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Feedback and Warm-Up Review

Feedback and Warm-Up Review. Feedback of your requests Cash Flow Cash Flow Diagrams Economic Equivalence. Feedback. Feedback 1: Power point on line to save toner $$$ -- done; background changed; PPT: there is a non-background option

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Feedback and Warm-Up Review

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  1. Feedback and Warm-Up Review • Feedback of your requests • Cash Flow • Cash Flow Diagrams • Economic Equivalence

  2. Feedback • Feedback 1: Power point on line to save toner $$$ -- done; background changed; • PPT: there is a non-background option • Feedback 2: More examples in class------------- yes, we also have tutorial class; • Feedback 3: Arrange projects early----------------yes, quiz review changed to project and quiz review, starting this Friday. • Important: Homepage updates……

  3. Cash Flows • The expenses and receipts due to engineering projects.

  4. Cash Flow Diagrams • The costs and benefits of engineering projects over time are summarized on a cash flow diagram. • Cash flow diagram illustrates the size, sign, and timing of individual cash flows

  5. Cash Flow Diagrams $15,000 $2000 Positive net Cash flow (receipts) $13,000 is net positive cash flow 2 1 4 5 3 Time (# of interest periods) 0 Negative net Cash Flow (payments)

  6. Economic Equivalence • We need to compare the economic worth of $. • Economic equivalence exists between cash flows if they have the same economic effect. • Convert cash flows into an • equivalent cash flow at • any point in time and compare.

  7. Topics Today • Single Sum Compounding • Annuities • Conversion for Arithmetic Gradient Series • Conversion for Geometric Gradient Series

  8. Simple Interest • The interest payment each year is found by multiplying the interest rate times the principal, I = Pi. After any n time periods, the accumulated value of money owed under simple interest, Fn, would be: • For example, $100 invested now at 9% simple interest for 8 years would yield • Nobody uses simple interest. Fn = P(1 + i*n) F8 = $100[1+0.09(8)] = $172

  9. Compound Interest • The interest payment each year, or each period, is found by multiplying the interest rate by the accumulated value of money, both principal and interest.

  10. Compound Interest • Consequently, the value for an amount P invested for n periods at i rate of interest using compound interest calculations would be: • For example, $100 invested now at 9% compound interest for 8 years would yield: • Compound interest is the basis for practically all monetary transactions. Fn = P( 1 + i )n F8 = $100( 1 + 0.09 )8 = $199

  11. Future/Present Value • FV = PV(1 + i)n. • PV = FV / (1+i)n. • Discounting is the process of translating a future value or a set of future cash flows into a present value.

  12. Calculating Present Value If promised $500,000 in 40 years, assuming 6% interest, what is the value today? (Discounting) FVn= PV(1 + i)n PV = FV/(1 + i)n PV = $500,000 (.097) PV = $48,500

  13. The Rule of 72 • Estimates how many years an investment will take to double in value • Number of years to double = 72 / annual compound interest rate • Example -- 72 / 8 = 9 therefore, it will take 9 years for an investment to double in value if it earns 8% annually • Challenge: Prove it!!!!!!!!!!!!!!!!!!!!!

  14. Example: Double Your Money!!! Quick!How long does it take to double $5,000 at a compound rate of 12% per year? Key “Rule-of-72”.

  15. Example: Double Your Money!!! Quick!How long does it take to double $5,000 at a compound rate of 12% per year? Approx. Years to Double = 72/ i% • 72 / 12% = 6 Years • [Actual Time is 6.12 Years]

  16. Single Sum Problems: Future Value Given: • Amount of deposit today (PV):$50,000 • Interest rate: 11% • Frequency of compounding: Annual • Number of periods (5 years): 5 periods What is the future value of this single sum? FVn = PV(1 + i)n $50,000 x (1.68506) = $84,253

  17. Single Sum Problems: Present Value Given: • Amount of deposit end of 5 years: $84,253 • Interest rate (discount) rate: 11% • Frequency of compounding: Annual • Number of periods (5 years): 5 periods What is the present value of this single sum? • FVn = PV(1 + i)n $84,253 x (0.59345) = $50,000

  18. Annuities • Definition -- a series of equal dollar payments coming at the end of a certain time period for a specified number of time periods. • Examples -- life insurance benefits, lottery payments, retirement payments.

  19. Annuity Computations An annuity requires that: • the periodic payments or receipts (rents) always be of the same amount, • the interval between such payments or receipts be the same, and • the interest be compounded once each interval.

  20. If one saves $1,000 a year at the end of every year for three years in an account earning 7% interest, compounded annually, how much will one have at the end of the third year? Example of Annuity End of Year 0 1 2 3 4 7% $1,000 $1,000 $1,000 $1,070 $1,145 $3,215 = FVA3 FVA3 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0 = $3,215

  21. Derivation of Equation A A A A A A A A 1 2 4 n-2 n-1 n 3 ? Year n n-1 n-2 . . 1 Future Value of Annuity A A(1+i) A(1+i)2 . . A(1+i)n-1 Total Future Value (F) = A + A(1+i) + A(1+i)2 + ... + A(1+i)n-1

  22. Derivation (cont.) F = A + A(1+i) + A(1+i)2 + ... + A(1+i)n-1 :Eqn 1 Multiply both sides by (1+i) to get: F(1+i) = A(1+i) + A(1+i)2 + ...+ A(1+i)n :Eqn 2 Subtract Eqn 2 from Eqn 1 to get: F = A[(1+i)n - 1] / i = A (F/A,i,n)

  23. Annuities: Future Value Given: • Deposit made at the end of each period: $5,000 • Compounding: Annual • Number of periods: Five • Interest rate: 12% What is future value of these deposits? F = A[(1+i)n - 1] / i $5,000 x (6.35285) = $ 31,764.25

  24. Annuities: Present Value Given: • Rental receipts at the end of each period: $6,000 • Compounding: Annual • Number of periods (years): 5 • Interest rate: 12% What is the present value of these receipts? F = A[(1+i)n - 1] / i $6,000 x (3.60478) = $ 21,628.68

  25. Annuities: Future Value Given: Deposit made at the beginning of each period: $ 800 • Compounding: Annual • Number of periods: Eight • Interest rate 12% What is the future value of these deposits?

  26. Annuities: Future Value First Step: Convert future value of ordinary annuity factor to future value for an annuity due: • Ordinary annuity factor: 8 periods, 12%: 12.29969 • Convert to annuity due factor: 12.29969 x 1.12: 13.77565 Second Step: Multiply derived factor from first step by the amount of the rent: • Future value of annuity due: $800 x 13.77565 = $11,020.52

  27. Annuities: Present Value Given: • Payment made at the beginning of each period: $ 4.8 • Compounding: Annual • Number of periods: Four • Interest rate 11% What is the present value of these payments?

  28. Annuities: Future Value First Step: Convert future value of ordinary annuity factor to future value for an annuity due: • Ordinary annuity factor: 4 periods, 11%: 3.10245 • Convert to annuity due factor: 3.10245 x 1.11 3.44372 Second Step: Multiply derived factor from first step by the amount of the rent: • Present value of annuity due: $4.8M x 3.44372: $16,529,856

  29. Key of Annuity Calculation Fv = Pv[(1+i)n - 1] / i

  30. Summary • Single Sum Compounding • Annuities • Key: Compound Interests Calculation

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