© 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 1 Lecture slides to accompany Basics.

Slide to accompany Blank and TarquinBasics of Engineering Economy, 2008 3 - 1 © 2008 McGraw-Hill

All rights reserved

Lecture slides to accompanyLecture slides to accompany

Basics of Engineering EconomyBasics of Engineering Economybyby

Leland Blank and Anthony Tarquin Leland Blank and Anthony Tarquin

Chapter 3Chapter 3

Nominal and Effective Nominal and Effective

Interest RatesInterest Rates

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Chapter 3 – Nominal & Effective Interest

PURPOSE

Perform calculations for interest rates and

cash flows that occur on a time basis

other than yearly

TOPICS

Recognize nominal and effective rates

Effective interest ratesPayment period (PP) and

compounding period (CP)Single amounts with

PP ≥ CPSeries with PP ≥ CPSingle and series with

PP < CPSpreadsheet use

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Sec 3.1 – Nominal and Effective Rate Statements

Nominal rates• Interest rate per time

period without regard to compounding frequency

• Some nominal statements:– 8% per year compounded

monthly– 2% per month compounded

weekly– 8% per year compounded

quarterly– 5% per quarter compounded

monthly

Effective rates• Interest rate is compounded

more frequently than once per year

• Some statements indicating an effective rate:– 15% per year– effective 8.3% per year

compounded monthly– 2% per month compounded

monthly– effective 1% per week

compounded continuously

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Sec 3.2 – Effective Interest Rate Formula

• i = effective rate per some stated period, e.g., quarterly, annually

• r = nominal rate for same time period

• m = frequency of compounding per same time period

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Sec 3.2 – Effective Interest Rate

Compounding frequency

Period for effective i

Time period for r

m must equal

Annual annual year 1

Semi-annual annual year 2

Quarterly annual year 4

Monthly annual year 12

Daily annual year 365

Monthly semi-annual 6 months 6

Weekly quarterly quarter 12

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Sec 3.2 – Effective Interest Rate

Example: Find i per year, if m = 4 for quarterly compounding, and

r = 12% per year

Stated period for i is YEAR

i = (1 + 0.12/4)4 - 1 = 12.55%

rEffective i = (1+ ) 1

mm

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Sec 3.2 – Nominal and Effective RatesNominal

r = rate/period × periods

Example: Rate is 1.5% per month. Determine nominal rate per quarter, year, and over 2 years

Qtr: r = 1.5 × 3 mth = 4.5%

Year: r = 1.5 ×12 mth = 18% = 4.5 × 4 qtr = 18%

2 yrs: r =1.5 × 24 mth = 36% = 18 × 2 yrs = 36%

Effective

Example: Credit card rate is 1.5% per month compounded monthly. Determine effective rate per quarter and per year

Period is quarter: r = 1.5 × 3 mth = 4.5% m = 3 i = (1 + 0.045/3)3 – 1 = 4.57% per quarter

Period is year: r = 18% m = 12

i = (1 + 0.18/12)12 - 1) = 19.6% per year

rEffective i = (1+ ) 1

mm

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Sec 3.2 – Effective Interest Rate

As m → ∞, continuous compounding is approached

effective i = (℮r – 1)

Example: r = 14% per year compounded continuously

i = (℮ 0.14 - 1) = 15.03% per year

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Sec 3.2 – Nominal and Effective Rates

Using Excel functions to find rates

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Sec 3.3 – Payment Periods (PP)and Compounding Periods (CP)

• PP – how often cash flows occur• CP – how often interest in compounded• If PP = CP, no problem concerning effective i rate

Examples where effective i is involved: Monthly deposit, quarterly compounding (PP < CP)Semi-annual payment, monthly compounding (PP > CP)

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Sec 3.3 – Payment Periods (PP)and Compounding Periods (CP)

Initial things to observe about cash flows1. Compare length of PP with CP PP = CP PP > CP PP < CPPP = CP PP > CP PP < CP1. Determine types of cash flows present

• Only single amounts (P and F)• Series (A, G, g)

2. Determine correct effective i and n (same time unit on both)

Remember: An effective i rate must be used in all factors

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Sec 3.4 – Equivalence with Single Amounts

If only P and F cash flows are present, equivalence relations are

P = F(P/F, effective i per period, # of periods) [1] F = P(F/P, effective i per period, # of periods) [2]

Example: Find equivalent F in 10 years if P is $1000 now. Assume r = 12% per year compounded semi-annually.

- PP = year and CP = 6 months; period is 6 months - Only single amount cash flows - Use relation [2] above to find F

F = 1000(F/P, 6% semi-annually, 20 periods) = 1000(3.2071) = $3207

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Sec 3.5 – Equivalence with Series and PP ≥ CP

• Count number of payments. This is n• Determine effective i over same time

period as n• Use these i and n values in factors

Example: $75 per month for 3 years at 12% per year compounded monthlyPP = CP = monthn = 36 monthseffective i = 1% per month

Relation: F = A(F/A,1%,36)

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Sec 3.5 – Equivalence with Series and PP ≥ CP

• Count number of payments. This is n• Determine effective i over same time period as n• Use these i and n values in factors

Example: $5000 per quarter for 6 years at 12% per year compounded monthlyPP = quarter and CP = month → PP > CPn = 24 quartersi = 1% per month or 3% per quarterm = 3 CP per quartereffective i per quarter = (1 + 0.03/3)3 – 1 = 3.03%

Relation: F = A(F/A,3.03%,24)

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Sec 3.5 – Equivalence with Series and PP ≥ CP

0

P = $3M

• First step: Find P for n = 10 annual payments• Period is year• CP = 6 months; PP = year; PP > CP• Effective i per year = (1 + 0.08/2)2 – 1 = 8.16% Relation: P = 3M + 200,000(P/A,8.16%,10) = $4,332,400

(continued →)

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Sec 3.5 – Equivalence with Series and PP ≥ CP

0

P = $3M

• Second step: Find A for n = 20 semi-annual amounts• Period is six months• CP = 6 months; PP = 6 months; PP = CP• Effective i per 6 months = 8%/2 = 4% Relation: A = 4,332,400(A/P,4%,20) = $318,778

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Sec 3.6 – Equivalence with Series and PP < CP

Example: deposits monthly (PP) with interest compounded semi-annually (CP)

Result: PP < CP

Usually, interest is not paid on interperiod deposits

For equivalence computations: Cash flows are ‘moved’ to match CP time period

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Sec 3.6 – Equivalence with Series and PP < CP

APPROACH NORMALLY TAKEN

Move cash flows not at end of a compounding period: Deposits ( minus cash flows) - to end of period Withdrawals (plus cash flows) - to beginning of same

period (which is the end of last period)

Example (next slide): move monthly deposits to match quarterly compounding. Now, PP = CP = quarter

Find P, F or A using effective i per quarter

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Sec 3.6 – Equivalence with Series and PP < CPMoving cash flows turns top cash flow diagram into bottom

Qtr 1 Qtr 2 Qtr 3 Qtr4

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Sec 3.7 – Spreadsheet UsageSpreadsheet function format and structure:

Fine effective rate: = EFFECT(nom r%, m)Nominal r is over same time period as effective i

Find nominal rate: = NOMINAL(eff i%, m)Result of nominal is always per year

Example: Deposits are planned as follows: $1000 now, $3000 after 4 years, $1500 after 6 years. Find F after 10 years. Interest is 12% per year compounded semiannually

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Sec 3.7 – Spreadsheet Usage