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EGR 312 - 81 Nominal -vs- Effective Interest Rates Nominal interest rate, r, is an interest rate that does not include any consideration of compounding. This rate is often referred to as the Annual Percentage Rate (APR). r = interest rate per period x number of periods Effective interest rate is the actual rate that applies for a stated period of time. The effective interest rate is commonly expressed on an annual basis as the effective annual interest, i a. This rate is often referred to as the Annual Percentage Yield (APY).

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EGR 312 - 82 Nominal -vs- Effective Interest Rates The following are nominal rate statements: Nominal Rate (r) Time Period (t) Compounding Period (CP) 1)12% interest per year, compounded monthly. 2)12% interest per year, compounded quarterly. 3)3% interest per quarter, compounded monthly. What are the corresponding effective annual interest rates?

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EGR 312 - 83 Nominal -vs- Effective Interest Rates Corresponding effective annual interest rates: Let compounding frequency, m, be the number of time the compounding occurs within the time period, t. 1)12% interest per year, compounded monthly. m = 12 Effective rate per CP, i CP = r/m = 1% (per month). Effective annual rate, i e = ________________ In general, i e = __________________

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EGR 312 - 84 Nominal -vs- Effective Interest Rates Corresponding effective annual interest rates: 2)12% interest per year, compounded quarterly. m = 4 Effective rate per CP, i CP = r/m = _____________ Effective annual rate, i e = _____________________ 3)3% interest per quarter, compounded monthly. m = _____ Effective rate per CP, i CP = __________________ Effective annual rate, i e = ___________________________

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EGR 312 - 85 Nominal -vs- Effective Interest Rates Example: You are purchasing a new home and have been quoted a 15 year 6.25% APR loan. If you take out a $100,000 mortgage using the above rates, what is your monthly payment? Compound period __________ i CP = __________________________ n = ____________________________ A = ___________________________

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EGR 312 - 86 Determining m Given a stated APR and APY can you determine the compounding frequency? Example: A Certificate of Deposit has a stated APR of 8% with an Annual Yield of 8.3%. What is the compounding period? Compound PeriodEffective Annual Interest 1 day________________________ 1 week________________________ 1 month________________________ 6 months________________________

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EGR 312 - 87 Effective interest rates for any time period Let PP represent the payment period (period of time between cash flows.) And m is the number of compounding periods per payment period. –Effective i = (1+r/m) m –1 –Where, r = nominal interest rate per payment period, PP. m = number of compounding periods per payment period.

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EGR 312 - 88 Other Examples If cash flows are received on a semi-annual basis, what is the effective semi-annual interest rate under the following conditions: a)9% per year, compounded quarterly: Effective i sa = ____________________ b)3% per quarter, compounded quarterly: Effective i sa = ____________________ c)8.8% per year, compounded monthly. Effective i sa = ____________________

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EGR 312 - 89 Equivalence Relations Example : Consider the following cash flow. Find the present worth if the cash flows earn a) 10% per year compounded quarterly, or b) 9% per year compounded monthly. (Time in Years) 1 2 3 4 5 6 7 $500 $700 $300

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EGR 312 - 810 Equivalence Relations Example: a) 10% per year compounded quarterly (Time in Years) i a = _______________________ P = _______________________________ 1 2 3 4 5 6 7 $500 $700 $300

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EGR 312 - 811 Equivalence Relations Example: b) 9% per year compounded monthly. (Time in Years) i a = ____________________ P = _____________________ 1 2 3 4 5 6 7 $500 $700 $300

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EGR 312 - 812 Equivalence Relations (PP > CP) Find P for the following in standard factor expressions : Cash Flow Interest RateStandard Notation $500 semi–annually 16% per year, P = for 5 years compounded semi-annually $75 monthly for 24% per year, P = 3 years compound monthly $180 quarterly for 5% per quarter P = 15 years $25 per month 1% per month P = increase for 4 years $5000 per quarter 1% per month P = for 6 years

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EGR 312 - 813 Equivalence Relations (PP < CP) If payments occur more frequently than the compounding period, do these payments compound within the compounding period? Answer: Depends

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EGR 312 - 814 Equivalence Relations (PP < CP) No compounding within compound period: All deposits are regarded as occurring at the end of the compounding period. All withdrawals are regarded as occurring at the beginning of the period.

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EGR 312 - 815 Equivalence Relations (PP < CP) No compounding within compound period: Example: A company has the following monthly cash flows. If the company expects an ROR of 12% per year, compounded quarterly, what is the present value of the cash flows? 1 2 3 4 5 6 7 8 9 10 11 12 $500 $700 $300 P $800 $600 $400 $500 $700 $500 i = 12%/yr cpd. qtly withdrawals deposits

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EGR 312 - 816 Equivalence Relations (PP < CP) No compounding within compound period: => $500 $700 $300 P $800 $600 $400 $500 $700 $500 P 1 2 3 4 5 6 7 8 9 10 11 12 i = 12%/yr cpd. qtly withdrawals deposits

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EGR 312 - 817 Equivalence Relations (PP < CP) No compounding within compound period: P = ____________________________________ (Hint: we can now look at quarters …) 1 2 3 4 5 6 7 8 9 10 11 12 $500 $700 P $400 $500 $1000 $500 $1400 i = 12%/yr cpd. qtly

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EGR 312 - 818 Equivalence Relations (PP < CP) With interperiod compounding: If interest is compounded within the period, treat interest on cash flows the same as the treatment of nominal interest rates.

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EGR 312 - 819 Equivalence Relations (PP < CP) With interperiod compounding: Example: A company has the following monthly cash flows. If the company expects an ROR of 12% per year, compounded quarterly, what is the present value of the cash flows? 1 2 3 4 5 6 7 8 9 10 11 12 $500 $700 $300 P $800 $600 $400 $500 $700 $500

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EGR 312 - 820 Equivalence Relations (PP < CP) With interperiod compounding: For our example: Interest is compounded monthly at the rate of 1%. P = $800(P/F,1%,1) + $600(P/F,1%,2) - $500(P/F,1%,3) + etc…. $500 $700 $300 P $800 $600 $400 $500 $700 $500 1 2 3 4 5 6 7 8 9 10 11 12

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EGR 312 - 821 Continuous compounding Recall, Effective i = (1+r/m) m –1 Where m = number of compounding periods per payment period. As m approaches infinity, –i = e r – 1 Example: A 15% APR compounded continuously is effectively: i =_____________________

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