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ENGINEERING ECONOMICS
Simple Interest
I = Pin
F = P + Pin
F = P (1 + in)
where: I = interest
P = principal present worth
n = no of interest periods
i = rate of interest per interest period
F = accumulated amount or future amount worth
Types of Simple Interest
1. Ordinary Simple Interest – based on 30 days per month or 360 days per year
(also known as the banker’s year).
2. Exact Simple Interest – based on the exact number of days in a year, 365 days
for an ordinary year and 366 days for a leap year.
Note: A year is a leap year if it is divisible by 4 except when it is divisible by 100
but not by 400.
Examples:
1996 (leap year)
1900 (not leap year)
2000 (leap year)
Compound Interest
In calculations of compound interest, the interest for an interest period is calculated
on the principal plus total amount on interest accumulated in previous periods.
Thus compound interest means “interest on top of interest”.
F = P (F/P, i%, n)
or
F = P(1 + i)n
P = F(1 + i)-n
or
P = F(P/F, i%, n)
where: (F/P, i%, n) = (1 + i)n – single payment compound factor (SPCAF)
(P/F, i%, n) = (1 + i)-n – single payment present worth factor (SPPWF)
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Rate of Interest
Nominal Rate of Interest – it specifies the rate of interest and a number of interest
periods in one year. It is the annual interest rate ignoring the effect of any
compounding.
r
r  im or i 
m
where: i = rate of interest per interest period
r = nominal interest rate
m = no. of compounding periods per year
Example 1: The nominal rate of interest is 20% compounded quarterly, therefore,
20%
i
 5% per period
4
Or
1 year
20%
i
compounded quarterly X
 5% per quarter
year
quarter
Example 2: 12% compounded semi-monthly
12%
i
 0.5% per semi  monthly compounded semi  monthly
24
Or
1 year
12%
i
X
 0.5% per semi  monthly
year
24 semi  monthly
Effective Rate of Interest (ERI) – is the actual rate of interest on the principal for
one year.
Formula:
r

ERI  1  i m  1   1  
m

m
1
Illustration:
If P1.00 is invested at a nominal rate of 10% compounded quarterly, after one year
this will become, F = P(1 + r/m)m = 1(1 + 0.1/4)4 or P 1.1038128 then the actual
interest earned is, I = F – P = 1.1038128 -1 = P 0.1038128 therefore the actual rate
of interest after one year is, ERI = i eff = (P 0.1038128/P 1) x 100% = 10.38128% it
means,
F  P  I OR I  F  P
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For n = 1,
r

P i eff  P 1  
m

m
P
r

ERI  i eff   1  
m

m
1
Analysis by Theory of Equivalence
F
r

F  P 1  
m


m
n = 1 yr
P
F'  P 1  i eff  1
F
But F = F’, therefore,
m
r

P  1    P1  i eff  1
m

ERI  i eff
Note:
r

 1  
m

n = 1 yr
P
m
1
For two or more nominal rates to be equivalent, their corresponding
effective rates must be equal.
Discount – is interest deducted in advance.
D=F–P
where: D – Discount
P – Present worth
F – Future worth
The rate of discount is the discount on one unit of principal for one unit of time.
DFP
Fd  F  F 1  i  1
P = F(1 + i)-1
d  1  1  i  1
i
1
0
d
1d
F
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where: d – rate of discount
i – rate of interest
Note:
In general,
D = Fdn
where: n = no. of periods
Equation of Value
An equation of value is obtained by setting the sum of the values on a certain
comparison or focal date of one set of obligations equal to the sum of the values on
the same date of another set of obligations.
Annuities
Annuity – is a series of equal payments occurring at equal periods of time.
1.
Ordinary Annuity – is one where the payments are made at the end of each
period.
P
0
1
2
3
n-1
n
A
A
A
A
A
 1  1  i   n 
 1  i  n  1 
P  A
  A
n 
i


 i 1  i  
OR
P = A (P/A, i%, n)
where: (P/A, i%, n) – uniform series present worth factor
Also,


i
A  P
 P A / P, i%, n 
n 
 1  1  i  
where: (A/P, i%, n) – capital recovery factor
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To find F in terms of A, i, and n
F
0
1
2
3
n-1
n
A
A
A
A
A
a series of periodic, equal amount of money
From,
F  P 1  i  n
OR
 1  1  i   n 
n
F  A
 1  i 
i


 1  i  n  1 
F  A

i


F  A F / A, i%, n 
Also,
OR


i
A  F

n
 1  i   1 
A  F A / F, i%, n 
where: (F/A, i%, n) – uniform series compound factor
(A/F, i%, n) – sinking fund factor
Note:
Also,
(A/F, i%, n) + i = (A/P, i%, n)
Sinking fund factor + i = capital recovery factor
Annuity Due – is one where the payments are made at the beginning of each
period.
P
-1
0
A
F
1
A
2
3
n-1
A
A
A
Annuity due , n periods
(n + 1) periods
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 1  1  i  1  n 
P  A 1 

i


 1  i  1  n  1

F  A
 1
i


where: n – number of equal payments
Deferred Annuity – is one where the first payment is made several periods after
the beginning of the annuity.
P
0’
n periods
m periods
1
2
0
m
Deferment, m periods
A(P/A, i%, n)(P/F, i%, m)
1
2
A
A
n–1 n
A
A
A(P/A, i%, n)
P  AP / A, i%, n P / F, i%, m 
 1  1  i   n 
m
P  A
 1  i 
i


Perpetuity – is an annuity in which the payments continue indefinitely.
P
0
1
2
3
A
A
A
n→∞
 1  1  i   n 
 1  1  i    
P  A
  A

i
i




A
P
i
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Depreciation – is the decrease in the value of physical property with the passage
of time.
Purposes of Depreciation
1. To provide for the recovery of capital which has been invested in physical
property.
2. To enable the cost of depreciation to be charged to the cost of producing
products or services that results from the use of the property.
Types of Depreciation
1. Normal Depreciation
a. physical
b. functional
2. Depreciation due to changes in price levels
3. Depletion – refers to the decrease in the value of a property due to the
gradual extraction of its contents.
Depreciation Methods
We shall use the following symbols for the different depreciation methods.
L – useful life of the property in years
Co – the original cost
CL – the value at the end of the life, the scrap value (including gain and
loss due to removal)
d – the annual cost of depreciation
Cn – the book value at the end of n years
Dn – depreciation up to age n years
The Straight Line Method
This method assumes that the loss in value is directly proportional to the age of the
property.
Co  C L
L
n Co  C L 
Dn 
L
Cn  Co  Dn
d
Also,
Dn  n d
d
Co  Cn
n
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The Sinking Fund Method
This method assumes that a sinking fund is established in which funds will
accumulate for replacement. The total depreciation that has taken place up to any
given time is assumed to be equal to the accumulated amount in the sinking fund at
that time.
Dn Co – CL
0
1
2
3
d
d
d
n
L
d
d
 C C 
Co  CL
  o L L i
F / A, i%, L  1  i   1 
 1  i  n  1 
Dn  d F / A, i%, n   d 

i


Cn  Co  Dn
d
Declining Balance Method – sometimes called the constant percentage method or
the Matheson Formula.
dn  Co 1  k  n  1 k
where: dn – depreciation during the nth year
k – the rate of depreciation
n
Cn  Co 1  k 
C L
 Co  L 
 Co 
n
CL  Co 1  k  L
k 1 n
Note:
Cn
C
1 L L
Co
Co
This method does not apply, if the salvage value is zero because k will be
equal to one and d1 will be equal to Co.
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Double Declining Balance (DDB) Method
This method is very similar to the declining balance method except that the rate of
depreciation k is replaced by 2/L.
2

dn  Co  1  
L


Note:
n 1
2

Cn  Co  1  
L


n
2

CL  Co  1  
L


L
2
 
L
When the DDB method is used, the salvage value should not be subtracted
from the first cost when calculating the depreciation charge.
The Sum of the Years Digit (SYD) Method
dn – depreciation change during the n th year
dn – depreciation factor x total depreciation
reverse digit
Co  CL 
dn 
sum of digits
For example, for a property whose life is 3 years
Year
Year in Reverse
Order
1
3
2
2
3
1
Depreciation
Factor
3 1

6 2
2 1

6 3
1
6
Sum of Year = 6
The Service-Output Method
Depreciation per unit output 
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Co  C L
T
Depreciation
during the Year
1
Co  CL 
2
1
Co  CL 
3
1
Co  CL 
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 C  CL 
dn   o
 Qn 
T


where: T – total units of output to the end of life
Qn – total no. of units of output during the nth year.
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