# Engineering Economics

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Engineering economics is one of the important topics for GATE and other competitive exams, and this topic is more important for designing and working on different projects. The topic of engineering economics tells us the value of money and the value of assets at present or future times.

With the help of the principle of engineering economics, an engineering plan and schedule for the different activities of the project so that the total cost of the project remains as minimum as possible. The article contains fundamental notes on the “Engineering Economics” topic of the “Construction Planning and Management” subject.

Table of content

## What is the Meaning of Engineering Economics?

Engineering economics is one of the branches of engineering in which engineering projects are executed with an optimum cost. It can be either macroeconomics or microeconomics. With the help of the principle of engineering economics, the future value of money can be estimated.

With the help of the principles of engineering economics, the future value of money is calculated, and it will then be invested accordingly. These principles are based on some interest factors, which are described below.

## Interest and Interest Formulae

The term interest is used to designate a rental for the use of money. It can be classified into different types, which are mentioned below

• Simple Interest

When the total interest earned or charged is directly proportional to the principal involved, the interest rate and the no. of interest periods for which the principal is committed, the interest is called simple

l = (P n i)

Where l = total interest

P = principal amount rent or borrow

n = number of years (interest periods)

i = interest rate per year (per interest period)

• Compound Interest

Whenever the interest charge for any interest period (a year, for example) is based on the remaining principal amount plus any accumulated interest charges upto the beginning of that period. The interest is said to be compound. It can be mathematically expressed as I = P(1+i)n for n years.

## What is Cash Flow Diagram

A cash flow diagram is a diagram with the help of which the interest of particular money is calculated for a particular year.

Notations and Cash Flow

i = the annual interest rate

n = the number of annual interest periods

P = Present sum of money i.e. Present worth at zero time

F = future sum of money i.e. Future worth equal to the compound amount at the end of n years

A = a single payment, in a series of n equal payments, made at the end of each annual interest period

G = uniform period-by-period increase or decrease in amount (the arithmetic gradient)

The graphic presentation of each value plotted at the appropriate time is called a cash flow diagram. The normal conventions for cash flow diagrams are as follows;

The horizontal line is a time scale with a progression of time moving from left to right. The value indicated on the time scale (viz., 0, 1,2,….n) indicates the end of the respective period.

The arrows signify cash flow; normally, downward arrows represent disbursement or costs, and upward arrows represent receipts or benefits.

### Interest Formulae for Single Payment Series

The figure shows a cash flow diagram involving a present single sum (P), and a future single sum (F), separated by n years with interest rate I per year;

• Case I: Finding F when P is given

At the end of n year,

The quantity (1 + i)n is commonly called the single payment compound amount factor indicated by the functional notations as [F/P, i, n]

∴ [F/P, i, n] = Single payment compound amount factor = (1 + i)n.

• Case II: Finding P when F is given

From the equation, F = P (1 + i)n, solving this for P gives the relation P = F[1/(1+i)n]

The quantity [1/(1+i)n] is called single payment present worth factor, indicated by the functional symbol as [P/F, i, n]

∴ [P/F, i, n] = Single payment present worth factor = [1/(1+i)n]

### Interest Formulae for Equal Payment Series

The figure shows a general cash flow diagram involving a series of uniform (equal) payments, each of amount A, occurring at the end of each year with interest rate i per year.

• Case III: Finding P when A is given

If A exists at the end of each year for n years with the i rate of interest, the present worth P is obtained by summing the present worth of each payment of amount A

The series in the bracket is in the geometric progression whose first term (a) is given by 1/(1+i), and geometric ratio (r) is 1/(1+i); Hence sum is given by a(1-rn)/(1-r).

So,

The quantity [{(1+i)n-1}/i(1+i)n] is called uniform series present worth factor indicated by the function notation as [P/A, i, n]

[P/A, i, n] = uniform (equal) series present worth factor = {(1+i)n-1}/i(1+i)n

Some of these interest factors are given below in tabular form, these can also be derived as above.

### Interest Formulae for Uniform Gradient Payment Series

Some economic analysis problems involve receipts or disbursements projected to increase or decrease by a uniform amount each period, thus contributing an arithmetic series. Generally, a uniformly increasing series of payments for n interest periods may be expressed as A1, A1 + g, A1 + 2g, A1 + 3g … A1 +(n-1)g as shown in Fig. Where A1 denotes the first year-end payment in the series and ‘g,’ is the annual change in magnitude called gradient amounts.

A1 = payment at the end of the first year

g = annual change in gradient

n = number of years

A = the equivalent annual payment of the series

A2 = the equivalent annual payment of the gradient series [0, g, 2g, … (n – 1)g] at the end of successive years.

A = A1 + A2

Where  and F is the future amount equivalent to the gradient series. The future amount equivalent to the gradient series can be derived from the table as follows,

Table: Gradient Series and an equivalent Set of series

The terms in the brackets consist of n terms, 1st term being (1 + i)0 and the ratio being (1 + i) of geometric progression.

The resulting factor [(1/i) – (n/i){A/f, i,n}] is called the gradient factor for annual compounding interest and will be designated (A/G, i,n)

∴ A2 = Equivalent annual cost of a set of gradient series = g(A/G, i,n)

### Nominal and Effective Interest Rates

Effective interest rate =(1+r/c)c – 1

Where r is the nominal interest rate and c is no. of interest periods per year if the compounding is annual, then r = i.

## What is Depreciation?

Depreciation is the loss in value of an asset with the passage of time. The main purpose of depreciation is to recover capital involved in possessing the physical property. Depreciation of assets can be related to the following parameters.

• Salvage Value (or Resale Value)

It is the property’s value at the end of its utility period without being dismantled. Salvage value implies that the property has further utility.

• Scarp Value

The value of a property is realized when it becomes useless except for sale, as junk is its scrap value.

• Book Value

It is defined as the value of the property shown in the account books in that particular year i.e. the original cost less total depreciation till that year.

## Methods for Calculating Depreciation

There are several methods of calculating depreciation. The following notation has been used;

Ci = Initial cost of an asset at zero time or original cost (This will include the cost of asset + transporting charge + installation and other charges spent initially)

Cs = Salvage value (or scrape value) to be estimated at the end of the utility period or scrap cost

n = the life of the asset

Bm = Book value at the end of the period ‘m’

(i) Straight Line Method

In the method, the property is assumed to lose value by a constant amount yearly. The salvage value (or scrap value) is left at the end of the life.

Dm = (Ci-Cs)/n

Dm=D1=D2=D3=Dn

This method is recommended for all equipment/assets with constant demand and does not face any obsolescence during their useful life. It is widely used in the case of all civil engineering appliances and applications.

(ii) Declining Balance Method (or Constant Percentage Method)

In this method, the property is assumed to lose value annually at a constant percentage of its book value.

Fixed Declining Balance (FDB) = 1- (Cs/Ci)1/n

(iii) Fixed Double Declining Balance Method

In this method, the property is also assumed to lose value annually by a fixed factor of the book value.

FDDB = Fixed factor for double declining balance method.

FDDB is taken as double the straight-line rate. i.e. FDDB = 2/n

FDDB = 1- (Cs/Ci)1/n

(iv) Sum of the Years Digit Method

In this method, the digits corresponding to the number of each year of life are listed in reverse order. The general expression for the annual depreciation for any year (m) when the life is n years is expressed as

m = No. of years of which depreciation is calculated

(v) Sinking Fund Method

The sinking fund depreciation model assumes that the value of an asset decreases at an increasing rate. An equal amount (D) is assumed to be deposited into a sinking fund at the end of each year of the asset’s life.

Depletion

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