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

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? 1 2 3 4 $25,000 $3,000 $5,000 P

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.”

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

Solution

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4 annual deposits of $100 in your savings account, which earns 10% annual interest, what equal annual amount (A) can be withdrawn over 4 subsequent years?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%?

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

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

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.

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)

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

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

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.

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

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

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

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.