Time Value of Money • Interest rates reflects the fact that money has a time value. • You can put money to work and earn more money after a period of time. • Money has an Earning Power • A dollar today has a greater value than a dollar received in the future.
ENGR 301 Lecture 9 Engineering Economics Time Value of Money
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Example
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Economical Decisions
• If you put $1000 today in a bank and receive 10% per year interest rate. How much will it be after one year?
• • • •
Investment alternatives. Equipment replacement Equipment selection & improvements Process improvement
1000 + (1000 * 10%) = 1000 + 100 = $1100
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Scope of Economics 1. 2. 3. 4. 5. 6. 7.
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Definitions 1. Principal: Initial amount of money involved in debt or investment. 2. Interest rate: The cost or price of money and is expressed in percentage per period of time 3. Interest period: Is the period of time that determine how interest is calculated. (monthly, quarterly, yearly) 4. Number of interest periods: Specifies the duration of transaction. 5. A plan for receipts or payments: Cash flow pattern over a specified length of time. 6. Future amount of money: The end value after the cumulative effects of interest rate over a number of interest periods.
Interest rate. Types of interest rates. Project evaluation. Comparison between alternatives. Inflation. Depreciation. Taxation.
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Definition of variables
Example
• A n = the value of payment or receipt at the end of some interest period. • i = interest rate per interest period. • N = the total number of payments or interest periods. • P = the sum of money at time zero of the analysis period. Also called (Present Value or Present Worth). • F = the sum of money at the end of the analysis period. Also called (Future Value or Future Worth). • A = the value of payment or receipt at a uniform series that continues over a N periods. ( A1 = A2 = ......... = An ) • Vn = is an equivalent sum of money at a specified period of time V0 = P = ......... = Vn = F S. El-Omari
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Example cont. • • • •
• An electronics manufacturing company buys a machine for $25,000 and borrows $20,000 from a bank at 9% annual interest rate. The bank offers two repayments plans, one with equal payments made at the end of every payment year for the next 5 years, and the other with a single payment made at the end of the loan period of 5 years.
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Example of Interest Transaction Receipts
Cost of machine = $25,000 Loan = $20,000 interest rate i = 9% Interest period = 1 year Transaction duration = 5 years = 5 interest periods = N
Year 0 Year 1 Year 2 Year 3
Payments
$20,000.00
Year 4 Year 5
Plan 1
Plan 2
0 5141.85 5141.85 5141.85
0 0 0 0
5141.85 5141.85
0 30,772.48
P = $20,000.00 A = $5141.85 N=5 i = 9% S. El-Omari
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Cash Flow Diagrams • Represent time by a horizontal line. • Interest periods are marked on the line. • Upward arrows represent positive net cash flow (receipt) • Down ward arrows represent negative net cash flow (payment). S. El-Omari
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F = $30,772.48
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Methods of calculating interest • Simple interest
$20,000
– Charging interest rate only to the initial sum
• Compound interest 1
2
3
4
– Charging interest rate to the initial sum and to any previously accumulated interest that has not been withdrawn.
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$5141.85
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Simple Interest
Simple Interest Period
• If P is the initial sum, i the interest rate, N the number of interest periods, F the future value and I the value of the interest earned then: The interest earned is The Future value is
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Amount at beginning of period P
Interest earned at period Pi
Amount at end of interest period P(1+i)
2
P(1+i)
Pi
P(1+2i)
3
P(1+2i)
Pi
P(1+3i)
……
…..
….
….
N
P(1+(N-1)i)
Pi
P(1+Ni)
I = (i*P)N F = P + I = P(1+iN)
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Simple Interest P = 1000
i = 8%
Period
Amount at beginning of period 1000 1080 1160
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Compound Interest
N=3
• If P is the initial sum, i the interest rate, N the number of interest periods, and F the future value then:
Interest earned at period 80 80 80
Amount at end of interest period 1080 1160 1240
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The sum with interest after one year is: P + iP = P(1+i) The sum with interest after two years is: 2 P(1+i) + i[P(1+i)] = P(1+i)(1+i) = P(1+i) The sum with interest after N years is: N F = P(1+i) S. El-Omari
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Compound Interest Period
1 2
Amount at beginning of period P P(1+i)
3
P(1+i)
……
…..
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P(1+i)
Interest earned at period Pi
N-1
Amount at end of interest period P(1+i) P(1+i)
[P(1+i) ]i
P(1+i)
3
….
….
2
[P(1+i)
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]i
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Compound Interest
2
P(1+i)i
2
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P(1+i)
P = 1000
i = 8%
Period
Amount at beginning of period 1000 1080 1166.4
1 2 3
N
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N=3
Interest earned at period 80 86.4 93.31
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Amount at end of interest period 1080 1166.4 1259.71 18
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Economic Equivalence • It exists between cash flows that have the same economic effect and could be traded for one another. • It can convert a cash flow to an equivalent one at any point in time. • Equivalence depends on interest rate, changing the interest rate destroy the equivalence
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Example
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Repayments Plan 2 0 0
Year1 Year 2
Plan 1 5141.85 5141.85
Year 3 Year 4 Year 5 Payments Interest
5141.85 5141.85 5141.85 25,709.25 5709.25
0 0 30,772.48 30,772.48 10772.48
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Plan 3 1800.00 1800.00 1800.00 1800.00 21,800.00 29,000.00 9000.00
P = $20,000.00 N=5 i = 9% Plan 1: Equal annual installments Plan 2: End of loan repayment of principal and interest Plan 3: Annual repayment of interest and end of loan repayment of principal S. El-Omari
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Example • You are offered to receive either $3000 after 5 years from now or get P dollars today. If you deposit P in a bank for 8%, how much should P if you decide to receive the 3000 in 5 years. F=3000, N=5 years, N F = P(1+i)
i=8% find P N P = F/(1+i)
P = $2042 S. El-Omari
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Types of cash flows 1. 2. 3. 4. 5.
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Single cash flow Equal (uniform) series Linear gradient series Geometric gradient series Irregular series
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