Vectors: We will use the notation [V] to denote a vector V. A vector is an array of numbers where position is important. For example, a sales projection for years [1,2,3] of [100,130,200] is a vector where the first number indicates the sales projection for year 1 and the second the projection for year 2 and so on. The vector [200,130,100] is an altogether different sales projection although the same numbers appear in both.

The starting point of the CB process is the the determination of project life, n, and the sales vector, [S], which is a projection of sales over the life of the project. [S] = [S1, S2, ..., Sn]

Variables assumed to be constant over the life of the project:

• vcs = variable cost to sales ratio
• FC' = fixed costs not including depreciation
• wcs = working capital requirement to sales ratio

• n = project life in years
• FAo = initial investment in fixed assets
• FAn = salvage value of fixed assets at end of project lifet = the taxation rate

• Io = FAo + WCo
• Io = FAo + wcs*S1 initial investment required
• Assuming straight line depreciation we can compute: Dep = (FAo - FAn)/n
• [NOI] = [S] * (1 - vcs ) - FC' - Dep (a vector)
• [NIAT] = [NOI] * (1-t)
• [NIAT] = [S] * (1-vcs)*(1-t) - FC'*(1-t) - Dep*(1-t)
• Cash flows from operations: [CFO] = [NIAT] + Dep
• [CF] = [S] * (1-vcs)*(1-t) - FC'*(1-t) + t*Dep

• If we have higher sales next year, we will require an additional investment in working capital of; delta.WCj = (Sj+1 - Sj)*wcs
• In some years, usually the last, some fixed assets are fully depreciated and liquidated at salvage value = Recj.
• The NET cash flows from operations that are available for reinvestment is given by the [NCF] vector
• [NCF] = [CFO] - [delta.WC] + [REC].

An investment opportunity offers the sales potential of [S] = [1300,1600,2000,1900,1500] over the 5 year project life. A variable cost of 75% of sales is expected. The fixed costs are estimated to be $250 plus depreciation. Our tax rate is 40%. Fixed assets of $700 will be required. These assets will be depreciated to a salvage value of $100 using straight line depreciation. We plan to maintain working capital at 14% of sales. Estimate the initial outlay of capital required and the net cash flows that will be available for reinvestment as a result of operations funded by this investment.

Solution:

• vcs = 0.75, wcs = 0.14, t=0.40, (1-vcs)=0.25,(1-t)=0.6
• Dep = (700-100)/5 = 600/5 = 120, FC' = 250
• First compute the [CFO] vector
• [CFO] = [S]*0.25*0.6 - 250*0.6 + 0.4*120
• [CFO] = 0.15 * [S] - 102
• [S] = [1300,1600,2000,1900,1500]
• [CFO] = [0.15*1300-102, 0.15*1600-102, 0.15*2000-102, 0.15*1900-102, 0.15*1500-102]
• [CFO] = [93, 138, 198, 183, 123]
• Now compute the working capital requirements.
• [WC] = [0.14*1300, 0.14*1600, 0.14*2000, 0.14*1900, 0.14*1500]
• [WC] = [182, 224, 280, 266, 210]
• [delta.WC] = [224-182, 280-224, 266-280, 210-266, 0-210]
• [delta.WC] = [42, 56, -14, -56, -210]
• The Recovery vector consists of just the salvage value
• [REC] = [0,0,0,0,100]
• Compute Net Cash Flows [NCF] = [CFO] - [delta.WC] + [REC]
• [NCF] = [93-42, 138-56, 198+14, 183+56, 123+210+100]
• [NCF] = [51, 82, 212, 239, 433]
• Initial investment required:
• Io = FAo + wcs*S1 = 700 + 0.14*1300 Io = 882

Making the investment decision

Compute the PV of the [NCF] at the reuired rate. This is the value of the investment opportunity. If it exceeds Io then invest. Alternately, compute what the required rate would be to make the PV equal Io. This is called the internal rate or IRR. If the internal rate exceeds the required rate, invest.