Engineering Economics

Chapter 3

Interest and Equivalence

 Cash Flow Diagrams (CFD)

• Used to model the positive and negative cash flows.

• At each time at which cash flow will occur, a vertical
arrow is added, point down for costs and up for
revenues.
• Cash flow are drawn to relative scale
• Rent and insurance are beginning-of-period cash
flows; i.e. just put an arrow in where it occurs.
• O&M, salvages, and revenues are assumed to be end-
of-period cash flows.

2
 Example 3-1
Purchase a new $30,000 mixing machine. The machine
may be paid for in one of two ways
– A. Pay the full price now minus a 3% discount
– B. Pay $5000 now, $8000 at the end of 1st yr, and $6000 at
end of each following year
List the alternatives in the form of a table of cash flows

3
 Continue … Example 3-1
Cash flow table:
End of year Pay in Full Now Pay over 5 Yrs
0 (now) -$29,100 -$5000
1 0 -$8000
2 0 -$6000
3 0 -$6000
4 0 -$6000
5 0 -$6000

4
 Example 3-2
A man borrowed $1000 from a bank at 8% interest.
– At the end of 1st yr: Pay half of the $1000 principal amount
plus the interest.
– At the end of 2nd yr: Pay the remaining half of the principal
amount plus the interest for the second year.
Compute the borrower’s cash flow

End of Year Cash Flows

0 (Now) +$1000
1 -580
2 -540

5
 Time Value of Money
• If monetary consequences occur in a short period of
time → Simply add the various sums of money
• What if time span is greater?
• $100 cash today vs. $100 cash a year from now?
• Money is rented. The rent is called the interest
• If you put $100 in the bank today, and interest rate is
9% → $109 a year from now

6
 Interest

• Simple Interest
• Compound interest

7
 Simple Interest

• Interest that is computed only on the original sum and

not on accrued interest.
– e.g. if you loaned someone the amount of P at a simple
interest rate of i for a period of n years:
• Total interest earned = P × i × n = P i n
• The amount of money due after n years:
F=P+Pin
Or F = P(1+ i n)

8
 Example 3-3
You loaned a friend $5000 for 5 years at a simple
interest rate of 8% per year.
How much interest you receive from the loan?
How much will your friend pay you at the end of 5 yrs.

Total interest earned = P i n = (5000)(0.08)(5) = $2000

Amount due at the end of loan = P + P i n = 5000 + 2000
= $7000

9
 Compound Interest
• This is the interest normally used in real life
• Interest on top of interest
• Next year’s interest is calculated based on the unpaid
balance due, which includes the unpaid interest from
the preceding period.

10
Example 3-4

11
 … Compound Interest
Compound interest is interest that is charged on the original sum and
un-paid interest.

You put $500 in a bank for 3 years at 6% compound interest per year.
 At the end of year 1 you have (1.06)  500 = $530.
 At the end of year 2 you have (1.06)  530 = $561.80.
 At the end of year 3 you have (1.06)  $561.80 = $595.51.

Note:
$595.51 = (1.06)  561.80
= (1.06) (1.06) 530
= (1.06) (1.06) (1.06) 500 = 500 (1.06)3

12
 Single Payment Compound Amount Formula
If you put P in the bank now at an interest rate of i% for n years,
the future amount you will have after n years is given by
F = P (1+ i )n
i = interest rate per interest period (stated as decimal)
n = number of interest periods
P = a present sum of money
F = A future sum of money

The term (1+i)n is called the single payment compound factor.

F = P (1+i)n = P (F/P,i,n)
Also P = F (1+i)-n = F (P/F,i,n)
The factor (F/P,i,n) is used to compute F, given P, and given i and n.
The factor (P/F,i,n) is used to compute P, given F, and given i and n.

13
 Present Value
Example 3-6
If you want to have $800 in savings at the end of four years, and
5% interest is paid annually, how much do you need to put into
the savings account today?
We solve F = P (1+i)n for P with i = 0.05, n = 4, F = $800.
P = F/(1+i)n = F(1+i)-n
P = 800/(1.05)4 = 800 (1.05)-4 = 800 (0.8227) = $658.16.
F = 800
Alternate Solution
Single Payment Present Worth Formula

P = F/(1+i)n = F(1+i)-n P=?

P = F (P/F,i,n) , i = 5% and n = 4 periods
From tables in Appendix B, (P/F,i,n) = 0.8227
P = 800 x 0.8227 = $658.16

14
Factors in the Book (page 573 in 9-th edition)

15
 Present Value
Example: You borrowed $5,000 from a bank at 8%
interest rate and you have to pay it back in 5 years.
The debt can be repaid in many ways.

Plan A: At end of each year pay $1,000 principal

plus interest due.

Plan B: Pay interest due at end of each year and

principal at end of five years.

Plan C: Pay in five end-of-year payments.

Plan D: Pay principal and interest in one payment

at end of five years.
16
 …Example (cont’d)
You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back
in 5 years.
Plan A: At end of each year pay $1,000 principal plus
interest due.
a b c d e f
Int. Owed Total Owed
Year Amnt. Princip. Total
Owed int*b b+c Payment Payment
1 5,000 400 5,400 1,000 1,400
2 4,000 320 4,320 1,000 1,320
3 3,000 240 3,240 1,000 1,240
4 2,000 160 2,160 1,000 1,160
5 1,000 80 1,080 1,000 1,080
SUM 1,200 5,000 6,200
17
 …Example (cont'd)
You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back
in 5 years.
Plan B: Pay interest due at end of each year and principal at
end of five years.
a b c d e f
Int. Owed Total Owed
Year Amnt. Princip. Total
Owed int*b b+c Payment Payment
1 5,000 400 5,400 0 400
2 5,000 400 5,400 0 400
3 5,000 400 5,400 0 400
4 5,000 400 5,400 0 400
5 5,000 400 5,400 5,000 5,400
SUM 2,000 5,000 7,000
18
 … Example (cont'd)
You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back
in 5 years.
Plan C: Pay in five end-of-year payments.

a b c d e f
Int. Owed Total Owed
Year Amnt. Princip. Total
Owed int*b b+c Payment Payment
1 5,000 400 5,400 852 1,252
2 4,148 332 4,480 920 1,252
3 3,227 258 3,485 994 1,252
4 2,233 179 2,412 1,074 1,252
5 1,160 93 1,252 1,160 1,252
SUM 1,261 5,000 6,261
19
 … Example (cont'd)
You borrowed $5,000 from a bank at 8% interest rate and you have to pay it back
in 5 years.
Plan D: Pay principal and interest in one payment at end of
five years.
a b c d e f
Int. Owed Total Owed
Year Amnt. Princip. Total
Owed int*b b+c Payment Payment
1 5,000 400 5,400 0 0
2 5,400 432 5,832 0 0
3 5,832 467 6,299 0 0
4 6,299 504 6,802 0 0
5 6,802 544 7,347 5,000 7,347
SUM 2,347 5,000 7,347
20
 The four plans were
Year Plan 1 Plan 2 Plan 3 Plan 4
1 $1400 $400 $1252 0
2 1320 400 1252 0
3 1240 400 1252 0
4 1160 400 1252 0
5 1080 5400 1252 7347
Total $6200 $7000 $6260 $7347

How do we know whether these plans are equivalent or not?

→We won’t be able to know by simply looking at the cash flows,
therefore some effort should be made.

21
 Equivalence
• In the previous example, four payment plans were
described.
• The four plans were used to accomplish the task of
repaying a debt of $5000 with interest at 8%.
• All four plans are equivalent to $5000 now.
• i.e. all four plans are said to be equivalent to each
other and to $5000 now.

22
 Present Value
Example 3-8
If you want to have $800 in savings at the end of four years, and
5% interest is paid annually, how much do you need to put into
the savings account today?
We solve F = P (1+i)n for P with i = 0.05, n = 4, F = $800.
P = F/(1+i)n = F(1+i)-n
P = 800/(1.05)4 = 800 (1.05)-4 = 800 (0.8227) = $658.16.

Alternate Solution
Single Payment Present Worth Formula

P = F/(1+i)n = F(1+i)-n
F = 800
P = F (P/F,i,n) , i = 5% and n = 4 periods
From tables in Appendix B, (P/F,i,n) = 0.8227
P = 800 x 0.8227 = $658.16
P=?
23
 Example 3-8
In 3 years, you need $400 to pay a debt. In two more years,
you need $600 more to pay a second debt. How much
should you put in the bank today to meet these two needs if
the bank pays 12% per year?
P

0 1 2 3 4 5

$400
Interest is compounded yearly $600

P = 400(P/F,12%,3) + 600(P/F,12%,5) Alternate Solution

= 400 (0.7118) + 600 (0.5674) P = F(1+i)-n
= 284.72 + 340.44 = $625.16 P = 400(1+0.12)-3 + 600(1+0.12)-5
P = $625.17

24
 Example 3-8 (Interest Compounded monthly)
In 3 years, you need $400 to pay a debt. In two more years,
you need $600 more to pay a second debt. How much
should you put in the bank today to meet these two needs if
the bank pays 12% compounded monthly?
P

0 1 2 3 4 5

$400
$600
Interest is compounded yearly
Interest is compounded monthly

P = 400(P/F,12%,3) + 600(P/F,12%,5) P = 400(P/F,12%/12,3*12) + 600(P/F,12%/12,5*12)

= 400 (0.7118) + 600 (0.5674) = 400(P/F,1%,36) + 600(P/F,1%,60)
= 284.72 + 340.44 = $625.16 = 400 (0.6989) + 600 (0.5504)
= 279.56 + 330.24 = $609.80

25
 Points of view

Borrower point of view:You borrow money from the bank to start a

business.

Year Cash Year Cash

flow flow
0 -P 0 +P
1 0 1 0
2 0 2 0
3 +400 3 -400
4 0 4 0
5 +600 5 -600

Investors point of view:You invest your money in a bank and buy a

bond.

26
 Concluding Remarks

Appendix B in the text book tabulate:

Compound Amount Factor

(F/P,i,n) = (1+i)n

Present Worth Factor

(P/F,i,n) = (1+i)-n

These terms are in columns 2 and 3, identified as

Compound Amount Factor: “Find F Given P: F/P”
Present Worth Factor: “Find P Given F: P/F”

27