Single cash Flow
ENGR 301 Lecture 10 Engineering Economics Types of Cash Flows
Example: Compound amount factor find F, given i, N, P If you had $2000 and invested it at 10 %, how much would it be worth in 8 years? P = $2000 i = 10 % N = 8 years F = $2000(1 + 010 . )8 1
• F=$2000 (F/P,10%,8)
ENGR 301 Lecture 10
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4
5
6
7
8
F = $4287.2
• F=$2000(2.1436) S. El-Omari
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$2000
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ENGR 301 Lecture 10
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ENGR 301 Lecture 10
Types of cash flows 1. Single cash flow 2. Equal (uniform) series 3. Linear gradient series 4. Geometric gradient series 5. Irregular series S. El-Omari
ENGR 301 Lecture 10
Single cash flow N
F = P(1+i) = P(F/P,I,N)
Compound amount factor • The process of finding F is called compounding process • The process of finding P is called discounting process S. El-Omari
ENGR 301 Lecture 10
Uneven cash flow series Example: Present value Wilson technology, a growing machine shop, wishes to set a side money now to invest over the next 4 years in automating its customer service department. The company can earn 10% on a lump sum deposited now, and it wishes to withdraw the money in the following increments: Year 1: $25,000 to purchase computer and software Year 2: $3,000 for additional hardware. Year 3: 0 Year 4: $5,000 to purchase software upgrade. S. El-Omari
ENGR 301 Lecture 10
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25000
25000
Uniform Series
5000 3000
0 1
2
3
0
=
4
1
P = $28,622
P
P1
5000
3000 0 1
+
2
P2
+
0 1
2
3
Example: Compound amount factor Suppose you make an annual contribution of $3000 to your saving account at the end of each year for 10 years. If your savings account earns 7% interest annually, how much can be withdrawn at the end of ten years.
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P3
P = 25000(P/F,10%,1) + 3000(P/F,10%,2) + 5000(P/F,10%,4) S. El-Omari
ENGR 301 Lecture 10
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ENGR 301 Lecture 10
Uniform Series F=? i = 7% 0
1
2
3
4
5
6
7
8
9
10
A = $3000
F = $3,000 (F/A, 7%, 10) = $3,000 (13.8164)
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ENGR 301 Lecture 10
Uniform Series 0
1
2
A1
A2
N-1
F = A(1+i) F=A
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(1+i)N - 1 i
A N-1
N-2
ENGR 301 Lecture 10
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F
N-1
+ A(1+i)
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F = $41,449.2
N AN
+… .. + A(1+i) + A
= A (F/A, i, N)
ENGR 301 Lecture 10
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Time shifts in Uniform Series
Uniform series
Example: In the previous example the first deposit of the 10 deposit series was made at the end of period 1 and the remaining nine deposits were made at the end of the following period. Suppose that all deposits were made at the beginning of each period instead. How much would you compute the balance at the end of period 10?
Example: Sinking-fund factor–find A, given F, i, N To help you reach $5000 goal 5 years from now, your father offers to give you $500 now. You plan to get a part time job and make 5 additional deposits at the end of each year. (first deposit at the end of first year). If all your money is deposited in a bank that pays 7% interest, how large must your annual deposit be?
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ENGR 301 Lecture 10
ENGR 301 Lecture 10
Uniform Series
Uniform series F
$5000
i = 7% 0
1
2
3
$5000 - $500(F/P,7%,5)
i = 7% 4
5
6
7
8
9
10
0
1
2
3
4
5 i = 7% 0
A = $3000
A
A
A
A
2
3
4
A
A
A
A
ENGR 301 Lecture 10
A
A = [$5000 – $500(F/P,7%,5)](A/F,7%,5) = [$5000 – $500(1.4026)](0.1739)
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A
F = $41,449.2(1.07) = $44,350.64 Or = $41.449.2+ 3,000 (F/P, 7%, 10) – 3000
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ENGR 301 Lecture 10
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A = $747.55
Uniform Series A=F
i (1+i)N - 1
= F(A/F, i, N)
Sinking-fund factor
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Uniform series 1
2
A1
A2
N
N-1 A N-1
AN
P A=P
i (1+i)N
= P (A/P, i, N)
(1+i)N - 1
Capital Recovery factor N
P=A
(1+i)
-1
= A (P/A, i, N)
i (1+i)N S. El-Omari
Present worth factor
ENGR 301 Lecture 10
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ENGR 301 Lecture 10
Uniform series
Uniform series
Example: Find A, given P, i, N A small biotechnology firm, has borrowed $250,000 to purchase laboratory equipment for gene splicing. The loan carries an interest rate of 8 % per year and is to be repaid in equal installments over the next 6 years. Compute the amount of this annual installment.
Example: Deferred loan repayment Suppose that in the last example the bank agreed to defer the first loan repayment until the end of year 2.
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ENGR 301 Lecture 10
ENGR 301 Lecture 10
Uniform series
Uniform series
$250,000
$250,000 i = 8%
i = 8%
1
2
3
4
5
6
A
A
A
A
A
A
0
A = $250,000(A/P,8%,6)
A = $54,075
ENGR 301 Lecture 10
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$250,000(F/P,8%,1)
0
1
2
i = 8% 3
A
4
5
6
7
A
A
A
A
3
4
5
6
7
A
A
A
A
A
A=[250000(F/P,8%,1)]* (A/P,8%,6)
A=$58,401
= $250,000(0.2163) S. El-Omari
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0
A S. El-Omari
A
ENGR 301 Lecture 10
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Uniform series
Linear Gradient Series
Example: Find P, given A, i, N John is a factory worker who won a lottery and will receive his winning ticket in 21 annual payments of about $24,000. He decided to quit the factory and start his own business, which required him to secure a $250,000 bank loan. John offered to put up his future lottery earning to secure the loan. If the bank’s interest rate is 10% per year, how much can john borrow against his future lottery earning? S. El-Omari
ENGR 301 Lecture 10
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Uniform series A = $24,000
Linear Gradient Series
i = 10%
0 5
15
10
20
P= $24000(P/A,10%,21) = $24,000(8.6487)
P= $207,569 < $250,000 S. El-Omari
ENGR 301 Lecture 10
ENGR 301 Lecture 10
Example: John and Barbara have just opened two saving accounts at their credit union. The accounts earn 10% annual interest. John wants to deposit $1000 in his account at the end of the first year and increase this amount by $300 for each of the following 5 years. Barbara wants to deposit an equal amount at the end of each year for the next 6 years. What should the size of Barbara’s annual so that the two accounts would have the equal balance at the end of 6 years? S. El-Omari
Linear Gradient Series
ENGR 301 Lecture 10
Linear Gradient Series
4G G 0
1
A1 0 S. El-Omari
1
2
A 1 +G 2
3G
2G 3
A 1+2G
3
4 A 1+3G
P = A(P/A,i,N) +G(P/G,i,N) 4
ENGR 301 Lecture 10
0 1 2 3 4
5 6
A 1 =1000
5 A 1+4G
5
5 6
0 1 2 3 4
Cash flow increases Or decreases by a Fixed amount, G
1000 = 1300 1600 1900 2200 2500
+ 0 1 2 3 4 300
A =$1000 + 300(A /G,10%,6) =1000 + 300(2.2236) S. El-Omari
600
5 6
900 1200 1500
A = $1667.08 ENGR 301 Lecture 10
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Geometric Gradient Series
Cash flow increases Or decreases by a Fixed rate, g S. El-Omari
ENGR 301 Lecture 10
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