10.2 Engineering Economics

Standards and rules: these are study notes, not a current code or legal reference. Check the applicable edition, amendments, units and design assumptions before using a numerical limit or rule.

Engineering economics is the application of economic principles to the evaluation of engineering projects and investments. It involves assessing the financial viability and risks associated with engineering decisions. This section covers key concepts such as project cash flow, time value of money, and various methodologies used to analyze and compare engineering projects.

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1. Understanding Project Cash Flow

Project cash flow represents the inflow and outflow of money during the life cycle of a project. It helps in assessing the financial feasibility of a project by determining whether the revenues generated by the project outweigh the costs.

• Inflow: The cash received from the project, such as sales, investments, or cost savings.
• Outflow: The cash spent on the project, such as operational costs, equipment, labor, and other capital expenses.
• Net Cash Flow: The difference between the inflows and outflows at any given point in time.

To evaluate the project’s viability, these cash flows are typically projected over a specific time period and discounted to their present value using an appropriate discount rate.

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2. Discount Rate, Interest, and Time Value of Money

Time value of money is the concept that money available now is worth more than the same amount in the future due to its potential earning capacity. This principle underlies most financial calculations and decision-making.

• Discount Rate: The rate at which future cash flows are adjusted to reflect their present value. It reflects the opportunity cost of capital, inflation, and risk.
• Interest: The cost of borrowing money or the return on invested capital. Interest is calculated based on the principal amount and the interest rate.
• Present Value (PV): The current value of a future sum of money, discounted at a certain rate over time.
• Future Value (FV): The value of an amount of money at a future time, after accounting for interest or discounting.

Formula for calculating the present value (PV) of a future cash flow:
$PV = \frac{FV}{(1 + r)^t}$

Where:

• $FV = \text{Future value,}$
• $r = \text{Discount rate,}$
• $t = \text{Time period (years).}$

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Effective Interest Rate Calculations

Problem 1: Effective Interest Rate (EIR) Per Year
Question: A bank has a nominal interest rate of 9% per annum, compounded quarterly. Calculate the effective interest rate per year.
Given:

• Nominal interest rate $r = 9\% = 0.09$
• Compounding frequency $n = 4$ (quarterly)

Formula:
$EIR = \left(1 + \frac{r}{n}\right)^n - 1$

Solution:
$EIR = \left(1 + \frac{0.09}{4}\right)^4 - 1 = 0.09308$

Answer: The effective interest rate per year is 9.308%.

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Problem 2: Effective Interest Rate Per Semi-Annual
Question: A bank has a nominal interest rate of 9% per annum, compounded quarterly. Calculate the effective interest rate per semi-annual period.
Given:

• Nominal interest rate $r = 9\% = 0.09$
• Compounding frequency $n = 4$ (quarterly)
• Semi-annual compounding means $n = 2$ periods per year.

Formula:
$EIR_{\text{semi-annual}} = \left(1 + \frac{r}{n}\right)^{n/2} - 1$

Solution:
$EIR_{\text{semi-annual}} = \left(1 + \frac{0.09}{4}\right)^2 - 1 = 0.04550625$

Answer: The effective interest rate per semi-annual period is 4.551%.

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Problem 3: Effective Interest Rate Per Month
Question: A bank has a nominal interest rate of 9% per annum, compounded quarterly. Calculate the effective interest rate per month.
Given:

• Nominal interest rate $r = 9\% = 0.09$
• Compounding frequency $n = 4$ (quarterly)
• Monthly compounding means $n = 12$ periods per year.

Formula:
$EIR_{\text{monthly}} = \left(1 + \frac{r}{n}\right)^{n/12} - 1$

Solution:
$EIR_{\text{monthly}} = \left(1 + \frac{0.09}{4}\right)^{1/3} - 1 = 0.007448$

Answer: The effective interest rate per month is 0.7448%.

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Some Important Financial Formulas

• Single-Payment Compound Amount:
  $F = P(1 + i)^n$
• Single-Payment Present Worth Amount:
  $P = F \cdot (P/F, i, n) = \frac{F}{(1 + i)^n}$
• Equal-Payment Series Compound Amount:
  $F = A \cdot (F/A, i, n) = A \cdot \frac{(1 + i)^n - 1}{i}$
• Equal-Payment Series Sinking Fund:
  $A = F \cdot (A/F, i, n) = F \cdot \frac{i}{(1 + i)^n - 1}$
• Equal-Payment Series Present Worth Amount:
  $P = A \cdot (P/A, i, n) = A \cdot \frac{(1 + i)^n - 1}{i(1 + i)^n}$
• Equal-Payment Series Capital Recovery Amount:
  $A = P \cdot (A/P, i, n) = P \cdot \frac{i(1 + i)^n}{(1 + i)^n - 1}$
• Uniform Gradient Series Annual Equivalent Amount:
  $A = A_1 + G \cdot (A/G, i, n) = A_1 + G \cdot \frac{(1 + i)^n - i \cdot n - 1}{i(1 + i)^n - i}$

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3. Basic Methodologies for Engineering Economics Analysis

Engineering economic analysis is a critical aspect of evaluating the feasibility and financial viability of engineering projects. It combines economic and financial principles with engineering knowledge to assess costs, benefits, and risks. Below are some commonly used methodologies in this field:

1. Payback Period

The payback period is the time required to recover the initial investment from the cash inflows generated by a project.

Simple Payback Period:
$\text{Simple Payback Period} = \frac{\text{Initial Investment}}{\text{Annual Cash Inflow}}$

Decision Rule:

• Accept: If the payback period is less than or equal to the required period.
• Reject: If the payback period is greater than the required period.

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2. Discounted Payback Period:

Time taken for the sum of discounted cash inflows to equal the initial investment.

Formula:
$\text{Discounted Cash Flow} = \frac{C_t}{(1 + i)^t}$

Decision Rule:

• Accept: If the discounted payback period is less than or equal to the required period.
• Reject: If the discounted payback period is greater than the required period.

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3. Equivalent Worth Method

This method involves converting cash flows to a common point in time using the time value of money. Sub-methods include:

• Present Worth (PW):

  
  $\text{PW} = \sum \frac{C_t}{(1 + i)^t}$

Decision Rule:

• Accept: If the present worth is greater than or equal to zero (NPV ≥ 0).
• Reject: If the present worth is less than zero (NPV < 0).

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• Future Worth (FW):

  
  $\text{FW} = \text{PW} \times (1 + i)^n$

Decision Rule:

• Accept: If the future worth is greater than or equal to the initial investment.
• Reject: If the future worth is less than the initial investment.

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• Annual Worth (AW):

  
  $\text{AW} = \text{PW} \times \frac{i(1 + i)^n}{(1 + i)^n - 1}$

Decision Rule:

• Accept: If the annual worth is greater than or equal to the required minimum annual return.
• Reject: If the annual worth is less than the required minimum annual return.

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4. Rate of Return Method

This method calculates the rate of return that a project earns over its life. Here the project cashflow reinvested at the IRR. It includes:

• Internal Rate of Return (IRR):

  
  $\text{NPV} = \sum \frac{C_t}{(1 + \text{IRR})^t} - \text{Initial Investment} = 0$

Decision Rule:

• Accept: If the IRR is greater than or equal to the required rate of return.
• Reject: If the IRR is less than the required rate of return.

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• Minimum Attractive Rate of Return (MARR):

The MARR is the minimum return that an investor or organization requires from a project, typically based on the cost of capital or opportunity cost of funds. It serves as a benchmark to compare with the IRR to determine the project’s acceptability.

Decision Rule:

• Accept: If IRR ≥ MARR.
• Reject: If IRR < MARR.

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• External Rate of Return (ERR):

Also known as modified internal rate of return. On ERR, here the project cashflow reinvested at the cost of capital not the IRR. Here the slight modification makes the MARR more effective than that of IRR.

Decision Rule:

• Accept: If the ERR exceeds the cost of capital or required return rate.
• Reject: If the ERR is less than the cost of capital or required return rate.

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5. Benefit-Cost Ratio (BCR):

The Benefit-Cost Ratio compares the present value of benefits to the present value of costs.

Formula:

$\text{BCR} = \frac{\text{PV of Benefits}}{\text{PV of Costs}}$

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• Conventional B/C Ratio:

  
  $B/C = \frac{\text{PV of (B)}}{1 + \text{PV of (O\&M)} - \text{PV of (Sv)}}$

Where:

• PV of Benefits is the present value of the benefits over the project's life.
• PV of O&M Costs is the present value of the operating and maintenance costs.
• PV of Salvage Value is the present value of the value at the end of the project's life.
• PV of Costs is the present value of the total project costs (including initial investment and operating costs).

Decision Rule:

• Accept: If B/C ≥ 1
• Reject: If B/C < 1

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• Modified B/C Ratio:

  
  $B/C = \frac{\text{PV of Benefits} - \text{PV of O\&M Costs}}{1 - \text{PV of Salvage Value}}$

Decision Rule:

• Accept: If B/C ≥ 1
• Reject: If B/C < 1

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4. Comparison of Alternatives

When comparing multiple project alternatives, various factors are considered, such as:

• Cost-benefit analysis: Evaluating the costs and benefits of each alternative.
• NPV Comparison: Projects with higher NPVs are generally preferred as they add more value.
• IRR Comparison: Projects with higher IRRs are typically more attractive, but the MARR must also be considered.
• Payback Period: A shorter payback period is typically more desirable, especially for companies with liquidity concerns.

The comparison helps in choosing the alternative that offers the best financial returns and risk profile.

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5. Depreciation System in Nepal

Depreciation is the process by which a company allocates the cost of a long-term asset over its useful life. This reflects the decrease in the asset's value as it is used in the business. In Nepal, the rules regarding depreciation are governed by the Income Tax Act, and companies are allowed to deduct depreciation expenses from their taxable income, which helps reduce their tax burden.

Methods of Depreciation:

1. Straight-Line Method of Depreciation:

• Depreciation Charge (Dn) per Year:

  
  $D_n = \frac{P - S}{N}$

• Where:

  • $P = \text{Cost of the asset, including installation expenses.}$
  • $S = \text{Salvage value at the end of the asset’s useful life.}$
  • $N = \text{Useful life of the asset (in years).}$
  • $D_n = \text{Depreciation charge for each year (uniformly distributed).}$

• Book Value after n Years (Bn):

  
  $B_n = P - (D_1 + D_2 + \dots + D_n)$

  Where:

  • $B_n = \text{Book value of the asset after n years.}$
  • $D_1, D_2, \dots, D_n$ = Depreciation charges for the years $1, 2, ..., n$ (which are all the same using the straight-line method

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2. Declining Balance Method:

• Depreciation Charge (D) per Year:

  
  $D = C \times \left( 1 - \left( \frac{V}{C} \right)^n \right)$

  Where:

  • $D = \text{Depreciation charge for the year.}$
  • $C = \text{Original cost of the asset.}$
  • $V = \text{Scrap (or salvage) value of the asset.}$
  • $n = \text{Useful life of the asset (in years).}$

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3. Sum-of-Years-Digits (SOYD) Method:

• Sum of the Years' Digits (SOYD):

  
  $SOYD = 1 + 2 + \dots + N = \frac{N(N + 1)}{2}$

  Where:

  • $N = \text{Useful life of the asset (in years).}$

• Depreciation Charge (Dn) per Year:

  
  $D_n = \frac{(N - n + 1)(P - S)}{SOYD}$

  Where:

  • $D_n = \text{Depreciation charge for the nth year.}$
  • $P = \text{Cost of the asset, including installation expenses.}$
  • $S = \text{Salvage value at the end of the asset’s useful life.}$
  • $N = \text{Useful life of the asset (in years).}$
  • $n = \text{Year in which the depreciation is being calculated (e.g., for year 1, n = 1).}$
  • $SOYD = \text{Sum of the years' digits.}$

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4. Sinking Fund Method:

The Sinking Fund Method of depreciation is used to accumulate funds over time to replace an asset once its useful life has ended. It assumes that a fixed annual deposit is made into a sinking fund, and the interest earned on this fund will cover the depreciation of the asset.

• Depreciation Charge (Dn) per Year:

  
  $D_n = \frac{P - S}{\text{Present Value of Annuity Factor for N Years}}$

  Where:

  • $P = \text{Cost of the asset, including installation expenses.}$
  • $S = \text{Salvage value at the end of the asset’s useful life.}$
  • $D_n = \text{Depreciation charge for the nth year.}$
  • $N = \text{Useful life of the asset (in years).}$

• Present Value of Annuity Factor:

  
  $\text{Present Value of Annuity Factor} = \frac{1 - (1 + r)^{-N}}{r}$

  Where:

  • $r = \text{Interest rate per period (usually annual).}$
  • $N = \text{Useful life of the asset (in years).}$

• Book Value after n Years (Bn):

  
  $B_n = P - (D_1 + D_2 + \dots + D_n)$

  Where:

  • $B_n = \text{Book value of the asset after n years.}$
  • $D_1, D_2, \dots, D_n = \text{Depreciation charges for each year.}$

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6. Taxation and depreciation in project analysis

Distinguish accounting profit from cash flow, and accounting depreciation from a tax deduction. A non-cash depreciation charge may affect taxable income and therefore after-tax cash flow; it does not represent a second cash purchase of the asset.

For a stated hypothetical tax rate, calculate the tax effect using the question's assumptions. In a real Nepal tax calculation, use the Income Tax Act, applicable Finance Act and current IRD guidance for asset classes, deductions, rates and allowances. Tax depreciation need not match straight-line depreciation in an engineering-economics example. The imported source's undated tax examples are omitted rather than presented as current requirements.

Conclusion

Engineering economics helps in making informed decisions by providing quantitative methods to evaluate the financial viability of projects. By considering cash flow, the time value of money, and applying methodologies like NPV, IRR, and the discounted payback period, engineers can assess the profitability of projects. Additionally, understanding depreciation systems and taxation is crucial in optimizing the financial outcomes of projects, especially in specific regions like Nepal.