1. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 1 Lecture slides to accompany Basics of Engineering Economy by Leland Blank and Anthony Tarquin Chapter 3 Nominal and Effective Interest Rates

2. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 2 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 rates  Payment period (PP) and compounding period (CP)  Single amounts with PP ≥ CP  Series with PP ≥ CP  Single and series with PP < CP  Spreadsheet use

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

4. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 4 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

5. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 5 Sec 3.2 – Effective Interest Rate Compounding frequency Period for effective i Time period for r m must equal Annualannualyear1 Semi-annualannualyear2 Quarterlyannualyear4 Monthlyannualyear12 Dailyannualyear365 Monthlysemi-annual6 months6 Weeklyquarterlyquarter12

6. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 6 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%

7. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 7 Sec 3.2 – Nominal and Effective Rates Nominal 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

8. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 8 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

9. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 9 Sec 3.2 – Nominal and Effective Rates Using Excel functions to find rates

10. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 10 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)

11. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 11 Sec 3.3 – Payment Periods (PP) and Compounding Periods (CP) Initial things to observe about cash flows 1.Compare length of PP with CP PP = CP PP > CP PP CP PP < CP 1.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

12. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 12 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

13. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 13 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 monthly PP = CP = month n = 36 months effective i = 1% per month Relation: F = A(F/A,1%,36)

14. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 14 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 monthly PP = quarter and CP = month → PP > CP n = 24 quarters i = 1% per month or 3% per quarter m = 3 CP per quarter effective i per quarter = (1 + 0.03/3) 3 – 1 = 3.03% Relation: F = A(F/A,3.03%,24)

15. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 15 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 →)

16. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 16 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

17. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 17 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

18. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 18 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

19. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 19 Sec 3.6 – Equivalence with Series and PP < CP Moving cash flows turns top cash flow diagram into bottom Qtr 1 Qtr 2 Qtr 3 Qtr4

20. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 20 Sec 3.7 – Spreadsheet Usage Spreadsheet 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

21. © 2008 McGraw-Hill All rights reserved Slide to accompany Blank and Tarquin Basics of Engineering Economy, 2008 3 - 21 Sec 3.7 – Spreadsheet Usage