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 Time (# of interest periods) Positive net Cash flow ( receipts) Negative net Cash Flow (payments) $15,000 $2000 $13,000 is net positive cash flow

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

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, F n, would be: For example, $100 invested now at 9% simple interest for 8 years would yield Nobody uses simple interest. F n = P(1 + i*n) F 8 = $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. F n = P( 1 + i ) n F 8 = $100( ) 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) FV n = 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 / 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? “Rule-of-72”. Key “Rule-of-72”.

15 72 Approx. Years to Double = 72 / i% 726 Years 72 / 12% = 6 Years [Actual Time is 6.12 Years] Quick! How long does it take to double $5,000 at a compound rate of 12% per year? Example: Double Your Money!!!

16 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? FV n = PV(1 + i) n $50,000 x ( ) = $84,253 Single Sum Problems: Future Value

17 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? FV n = PV(1 + i) n $84,253 x ( ) = $50,000 Single Sum Problems: Present Value

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 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. Annuity Computations

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 $1,000 $1,000 $1, $3,215 = FVA 3 End of Year 7% $1,070 $1,145 FVA3 = $1,000(1.07) 2 + $1,000(1.07) 1 + $1,000(1.07) 0 = $3,215

21 Derivation of Equation AAAAAAAA ? n-2n-1n 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) A(1+i) n-1

22 Derivation (cont.) F = A + A(1+i) + A(1+i) A(1+i) n-1 :Eqn 1 Multiply both sides by (1+i) to get: F(1+i) = A(1+i) + A(1+i) 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 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 ( ) = $ 31, Annuities: Future Value

24 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 ( ) = $ 21, Annuities: Present Value

25 Given: Deposit made at the beginning of each period: $ 800 Compounding:Annual Number of periods:Eight Interest rate12% What is the future value of these deposits? Annuities: Future Value

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

27 Given: Payment made at the beginning of each period: $ 4.8 Compounding:Annual Number of periods:Four Interest rate11% What is the present value of these payments? Annuities: Present Value

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

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

30 Summary Single Sum Compounding Annuities Key: Compound Interests Calculation