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Decision-Making Steps 1. Define problem 2. Choose objectives 3. Identify al
Clayton College & State University, US has reference to this Academic Journal, Decision-Making Steps 1. Define problem 2. Choose objectives 3. Identify alternatives 4. Evaluate consequences 5. Select 6. Implement 7. Audit EGR 403, Jan 99 Engineering Economy uses mathematical formulas so that account in consideration of the time value of money in addition to so that balance current in addition to future revenues in addition to costs. Cash flow diagrams depict the timing in addition to amount of expenses (negative, downward) in addition to revenues (positive, upward) in consideration of engineering projects. EGR 403, Jan 99 Example on Cash Flow Diagram Draw the cash flow diagram in consideration of a Corolla alongside the following cash flow: Down payment $1000, refundable security deposit $ 225, in addition to first month?s payment $189 which is due at signing (total $1414). Monthly payment $ 189 in consideration of 36 month lease (total $ 6804). Lease-end purchase option $ 7382. EGR 403, Jan 99
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Interest is the return on capital or cost of using capital. Simple Vs Compound interest rate Equivalence (page 45) When we are indifferent as so that whether we have a quantity of money now or the assurance of some other sum of money in the future, or series of future sums of money, we say that the present sum of money is equivalent so that the future sum or series of future sums. Equivalence depends on interest rate EGR 403, Jan 99 Notation: i = Interest rate per payment period n = Number of payment periods P = 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 interest Single 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 Example on P in addition to F 1- An antique piece is purchased in consideration of $10,000 today.ÿ How much will 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 you can pay back $6,000 in five years? 3- How long will it take in consideration of an investment of $300 so that double considering 8% interest rate? 4- An investment of $2,000 has been cashed in as $3,436 eight years later.ÿ What was the interest rate? 5- How much will a deposit of $400 in a bank be six years from now, if the interest rate is 12% compounded quarterly?
Different Methods in consideration of 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 In most Eng. Econ. Formula, there are four parameters , i.e. F, P, i, in addition to n. If three of these are known you can find the fourth one. Specific Situations: * The required number of years is not in the compound interest table * There is no compound interest table in consideration of the required interest rate * The interest is compounded in consideration of some period other than annually EGR 403, Jan 00 Different Skills (Tricks) so that Solve Cash Flow Diagram Shift origin (time 0) so that an imaginary point of time Add in addition to subtract imaginary cash flows Dissect cash flow diagram EGR 403, Jan 04
Notation: A = A series of n uniform payments at the end-of-period Considering Compound interest Uniform 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 Example on A 1- You make 12 equal annual deposits of $2,000 each into a bank account paying 4% interest per year.ÿ The first deposit will be made one year from today.ÿ How much money can be withdrawn from this bank account immediately after the 12th deposit? 2- Your parents deposit $7,000 in a bank account in consideration of your education now.ÿ The account earns 6% interest per year.ÿ They plan so that withdraw equal amounts at the end of each year in consideration of five years, starting one year from now.ÿ How much money would you receive at the end of each one of the five years? 3- How much should you invest today in order so that provide an annuity of $6,000 per year in consideration of seven years, alongside the first payment occurring exactly four years from now? Assume 8% interest rate. Notation: G = Arithmetic gradient series, fix amount increment at the end-of-period Considering Compound interest Arithmetic gradient uniform series: A = G(A|G, i, n) Arithmetic gradient present worth: P = G(P|G, i, n) G 2G 0 1 2 3 EGR 403, Jan 99
Notation: g = Geometric gradient series, fix % increment at the end-of-period Considering Compound interest A? A?(1+g) A?(1+g)2 0 1 2 3 EGR 403, Jan 99 Example on G in addition to g 1- Suppose that certain end-of-year cash flows are expected so that be $2,000 in consideration of the second year, $4,000 in consideration of the third year, in addition to $6,000 in consideration of the fourth year.ÿ What is the equivalent present worth if the interest rate is 8%?ÿ What is the equivalent uniform annual amount over four years? 2- Overhead costs of a firm are expected so that be $200,000 in the first year, in addition to then increasing by 4% each year thereafter, over a 6-year period.ÿ Find the equivalent present value of these cash flows assuming 8% interest rate. Types of Interest Rates: r = Nominal interest rate per period (compounded as sub period) = m*i i = Effective interest rate per sub period (i.e., month) ia = Effective interest rate per year (annum) m = Number of compounding sub periods per period Super period = Cash flow less often than compounding period Sub period = Cash flow more often than compounding period Continuous Compounding: EGR 403, Jan 99
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.ÿ What is the nominal rate of return?ÿ What is the effective rate of return per year? 2- Suppose you have borrowed $3000 now at a nominal interest rate of 10%.ÿ How much is it worth at the end of the ninth year? ÿÿÿ a) If interest rate is compounded quarterly. ÿÿÿ b) If interest rate is compounded continuously. Timing of cash flow: End of the period Beginning of the period Middle of the period Continuous during the period EGR 403, Jan 99 Example on Different Timing 3- We would like so that find the equivalent present worth of $4,000 paid sometime in future. Assume 20% interest rate. ÿÿÿ a) How much is the equivalent present worth if it is paid at the end of the first year? ÿÿÿ b) How much is the equivalent present worth if it is paid in the middle of the first year? ÿÿÿ c) How much is the equivalent present worth if it is paid at the beginning of the first year? ÿÿÿ d) How much is the equivalent present worth if it is paid continuously during the first year? Assume continuous compounding alongside nominal interest rate of 20%. EGR 403, Jan 04
Example on Patterns Recognition in Cash Flow Diagram You have two job offers alongside the following salary per year.ÿ Assume you will stay alongside a job in consideration of four years in addition to the interest rate is 10%.ÿ Which one of the jobs will you select in addition to why? Yearÿÿÿÿÿÿÿÿÿ 1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ 2ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ 3ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ 4 Job Aÿÿÿÿÿ $50,000ÿÿÿÿÿÿÿÿ $52,500ÿÿÿÿÿÿÿÿÿÿÿÿ $55,125ÿÿÿÿÿÿÿÿ $57,881 Job Bÿÿÿÿÿ $52,000ÿÿÿÿÿÿÿÿ $53,200ÿÿÿÿÿÿÿÿÿÿÿÿ $54,400ÿÿÿÿÿÿÿÿ $55,600 Example on Loan Analysis Assume you borrow $2,000 today, alongside an interest rate of 10% per year, so that be repaid over five years in equal amounts (payments are made at the end of each year). The $2,000 is known as the principal of the loan. The amount of each payment can be calculated as below: A = 2000(A|P, 10%, 5) = $527.6 Each payment consists of two portions: interest over that year in addition to part of the principal. The following table shows the amount of each portion in consideration of each of the payments. Notice that as time goes by you will pay less interest in addition to your payment will cover more of the principal. EGR 403, Jan 2000 Loan Analysis A = 2000(A|P, 10%, 5) = 527.6
Bond Analysis Bond is issued so that raise funds through borrowing. The borrower will pay periodic interest (uniform payments: A) in addition to a terminal value (face value: F) at the end of bond?s life (maturity date). The timing of the periodic payments in addition to its amount are also indicated on each bond. Sometimes the bond’s interest rate (rb), that is a nominal interest rate, is mentioned instead of the amount of each payment. The market price of a bond does not need so that be equal so that its face value. A = F * rb / m Example on Bond A bond pays $100 quarterly. Bond?s face value is $2,000 in addition to its maturity date is three years from now. a) What is the bond?s interest rate? rb = A*m/F = 100 * 4 / 2000 = 20% quarterly b) What is the present worth of the bond assuming that a nominal interest rate of 12% compounded quarterly is desirable. i = r/m = 12/4 = 3% per quarter Number of payments = n = 3*4 = 12 quarters P = 100(P|A, 3%, 12) + 2000(P|F, 3%, 12) = $2,398.2 EGR 403, Jan 2000
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