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