In this chapter, students will be able to understand: Calculating future values from annual amounts, calculating present values from annual amounts, calculating future and present values from gradient amounts, calculating present value of a future perpetual amounts, calculating deferred annuities.
Chapter 5 – Unit 1 Annual Amount and Gradient Functions IET 350 Engineering Economics Learning Objectives – Chapter 5 Upon completion of this chapter you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts Calculating deferred annuities Learning Objectives – Unit 1 Upon completion of this unit you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts Calculating deferred annuities Introduction The prior chapter covered single‐payment functions where a cash inflow occurred at one point in time and a cash outflow occurred at a second point in time Many financial transactions have elements that occur at y multiple points in time. These can include: Equal annual cash flow Linear gradient cash flow Non‐linear gradient cash flow Mixed annual cash flow Introduction This chapter covers three types of multiple‐payment situations: Equal annual amounts (A) – equal dollar amounts flow into or out of an investment or project each year p j y Linear gradient amounts (G) – dollar amounts flowing into or out of an investment or project increase/decrease each year by a constant amount (linear) Mixed annual amounts – differing dollar amount flow into and/or out of an investment or project each year Equal Annual Amounts Assumptions for equal annual amount analysis include: Cash flow occurs at the end of each year All cash flows are equal and occur each year Note that most interest table such as those in appendix B of the Bowman text are based on end of year transactions. Interest table are available that use the beginning or middle of time periods. If using a time value of money function on your calculator, check the manual to determine if the time basis is end of period (year) or some other basis Equal Annual Amounts Notations used for time value of money calculation Future Value (one‐time occurrence) → F Present Value (one‐time occurrence) → P Equal Annual Amount → A Cash flow diagrams represent annual amounts as equal length lines as illustrated in Figure 5‐1: Future Value Calculations Future value for an equal annual amount is determined by the following equation: ⎡ (1 + i)n ‐ 1 ⎤ F = A × ⎢ ⎥ i ⎣ ⎦ Where: F = Future Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) Future Value Calculations Solution methods for finding future values: Use the F/A column on a Interest Factors table (Bowman text appendix B, page 580). Notation F/A is interpreted as → Find F given A Notation F/A is interpreted as → Find F given A Notation (F/A, n, i) is interpreted as → Find F given A for n years at i interest rate. Use the Excel function1 → FV(rate, nper, pmt, pv, type) Use the formula and calculator 1Note that the cash outflows are entered as a negative number Future Value/Annual – Example Your plan is to save $100 at the end of each year at 8% interest What will be the size of the account in 10 years? F = A × (F/A, / 10, 8%) = $100ì(14.487) F=$1,448.70 10 FValue/Annual Example(continued) SolutionusingExcelđ: Notethatthe annual amount was entered as a negative number which indicates a cash outflow 11 FValue/Annual – Example (continued) Solution using formula: F = Future Value = ? A = Annual Amount = $100 n = Time = 10 years i = Interest = 8% per year The slight difference between this amount and the amount determined by the factor from the tables is due to rounding ⎡ (1 + i)n ‐ 1 ⎤ F = A × ⎢ ⎥ i ⎣ ⎦ ⎡ (1 + 0.08)10 ‐ 1 ⎤ = $100 $ × ⎢ ⎥ 0.08 ⎦ ⎣ ⎡ 2.158925 ‐ 1 ⎤ = $100 × ⎢ ⎥⎦ 0.08 ⎣ = $100 × [14.48656] F = $1,448.66 12 Future Value Calculations Equal annual amounts for a future value is determined by the following equation: Note that this formula is the inverse of the formula to find F given A ⎡ ⎤ i A = F × ⎢ n ⎥ ( ) + i ‐ ⎣ ⎦ Where: F = Future Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) 13 Future Value Calculations Solution methods for finding annual amounts: Use the A/F column on a Interest Factors table (Bowman text appendix B, page 580). Notation A/F is interpreted as → Find A given F Notation A/F is interpreted as → Find A given F Notation (A/F, n, i) is interpreted as → Find F given A for n years at i interest rate. Use the Excel function1 → PMT(rate, nper, pv, fv, type) Use the formula and calculator 1Note that the cash outflows are entered as a negative number 14 Annual/Future Value – Example Your goal is to save $7,500 for a car down payment in 4 years by investing part of your end‐of‐year bonus. How much to you need to save annually at 4% interest? A = F × (A/F, 4, 4% ) = $7,500 × (0.2355) A = $1,766.25/year 15 Annual/FValue – Example (continued) Solution using Excel®: Note that the function returns a negative ti number which indicates a cash outflow 16 Annual/FValue – Example (continued) Solution using the formula: F = Future Value = $7,500 A = Annual Amount = ? n = Time = 4 years i = Interest = 4% per year ⎡ ⎤ i A = F × ⎢ ⎥ n ⎣ (1 + i) ‐ 1 ⎦ 0.04 ⎡ ⎤ = $7,500 × ⎢ ⎥ ⎣ (1 + 0.04) ‐ 1 ⎦ The slight difference between this amount and the amount determined by the factor from the tables is due to rounding 0.04 ⎡ ⎤ = $7,500 × ⎢ ⎣ 1.169859 ‐ 1 ⎥⎦ = $7,500 × [0.235490] A = $1,766.17 17 End Unit 1 Material Additional Reading Ö Financial Functions: http://www.functionx.com/excel/Lesson12.htm Go to Unit 2 Present Value Amounts 18 Chapter 5 – Unit 2 Present Value Amounts IET 350 Engineering Economics Learning Objectives – Unit 2 Upon completion of this unit you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts Calculating deferred annuities 20 Present Value Calculations Present value for an equal annual amount is determined by the following equation: ⎡ (1 + i)n ‐ 1 ⎤ P = A × ⎢ n ⎥ ⎣ i (1 + i) ⎦ Where: P = Present Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) 21 Present Value Calculations Solution methods for finding present values: Use the P/A column on a Interest Factors table (Bowman text appendix B, page 580). Notation P/A is interpreted as → Find P given A Notation P/A is interpreted as → Find P given A Notation (P/A, n, i) is interpreted as → Find P given A for n years at i interest rate. Use the Excel function1 → PV(rate, nper ,pmt, fv, type) Use the formula and calculator 1Note that the cash outflows are entered as a negative number 22 Present Value/Annual – Example You scheduled to receive $15,000 at the end of the next 7 years If the current interest rate is 6%, what is the equivalent amount today? P = A × (P/A, 7, 6% ) = $15,000ì(5.5824 ) P=$83,736 23 PValue/Annual Example(continued) SolutionusingExcelđ: Note that the function returns a negative number which indicates a cash outflow 24 PValue/Annual – Example (continued) Solution using the formula: P = Present Value = ? A = Annual Amount = $15 000 A = Annual Amount = $15,000 n = Time = 7 years i = Interest = 6% per year ⎡ (1 + i)n − ⎤ P = A × ⎢ n ⎥ ⎣ i(1 + i) ⎦ ⎡ (1 + 0.06)7 − ⎤ = $15,000 × ⎢ 7⎥ ⎣ 0.06 × (1 + 0.06) ⎦ ⎡ 1.50363 − ⎤ = $15,000 × ⎢ ⎥ ⎣ (0.06) × (1.50363) ⎦ = $15,000 × [5.582381] P = $83,735.72 25 Present Value Calculations Equal annual amounts for a present value is determined by the following equation: Note that this formula is the inverse of the formula to find P given A ⎡ i (1 + i)n ⎤ A = P × ⎢ ⎥ n ⎣ (1 + i) ‐ 1 ⎦ Where: P = Present Value ($) A = Annual Amount ($) n = Time (years) i = Interest (% per year) 26 Present Value Calculations Solution methods for finding present values: Use the A/P column on a Interest Factors table (Bowman text appendix B, page 580). Notation A/P is interpreted as → Find A given P Notation A/P is interpreted as → Find A given P Notation (A/P, n, i) is interpreted as → Find A given P for n years at i interest rate. Use the Excel function1 → PMT(rate, nper, pv, fv, type) Use the formula and calculator 1Note that the cash outflows are entered as a negative number 27 Annual/Present Value – Example You $5,000 invest in an account that returns 6% annual interest How much can you withdraw each semester (twice/year) over the next 4 years for books and supplies? A = P × (A/P, A/P 8, 3% ) = $5,000 × (0.1425) A = $712.50/semester Time periods other than a year can be used, however, the tabulated interest rate is a yearly rate so it must be adjusted to match the number of periods/ year → 6% per year/2 periods per year = 3% per period . Also the total number of periods is used → 4 yrs x 2 periods/yr = 8 periods 28 Annual/PValue – Example (continued) Solution using Excel®: Remember that i and n must be and n must be adjusted for time periods other than yearly 29 Annual/PValue – Example (continued) Solution using the formula (see page 187 Bowman text): P = Present Value = $5,000 P = Present Value = $5 000 A = Annual Amount = ? n = Time = 4 years M = #Periods/year = 2 r = Annual Interest = 6% nm ⎡ r⎛ r⎞ ⎤ ⎢ ⎜ 1 + ⎟ ⎥ A m m ⎠ ⎥ = P × ⎢ ⎝ nm m ⎥ ⎢⎛ r⎞ ⎢ ⎜ 1 + m ⎟ ‐ 1 ⎥ ⎠ ⎦ ⎣⎝ 0.06 4×2 ⎤ ⎡ 0.06 (1 + ) ⎥ × ⎢ = $5,000 × ⎢ ⎥ 0.06 4×2 ) ‐ 1 ⎥ ⎢ (1 + ⎣ ⎦ ⎡ 0.03 × 1.26677 ⎤ = $5,000 × ⎢ ⎣ 1.26677 ‐ 1 ⎦⎥ = $5,000 × [0.142456] A = $712.28/semester m 30 10 Finding Unknown n or i Values Occasionally an engineering economic analysis will occur when the number of years (n) or the interest rate (i) is unknown Like single‐payment calculations, if three of the four factors g p y , are known, we can solve for the unknown Future value factors → F, A, i, n Present value factors → P, A, i, n 31 Unknown i and n Calculations Solution methods for finding interest or time period values: Interpolate using the appropriate column on a Interest Factors table (Bowman text appendix B, page 580). Use the Excel functions Use the Excel functions1: : RATE(nper, pmt, pv, fv, type, guess) → returns the interest rate per period for a cash flow NPER(rate, pmt, pv, fv, type) → returns the number of periods for a cash flow with a constant interest rate Rearrange the appropriate formula and solve with your calculator 1Note that the cash outflows are entered as a negative number 32 Unknown Interest – Example You have $5,000 to invest in an account and would like to withdraw $1,550 per year for the next four years What interest rate will be required to meet the needs? You will need to invest the $5,000 at 9.2% annual interest rate 33 11 Unknown Time – Example You plan to invest $1,250 per year in a security with a 4.75% annual return rate. How many years before the account grows to $12,500? Time Required 8 years 4 months 15 days 34 End Unit 2 Material Go to Unit 3 Gradient Amounts 35 Chapter 5 – Unit 3 Gradient Amounts IET 350 Engineering Economics 12 Learning Objectives – Unit 3 Upon completion of this unit you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts Calculating deferred annuities 37 Gradient Amounts Unlike equal annual amounts, gradient amounts increase or decrease each time period. Types: Linear – change in cash flow is by an equal amount for each time period. Gradient factors tabulated in the p interest tables or determined by formula Non‐linear – change is cash flow varies between time periods. Non‐linear gradient functions must be calculated with a series of P/A or A/P for each time period 38 Gradient Amounts Assumptions for linear gradient amount analysis include: Cash flow occurs at the end of each year Change in cash flow year to year is at a constant rate. The g g G amount of change is designated → Initial cash flow is designated → A’ 39 13 Gradient Calculations Solution methods for finding present values: Use the A/G column on a Interest Factors table (Bowman text appendix B, page 580). Future or present values can then be determined using the annual amount (A): Notation A/G is interpreted as → Find A given G Notation (A/G, n, i) is interpreted as → Find A given P for n years at i interest rate. Use the Excel function1 → XNPV(rate, values, dates) Use the formula and calculator 1Note that the cash outflows are entered as a negative number 40 Gradient Calculations Equal annual amounts for a linear gradient values is determined by the following equation: ⎡1 ⎤ n A = A′ ± G × ⎢ − (1 + i)n ‐ 1 ⎥⎦ ⎣i When the change is increasing between time periods, the gradient is added (+) to the initial value and subtracted (‐) when decreasing Where: A = Annual Amount ($) A′ = Initial Cash Flow($) G = Gradient Amount ($) n = Time (years) i = Interest (% per year) 41 Gradient – Example Your 1st year’s salary is $45,000. Your contract states that your raise will be $5,000/year in years 2 through 6 What is the present value of the contract at 5% interest? Cash Flow Diagram: Cash Flow Diagram: 42 14 Gradient – Example (continued) Solution method: Find annual value (A) of the gradient (G) Convert the annual amount (A) into the present value (P) A = A′ + G (A/G, / 6, 5% ) = $45,000 + $5,000 × (2.3579) A = $56,789.50/year P = A(P/A, 6, 5%) = $56,789.50 × (5.0757) P = $288,246.50 43 Gradient – Example (continued) Solution using Excel®. You must create a schedule of amounts with a date. The schedule must start at time = 0 (today) Non‐linear gradients can be solved with this method 44 Gradient – Example (continued) Solution using the formulas: P = Present value = ? A = Annual Amount = ? A′ = Initial Amount = $45,000 G = Gradient = $5,000 n = Time = 6 years i = Interest = 5% per year ⎤ ⎡1 n A = A′ ± G × ⎢ − (1 + i)n ‐ 1 ⎥⎦ ⎣i ⎤ ⎡ − = $45,000 + $5,000 × ⎢ ⎥ ⎣ 0.05 (1 + 0.05) ‐ 1 ⎦ ⎡ ⎤ = $45,000 + $5,000 × ⎢20 ‐ 1.340096 ‐ 1 ⎥⎦ ⎣ = $45,000 + $5,000 × [2.357922] A = $56,789.61/year ⎡ (1 + i)n − ⎤ P = A × ⎢ n ⎥ ⎣ i(1 + i) ⎦ ⎡ (1 + 0.05)6 − ⎤ = $56,789.61 × ⎢ 6⎥ ⎣ 0.05 × (1 + 0.05) ⎦ = $56,789.61× [5.075697] P = $288,246.90 45 15 End Unit 3 Material Go to Unit 4 Perpetual Amounts and Deferred Annuities 46 Chapter 5 – Unit 4 Perpetual Amounts and Deferred Annuities IET 350 Engineering Economics Learning Objectives – Unit 4 Upon completion of this unit you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts Calculating deferred annuities 48 16 Perpetual Gradient Amounts Perpetual gradient amounts increase or decrease each time period. Assumptions: Cash flow occurs at the end of each year with change at a g g G constant rate. The amount of change is designated → Project life is considered infinite: n → ∞ 49 Perpetual Gradient Amounts Present value for project with an infinite life and a gradient increase in cash flow is determined by the following equation: PTotal = P0 + PAnnual + PGradient A G PTotal = P0 + + i i Where: P0 = Initial Project Cost ($) A = Initial Annual Amount ($) G = Gradient Amount ($) i = Interest (% per year) 50 Perpetual Gradient – Example A warehouse was constructed at an initial cost of $500,000 and is expected to last forever. First year maintenance cost is $5,000 and is expected to increase at a constant $500/year If the firm uses a 7.5% interest rate, what is the total present , p value of the project? A G P = Present value = ? PTotal = P0 + + i i P0 = $500,000 $5000 $500 A = Maintenance = $5000/yr = $500,000 + + 0.075 0.0752 G = Gradient = $500/yr = $500,000 + $66,666.67 + $88.888.89 i = 7.5% per year PTotal = $655,556 51 17 Deferred Annuities An annuity is equivalent to an annual amount. A deferred annuity is a set of annual cash flows that will occur in the future (deferred) rather than immediately Figure 5‐18 from the Bowman text illustrates an investment Figure 18 from the Bowman text illustrates an investment made a t0 (present time) with annual disbursement beginning in the 4th year. 52 Deferred Annuities Solution steps for deferred annuities: Determine the future value (F) of the initial investment at the beginning time point of the deferred annuity Set the future value (F) to the deferred present value (P′) ( ) p ( ) Determine the annual amount (A) using the deferred present value (P′) 53 Deferred Annuity – Example For your 8th birthday your rich uncle gave you $25,000 for college which you invested in a deferred annuity at 4% You begin drawing on the annuity at age 18 for 4 years. How much will you receive each year? y y F = P (F/P, 10, 4%) = $25,000(1.480) F = $37,000 A = P′(A/P, 4, 4%) with P′ = F = $37,000(0.2755) A = $10,193.50/year 54 18 End Chapter 5 Material Student Study Guide Ö Chapter 5 Homework Assignment Ö Problem Set 5 55 19 ... each year by a constant amount (linear) Mixed annual amounts – differing dollar amount flow into and/ or out of an investment or project each year Equal Annual Amounts Assumptions for equal annual amount analysis include:... Upon completion of this unit you should understand: Calculating future values from annual amounts Calculating present values from annual amounts Calculating future and present values from gradient amounts Calculating present value of a future perpetual amounts... Go to Unit 3 Gradient Amounts 35 Chapter 5 – Unit 3 Gradient Amounts IET 350 Engineering Economics 12 Learning Objectives – Unit 3 Upon completion of this unit you should understand: Calculating future values from annual amounts