One More Time for the Next Generation: How to do Quick Estimates of Project and
ID: 2754221 • Letter: O
Question
One More Time for the Next Generation:
How to do Quick Estimates of Project and Business Acquisition Economics
It is often necessary to be able to quantitatively assess the economics of projects, new ventures and business acquisitions when they are in their early conceptual stages. It is then that there is a sufficiently large uncertainty in contributing factors that rough “back of the envelope” calculations are all that is necessary.
With the easy availability of spreadsheets, this capability among practitioners is at risk. It is very easy to build a spreadsheet model. The problem with this approach is that the sophistication of the models often greatly exceeds the information available at the early stages of estimate. Moreover the models often take time to build and they are often difficult for others to follow. Perhaps most significantly, spreadsheets often don’t lend itself to their users building useful quantitative intuition about the contributing factors to the economics. And yet this intuition can allow for quick estimates and insights into economic drivers.
Let’s take a simple example of a common project evaluation where a facility is being expanded by, say, 10,000 units per day of product for $200 million. The project life is estimated at 20 years. Feedstock and operating costs are $100/unit and $50/unit, respectively. What price does one need to charge for the product for the return on capital to be 15% ? This can be quickly estimated. Moreover, the estimate is relatively robust to different assumptions about the project life and the tax rate. Furthermore, the required product price (breakeven price) is made up of factors that are additive meaning that one can readily see their contributions.
The required price is the sum of a contribution needed to cover the operating and feedstock costs (per annual units of production) and a second term that accounts for the return on capital. The latter term will be different depending on whether the return is before or after taxes but the difference is important at early stages of the evaluation relative to the uncertainties in the capital itself, feedstock and operating costs. Stated another way, the “go/no go” decision to move from a conceptual phase the next step should not be sensitive to these factors.
The required price, referred to here as the breakeven (p BE ) price, can be estimated from the equation:
p BE = (capital/annual product volume)(CRF) + (operating and feedstock cost)/annual product volume
CRF is the capital recovery factor. It is a function of return on capital and the project life. It is also a function of the incremental tax rate and depreciation schedule when the return desired is after taxes. Once these conditions are set the CRF is relatively insensitive to changes in these parameters. For example, for an incremental tax rate of 35% and straight line depreciation over a 15 year project life the CRF is 0.23. It stays within 10% of this number as the project life varies moves down to 10 years and then out to infinity. This simplifies the calculations considerably and allows quick, “back of the envelope”, estimates to be done without spreadsheets.
For the above problem then the required price is:
p BE = ($200 million/(10,000 units/day x 365 days)(0.23) + ($70/unit + $50/unit)
= 13 + 120
= $133/unit
Where a 15% after tax (a/t) return, 20 year project life, 35% tax rate have been assumed.
From the above you can see that $120/unit is needed to cover feedstock and operating costs and $13/unit is needed to cover capital and achieve a 15% a/t return.
The p BE can be compared to the market price. If the market price is above p BE then the return on capital will be greater than assumed when calculating the capital recovery factor. Conversely if the price is below the market price, the project will not give the expected return on capital.
One insight from this approach is that very competitive markets often will work such that feedstock and operating costs are covered but capital recovery is “competed away” before the project life is achieved. This market equilibrium price will move toward $120/unit plus a small profit margin if the competitors all face similar economics. This situation is especially true in commodity markets at the end of an investment cycle.
Basis for CRF Approach
Model the cash flows as:
Where
I Initial investment, $
A After tax cash flow, $
n Project life, years
Note that the future after tax cash flows in years 1 through n are modelled as constant.
A = (R-O)(1-t) + tD
Where
R Revenue, $
O Operating Cost, $
t marginal tax rate, taken to be 35%
D Annual depreciation, $
Furthermore, take R = v p where
v Annual production units, i.e. gallons, tons, lbs or bbl’s
p Price of the product, $/unit i.e. $/gallons, $/tons, $/lbs or $/bbl’s
Furthermore assume D = I/n for straight line depreciation over the project life n. At the early stages of an economic evaluation this simple schedule is good enough. After all…the decision to go to the next step should not depend strongly on the depreciation method.
Using the above, we can arrive at a breakeven price for the product as follows:
NPV = - I + PVfuture cash flows
Let
NPV = 0
And re-write PVfuture cash flows as
PVfuture cash flows = [(R-O)(1-t)+tD] (PV/A, i, n)
The factor (PV/A, i, n) is a term commonly found in standard engineering economics or finance texts. It is defined as:
(PV/A, i, n) = [i (1+i) n / [(1+i) n – 1]]-1
i is the interest rate or in this case the rate of return
The above equations and definitions along with some algebra can be used to arrive at the following expression for p BE, the breakeven price of the product with the tax rate assumed to be zero.
0 = - I + ( v p BE – O ) (PV/A, i, n)
Solving for p BE gives
p BE = (I/v) (A/PV, i, n) + O/v
From this expression it is clear that the breakeven product price consists of two components, capital recovery I/v) (A/PV, i, n) and recovery of operating costs O/v.
Note that the equality (PV/A, i, n) = (A/PV, i, n)-1 has been used.
For the case where the tax rate is non-zero the corresponding expressions are:
0 = - I + [( v p BE – O ) (1-t) + t I/n] (PV/A, i, n)
Solving for p BE gives
p BE = (I/v) [(A/PV, i, n)/(1-t) - (t/(1-t))(1/n)] + O/v
As for the before tax case there is a capital recovery component which in this case is (I/v) [(A/PV, i, n)/(1-t) - (t/(1-t))(1/n)] and reduces to the before tax term when t = 0 and the same operating cost recovery term O/v.
We can define the “capital recovery factor” (CRF) to be:
CRF = (A/PV, i, n)/(1-t) - (t/(1-t))(1/n)
This leads to:
p BE = (I/v) CRF + O/v
The following two tables show calculated values of the CRF for a range of project lives from 10 out to an infinite life and a range of rates of return from 0 to 20%. One table is for the before tax case and the other is for the case of a 35% marginal tax rate.
CRF, before tax Project life,n
10
15
20
infinite
0%
0.1
0.07
0.05
0
10%
0.16
0.13
0.12
0.1
15%
0.2
0.17
0.16
0.15
20%
0.24
0.21
0.21
0.2
Rate of return, i
CRF, after tax project life, n
10
15
20
infinite
0%
0.1
0.07
0.05
0
10%
0.20
0.17
0.15
0.15
15%
0.25
0.23
0.23
0.23
20%
0.31
0.29
0.29
0.31
rate of interest, i
In the above tables, the fact that (A/PV, i, n) approaches i as n goes to infinity has been used. Also that (A/PV, i, n) approaches 1/n as i goes to 0.
Looking at the above tables one can use a CRF of say 0.2 when doing “back of the envelope” estimates of the p BE for a 15% rate of return and a very wide range of project lives and either before or after taxes. The expression for p BE becomes:
p BE = (I/v) 0.2 + O/v
Problems for Credit
1) Calculate the CRF for an incremental tax of 40%, a depreciation life of 20 years, a depreciation schedule that is straight line over the same 20 years and a 15% after tax rate of return
2) If the capital for a project was $400 million to produce 10,000 units per year, what is the breakeven price that you need to charge to charge using the CRF from above to calculate the contribution needed to recover the cost of capital and the and an operating cost was $50/unit
3) Your boss has read the above paper and thinks that the equation p BE = (I/v) CRF + O/v has to be in error since there is no term in it that breaks out the profit. You disagree because:
a) Profit is assumed in the depreciation schedule
b) The profit is really included in the rate of return. As long as it is non-zero it means there is a return capital and therefore a profit
c) Profit is assumed in the tax rate
Carefully think about O/v and I/v. O is the annual operating cost and v is the number of units produced PER YEAR. You might be given O/v as a ratio where the annual operating cost has already been calculated (i.e. a co-worker says it is $10/unit) or you may have to calculate it (instead the co-worker says the operating costs are $10 million per year and the number of units produced per year is 1 million).
With respect to I/v. The symbol I is the investment at time 0 in $ and v is as above. You might be told v (i.e. 1 million units per year) or you may have to calculate it (i.e. 2739 units per day are produced which means over a year 2739 x 365 = 1 million units are produced which is v).
Think about the above when doing #2
10
15
20
infinite
0%
0.1
0.07
0.05
0
10%
0.16
0.13
0.12
0.1
15%
0.2
0.17
0.16
0.15
20%
0.24
0.21
0.21
0.2
Explanation / Answer
!) CRF is given by the formula
CRF = (A/PV i,n)/(1-T) - (t/1-t)*(1/n)
substituting values we have, (0.160/0.6) - (0.4/0.6)(1/20) = 0.26667 - 0.03333 = 0.23334
2) p BE = I/V*CRF + O/V = (400/3.65) *0.23334 + 50 = 25.58 + 50 = $75.58
Note: It appears one item of cost is not given in the requirement; only the operating cost is given; if other costs are there add it to 75.58 to get the p BE.
3) Option b. Profit is included in the rate of return. As long as it is not zero, there is return and a profit.
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