1. Irregular Payment Series and Unconventional Equivalence Calculations
Lecture No. 9
Chapter 3
Contemporary Engineering Economics
Copyright © 2016

2. Example 3.23: Uneven Payment Series
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?
$25,000
$3,000
$5,000 P

3. Check to see if $28,622 is indeed sufficient.1 2 3 4
Beginning Balance
28,622
6,484.20
4,132.62
4,545.88
Interest Earned (10%)
2,862
648.42
413.26
454.59
Payment
+28,622
−25,000
−3,000
−5,000
Ending Balance
$28,622
0.47
Rounding error.
It should be “0.”

4. Given: Deposit series as given over 5 years
Example 3.25: Future Value of an Uneven Series with Varying Interest Rates
Given: Deposit series as given over 5 years
Find: Balance at the end of year 5

5. Solution

6. Composite Cash Flows
Situation 1: If you make 4 annual deposits of $100 in your savings account, which earns 10% annual interest, what equal annual amount (A) can be withdrawn over 4 subsequent years?
Situation 2: What value of A would make the two cash flow transactions equivalent if i = 10%?

7. Establishing Economic Equivalence
Method 1: At n = 0
Method 2: At n = 4

8. Example 3.26: Cash Flows with Sub-patterns
Given: Two cash flow transactions, and i = 12%
Find: C

9. Solution
Strategy: First select the base period to use in calculating the equivalent value for each cash flow series (say, n = 0). You can choose any period as your base period.

10. Example 3.27: Establishing a College Fund
Given: Annual college expenses = $40,000 a year
for 4 years, i = 7%, and N = 18 years
Find: Required annual contribution (X)

11. Solution
Strategy: It would be computationally efficient if you chose n = 18 (the year she goes to college) as the base period.

12. Cash Flows with Missing Payments
Given: Cash flow series with a missing payment, i = 10%
Find: P

13. Solution
Strategy: Pretend that we have the 10th missing payment so that we have a standard uniform series. This allows us to use (P/A,10%,15) to find P. Then, we make an adjustment to this P by subtracting the equivalent amount added in the 10th period.

14. Example 3.28: Calculating an Unknown Interest Rate
Given: Two payment options
Option 1: Take a lump sum payment in the amount of $192,373,928.
Option 2: Take the 30-installment option ($9,791,667 a year).
Find: i at which the two options are equivalent
Figure: 03-41

15. Solution Excel Solution: Figure: 03-41
Contemporary Engineering Economics, 6th edition, ©2015

16. Example 3.29: Unconventional Regularity in Cash Flow Pattern
Given: Payment series given, i = 10%, and N = 12 years
Find: P

17. Solution Equivalence Calculations for a Skipping Cash Flow Pattern
Strategy: Since the cash flows occur every other year, find out the equivalent compound interest rate that covers the two-year period.
Solution
Actually, the $10,000 payment occurs every other year for 12 years at 10%.
We can view this same cash flow series as having a $10,000 payment that occurs every period
at an interest rate of 21% over 6 years.