Uniform Series Compound Amount. Uniform Series Sinking Fund. Uniform Series Capital Recovery P. Uniform Series Present Worth ( ) A A A A A A A

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1 ENGM 401 & 620 X1 Fudametals of Egieerig Fiace Fall 2010 Lecture 22: Iterest Calculatios (3) Moey, so they say Is the root of all evil today But if you ask for a rise It's o surprise that they're givig oe away. - Pik Floyd M.G. Lipsett Departmet of Mechaical Egieerig Uiversity of Alberta Uiform Series Iterest Cases (Review) Uiform Series Compoud Amout A A A A A A A Uiform Series Sikig Fud F Uiform Series Capital Recovery P Uiform Series Preset Worth A A A A A A A A A A A A A A A A A A A A A P F ( i ) F = A i F = A ( F / A, i, ) i A = F ( + ) 1 i 1 A = F ( A / F, i, ) i ( 1 + i ) ( 1 + i ) 1 ( ) A = P A = P A / P, i, ( + i ) P A i ( 1 + i ) P = A ( P / A, i, ) 1 1 = MG Lipsett,

2 Market Value of Bod Example For a twety-year $10,000 bod with a coupo rate of 5%, what is the market value whe the iterest rate is 5.5% ad te years remaiig? MG Lipsett, Arithmetic Gradiet Iterest Formulas P/G If you deposit a uiformly icreasig series of sums, G, 2G, etc., for -1 years as show below, what is it worth today? 0 G 2 G 3 G (-1) G 4 G 5 G P ( ) 2 i ( 1+ i ) 1+ i i 1 P = G Or equivaletly, usig iterest tables: P = G ( P / G, i, ) Arithmetic Gradiet Preset Worth Factor Notice how this series works. The first amout G occurs at the ed of the SECOND period, ot the first. MG Lipsett,

3 Arithmetic Gradiet Preset Worth Tables (P/G, 4%, ) (P/G, 5%, ) (P/G, 6%, ) (P/G, 7%, ) (P/G, 8%, ) (P/G, 9%, ) (P/G, 10%, ) 1 / / / / / / / ( ) ( ) Here is the tabular represetatio MG Lipsett, Example: Arithmetic Gradiet P/G If you deposit $50 two years from ow, $100 a year after that, $150 the ext year, ad so o, up to year 10, what is that worth today with 10% iterest? MG Lipsett,

4 Arithmetic Gradiet Iterest Formulas F/G If you deposit a uiformly icreasig series of sums, G, 2G, etc., as show below, what is its future worth, F, i years? 0 G 2 G 3 G (-1) G 4 G 5 G F ( ) 1+ i i 1 F = G 2 i Or equivaletly, usig iterest tables: F = G ( F / G, i, ) Arithmetic Gradiet Future Worth Factor MG Lipsett, Example: Arithmetic Gradiet F/G If you deposit $50 two years from ow, $100 a year after that, $150 the ext year, ad so o, what is it worth 10 years from ow with a 5% iterest rate? MG Lipsett,

5 Aother Arithmetic Gradiet Example: P/G + P/A You lease a ew car ad expect maiteace over the fiveyear cotract to be as follows: Year Cost 1 $120 2 $150 3 $180 4 $210 5 $240 How much moey do you eed to put aside today i a accout with 5% iterest to pay for that maiteace? We ca t use oe of our stadard formulae for this case. So what do we do? MG Lipsett, Aother Arithmetic Gradiet Example: P/G + P/A These formulæ ca be used i combiatio, for more complex time series cases Year Cost 1 $120 2 $150 3 $180 4 $210 5 $240 MG Lipsett,

6 Yet Aother Arithmetic Gradiet Example: A/G The expected maiteace costs of a piece of machiery are as follows: Year Cost 1 $100 2 $200 3 $300 4 $400 If you wish to budget a uiform aual maiteace allowace with a 6% discout rate, what would that aual allowace be? MG Lipsett, Yet Aother Arithmetic Gradiet Example: A/G Year Cost 1 $100 2 $200 3 $300 4 $400 MG Lipsett,

7 Geometric Series Iterest Formulas If you deposit a icreasig series of sums, A, (1+g) A, etc., for years as show below, what is it worth today? A (1+g) 2 A (1+g) 3 A (1+g) 4 A (1+g) A (1+g) -1 A P This is a pretty good model for maiteace costs MG Lipsett, Example: Geometric Series You estimate the maiteace costs of ew equipmet to be $100 i the first year, icreasig by 10% per year. What is the preset worth of the first five years of maiteace if you estimate a 8% iterest rate? A (1+g) A (1+g)2 A (1+g)3 A (1+g) 4 A MG Lipsett,

8 Example: Geometric Series (2) You sig a ew Executive VP to a 5-year cotract that pays $ i the first year, icreasig by 8% each year after that. At the ed of the first year, the board of directors decides that the he is t workig out ad decides to buy him out of his cotract. How much will they eed to pay him if they assume iterest rates are 12%? MG Lipsett, Preset Worth Equivalece says that we ca shift ay sum to a equivalet sum at some other poit i time The equivalet sum at the preset time is called Preset Worth (PW) PW is the comparable equivalet value at the preset time of a future sum (or set of sums) Preset Worth is also called Net Preset Value Net Preset Value (NPV), Although that term is more ofte used whe referrig to the total preset worth of a series of sums PW or NPV ca be thought of as the differece betwee a future set of cash iflows ad cash outflows shifted to the preset MG Lipsett,

9 Preset Worth Aalysis Preset worth aalysis is whe we compare the et preset value of multiple mutually exclusive optios Preset worth aalysis cosiders oly future icomes ad expeditures. Fiacig ad ivestmet activities are cosidered separate activities ad the their NPVs combied The goal: maximize the PW of ivestmet beefits & miimize the PW of fiacig costs MG Lipsett, Preset Worth Aalysis Example Cosider two types of equipmet for your compay, each with differet costs, et aual beefits, useful lives, ad salvage values: #1 #2 Purchase cost $ 200k $ 700k Aual beefit $ 95k $ 120k Useful life 6 yrs 12 yrs Salvage value $ 50k $ 150k Which optio is best (assumig a 10% iterest rate)? #1: MG Lipsett,

10 Preset Worth Aalysis Example (2) #2: Best iterest? #1 #2 Purchase cost $ 200k $ 700k Aual beefit $ 95k $ 120k Useful life 6 yrs 12 yrs Salvage value $ 50k $ 150k MG Lipsett, Summary Market value of a ivestmet is its preset value, usig the market iterest rate More complex cash flow series calculatios ca be doe usig liear combiatios of stadard formulae A spreadsheet is a effective tool for realistic (complicated) ivestmet cash flow series Midterm #2 will cover lectures 11 to 22 iclusive, ad chapters 4, 5, ad 6 of the text, as well as material o timevalue of moey i Chapter 7 (sectios 7.1 ad 7.2) Ope book, ope otes, brigs a calculator, o computers or wireless devices, absolutely o talkig durig the test Same format as Midterm #1, o-cumulative, but cocepts from Midterm #1 (such as icome statemets) remai relevat MG Lipsett,

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