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Lecture slides to accompany Basics of Engineering Economy by Leland Blank and Anthony Tarquin

Lecture slides to accompany Basics of Engineering Economy by Leland Blank and Anthony Tarquin Chapter 3 Nominal and Effective Interest Rates. PURPOSE Perform calculations for interest rates and cash flows that occur on a time basis other than yearly. TOPICS

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Lecture slides to accompany Basics of Engineering Economy by Leland Blank and Anthony Tarquin

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  1. Lecture slides to accompany Basics of Engineering Economy by Leland Blank and Anthony Tarquin Chapter 3 Nominal and Effective Interest Rates

  2. 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 Chapter 3 – Nominal & Effective Interest

  3. 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 Sec 3.1 – Nominal and Effective Rate Statements

  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. Sec 3.2 – Effective Interest Rate

  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. 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 Sec 3.2 – Nominal and Effective Rates

  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. Sec 3.2 – Nominal and Effective Rates Using Excel functions to find rates

  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. Sec 3.3 – Payment Periods (PP)and Compounding Periods (CP) Initial things to observe about cash flows • Compare length of PP with CP PP = CP PP > CP PP < CP • Determine types of cash flows present • Only single amounts (P and F) • Series (A, G, g) • Determine correct effective i and n (same time unit on both) Remember: An effective i rate must be used in all factors

  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. 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. 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. 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. 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. 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. 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. 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

  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. Sec 3.7 – Spreadsheet Usage

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