Choice In Capital Budgeting

When capital budgeting occurs

Capital budgeting is a restriction on the unconstrained choice environment. Instead of having an infinite amount of funds available you have only a limited amount in period zero. While it is possible to analyze capital budgeting problems when there are cash flow limitations in all years the general problem is beyond the scope of this course. You will find these kinds of problems in an operations research course.

Capital budgeting occurs:

  1. When a business is starting up and only has retained earnings to work with. (This is fairly common since access to credit can be difficult in the early stages of a business.)
  2. When part of a larger enterprise is given separate operating and capital budgets.

Procedure with capital budgeting

This is a combinatorial optimization problem. There are ways of speeding up this process if there is enough structure to the problem; the increasing cost of funds algorithm is an example of a short cut. What is described below is a brute force search of all the combinations. This algorithm is impractical with even a moderate number of choices. The number of evaluations is $2^N -1$. Any time you can prune an option, you can cut your computation time in half.

  1. Identify all feasible combinations that can be made with the allowed budget, eliminate any options that are outside of the allowed budget.
  2. Find the present worth of these combinations.
  3. Choose combination with highest present worth.
          • Often time there will be money left over in the budget
          • If the money rolls over into the next capital budgeting period, don't worry about it.
          • If the money does not roll over, there is an incentive to try to use all of it up in a given period.

Choice in Capital Budgeting Video Transcription

Capital budgeting algorithm and examples

Assuming you have an x number of items you want to decide on purchasing. The idea is to maximize and focus on the items that will give you the most in return for your investment.

Algorithm

  1. First we need to figure out the items that we need to decide on
          • We need to figure out the Cost and the Present Worth for each item
          • Discount/ remove any items from the decision that has a negative Present Worth
          • This means that the item being removed will only cost you money in the future, therefore it's not a worth while investment
  2. Figure out the limit for the Capital Budget
          • The idea is that there is only a limited availability of cash on hand
          • We need to figure out how the maximize the available funds to get the best return of investment
  3. Create combinations of items that the sum of the cost of the items does not exceed the total Capital Budget
          • The idea is to create decisions based on the limited budget
          • Figure out which combination of items will yield the most return on investment
          • It is possible to create a combination where the Cost is well below the constraints of Capital Budget, the whole idea is to not exceed
          • Not spending all the available funds might be optimal situation to get the best return for your investment

Example 1:

Ideal Scenario
Given a Capital Budget of $40 and the following items

Item Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5
  1. We determined our items and their associated Cost and Present Worth
    • We find that none of the above 5 items have a negative Present Worth
  2. We determined that given the circumstance that we have a Capital Budge limit of $40
  3. We then create combination of each item that the sum of each item will not exceed the Capital Budget limit

Using single combinations

Combo Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5

Using combinations of 2

Combo Cost PW
AB 23 23
AC 34 12
AD 12 19
AE 29 14
BC 37 17
BE 32 19
CD 26 13
CE 43 8
DE 21 21

We see that we have a CE combination that exceeds the capital budget.

Using combinations of 3

Combo Cost PW
ABC 47 26
ABD 25 33
ABE 42 28
ACD 36 22
ACE 53 17
ADE 31 24
BCD 39 27
BCE 56 22
CDE 45 18

We see that the combinations of ABC and ABE exceed our limited Capital Budget. Also note that any combination that included CE exceeded the Capital Budget. Therefore any combination that included ABC, ABE, and CE will always exceed the Capital Budget.

Using the combination of 4

Combo Cost PW
ABCD 49 36
ABCE 66 31
BCDE 58 32

Using the combination of 5

Combo Cost PW
ABCDE 68 41

We see using the combination of 4 and 5 exceeds the Capital Budget and will not be used in determining the combination of items that will give you the maximum return of investment.
Looking at all the different combinations, the combination of ABD gives us the highest Present Worth without exceeding the Capital Budget.

Example 2:

Removing an item with a negative Present Worth from the decisions.
Given a Capital Budget of $40 and the following items

Item Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5
F 5 -2
  1. We determined our items and their associated Cost and Present Worth.
          • We find that one of the six items has a value with a negative Present Worth
Item Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5
F 5 -2

We will have to discount that item, and the possible items will now look like:

Item Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5

2. We determined that given the circumstance that we have a Capital Budge limit of $40
3. We then create combination of each item that the sum of each item will not exceed the Capital Budget limit.

Using single combinations

Combo Cost PW
A 10 9
B 13 14
C 24 3
D 2 10
E 19 5

Using combinations of 2

Combo Cost PW
AB 23 23
AC 34 12
AD 12 19
AE 29 14
BC 37 17
BD 15 24
BE 32 19
CD 26 13
CE 43 8
DE 21 21

We see that we have a CE combination that exceeds the capital budget. Any permutation that includes CE should be excluded.
Using combinations of 3, with the exclusion of CE

Combo Cost PW
ABC 47 26
ABD 25 33
ABE 42 28
ACD 36 22
ADE 31 24
BCD 39 27

We see that the combinations of ABC and ABE exceed our limited Capital Budget. Also, increasing the combination further will have ABC, ABE, and CE as an element in any of the possible combinations. Therefore it is not necessary to do any further combiantions
Looking at all the different combinations, the combination of ABD gives us the highest Present Worth without exceeding the Capital Budget.

Example 3: Over the summer you decide to do some home improvement projects to improve the value of your home. You have a budget of $100. MARR = 10%. Your options are:

First Year Year 2 Present Worth
A: Paint the fence -25 30 2.27
B: Scrape moss off the roof -50 65 9.09
C: Get the carpets steamed -80 100 10.91
D: Build a deck -300 500 154.545

We can immediately eliminate option (D). Despite it having the largest present worth, it is out of our budget.

Our feasible combinations and their present worths are as follows:
(A) = 2.27
(B) = 9.09
(C) = 10.91
(A) & (B) = 11.36

In this case the combination of doing (A) & (B) offers us the highest present worth and thus is our best option. To maximize our benefits we should paint the fence and scrape moss off of the roof this summer.

*Note that if our budget was $105 we would benefit more from the combination of (A) & (C). To persuade clients towards one option over another you want to persuade them to expand the budget by $5, you do not want to argue against doing another combination.
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