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, ia. This rate is often referred to as the Annual Percentage Yield (APY).

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?

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. 12% interest per year, compounded monthly. m = 12 Effective rate per CP, iCP = r/m = 1% (per month). Effective annual rate = (1+ iCP)12 –1 = 12.68%

Nominal -vs- Effective Interest Rates Corresponding effective annual interest rates: 12% interest per year, compounded quarterly. m = 4 Effective rate per CP, iCP = r/m = 3% (per quarter). Effective annual rate = (1+ .03)4 –1 = 12.55% 3% interest per quarter, compounded monthly. m = 3 Effective rate per CP, iCP = r/m = 1% (per month). Effective annual rate = (1+ .01)12 –1 = 12.68%

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 – monthly iCP = 6.25% / 12 = .521% (per month) n = 15(years) x 12(months/year) = 180 months A = $100,000(A/P, .521%, 180).

Nominal -vs- Effective Interest Rates Determining 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 Interest 1 day 8% / 365 = .022% / day ia = (1+.00022)365 – 1 = 8.36% 1 week 8% / 52 = .15% / week ia = (1+.0015)52 – 1 = 8.322% 1 month 8% / 12 = .67% / month ia = (1+.0067)12 – 1 = 8.30% 6 months 8% / 2 = 4% / semi-annual ia = (1+.04)2 – 1 = 8.16%

Nominal -vs- Effective Interest Rates 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 r = nominal interest rate per payment period, PP. m = number of compounding periods per payment period.

Nominal -vs- Effective Interest Rates Effective 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: 9% per year, compounded quarterly: Effective isa = (1+4.5%/2)2 – 1 = 4.55% 3% per quarter, compounded quarterly: Nominal is 6% per semi-annual. Effective isa = (1+6%/2)2 – 1 = 6.09% 8.8% per year, compounded monthly. Effective isa = (1+4.4%/6)6 – 1 = 4.48%

Nominal -vs- Effective Interest Rates 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 $300 $500 $700

Nominal -vs- Effective Interest Rates Equivalence 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 $300 $500 $700

Nominal -vs- Effective Interest Rates Equivalence 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 $300 $500 $700

Nominal -vs- Effective Interest Rates Equivalence 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 years

Nominal -vs- Effective Interest Rates Equivalence 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 years

Nominal -vs- Effective Interest Rates Equivalence Relations (PP < CP): If payments occur more frequently than the compounding period, do these payments compound within the compounding period? Answer: Depends

Nominal -vs- Effective Interest Rates 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.

Nominal -vs- Effective Interest Rates 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? $800 $500 P $600 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $300 $500 $500 $500 $700 $700

Nominal -vs- Effective Interest Rates Equivalence Relations (PP < CP): No compounding within compound period: => $800 $500 P $600 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $300 $500 $1400 $500 $500 $700 $700 P $500 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $500 $700 $1000 $1000

Nominal -vs- Effective Interest Rates Equivalence Relations (PP < CP): No compounding within compound period: P = $1400 - $100(P/F,3%,1) - $200(P/F,3%,2) - $500(P/F,3%,3)- - $1000(P/F,3%,4) $1400 P $500 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $500 $700 $1000 $1000

Nominal -vs- Effective Interest Rates 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.

Nominal -vs- Effective Interest Rates 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? $800 $500 P $600 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $300 $500 $500 $500 $700 $700

Nominal -vs- Effective Interest Rates Equivalence Relations (PP < CP): With interperiod compounding: 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…. $800 $500 P $600 $500 $400 1 2 3 4 5 6 7 8 9 10 11 12 $300 $500 $500 $500 $700 $700

Nominal -vs- Effective Interest Rates Continuous compounding: Recall, Effective i = (1+r/m)m –1 Where m = number of compounding periods per payment period. As m approaches infinity, i = er – 1 Example: A 15% APR compounded continuously is effectively: i =e0.15 – 1 = 16.183%