Nominal vs Effective Interest Rates Nominal interest rate

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Nominal -vs- Effective Interest Rates Nominal interest rate, r, is an interest rate that

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 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). EGR 312 - 8 1

Nominal -vs- Effective Interest Rates The following are nominal rate statements: Nominal Rate (r)

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? EGR 312 - 8 2

Nominal -vs- Effective Interest Rates Corresponding effective annual interest rates: Let compounding frequency, m,

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, ie = ________ In general, ie = _________ EGR 312 - 8 3

Nominal -vs- Effective Interest Rates Corresponding effective annual interest rates: 2) 12% interest per

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 , ie = ___________ 3) 3% interest per quarter, compounded monthly. m = _____ Effective rate per CP, i. CP = _________ Effective annual rate, ie = ______________ EGR 312 - 8 4

Nominal -vs- Effective Interest Rates Example: You are purchasing a new home and have

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 = ______________ EGR 312 - 8 5

Determining m • Given a stated APR and APY can you determine the compounding

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 Period Effective Annual Interest 1 day ____________ 1 week ____________ 1 month ____________ 6 months ____________ EGR 312 - 8 6

Effective interest rates for any time period • Let PP represent the payment period

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. – – EGR 312 - 8 Effective i = (1+r/m)m – 1 Where, r = nominal interest rate per payment period, PP. m = number of compounding periods per payment period. 7

Other Examples • If cash flows are received on a semi-annual basis, what is

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 isa = __________ b) 3% per quarter, compounded quarterly: Effective isa = __________ c) 8. 8% per year, compounded monthly. Effective isa = __________ EGR 312 - 8 8

Equivalence Relations • Example: Consider the following cash flow. Find the present worth if

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. 1 2 3 4 5 6 7 (Time in Years) $300 $500 $700 EGR 312 - 8 9

Equivalence Relations Example: a) 10% per year compounded quarterly 1 2 3 4 5

Equivalence Relations Example: a) 10% per year compounded quarterly 1 2 3 4 5 6 7 (Time in Years) $300 $500 $700 ia = ____________ P = ________________ EGR 312 - 8 10

Equivalence Relations Example: b) 9% per year compounded monthly. (Time in Years) 1 2

Equivalence Relations Example: b) 9% per year compounded monthly. (Time in Years) 1 2 3 4 5 6 7 $300 $500 $700 ia = __________ P = ___________ EGR 312 - 8 11

Equivalence Relations (PP > CP) Find P for the following in standard factor expressions

Equivalence Relations (PP > CP) Find P for the following in standard factor expressions : Cash Flow $500 semi–annually for 5 years $75 monthly for 3 years $180 quarterly for 15 years $25 per month increase for 4 years $5000 per quarter for 6 years EGR 312 - 8 Interest Rate Standard Notation 16% per year, P= compounded semi-annually 24% per year, P= compound monthly 5% per quarter P= 1% per month P= 12

Equivalence Relations (PP < CP) • If payments occur more frequently than the compounding

Equivalence Relations (PP < CP) • If payments occur more frequently than the compounding period, do these payments compound within the compounding period? Answer: Depends EGR 312 - 8 13

Equivalence Relations (PP < CP) No compounding within compound period: • All deposits are

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. EGR 312 - 8 14

Equivalence Relations (PP < CP) No compounding within compound period: Example: A company has

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? i = 12%/yr cpd. qtly $800 P $600 1 2 3 $400 4 5 $500 6 7 8 9 10 11 $300 $500 $700 EGR 312 - 8 $500 withdrawals 12 $500 deposits $700 15

Equivalence Relations (PP < CP) No compounding within compound period: i = 12%/yr cpd.

Equivalence Relations (PP < CP) No compounding within compound period: i = 12%/yr cpd. qtly $800 P $600 1 2 3 $400 4 5 $500 withdrawals 6 7 8 9 10 11 $300 $500 $700 => $500 deposits $700 P 1 EGR 312 - 8 12 2 3 4 5 6 7 8 9 10 11 12 16

Equivalence Relations (PP < CP) No compounding within compound period: $1400 i = 12%/yr

Equivalence Relations (PP < CP) No compounding within compound period: $1400 i = 12%/yr cpd. qtly P $500 $400 1 2 3 4 5 6 $500 7 8 9 10 11 12 $500 $700 $1000 P = __________________ (Hint: we can now look at quarters …) EGR 312 - 8 17

Equivalence Relations (PP < CP) With interperiod compounding: • If interest is compounded within

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. EGR 312 - 8 18

Equivalence Relations (PP < CP) With interperiod compounding: Example: A company has the following

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 P $600 1 2 3 $400 4 5 $500 6 7 8 9 10 11 $300 $500 $700 EGR 312 - 8 $500 12 $500 $700 19

Equivalence Relations (PP < CP) With interperiod compounding: For our example: Interest is compounded

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…. $800 P $600 1 2 3 $400 4 5 $500 6 7 8 9 10 11 $300 $500 EGR 312 - 8 $700 $500 $700 12 $500 20

Continuous compounding • Recall, Effective i = (1+r/m)m – 1 Where m = number

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 =___________ EGR 312 - 8 21