Download presentation
Presentation is loading. Please wait.
Published byGary Carter Modified over 9 years ago
1
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-1 Developed By: Dr. Don Smith, P.E. Department of Industrial Engineering Texas A&M University College Station, Texas Executive Summary Version Chapter 4 Nominal and Effective Interest
2
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-2 LEARNING OBJECTIVES 1.Nominal and effective interest rates 2.Effective annual interest rates 3.Effective interest rates 4.Compare PP and CP 5.Single amounts with PP ≥ CP 6.Series with PP ≥ CP 7.Single and series with PP < CP 8.Continuous compounding 9.Varying interest rates Notation: CP = Compounding Period PP = Payment Period
3
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-3 Sct 4.1 Nominal and Effective Interest Rate Statements Review simple interest and compound interest definitions (from Chapter 1) Compound Interest – Interest computed on interest For a given interest period The time standard for interest computations – One Year
4
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-4 Nominal Rate of Interest Nominal interest rate definition: An interest rate that does not include any consideration of compounding For example, 8% per year is a nominal rate
5
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-5 Effective Interest Rate Definition: The effective interest rate is the actual rate that applies for a stated period of time. The compounding of interest during the time period of the corresponding nominal rate is accounted for by the effective interest rate. The effective rate is commonly expressed on an annual basis denoted as “ i a ” All interest formulas, factors, tabulated values, and spreadsheet relations must have the effective interest rate to properly account for the time value of money.
6
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-6 Three Time Based Units Time Period – The period over which the interest is expressed (always stated). Ex: “1% per month” Compounding Period (CP) – The shortest time unit over which interest is charged or earned. Ex: “8% per year, compounded monthly” Compounding Frequency – The number of times m that compounding occurs within time period t. Ex: “1% per month, compounded monthly” has m = 1 Ex: “10% per year, compounded monthly” has m = 12 One Year is segmented into: 365 days, 52 weeks, 12 months One quarter is: 3 months with 4 quarters/year
7
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-7 The Effective Rate Per CP The Effective rate per compounding period (CP) is: Ex: r = 9% per year, compounded monthly: m = 12 …….(12 months in a year) i per month = 0.09/12 = 0.0075 or 0.75%/month
8
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-8 Two Common Forms of Quotation Two types of interest quotation: 1. Quotation using a Nominal Interest Rate 2. Quoting using an Effective Interest Rate Nominal and Effective interest rates are common in business, finance, and engineering economy Each type must be understood in order to solve problems where interest is stated in various ways
9
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-9 Definition of a Nominal Interest Rate Nominal interest rate definition: An interest rate that does not include any consideration of compounding Means “in name only”, “not the true, effective rate” … 8% per year, compounded monthly 8% is NOT the true effective rate (per year) 8% represents the nominal rate Effective rate will consider the monthly compounding
10
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-10 Examples of Nominal Interest Rates 1.5% per month for 24 months Same as: (1.5%)(24) = 36% per 24 months 1.5% per month for 12 months Same as: (1.5%)(12 months) = 18%/year 1.5% per 6-month period for 1 year Same as: (1.5%)(2 six-month periods) = 3% per year 1% per week for 1 year Same as: (1%)(52 weeks) = 52% per year
11
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-11 Sct 4.2 Effective Annual Interest Rate r = nominal interest rate per year m = number of compounding periods per year i = effective interest rate per compounding period (CP) = r/m i a = effective interest rate per year r/year = eff i / CP ) X (CP / year) =(i)X(m)
12
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-12 Sct 4.3 Effective Interest Rates for Any Time Period How to calculate true, effective, annual interest rates. We assume the year is the standard of measure for time. The year can be comprised of various numbers of compounding periods (within the year). Equation [4.8] in the text is the effective interest rate relation
13
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-13 Example: Calculating the Effective Rate Example: Calculating the Effective Rate Interest is 8% per year, compounded quarterly What is the effective annual interest rate? Use Equation [4.8] with r = 0.08, m = 4 Effective i = (1 + 0.08/4) 4 – 1 = (1.02) 4 – 1 = 0.0824 or 8.24%/year
14
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-14 Sct 4.4 Equivalence Relations: Lengths of Payment Period (PP) and Compounding Period (CP) To be considered: Frequency of cash flows may or may not equal the frequency of interest compounding If the frequency of the cash flow equals the frequency of the interest compounding – No Problem! If not, must make adjustments
15
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-15 Situations Situation Text Reference PP = CPSections 4.5 and 4.6 PP > CPSections 4.5 and 4.6 PP < CPSection 4.7
16
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-16 Sct 4.5 Equivalence Relation: Single Amounts with PP ≥ CP There are only single amount cash flows, that is, P and F values To determine P or F using P = F(P/F,i,n)or F = P(F/P,i,n), there are two equivalent methods to determine i and n in the factors. Method 1. For the effective interest rate, i, in the factor: Determine i over the CP using i= r/m For the total number of periods, n, in the factor: Determine the number of CP between occurrence of P and F values using n = (m)(number of payment periods)
17
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-17 Sct 4.5 Equivalence Relation: Single Amounts with PP ≥ CP There are only single amount cash flows, that is, P and F values To determine P or F using P = F(P/F,i,n) or F = P(F/P,i,n), there are two equivalent methods to determine i and n in the factors. Method 2. Find the effective interest rate for the time period of the nominal rate using effective i formula, Eq. [4.8] Set n to the number of periods in the nominal rate statement
18
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-18 Single Amounts: Numerical Example Using Method 1 Find future worth in 5 years if $5000 now earns interest at 6% per year, compounded monthly. Effective i per month is i = 6%/12 = 0.5% Total number of CP for year and m = 12 times per year is n = (12)(5) = 60 periods F = 5000(F/P,0.5%,60) = 5000(1.3489) = $6744
19
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-19 Sct 4.6 Equivalence Relations: Series with PP ≥ CP When cash flows involve a series (A, G, or g) the PP is defined by the frequency of the cash flows IF PP ≥ CP… Calculate the effective i per payment period Apply the correct n for the total number of payments periods.
20
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-20 Series: Numerical Example A = $500 every 6 months F 7 = ? PP > CP since PP = 6-months and CP = quarter Calculate effective i per PP of 6-months i 6-months means adjusting r to fit the PP r = 20% per year, compounded quarterly 0 1 2 3 4 5 6 7 Years
21
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-21 Series: Numerical Example Adjusting the interest rate r = 20% per year, compounded quarterly i/qtr = 0.20/4 = 0.05 = 5% per quarter 2 quarters in the 6-month payment period Effective i = (1.05) 2 – 1 = 10.25% per 6-month Now, the interest matches the payment period Finding F year 7 = F period 14 F = 500(F/A,10.25%,14) = 500(28.4891) = $14,244.50
22
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-22 Sct 4.7 Single Amounts and Series with PP < CP This situation is different from the last where PP ≥ CP Here, PP is less than the compounding period, CP Raises questions of how interperiod compounding is handled Study Example 4.10
23
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-23 Sct 4.8 Effective Interest Rate for Continuous Compounding Recall that effective i = (1 + r/m) m – 1 What happens if the compounding frequency, m, approaches infinity? This means an infinite number of compounding periods within a payment period, and The time between compounding approaches “0” A limiting value of i will be approached for a given value of r
24
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-24 Derivation of Continuous Compounding Effective Rate Rewrite the effective i relation as Now, examine the impact of letting “m” approach infinity. This requires taking the limit of the above expression as m ∞ Remember the definition of the number e
25
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-25 Derivation of Continuous Compounding Effective Rate So that: The effective i when interest is compounded continuously is then: Effective i = e r – 1 To find the equivalent nominal rate given i when interest is compounded continuously, apply:
26
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-26 Sct 4.9 Interest Rates That Vary Over Time In practice, interest rates do not stay the same over time unless by contractual obligation. There can exist “variation” of interest rates over time – quite normal! If required, how is this situation handled?
27
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-27 Varying Rates: Finding the Present Worth To find the present worth: Bring each cash flow amount back to the desired point in time at the interest rate for each period according to: P = F 1 (P/F,i 1,1) + F 2 (P/F,i 1,1)(P/F,i 2,1) + … + F n (P/F,i 1,1)(P/F,i 2,1)(P/F,i 3,1)…(P/F,i n,1) This process can get computationally involved!
28
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-28 Varying Rates: Observations We seldom evaluate problem models with varying interest rates except in special cases. If required, it is best to build a spreadsheet model It can be a cumbersome task to perform
29
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-29 Chapter Summary Many applications use and apply nominal and effective compounding Given a nominal rate – must get the interest rate to match the frequency of the payments Apply the effective interest rate per payment period When comparing interest rates over different payment and compounding periods, must calculate the effective i to correctly compare P, F or A values
30
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-30 Chapter Summary - continued All time value of money interest factors require use of an effective (true) periodic interest rate The interest rate, i, and the payment or cash flow periods must have the same time unit One may encounter varying interest rates. An exact answer requires a sequence of interest rates for each period
31
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 4-31 Chapter 4 End of Slide Set
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.