate Note: r = ln(1+EAR) Jacoby, Stangeland and Wajeeh, 2000

  17. 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 0.08 - 1 = 0.08328707 = 8.328707%. Time: 0 1 2 3 r = 8% c.c. Cashflow: 100 100 100

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