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

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

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

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

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.

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

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 = or 0.75%/month

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

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

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

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)

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

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 = ( /4) 4 – 1 = (1.02) 4 – 1 = or 8.24%/year

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

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

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)

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

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

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.

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 Years

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( ) = $14,244.50

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

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

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

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:

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?

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!

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

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

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

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