Chapter 9 – Engineering Economic Analysis

Outline Interest Rates Cash Flow Diagrams Annuities and Discount Factors Depreciation Taxation, Cash Flow, and Profit Inflation

Definitions P – Principal or Present Value (of an investment) Fn – Future Value (of an investment) n – Years (or other time unit) between P and F i – Interest Rate (based on time interval of n) per anum Basis premise: Money when invested earns money $1 today is worth more than $1 in the Future

Interest Simple Interest – Annual Basis Interest paid in any year = Pis Pis – Fraction of investment paid as interest per year After n years total interest paid = Pisn Total investment is worth = P + Pisn What is the drawback of simple interest? We can earn interest on earned interest

Interest Compound Interest At time 0 we have P At the end of Year 1, we have F1 = P (1 + i) At the end of Year 2, we have F2 = P (1 + i)2 At the end of Year n, we have Fn = P (1 + i)n or P = Fn / (1 + i)n

Example How much would i need to invest at 8 % p.a. to yield $5000 in 10 years

What if Interest Rate Changes with Time? Eq. (9.7)

Different Time Basis for Interest Calculations Relates to statement “ Your loan is 6 % p.a. compounded monthly” Define actual interest rate per compounding period as r inom = Nominal annual interest rate m = Number of compounding periods per year (12)

Different Time Basis for Interest Calculations cont. ieff = Effective annual interest rate Look at condition after 1 year

Example Invest $1000 at 10 % p.a. compounded monthly. How much do I have in 1 year, 10 years?

Example cont. As m increases ieff increases Is there a limit as m goes to infinity Yes – continuously compounded interest Derivation – in Chapter 9 ieff (continuous) = e inom – 1

Cash Flow Diagrams Represent timings and approximate magnitude of investment on a cfd x-axis is time and y-axis is magnitude both positive and negative investments are possible. In order to determine direction (sign) of cash flows, we must define what system is being considered.

Consider a Discrete Cash Flow Diagram Discrete refers to individual cfds that are plotted

Example I borrow $20k for a car and repay as a $400 monthly payment for 5 years. For Bank For Me $20,000 $400 1 2 3 60 1 2 3 60 $400 $20,000

Cumulative CFD Cumulative CFD

Annuities 1 2 3 n Uniform series of equally spaced – equal value cash flows Note: The first payment is at the end of year 1 not at t = 0

Annuities What is future value Fn = ? Geometric progression

Calculations with Cash Flow Diagrams Invest $5k, $1k, $2k at End of Years 0, 1, 3, and take $3k at End of Year 4 $3,000 $2,000 $1,000 $5,000 3 7 4 1

Example 1 How much in account at end of Year 7 if i = 8% p.a. What would investment be at Year 0 to get this amount at Year 7

Example 2 What should my annual monthly car payment be if interest rate is 8% p.a. compounded monthly? A $20,000

Example 2 cont’d Compare at n = 60 Interest paid = $4,331.80

Discount Factors Just a shorthand symbol for a formula in i and n See Table 9.1

More Examples on Cash Flow Calculations You decide to start saving for a child’s college education as soon as your child is born. You estimate that it will cost a total of $18,000 per year for 4 years! How much should you put away into an account that bears an average interest rate of 7% pa in order to cover the costs of education? Assume that your child will start college at age 18, your first investment occurs at year 1, and that payments are made at the end of years 18, 19, 20, and 21. Also assume that you continue to invest until he/she graduates.

More Examples on Cash Flow Calculations 0 1 2 3 18 19 20 21 $ 18,000

More Examples on Cash Flow Calculations How much would you need to invest in a lump sum at the time of the birth of your child to pay for the education? What are the balances in the investment account in years 18, 19, 20, and 21 just before the $18,000 withdrawals have been made?

More Examples on Cash Flow Calculations 0 1 2 3 18 19 20 21 $ 18,000

More Examples on Cash Flow Calculations 0 1 2 3 18 19 20 21 $ 18,000

Depreciation Total Capital Investment = Fixed Capital + Working Capital Fixed Capital – All costs associated with new construction, but Land cannot be depreciated Working Capital – Float of material to start operations cannot depreciate

Definitions Salvage Value, S Life of Equipment Value of FCIL at end of project Often = 0 Life of Equipment n – Set by IRS Not related to actual equipment life Total Capital for Depreciation FCIL - S

3 Basic Methods for Depreciation Straight Line Sum of Years Digits (SOYD) Double Declining Balance (DDB) Modified Accelerated Cost Recovery System (MACRS)

n = # of years over which depreciation is taken Straight Line n = # of years over which depreciation is taken

Sum of Years Digits (SOYD)

Double Declining Balance (DDB) Adjust final year so that total depreciation is FCIL - S Book value

Example 9.21 Same for Years 1-7

Example 9.21 (cont’d) Sum of Year’s Digits

Example 7.21 (cont’d) Double Declining Balance

MACRS ½ year convention Double declining balance method over 5, 7, 9 years (depends on equipment use – most chemical processes are 5 yrs) with a ½ year convention and switching to SL over remaining years when depreciation for SL method is greater than DDB. Example for 5 year life – using a basis of 100 k dkDDB dkSL 1 2/5(100 – 0)(0.5) = 20 2 2/5(100 – 20) = 32 (100-20)/4.5 = 17.78 2/5(100 – 52) = 19.2 (100 – 52)/3.5 = 13.71 4 2/5(100 – 71.2) = 11.52 (100 – 71.2)/2.5 = 11.52 2/5(100 – 82.72) = 6.91 (100 – 82.72)/1.5 = 11.52 6 (0.5)(100 – 94.24)/0.5 = 5.76 ½ year convention

MACRS Year, i diMACRS 1 20.00 % 2 32.00 % 3 19.20 % 4 11.52 % 1 20.00 % 2 32.00 % 3 19.20 % 4 11.52 % 5 11.52 % 6 5.76 %

Taxation, Cash Flow, and Profit Tables 9.3 – 9.4 Expenses = COMd + dk Income Tax = (R – COMd - dk)t After Tax (net)Profit = (R – COMd –dk)(1 – t) After Tax Cash Flow = (R – COMd – dk)(1 – t) + dk (+ Investment)

Inflation $ Now Net Worth vs. $ Next Year f = Average inflation rate between Years j and n

Inflation Example

Inflation Effect of Inflation on Interest Rate f affects the purchasing power of the $ Look at the purchasing power of future worth, then If this future worth was obtained by investing at a rate i, then the inflation adjusted interest rate, i ’ is given by

Summary Investment involves principal, interest rate, and time The timing of transactions is conveniently shown on a Cash Flow Diagram Depreciation of equipment value impacts the amount of tax that one pays Inflation reduces the power of investment and decreases the buying power of future investments.