Engineering Economics Essentials

Cheatsheet Content

Fundamentals of Engineering Economics Definition: Involves systematic evaluation of economic merits of proposed solutions to engineering problems. Time Value of Money: A dollar today is worth more than a dollar in the future due to earning potential. Interest Rate ($i$): The cost of borrowing money or the return on an investment, usually expressed as a percentage per period. Cash Flow Diagram: Visual representation of money received (up arrow) and money spent (down arrow) over time. Interest Formulas (Single Payment) Future Value (FV) of a Single Payment Formula: $F = P(1+i)^n$ Where: $F$: Future value (or compound amount) $P$: Present value (or principal) $i$: Interest rate per period $n$: Number of periods Factor: $(F/P, i, n)$ - Single Payment Compound Amount Factor Present Value (PV) of a Single Payment Formula: $P = F(1+i)^{-n} = F / (1+i)^n$ Factor: $(P/F, i, n)$ - Single Payment Present Worth Factor Interest Formulas (Uniform Series) Future Value (FV) of a Uniform Series Formula: $F = A \left[ \frac{(1+i)^n - 1}{i} \right]$ Where: $A$: Uniform series amount per period Factor: $(F/A, i, n)$ - Uniform Series Compound Amount Factor Present Value (PV) of a Uniform Series Formula: $P = A \left[ \frac{(1+i)^n - 1}{i(1+i)^n} \right]$ Factor: $(P/A, i, n)$ - Uniform Series Present Worth Factor Capital Recovery (A given P) Formula: $A = P \left[ \frac{i(1+i)^n}{(1+i)^n - 1} \right]$ Factor: $(A/P, i, n)$ - Capital Recovery Factor Sinking Fund (A given F) Formula: $A = F \left[ \frac{i}{(1+i)^n - 1} \right]$ Factor: $(A/F, i, n)$ - Sinking Fund Factor Nominal vs. Effective Interest Rates Nominal Rate ($r$): Annual interest rate without considering compounding periods (e.g., 12% compounded monthly). Effective Annual Rate ($i_a$): Actual annual rate considering compounding periods. Formula: $i_a = \left(1 + \frac{r}{m}\right)^m - 1$ Where: $m$: Number of compounding periods per year Gradient Series Arithmetic Gradient (G) Series of payments that increase or decrease by a constant amount each period. Present Worth: $P_G = G \left[ \frac{1}{i} \left( \frac{(1+i)^n - 1}{i(1+i)^n} - \frac{n}{(1+i)^n} \right) \right]$ Or, convert to an equivalent uniform series: $A_G = G \left[ \frac{1}{i} - \frac{n}{(1+i)^n - 1} \right]$ Geometric Gradient Series of payments that increase or decrease by a constant percentage each period. Present Worth: If $g \neq i$: $P = A_1 \left[ \frac{1 - (1+g)^n (1+i)^{-n}}{i-g} \right]$ If $g = i$: $P = \frac{nA_1}{1+i}$ Where $A_1$ is the first payment, $g$ is the constant rate of change. Comparison Methods for Alternatives Present Worth (PW) Analysis Convert all cash flows of each alternative to their present worth. Choose the alternative with the highest PW (for benefits) or lowest PW (for costs). Must use a common analysis period (LCM or study period). Annual Worth (AW) Analysis Convert all cash flows of each alternative to an equivalent uniform annual series. Choose the alternative with the highest AW (for benefits) or lowest AW (for costs). No need for a common analysis period as AW is inherently uniform over any life. Rate of Return (ROR) Analysis Find the interest rate ($i^*$) at which PW or AW of an alternative equals zero. For mutually exclusive alternatives, compute incremental ROR ($\Delta ROR$) between pairs. Decision Rule: Accept alternatives where $i^* \geq MARR$ (Minimum Attractive Rate of Return). Benefit/Cost Ratio (B/C Ratio) Ratio of present worth of benefits to present worth of costs: $B/C = PW(Benefits) / PW(Costs)$ Decision Rule: Accept if $B/C \geq 1$. For mutually exclusive alternatives, use incremental B/C ratio. Depreciation Definition: Systematic allocation of the cost of a tangible asset over its useful life. Book Value (BV): Original cost minus accumulated depreciation. Salvage Value (SV): Estimated value of an asset at the end of its useful life. Cost Basis (B): Original cost of the asset plus all costs to get it ready for use. Straight-Line Depreciation Annual Depreciation: $D = (B - SV) / N$ Where $N$ is the useful life. Sum-of-Years'-Digits (SOYD) Depreciation SOYD = $N(N+1)/2$ Annual Depreciation for year $k$: $D_k = (B - SV) \times \frac{N - k + 1}{SOYD}$ Declining Balance (DB) Depreciation Rate: $R = 2/N$ (for Double Declining Balance, DDB) Annual Depreciation for year $k$: $D_k = BV_{k-1} \times R$ Cannot depreciate below salvage value. Switch to straight-line if it yields a higher depreciation amount in later years. MACRS (Modified Accelerated Cost Recovery System) Mandatory for tax purposes in the U.S. Uses specific recovery periods and depreciation percentages provided by the IRS. Usually assumes half-year convention in the first and last year. Inflation Definition: Rate at which the general level of prices for goods and services is rising. Inflation Rate ($f$): Annual percentage increase in prices. Real Interest Rate ($i'$): Interest rate adjusted for inflation. Market Interest Rate ($i$): Nominal interest rate observed in the market. Fisher Equation: $1 + i = (1 + i')(1 + f)$ or $i \approx i' + f$ (for small rates) Cost-Benefit Analysis Systematic process for calculating and comparing benefits and costs of a project or decision. Helps in decision-making by quantifying pros and cons. Break-Even Analysis Finds the point where total costs equal total revenues. Fixed Costs (FC): Costs that do not vary with production volume. Variable Costs (VC): Costs that vary directly with production volume. Total Cost (TC): $TC = FC + VC \times Q$ (where $Q$ is quantity) Total Revenue (TR): $TR = P \times Q$ (where $P$ is price per unit) Break-Even Quantity: $Q_{BE} = FC / (P - VC)$