1. Contemporary Engineering Economics
Economic Equivalence
Lecture No. 5
Chapter 3
Contemporary Engineering Economics
Copyright © 2016

2. Economic Equivalence What do we mean by “economic equivalence?”
Why do we need to establish an economic equivalence?
How do we measure and compare various cash payments received at different points in time?

3. What is “Economic Equivalence?”
Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another.
Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal in the economic sense.
The observation that money has a time value leads us to two important questions: • How do we measure and compare various cash payments received at different points in time? • How do we know, for example, whether we would prefer to have $20,000 today and $50,000 ten years from now or $8,000 each year for the next ten years?

4. Equivalence Example: Compounding Concept
If you deposit P dollars today for N periods at interest rate i, you will have F dollars at the end of period N.
We can view equivalence calculations as an application of the compound-interest relationships we developed in Section 3.1. Suppose that we invest $1,000 at 12% annual interest for five years. The formula developed for calculating compound interest, F = P(1 + i)N (Eq. 3.3), expresses the equivalence between some present amount P and a future amount F, for a given interest rate i and a number of interest periods N. Therefore, at the end of the investment period, our sums grow to
$1,000( )5 = $1,762.34
Thus, we can say that at 12% interest, $1,000 received now is equivalent to $1, received in five years and that we could trade $1,000 now for the promise of receiving $1, in five years. Example 3.3 further demonstrates the application of this basic technique.

5. Equivalence – Discounting Concept
F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i.

6. Equivalence Example 3.3
Given: If you deposit $2,042 today in a savings account that pays an 8% interest annually, how much would you have at the end of 5 years?
Find: At an 8% interest, what is the equivalent worth of $2,042 now in 5 years?

7. Solution
Various dollar amounts that will be economically equivalent to $3,000 in five years, at an interest rate of 8%
Our job is to determine the present amount that is economically equivalent to $3,000 in five years, given the earning potential of 8% per year. The “indifference” ascribed to you refers to economic indifference; that is, in a marketplace where 8% is the applicable interest rate, you could trade either cash flow for the other.
Sonra
In this example, it is clear that if P is anything less than $2,042, you would prefer the promise of $3,000 in five years to P dollars today; if P is greater than $2,042, you would prefer P.

8. Equivalent Example 3.4: Cash Flows
Given: $2,042 today was equivalent to receiving $3,000 in five years, at an interest rate of 8%.
Find: Are these two cash flows are also equivalent at the end of year 3?

9. Solution
Equivalent cash flows are equivalent at any common point in time, as long as we use the same interest rate (8%, in our example).

10. Equivalence Calculations: General Principles
Principle 1: Equivalence Calculations Made to Compare Alternatives Require a Common Time basis
When selecting a point in time at which to compare the value of alternative cash flows, we commonly use either the present time, which yields what is called the present worth of the cash flows, or some point in the future, which yields their future worth. The choice of the point in time often depends on the circumstances surrounding a particular decision, or it may be chosen for convenience.
Principle 2: Equivalence Depends on Interest Rate
The equivalence between two cash flows is a function of the amount and timing of individual cash flows and the interest rate or rates that operate on those flows.
Principle 3: Equivalence Calculations May Require the Conversion of Multiple Payment Cash Flows to a Single Cash Flow
For principle 2 use Example 3.5
In Example 3.3, we determined that, given an interest rate of 8% per year, receiving $2,042 today is equivalent to receiving $3,000 in five years. Are these cash flows also equivalent at an interest rate of 10%? If not, which option is more economical?
sOLutiOn Given: P = $2,042, i = 10% per year, and N = 5 years. Find: F and is it equal to $3,000? We first determine the base period under which an equivalence value is computed. Since we can select any period as the base period, let’s select N = 5. We then need to calculate the equivalent value of $2,042 today five years from now. F = $2,042( )5 = $3,289. Since this amount is greater than $3,000, the change in interest rate breaks the equivalence between the two cash flows. As you may have already guessed, at a lower interest rate, P must be higher to be equivalent to the future amount. For example, at i = 4%, P = $2,466.

11. Finding an Equivalent Value for Multiple Payments
Solution
Compute the equivalent value of the cash flow series at n = 3, using i = 10%. V3
$200
$150
$120
$100
$100
$80 1 2 3 4 5

12. Comparing Two Different Cash Flows
Find C, making the two cash flow transactions equivalent at i = 10%.
Approach
Step 1: Select a base period to use, say n = 2.
Step 2: Find the equivalent lump sum value at
n = 2 for both A and B.
Step 3: Equate both equivalent values and solve for the unknown, C.

13. Finding an Interest Rate that Establishes an Economic Equivalence
At what interest rate would you be indifferent choosing between the two cash flows?
Approach
Step 1: Select a base period to compute the equivalent value (say, n = 3).
Step 2: Find the equivalent worth of each cash flow series at n = 3.
$1,000
$500
$1,000 A
$500 A
i = 8%
$502 $502 $502
$ $ $502 B B