Nominal -vs- Effective Interest RatesSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Nominal -vs- Effective Interest RatesNominal 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 periodsEffective 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, ia. This rate is often referred to as the Annual Percentage Yield (APY).Nominal -vs- Effective Interest RatesThe 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?Nominal -vs- Effective Interest RatesCorresponding 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 = 12Effective rate per CP, iCP = r/m = 1% (per month).Effective annual rate = (1+ iCP)12 –1 = 12.68%Nominal -vs- Effective Interest RatesCorresponding effective annual interest rates:2) 12% interest per year, compounded quarterly.m = 4Effective rate per CP, iCP = r/m = 3% (per quarter).Effective annual rate = (1+ .03)4 –1 = 12.55%3) 3% interest per quarter, compounded monthly.m = 3Effective rate per CP, iCP = r/m = 1% (per month).Effective annual rate = (1+ .01)12 –1 = 12.68%Nominal -vs- Effective Interest RatesExample: 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 – monthlyiCP = 6.25% / 12 = .521% (per month)n = 15(years) x 12(months/year) = 180 monthsA = $100,000(A/P, .521%, 180).Nominal -vs- Effective Interest RatesDetermining n: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 Period Effective Annual Interest1 day 8% / 365 = .022% / dayia = (1+.00022)365 – 1 = 8.36%1 week 8% / 52 = .15% / weekia = (1+.0015)52 – 1 = 8.322%1 month 8% / 12 = .67% / monthia = (1+.0067)12 – 1 = 8.30%6 months 8% / 2 = 4% / semi-annualia = (1+.04)2 – 1 = 8.16%Nominal -vs- Effective Interest RatesEffective 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 –1r = nominal interest rate per payment period, PP.m = number of compounding periods per payment period.Nominal -vs- Effective Interest RatesEffective interest rates for any time period:Example: 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 isa = (1+4.5%/2)2 – 1 = 4.55%b) 3% per quarter, compounded quarterly:Nominal is 6% per semi-annual.Effective isa = (1+6%/2)2 – 1 = 6.09%c) 8.8% per year, compounded monthly.Effective isa = (1+4.4%/6)6 – 1 = 4.48%Nominal -vs- Effective Interest RatesEquivalence 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$300Nominal -vs- Effective Interest RatesEquivalence Relations:Example: a) 10% per year compounded quarterly (Time in Years)ia = (1+10%/4)4 –1 = 10.38%P = $500(P/F,10.38%,3) + $700 (P/F,10.38%,6) + $300 (P/F,10.38%,7)1 2 3 4 5 6 7$500$700$300Nominal -vs- Effective Interest RatesEquivalence Relations:Example: a) 9% per year compounded monthly. (Time in Years)ia = (1+9%/12)12 –1 = 9.38%P = $500(P/F,9.38%,3) + $700 (P/F,9.38%,6) + $300 (P/F,9.38%,7)1 2 3 4 5 6 7$500$700$300Nominal -vs- Effective Interest RatesEquivalence Relations (PP > CP):Find P for the following in standard factor expressions :Cash Flow Interest Rate Standard 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 yearsNominal -vs- Effective Interest RatesEquivalence Relations (PP > CP):Find P for the following in standard factor expressions :Cash Flow Interest Rate Standard Notation$500 semi–annually 16% per year, P = $500(P/A,8%,10) for 5 years compounded semi-annually $75 monthly for 24% per year, P = $75(P/A,2%,36) 3 years compound monthly$180 quarterly for 5% per quarter P = $180(P/A,5%,60) 15 years$25 per month 1% per month P = $25(P/G,1%,48) + increase for 4 years $25(P/A,1%,48) $5000 per quarter 1% per month P = $5000(P/A,3.03%,24) for 6 yearsNominal -vs- Effective Interest RatesEquivalence Relations (PP < CP):If payments occur more frequently than the compounding period, do these payments compound within the compounding period?Answer: DependsNominal -vs- Effective Interest RatesEquivalence 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.Nominal -vs- Effective Interest RatesEquivalence 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$500$500Nominal -vs-