Lecture Notes 23 - - Capital Budgeting: Replacement Chain Method and Equivalent Annual Annuity

The first three methods of capital budgeting; NPV/DCF, IRR* and Payback method are the standard methods used in capital budgeting. The next four methods are not used as often but are good tools to use when comparing different projects. The other four methods are 1.) Replacement Chain Method 2.) Equivalent Annual Annuity (EAA) 3.) Ascending Discount Rates 4.) Area Under the Curve.

We will look at Replacement Chain Method and EAA.

Replacement Chain Method and EAA are both tools for comparing projects with unequal lives.

For example if Project A has a life of five(5) years and Project B has a life of three (3) years it would be unfair to compare their NPV. It is like comparing apples and oranges, they have unequal lives, which means they have an unequal number of cash flows. To compensate for this problem we can use either the Replacement Chain Method or the EAA.

Replacement Chain Method

Replacement Chain Method takes two or more projects with life spans that are unequal and finds the lowest common denominator and carries the projects out to that lowest common denominator or terminal year. For example, Project A has a life span of 5 years and Project B has a life span of 3 years. The lowest common denominator for these two projects is 15 years. So, taking the following cash flows we can illustrate how this is done.

PROJECT A

Y0 Y1 Y2 Y3 Y4 Y5

CF <10> 2 4 6 8 11

PROJECT B

Y0 Y1 Y2 Y3

CF <8> 2 5 7

Now to carry this out to 15 years, it means that at the end of Y5 for Project A another negative cash flow of <10> ( for example a new purchase ) would have to happen therefore making Y5 cash flow 1 (Y5 + Y0) or (11 + <10>). This would also happen at Y10, but Y15 the end of the project the cash flow would be 11, because it is the end and no new purchase has to occur. The same applies to Project B but it would happen on Y3, Y6, Y9, Y12 and Y15 would be the end of the project.

PROJECT A

Y0 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12

CF <10> 2 4 6 8 11 2 4 6 8 11 2 4

<10> <10>

ADJUSTED CF 1 1

Y13 Y14 Y15

CF 6 8 11

PROJECT B

Y0 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12

CF <8> 2 5 7 2 5 7 2 5 7 2 5 7

<8> <8> <8> <8>

ADJUSTED CF <1> <1> <1> <1>

Y13 Y14 Y15

CF 2 5 7

Using these extended adjusted cash flows you would then determine the discount factors and the PV, which you then could determine, the NPV. The project with the largest NPV would be the best project.

Lets take a look at a different set of projects:

Project Years

A 17

B 23

C 13

D 22

E 7

F 14

G 24

What is the lowest common factor (denominator)? You would have to factor out and then multiply by the factors for each project. (Note: if a number appears more than one time in the factor of one number you must multiply by that number as many times as it appears, if a factor appears in more than one number then you only have to multiply one time. For further help with factors, please see Prof. Harding).

Project Years Factor

A 17 1 * 17

B 23 1 * 23

C 13 1* 13

D 22 1 * 2 * 11

E 7 1 * 7

F 24 1 * 2 * 2 * 2 * 3

G 14 1 * 2 * 7

17 * 23 * 13 * 11 * 7 * 2 * 2 * 2 * 3 = 9,393,384

Note: the 2 from project D and project G are not used because 2 has already been used in project F and it appears three times in project F therefore it must be multiplied three times. Also note that the 7 in project G is not used because it was already multiplied in project E.

So in order to find the NPV for all 7 of the projects, you would have to extend the projects out to 9,393,384 years, which is the lowest common denominator, and the terminal year for all 7 projects. This would take a very long time to calculate, so instead of using the Replacement Chain Method we can use the EAA..

Equivalent Annual Annuity (EAA)

This method finds the annual annuity that is equivalent to the NPV. To find the EAA all you have to do is use the payment formula PMT = (PV)K/1-(1+K)^-n or EAA = (NPV)K/1-(1+K)^-n Where K is the discount rate and n is the life of the project. For example:

If we know that a projects has a NPV of $115.199 and a life span of 5 years with a discount rate of 10% we can determine the EAA.. The EAA would equal $30.3955.

We would calculate the EAA for all the projects and choose the project with the largest EAA..

The EAA and the NPV are financially equivalent. The cash flow of a project is financially equivalent to the NPV and the EAA. They are all financially equal and therefore one number is not better than another.

End of lecture notes 23

For further explanation of the Replacement Chain Method see handout given out in class.