Interest Rate and Economic Equivalence
1
Engineering Economics
Time Value of Money Money has a time value
because it can earn more money over time (earning power).
Money has a time value because its purchasing power changes over time (inflation).
Time value of money is measured in terms of market interest rate which reflects both earning and purchasing power in the financial market.
The Interest Rate Interest is the cost of
money—a cost to the borrower and an earning to the lender.
Elements of Transactions Involving Interest:Principal Interest rate Interest period Total number of
interest periods A plan for receipts or
disbursements A future amount of
money
◦ INVESTMENT INTEREST = VALUE NOW - ORIGINAL AMOUNT
◦ LOAN INTEREST = TOTAL OWED NOW - ORIGINAL
AMOUNT
4
Interest Rate
5
Interest ExampleExample You borrowed $10,000 for one full year. Must pay back $10,700 at the end of one year, find the interest amount and the interest rate:SolutionInterest Amount (I) = $10,700 - $10,000Interest Amount = $700 for the yearInterest rate (i) = 700/$10,000 = 7%/Yr
6
Interest Rate - Notations
• NotationsI = the interest amount in $i = the interest rate (%/interest period)N = No. of interest periods (1 for this problem)P = The actual borrowed amount or the principal ($10,000 in this problem)
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Interest – Example• If you borrow $20,000 for 1 year at 9%
interest per year calculate the total amount of money that you will pay by the end of that year.
SolutionP= $20,000 i = 0.09 per year and N = 1 YearInterest Amt (I) = i*P*N =(0.09)($20,000) = $1,800Total Amt Paid after one year (Future amount of money)= P + I = $21,800Create an Excel Sheet
8
Interest – Investing Perspective
• On the other hand if you invest $20,000 for one year in a venture that will return to you, 9% per year.
At the end of one year, you will have:Original $20,000 backPlus……..The 9% return on $20,000 = $1,800
We say that you earned 9%/year on the investment! This is your RATE of RETURN on the investment
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Inflation
• Where a country’s currency becomes worth less over time thus requiring more of the currency to purchase the same amount of goods or services in a time period
10
Inflation
• Inflation impacts:• Purchasing Power (reduces)• Operating Costs (increases)• Rate of Returns on Investments (reduces)
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Taxes
• Taxes represent a significant negative cash flow
• A realistic economic analysis must assess the impact of taxes• Called an AFTER-TAX cash flow analysis
• Not considering taxes is called a BEFORE-TAX Cash Flow analysis
Cash Flows◦ Inflows (Receipts) =====> Revenues or
Benefits (+ positive)◦ Outflows (Disbursements) =====> Costs (-
negative)
Cash Flow Diagram
T=0 t = 1 Yr
Positive +
Negative -
$20,000 is received here
$21,800 paid back here
Methods of Calculating Interest
• Simple interest: the practice of charging an interest rate only to an initial sum (principal amount).
• Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.
Simple Interest
• P = Principal amount
• i = Interest rate• N = Number of
interest periods• Example:
P = $1,000i = 8%N = 3 years
End of Year
Beginning
Balance
Interest earned
Ending Balance
0 $1,000
1 $1,000 $80 $1,080
2 $1,080 $80 $1,160
3 $1,160 $80 $1,240
( )where
= Principal amount = simple interest rate
= number of interest periods = total amount accumulated at the end of period
F P iP N
PiNF N
$1,000 (0.08)($1,000)(3)$1,240
F
Simple Interest Formula
Compound Interest
• Compound interest: the practice of charging an interest rate to an initial sum and to any previously accumulated interest that has not been withdrawn.
Compound Interest
• P = Principal amount
• i = Interest rate• N = Number of
interest periods• Example:
P = $1,000i = 8%N = 3 years
End of
Year
Beginning Balance
Interest earned
Ending Balance
0 $1,000
1 $1,000 $80 $1,080
2 $1,080 $86.40 $1,166.40
3 $1,166.40 $93.31 $1,259.71
Compounding Process
$1,000
$1,080
$1,080
$1,166.40
$1,166.40 $1,259.71
0
1
23
Compound Interest Formulan = 0: Pn = 1: F1 = P(1+i)n = 2: F2 = F1(1+i)= P(1+i)2
.
.
.n = N: FN = P(1+i)N
0
$1,000
$1,259.71
1 2
33$1,000(1 0.08)
$1, 259.71F
Compound Cash Flow Diagram
$1,259.71 in three years is the economic equivalence of $1,000 now with interest rate 8% with compound calculation
EQUIVALENCE
Example• You travel at 68 miles per hour• Equivalent to 110 kilometers per hour
• Thus:• 68 mph is equivalent to 110 kph• Using two measuring scales• Miles and Kilometers
EQUIVALENCE
• Is “68” equal to “110”?• No, not in terms of absolute numbers• But they are “equivalent” in terms of the two measuring scales• Miles• Kilometers
ECONOMIC EQUIVALENCE
• Economic Equivalence• Two sums of money at two different points in time can be made economically equivalent if:• We consider an interest rate and,• Number of Time periods between the two sums
Equality in terms of Economic Value
Equivalence Illustrated• Returning to the Example from previous slides• Diagram the loan (Cash Flow Diagram)• The company’s perspective is shown
T=0 t = 1 Yr
$20,000 is received here
$21,800 paid back here
$20,000 now is economically equivalent to $21,800 one year from now IF the interest rate is set to equal 9%/year
Equivalence Illustrated
• $20,000 now is not equal in magnitude to $21,800 1 year from now
• But, $20,000 now is economically equivalent to $21,800 one year from now if the interest rate in 9% per year.
Equivalence Illustrated
• To have economic equivalence you must specify:
• Timing of the cash flows• An interest rate (i% per interest period)
• Number of interest periods (N)
Practice Problem
0
1 2 33 4 5
$2,042
5
F
0
At 8% interest, what is the equivalent worth of $2,042 now 5 years from now?
If you deposit $2,042 today in a savingsaccount that pays 8% interest annually.how much would you have at the end of5 years?
=
5$2,042(1 0.08)$3,000
F
Solution
Example
$3,289$2,042
50
At what interest rate would these two amounts be equivalent in 5 years period of time?
i = ?
Equivalence Between Two Cash Flows
• Step 1: Determine the base period, say, year 5.
• Step 2: Identify the interest rate to use.
• Step 3: Calculate equivalence value.
$3,289$2,042
50
i F
i F
i F
6%, 042 1 0 06 733
8%, 042 1 0 08 000
10%, 042 1 0 10
5
5
5
$2, ( . ) $2,
$2, ( . ) $3,
$2, ( . ) $3,289
Example Find the present dollar amount that will be
economically equivalent to $5,877.32 in 5 years, given an interest rate of 8%.
4000)08.1(
32.5877)1( 5
Ni
FP
0 1 2 3 4 5
P F
$4,000 $5877.32$4,320 $4,665.6 $5,441.96$5,038.85
EXAMPLE OF EQUIVALENT CASH FLOWS
$2,042 today was equivalent to receiving $3,000 in five years, at an interest rate of 8%. Are these two cash flows are also equivalent at the end of year 3? Equivalent cash flows are equivalent at any common point in time, as long as we use the same interest rate (8%, in our example).
PRACTICE PROBLEM
Compute the equivalent value of the cash flow series at n = 3, using i = 10%.
Solution:
Contemporary Engineering Economics, 5th edition, © 2010
0 1 2 3 4 5
$100$80
$120$150
$200
$100
0 1 2 3 4 5
V=
0 1 2 3 4 5
$100$80
$120$150
$200
$100
V3
Compounding Process: $511.90
3 23
1 2
Discounting Process: $264.46
100(1 0.10) $80(1 0.10) $120(1 0.10) $150
$200(1 0.10) $100(1 0.10)
$511.90 $264.46$776.36
V
Practice Problem How many years
would it take an investment to double at 10% annual interest?
P
2P
0
N = ?
2 (1 0.10)
2 1.1 log 2 log1.1
log 2log1.17.27 years
N
N
F P P
N
N
Solution
P
2P
0
N = ?
Rule of 72Approximating how long it will take for a sum of money to double
72interest rate (%)72107.2 years
N
Interest Formulas for Single Cash Flows
Single Cash Flow Formula
Single payment compound amount factor (growth factor)
P
F
N
0
F P iF P F P i N
N
( )( / , , )1
Practice Problem
• If you had $2,000 now and invested it at 10%, how much would it be worth in 8 years?
$2,000
F = ?
80
i = 10%
8
Given:$2,00010%8 years
Find:
$2,000(1 0.10)$2,000( / ,10%,8)$4,287.18
EXCEL command:=FV(10%,8,0,2000,0)=$4,287.20
Pi
N
F
FF P
Solution
Single Cash Flow Formula
• Single payment present worth factor (discount factor)
• Given:
• Find: P
F
N
0
P F iP F P F i N
N
( )( / , , )1
iNF
12%5
000 years
$1,
PP F
$1, ( . )$1, ( / , )$567.40
000 1 0 12000 12%,5
5
You want to set aside a lump sum amount today in a savings account that earns 7% annual interest to meet a future expense in the amount of $10,000 to be incurred in 6 years. How much do you need to deposit today?
Practice Problem
6$10,000(1 0.07)$10,000( / ,7%,6)$6,663
PP F
Solution
0
6
$10,000
P
Multiple Payments How much do you
need to deposit today (P) to withdraw $25,000 at n =1, $3,000 at n = 2, and $5,000 at n =4, if your account earns 10% annual interest?
0
1 2 3 4
$25,000
$3,000 $5,000
P
0
1 2 3 4
$25,000
$3,000 $5,000
P
0
1 2 3 4
$25,000
P1
0
1 2 3 4
$3,000
P2
0
1 2 3 4
$5,000
P3
+ +
1 $25,000( / ,10%,1)$22,727
P P F
2 $3,000( / ,10%,2)$2,479
P P F
3 $5,000( / ,10%,4)$3,415
P P F
1 2 3 $28,622P P P P
Uneven Payment Series
Check0 1 2 3 4
Beginning Balance
0 28,622 6,484.20 4,132.62 4,545.88
Interest Earned (10%)
0 2,862 648.42 413.26 454.59
Payment +28,622 -25,000 -3,000 0 -5,000
Ending Balance
$28,622 6,484.20 4,132.62 4,545.88 0.47
Rounding error
Terminology and Symbols
•P = value or amount of money at a time designated as the present or time 0.
• Also P is referred to as present worth (PW), present value (PV), net present value (NPV), discounted cash flow (DCF), and capitalized cost (CC); dollars
Terminology and Symbols
• F = value or amount of money at some future time.
• Also F is called future worth (FW) and future value (FV); dollars
Terminology and Symbols
•A = series of consecutive, equal, end‑of‑period amounts of money.
• Also A is called the annual worth (AW) and equivalent uniform annual worth (EUAW); dollars per year, dollars per month
•n = number of interest periods; years, months, days
Terminology and Symbols
• i = interest rate or rate of return per time period; percent per year, percent per month
• t = time, stated in periods; years, months, days, etc
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