4.3 Engineering Economics & Financial Analysis

4.3 Engineering Economics & Financial Analysis

1. Principles of Engineering Economics & Interest Formulas

Engineering Economics combines engineering decision-making with financial evaluation to select the most cost-effective alternatives over a project's lifecycle.

Simple vs. Compound Interest

• Simple Interest: Interest is charged only on the principal amount $P$: $$I = P \cdot i \cdot n, \quad F = P(1 + i \cdot n)$$
• Compound Interest: Interest earned in each period is added to the principal to earn interest in subsequent periods: $$F = P(1 + i)^n$$ where $P$ is Present Worth, $F$ is Future Worth, $i$ is interest rate per period, and $n$ is number of compounding periods.

Nominal vs. Effective Interest Rates

When an annual nominal interest rate $r$ is compounded $m$ times per year:

• Period interest rate: $i = \frac{r}{m}$
• Total periods in $Y$ years: $n = m \cdot Y$
• Effective Annual Rate ($i_{\text{eff}}$): $$i_{\text{eff}} = \left(1 + \frac{r}{m}\right)^m - 1$$
• Continuous Compounding: As $m \to \infty$, the future worth becomes $F = P e^{r n}$, and $i_{\text{eff}} = e^r - 1$.

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2. Discrete Cash Flow Equivalence Factors

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3. Project Evaluation Methods

1. Present Worth (PW) Method

All cash inflows ($+R_t$) and outflows ($-C_t$) are discounted to $t=0$ using the Minimum Attractive Rate of Return (MARR):

$$\text{PW} = \sum_{t=0}^n \frac{R_t - C_t}{(1 + \text{MARR})^t}$$

Decision Criterion: Accept project if $\text{PW} \ge 0$.

2. Annual Worth (AW) Method

Converts all project cash flows into an equivalent uniform annual cash flow over the service life:

$$\text{AW} = \text{PW} \times (A/P, i, n)$$

Decision Criterion: Accept project if $\text{AW} \ge 0$. Useful for comparing mutually exclusive projects with unequal service lives.

3. Internal Rate of Return (IRR / ROR)

The discount rate $i^*$ at which the Present Worth of cash inflows equals the Present Worth of cash outflows (i.e., $\text{PW}(i^*) = 0$):

$$\sum_{t=0}^n \frac{R_t - C_t}{(1 + i^*)^t} = 0$$

Decision Criterion: Accept project if $\text{IRR} \ge \text{MARR}$.

4. Capitalized Cost ($CC$)

Capitalized cost is the present worth of an asset that is assumed to provide service perpetually (infinite life $n = \infty$):

$$CC = C_0 + \frac{A}{i} + \frac{RC}{(1 + i)^k - 1}$$

where $C_0$ is initial cost, $A$ is annual operating cost, $RC$ is replacement cost every $k$ years, and $i$ is interest rate.

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4. Asset Depreciation & Tax Accounting

Depreciation is the systematic allocation of an asset's initial cost $C$ over its useful recovery life $n$, leaving a salvage value $S$.

Straight-Line Method (SL)

Constant annual depreciation charge $d_t$:

$$d_t = d = \frac{C - S}{n}$$

Book value at end of year $t$: $BV_t = C - t \cdot d$.

Sum-of-the-Years-Digits Method (SYD)

Accelerated depreciation method where annual charge decreases linearly:

$$\text{SOYD} = \frac{n(n + 1)}{2}$$ $$d_t = (C - S) \times \frac{n - t + 1}{\text{SOYD}}$$

Declining Balance Method (DB / DDB)

Fixed percentage rate $k$ applied to the beginning book value $BV_{t-1}$:

• $k = \frac{1.5}{n}$ (150% Declining Balance) or $k = \frac{2.0}{n}$ (200% Double Declining Balance)
• $d_t = k \cdot BV_{t-1}$
• $BV_t = C(1 - k)^t$

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5. Break-Even Analysis & Inflation

Break-Even Analysis

Determines the production volume $Q$ where Total Revenue $TR$ equals Total Cost $TC$:

$$TR = P \cdot Q, \quad TC = FC + VC \cdot Q$$ $$Q_{\text{BE}} = \frac{FC}{P - VC}$$

where $FC$ is fixed costs, $VC$ is variable cost per unit, and $P$ is selling price per unit.

Inflation Adjustment

If market interest rate is $i$ and annual inflation rate is $f$, the real (inflation-free) interest rate $i'$ is:

$$(1 + i) = (1 + i')(1 + f) \implies i = i' + f + i' f$$