TABLE 1 A
Activity
B C D
E
F
G
H
I
J
K
Predecessors --- A B C D,G A F,J --- H I Duration
5
ENGR 301 Lecture 11
5
5
11 11
B
2
6 3 4 5
6
0
0
6
H
3
5
23 23 10
13 15 N 3
2
7
4 2
3
E
K
J
1
3
5
3
8
I
7 4
8
4 G
8
0
Engineering Economics Nominal & Effective Interest Rates
2 7 2
18 18
F
5
N K,M
D 7
3
A
3
14 14
C
3
8 3
L M
F,J H L
9
M L
6
7
15 20
4
5 10 16
S. El-Omari
Activity
ENGR 301 Lecture 11
1
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ENGR 301 Lecture 11
2
TABLE 1
Time in Days A
B
C
D
E
H
I
Predecessors
--- A
B
C
D,G
A C,F,J ---
H
I C,F,J H L
K,M
Duration
5
3
4
5
8
2
7 2
3
Activity
123456789012345678901234567890 A B
6
F
G 3
3
J
K
L
M
7
4
N
C 5 5
D E F
3
0
I
3
0
6
H
5
J
3
L
6
M
10 3
2
7
4
23 23
N
K
2
K
E 5
3
14 15
8
I 1
8
4 G
8
0
J
D 7
F
5
H
9
M L
7
4
5
16 20
10 16 ENGR 301 Lecture 11
3
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Compounding periods
4
What does it mean compounding more frequently than a year?
F = P(1+i) = P(F/P,I,N) i = interest rate compounded annually N = Number of years What if compounding is not annually? Quarterly, semi annually, monthly, weekly or daily
ENGR 301 Lecture 11
ENGR 301 Lecture 11
Compounding periods
N
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6
18 18
14 14
C
3
A
G
S. N El-Omari
11 11
B
2
5
F = P(1+i)
N
For a case where 18% compounded monthly Like credit cards
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ENGR 301 Lecture 11
6
1
Compounding periods 1
2
3
4
5
6
7
8
9
10
Nominal & Effective Interest Rates 11
Nominal Interest Rate: An interest rate per year but not the amount of rate that will accumulate per year. Effective Interest Rate: is the rate that truly represents the interest earned in a year or some other time.
12
1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
Interest rate
Annual Percentage Rate = 1.5 x 12 = 18% = Nominal Percentage Rate 1.5 % = Effective Interest Rate per month
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ENGR 301 Lecture 11
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Nominal & Effective Interest Rates
ENGR 301 Lecture 11
ENGR 301 Lecture 11
F = P(1+i)
12
(1+0.01) - 1 = 12.68 % = Effective annual interest rate 9
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ENGR 301 Lecture 11
1
2
3
4
5
6
7
8
9
10
11
12
1
1
1
1
1
1
1
1
1
1
1
i i
C
=
(1+r/M)
C
= (1+r/CK)
-1 -1
M = the number of compounding periods per year C = the number of compounding periods per payment period K = the number of payment periods per year
Annual Percentage Rate = 1x 12 = 12% = Nominal Percentage Rate 12.86 % = Effective annual Interest Rate
ENGR 301 Lecture 11
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Effective rates per payment period
1
S. El-Omari
N Interest earned
Compounding periods
Interest rate
8
Nominal & Effective Interest Rates
Example: If you obtain a loan from a bank at an interest rate of 12% compounded monthly. Nominal interest rate is 12% = r Number of compounding periods or compounding frequency = 12 = M Interest rate per month = r/M = 12% / 12 = 1%
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11
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ENGR 301 Lecture 11
12
2
Effective rates per payment period
Effective rates per payment period
Example: If you make a quarterly deposits in a savings account which earns 9 % interest compounded monthly. Compute the effective interest rate per quarter. r = 9% C = three compounding periods per quarter K = four quarterly payments per year M = 12 compounding periods per year S. El-Omari
ENGR 301 Lecture 11
1st Quarter
0.75% 0.75% 0.75%
2.27%
13
ENGR 301 Lecture 11
0.75% 0.75% 0.75%
2.27%
-1 3 = (1+0.09/3*4) - 1
2.27%
i = 2.27 %
ENGR 301 Lecture 11
$20,000
14
Effective interest rate i =8.5% compounded monthly 10
5
48
0
i = 8.5% / 12 = 0.7083% per month N = (12)(4) = 48 months A = $20,000(A/P, 0.7083%, 48) 15
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A = $492.97
ENGR 301 Lecture 11
16
Effective interest rate
Example: compounding more frequent than payment Suppose you make annual quarterly deposits of $1000 into a fund that pays interest at a rate of 12% compounded monthly. Find the balance at the end of year two. A = $1000 r = 12% per year M = 12 compounding periods per year N = 8 quarter ENGR 301 Lecture 11
0.75% 0.75% 0.75%
= (1+r/CK)
Effective interest rate
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0.75%
4th Quarter
2.27%
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Example: payment period = compounding period Suppose you want to buy a car and you go for the following: 8.5% Annual percentage rate compounded monthly and 48-month financing with total price $24,406.87. You make down payment = $4406.87 $20,000 to be financed What would be your monthly payment
0.75% 0.75%
3rd Quarter
C
i i
Effective interest rate
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2nd Quarter
17
F
0
1
2
24
3
1000
1000
1000
1000
M = 12 compounding periods per year K = 4 payments periods per year C = three compounding periods per payment period S. El-Omari
ENGR 301 Lecture 11
18
3
Effective interest rate F = $8901.81
i = 3.03 % per quarter 1
0
2
Example: compounding less frequent than payment Suppose you make $500 monthly deposits to a registered retirement savings plan that pays interest at a rate of 10%, compounded quarterly. Compute the balance at the end of 10 years r = 10% per year M = 4 quarterly compounding periods per year K = 12 payment periods per year A = $500 N = 10 x 12 = 120 months Find i and F
F
24
3
1000
i
Effective interest rate
1000
1000
= (1+0.12/3*4)
1000
3
-1
N = K (number of years) = 4(2) = 8 quarters F = $1000 (F/A, 3.030 %, 8) S. El-Omari
ENGR 301 Lecture 11
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ENGR 301 Lecture 11
Effective interest rate
20
Effective interest rate F
F
i = 0.826 % per month 0
1
2
3
12
M = 4 compounding periods per year K = 12 payments periods per year C = 1/3 compounding periods per payment period
1
0
i
2
F = $101,907.89
3
12
1/3
= (1+0.10/((1/3)12))
-1
N = K (number of years) = 12(10) = 120 quarters F = $500 (F/A, 0.826 %, 120)
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ENGR 301 Lecture 11
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