Time Value of Money in Engineering Economics
The fundamental principle that money available today is worth more than the same amount in the future due to its potential earning capacity. This core concept drives all engineering economic decisions.
Key Formulas
Future Value (Single Amount)
\[ F = P(1 + i)^n \]
Where:
F = Future value
P = Present value
i = Interest rate per period
n = Number of periods
Present Worth Factor
\[ P = F \frac{1}{(1 + i)^n} \]
Annuity Calculations
Series of equal payments at regular intervals:
Uniform Series Present Worth
\[ P = A \left[ \frac{(1 + i)^n - 1}{i(1 + i)^n} \right] \]
Capital Recovery Factor
\[ A = P \left[ \frac{i(1 + i)^n}{(1 + i)^n - 1} \right] \]
Used to calculate loan payments
Engineering Cost Concepts
Fixed vs Variable Costs
| Fixed Costs | Variable Costs |
|---|---|
| Constant regardless of production volume | Vary directly with production volume |
| Examples: Rent, salaries, insurance | Examples: Materials, labor, utilities |
Marginal Cost Analysis
The cost of producing one additional unit:
\[ MC = \frac{\Delta TC}{\Delta Q} \]
Where:
MC = Marginal cost
ΔTC = Change in total cost
ΔQ = Change in quantity
Depreciation Methods for Engineering Assets
Straight-Line Method
\[ D_j = \frac{C - S_n}{n} \]
Where:
Dj = Depreciation in year j
C = Initial cost
Sn = Salvage value
n = Useful life
Declining Balance Method
\[ D_j = d \times BV_{j-1} \]
Where:
d = Depreciation rate (e.g., 1.5/n for 150% DB)
BV = Book value at beginning of year
MACRS (Tax Depreciation)
Modified Accelerated Cost Recovery System with predefined recovery periods (3,5,7,10,15,20,27.5,39 years)
Project Evaluation Techniques
Net Present Value (NPV)
\[ NPV = \sum_{t=0}^n \frac{R_t - C_t}{(1 + i)^t} \]
Decision Rule: Accept if NPV ≥ 0
Internal Rate of Return (IRR)
The discount rate that makes NPV = 0
\[ 0 = \sum_{t=0}^n \frac{R_t - C_t}{(1 + IRR)^t} \]
Decision Rule: Accept if IRR ≥ MARR
Benefit-Cost Ratio (BCR)
\[ BCR = \frac{PV(\text{Benefits})}{PV(\text{Costs})} \]
Decision Rule: Accept if BCR ≥ 1.0
Break-Even Analysis for Engineering Projects
The point where total revenue equals total costs:
\[ Q_{BE} = \frac{FC}{P - VC} \]
Where:
QBE = Break-even quantity
FC = Fixed costs
P = Price per unit
VC = Variable cost per unit
Break-Even Graph Components
- Fixed cost line (horizontal)
- Total cost line (starts at FC, slopes upward)
- Revenue line (starts at origin, slopes upward)
- Intersection point = Break-even
Engineering Economics Case Studies
Highway Construction Project
Problem: Compare asphalt vs concrete pavement for a 20km highway with 20-year life
Analysis: Life-cycle cost analysis including initial construction, maintenance, and user delay costs
Equipment Replacement
Problem: Determine optimal replacement year for an excavator with increasing maintenance costs
Analysis: Marginal cost comparison between keeping existing vs purchasing new equipment
Types of Property Values
Salvage Value
It is the estimated value of the material at the end of its useful life, without being dismantled.
Scrap Value
It is the estimated value of the material at the end of its useful life after being dismantled. Scrap Value is counted in the calculation of depreciation of a property, generally at 10% of the cost of the structure.
Distress Value (Forced Sale Value)
In case a property is sold at a lower price than the market value at that time, it is said to have a distressed value.
Book Value
It is defined as the value of the property shown in the account book in that particular year, i.e., the original cost less the total depreciation till that year.
Market Value
The market value of a property is the amount that can be obtained at any particular time from the open market if the property is put for sale. The market value will differ from time to time according to demand and supply.