2 Engineering Economy uses mathematical formulas to account for the time value of money and tobalance current and future revenues and costs.Cash flow diagrams depict the timing and amountof expenses (negative, downward) and revenues(positive, upward) for engineering projects.EGR 403, Jan 99
3 Example on Cash Flow Diagram Draw the cash flow diagram for a Corolla with thefollowing cash flow:Down payment $1000, refundable security deposit$ 225, and first month’s payment $189 which isdue at signing (total $1414). Monthly payment $ 189for 36 month lease (total $ 6804). Lease-endpurchase option $ 7382.EGR 403, Jan 99
4 Equivalence depends on interest rate Interest is the return on capital or cost of using capital.Simple Vs Compound interest rateEquivalence (page 45)When we are indifferent as to whether we have aquantity of money now or the assurance of someother sum of money in the future, or series of futuresums of money, we say that the present sum ofmoney is equivalent to the future sum or series offuture sums.Equivalence depends on interest rateEGR 403, Jan 99
5 i = Interest rate per payment period n = Number of payment periods Notation:i = Interest rate per payment periodn = Number of payment periodsP = Present value of a sum of money (time 0)Fn = Future value of a sum of money in year n(end of year n)Considering Compound interestSingle Payment Compound Amount:(F|P,i,n) = (1+i)n = F/P, thus F = P(F|P, i, n)Single Payment Present Worth:(P|F,i,n) = 1/(1+i)n, thus P = F(P|F, i, n)EGR 403, Jan 99
6 Example on P and F1- An antique piece is purchased for $10,000 today. How muchwill it be worth in three years if its value increases 8% per year?2- What sum can you borrow now, at an 8% interest rate, if youcan pay back $6,000 in five years?3- How long will it take for an investment of $300 to doubleconsidering 8% interest rate?4- An investment of $2,000 has been cashed in as $3,436 eightyears later. What was the interest rate?5- How much will a deposit of $400 in a bank be six years fromnow, if the interest rate is 12% compounded quarterly?
7 Different Methods for Solving Engineering Economy Problems: 1- Mathematical formulas (use calculator)2- Engineering Economy functional notation(use compound interest tables)3- Software such as Excel (use defined functions)EGR 403, Jan 00
8 Specific Situations: In most Eng. Econ. Formula, there are four parameters , i.e. F, P, i, and n. If three of theseare known you can find the fourth one.Specific Situations:* The required number of years is not in thecompound interest table* There is no compound interest table for therequired interest rate* The interest is compounded for some periodother than annuallyEGR 403, Jan 00
9 Different Skills (Tricks) to Solve Cash Flow Diagram Shift origin (time 0) to an imaginary point of timeAdd and subtract imaginary cash flowsDissect cash flow diagramEGR 403, Jan 04
10 A = A series of n uniform payments at the end-of-period Notation:A = A series of n uniform payments at the end-of-periodConsidering Compound interestUniform series compound amount: F = A(F|A, i, n)Uniform series sinking fund: A = F(A|F, i, n)Uniform series capital recovery: A = P(A|P, i, n)Uniform series present worth: P = A(P|A, i, n)Deferred annuities:EGR 403, Jan 99
11 Example on A1- You make 12 equal annual deposits of $2,000 each into a bankaccount paying 4% interest per year. The first deposit will bemade one year from today. How much money can be withdrawnfrom this bank account immediately after the 12th deposit?2- Your parents deposit $7,000 in a bank account for youreducation now. The account earns 6% interest per year. They plan to withdraw equal amounts at the end of each yearfor five years, starting one year from now. How much moneywould you receive at the end of each one of the five years?3- How much should you invest today in order to provide anannuity of $6,000 per year for seven years, with the first paymentoccurring exactly four years from now? Assume 8% interest rate.
12 G = Arithmetic gradient series, fix amount Notation:G = Arithmetic gradient series, fix amountincrement at the end-of-periodConsidering Compound interestArithmetic gradient uniform series: A = G(A|G, i, n)Arithmetic gradient present worth: P = G(P|G, i, n)G GEGR 403, Jan 99
13 g = Geometric gradient series, fix % increment at the end-of-period Notation:g = Geometric gradient series, fix %increment at the end-of-periodConsidering Compound interestA’ A’(1+g) A’(1+g)2EGR 403, Jan 99
14 Example on G and g1- Suppose that certain end-of-year cash flows are expected tobe $2,000 for the second year, $4,000 for the third year, and$6,000 for the fourth year. What is the equivalent presentworth if the interest rate is 8%? What is the equivalent uniformannual amount over four years?2- Overhead costs of a firm are expected to be $200,000 in thefirst year, and then increasing by 4% each year thereafter, overa 6-year period. Find the equivalent present value of these cashflows assuming 8% interest rate.
15 Types of Interest Rates: r = Nominal interest rate per period (compoundedas sub period) = m*ii = Effective interest rate per sub period (i.e., month)ia = Effective interest rate per year (annum)m = Number of compounding sub periods per periodSuper period = Cash flow less often than compounding periodSub period = Cash flow more often than compounding periodContinuous Compounding:EGR 403, Jan 99
16 Example on Different Interest Rates 1- A credit card company charges an interest rate of 1.5%per month on the unpaid balance of all accounts. What isthe nominal rate of return? What is the effective rate ofreturn per year?2- Suppose you have borrowed $3000 now at a nominalinterest rate of 10%. How much is it worth at the end of theninth year? a) If interest rate is compounded quarterly. b) If interest rate is compounded continuously.
17 Timing of cash flow: End of the period Beginning of the period Middle of the periodContinuous during the periodEGR 403, Jan 99
18 Example on Different Timing 3- We would like to find the equivalent present worth of $4,000paid sometime in future. Assume 20% interest rate. a) How much is the equivalent present worth if it is paid at theend of the first year? b) How much is the equivalent present worth if it is paid in themiddle of the first year? c) How much is the equivalent present worth if it is paid at thebeginning of the first year? d) How much is the equivalent present worth if it is paidcontinuously during the first year? Assume continuouscompounding with nominal interest rate of 20%.EGR 403, Jan 04
19 Example on Patterns Recognition in Cash Flow Diagram You have two job offers with the following salary per year. Assume you will stay with a job for four years and the interestrate is 10%. Which one of the jobs will you select and why?Year 1 2 3 4 Job A $50,000 $52,500 $55,125 $57,881 Job B $52,000 $53,200 $54,400 $55,600
20 Example on Loan Analysis Assume you borrow $2,000 today, with an interest rate of 10%per year, to be repaid over five years in equal amounts(payments are made at the end of each year). The $2,000 isknown as the principal of the loan. The amount of eachpayment can be calculated as below:A = 2000(A|P, 10%, 5) = $527.6Each payment consists of two portions: interest over that yearand part of the principal. The following table shows the amountof each portion for each of the payments. Notice that as timegoes by you will pay less interest and your payment will covermore of the principal.
22 Bond AnalysisBond is issued to raise funds through borrowing. The borrowerwill pay periodic interest (uniform payments: A) and a terminalvalue (face value: F) at the end of bond’s life (maturity date).The timing of the periodic payments and its amount are alsoindicated on each bond. Sometimes the bond's interest rate (rb),that is a nominal interest rate, is mentioned instead of theamount of each payment. The market price of a bond does notneed to be equal to its face value.A = F * rb / m
23 Example on BondA bond pays $100 quarterly. Bond’s face value is $2,000 andits maturity date is three years from now.a) What is the bond’s interest rate?rb = A*m/F = 100 * 4 / 2000 = 20% quarterlyb) What is the present worth of the bond assuming that anominal interest rate of 12% compounded quarterly isdesirable.i = r/m = 12/4 = 3% per quarterNumber of payments = n = 3*4 = 12 quartersP = 100(P|A, 3%, 12) (P|F, 3%, 12) = $2,398.2EGR 403, Jan 2000