# Project Valuation
## Discounted Cash Flow

In Corporate Finance, it is often necessary to determine if a project is worthwhile to undertake. Sometimes, there will only be enough investment capital to undertake some projects, in which case, one must determine which project to do.
The standard approach is to use the Net Present Value, or NPV. The NPV is calculated by finding all of the cash flows which will occur, and discount them back to the present time. This also includes expenses, hence, net present value.
For example:
STDs Unlimited is trying to determine whether to purchase a new piece of machinery. This machine will cost $1,000,000, and save $170,000 per year for 10 years, at which point it will be sold for $50,000. After 5 years, the machine will require an overhaul which will cost $100,000. If the company uses a 7% discount rate for projects of this type, what is the NPV of purchasing the machine? Should the company do it?
The first thing we must do is find the total cost savings of this project. Each year there will be a positive cash flow of $170,000, which must be discounted each year back to the present time:
$\sum _{t=1}^{10}{\frac {170,000}{(1.07)^{t}}}=1,194,009$
So we can see that this project will bring in $1,194,009 worth of savings. In addition, it will bring in $50,000 in year 10 when it is sold, so we find the present value of that:
${\frac {50,000}{(1.07)^{10}}}=25,417$
So we add those up to get the total positive cash flow of $1,219,426.
From this amount, we must subtract the costs. We know the initially cash outlay is $1,000,000, but we must find the present value of the overhaul which will occur at year 5:
${\frac {100,000}{(1.07)^{5}}}=\$71,299$
So now we can calculate the present value of all costs: 1,000,000 + 71,299 = 1,071,299.
Then we simply subtract the negative cash flows from the positive and see that
$NPV=1,219,426-1,071,299=\$148,127$
So, since this project has a positive NPV, the company should go ahead and do it.

## Depreciation Tax Shield

In the previous example, we did not calculate an important benefit which must be considered in real life examples: taxes. When a company purchases a piece of equipment, they will depreciate it. This means that they do not have to pay taxes on an amount of income equivalent to what they are spending on capital investment.
Let's consider our example above. We arrived at the NPV of $148,127 without taxes, so lets look at what happens when we depreciate the initial investment:
Assume STDs unlimited decides to depreciate the $1,000,000 machine using straight line depreciation for 10 years. What is the value of the depreciation tax shield? What is the new NPV?
Straight line depreciation means an equivalent amount is depreciated each year, for 10 years. So in this example, depreciation is $100,000 per year. This means that on the books, the company will spend $100,000 per year on the machine, instead of $1,000,000 up front. This makes more sense logically, since the company will be using the machine for 10 full years. The tax shield is equal to the following:
$\sum _{t=1}^{n}{\frac {AnnualDepreciation*Taxrate}{(1+r)^{t}}}$
In this case it will be:
${\frac {(100,000*.35)}{1.07}}+{\frac {(100,000*.35)}{1.07^{2}}}+{\frac {(100,000*.35)}{1.07^{3}}}+...+{\frac {(100,000*.35)}{1.07^{10}}}=\$245,825$
As we can see, this is a significant savings.
There is one more step we must take in this example. Since we depreciated the entire value of the machine, we are basically telling the government that it is worthless at the end of 10 years. However, since we sell at at the end of year 10 for $50,000, it is clearly not worthless. In order to make this right (and avoid an audit and costly fines), we must pay taxes on that $50,000 in year 10. This amount is clearly .35 * 50,000 = 17,500. Since we don't pay this until year 10, we discount this amount by ten years (17,500/(1.07^10)) to get $8,896. We then subtract this amount from the $245,825 tax shield to get $236,929.
So we see that our total tax savings from depreciation is $236,929, which we add to our NPV of 148,127 to get a final NPV on this project of $385,056.

## Equivalent Annual Costs

When a company is considering two machines which will be used for a long period of time, it is sometimes necessary to use equivalent annual costs to determine which machine is a better investment.
For example:
Suppose an upscale vodka company is looking to purchase one of the two distilling machines. Machine A costs $45,000, lasts for 5 years before replacement, and costs $10,000 per year to operate. Machine B costs $50,000, lasts for 7 years, and costs $11,000 per year to operate. Now, if we were to take the NPV of the costs, it looks quite obvious that machine B's NPV will be lower (meaning it costs more). We shall use an 8% discount rate.
NPV$_A$=-$84,927
NPV$_B$=-$107,270
Now, if we were to just look at NPV, we would choose machine A, as it has lower costs. However, this fails to take into account that machine B has to be replaced less often. In order to correctly evaluate these options, we must use Equivalent Annual Cost.
$EAC={\frac {NPV}{n-yearannuityfactor}}$
An annuity factor is defined as the present value of $1 received every year for n years, at a discount rate of r.The formula to find the n-year annuity factor is as follows:
${\frac {1}{r}}-{\frac {1}{r(1+r)^{n}}}$
Using this formula, we can see that the 5 year annuity factor at 8% is 3.99, and the 7 year annuity factor is 5.2.
So, in order to find the equivalent annual costs of these projects, we shall divide their respective NPVs by their respective annuity factors:
| Machine A | Machine B |
| $EAC_{A}={\frac {84,927}{3.99}}=\$21,284$ | $EAC_{B}={\frac {107,270}{5.2}}=\$20,628$ |
And so, we can see that although machine A appears to have lower costs, machine B is actually cheaper in the long run, since it must be replaced less frequently.

## Synergy/Cannibalism

When valuing a project, it is necessary to take into consideration how it will affect other parts of your business. For example, say you are the CFO of a computer store that sells X1000 high end computers, and X500 budget computers. If you are considering introducing a new line of X750 mid-range computers, you must consider that this may cannibalize some business from the high end line. That is, some customers who might have otherwise purchased a more expensive X1000 may now settle for the X750.
On the other hand, some new projects may create synergies, or benefits, to other product lines. For example, if you are in the oil refining business, and you are considering opening up you own oil well, this may reduce the costs for your refinery. So this too, must be added in as an additional benefit in your NPV calculation.

## Sunk Costs

Sunk Costs are expenditures which a company has already spent on a project. It is important to realize that sunk costs are already spent, and are not factored into an NPV equation. For example, if your corporation hires a consulting firm to evaluate customer reaction to a new product line, that is not included in the NPV calculation. Whether or not you decide to initiate the new product line, you already spend money on the consultants.
It is also important to understand the difference between opportunity cost and sunk costs. Lets suppose XYZ Corp. buys a parcel of land for $100,000. They do not do anything with this land for a number of years, and then decide to build a factory on it. When calculating the NPV of the factory, the cost of the land is NOT a sunk cost. It is still possible to sell the land or do something else with it, and it therefore must be included.

## Internal Rate of Return

Another common yardstick of valuing projects is Internal Rate of Return, or IRR. IRR is the discount rate which, when applied to a project, will cause the NPV to be zero. Let us use a simple example:
Say you are going to lay out $100,000 for a project. This project will produce cash flows of $15,000 in year 1, and $50,000 in year 2, and $70,000 in year 3. What is the IRR?
In order to solve this problem, we must set up an equation for NPV and set it to zero:
$0=-100,000+{\frac {15,000}{1+r}}+{\frac {50,000}{(1+r)^{2}}}+{\frac {70,000}{(1+r)^{3}}}$
Then we simply solve for r. Of course, as there are 3 variables in this equation, we cannot solve it algebraically, and must use either a financial calculator, Excel, or some type of graphing program or calculator. Using a financial calculator, the answer can be found as 13.454%. So, if the company's cost of capital is 13%, then this project is good to take. However, if the company requires a 13.5% return, the project would not be wise.
There is one caveat to using IRR to evaluate projects. IRR is a rate, not an amount. So if your company must choose between two projects, one with an IRR of 8%, and another with an IRR of 11%, the 11% IRR project is not always the best choice.
| | Project A | Project B |
| Initial Investment | $100,000 | $1,000 |
| Return after 1 yr | $108,000 | $1,110 |
| Total Profit | $8,000 | $110 |
| IRR | 8% | 11% |
So you see, the project with the higher IRR ends up netting the company far less money, as the projects are mutually exclusive. An example of resource allocation maybe that 70 project Bs can be undertaken, vs 1 project A. Project B has a higher IRR and would result in $7700 with an outlay of $70000 , not accounting for administrative costs, whereas 1 Project A would result in 8000 for $100000 outlay, and be likely less costly administratively. For the effort, work or taking on risk, the rate of return would be 7.7% or 8% on capital vs 3% from term deposit with no effort, (assuming tax deductions, e.g. for depreciation of capital plant/equipment/property , are already accounted for by reduction in tax payment cash outflow, which are included in annual cash flow calculations ).

#### addendum: algorithm for irr calculation

irr is a guessing algorithm similar to binary searching for bugs in a program. For a project with a positive IRR, then two relatively prime factors could be used , one to increase the irr (multiple by factor > 1) if the npv is still positive and the other to decrease the irr when npv negative (divide by factor > 1), so that the tried values of the irr are unlikely to cycle, due to the lack of common factors in multiply and divide.
Below is a python algorithm for irr calculation, which also deals with going to a negative IRR when the guessing shows the irr is approaching zero.
This is a rather randomised algorithm; Newton's method has been mentioned as being used in open source spreadsheet programs.
Newton's method relies on the differentiation , or finding a derivative function to tell the slope of the graph of another function. The basic rule is that if there is a term which is ax$^n$ then the derivative has a term , n.ax$^{n-1}$.
The npv function is a function of R the rate of return, such that npv is y, and R is x in the general function y = f(x) . Hence the derivative of the npv function is dy/dx = d(npv)/d(R) , which then requires finding the derivatives of all the terms in the npv function.
In Newton's method, an arbitrary R which is x, gives a NPV , which is a y value , when divided by the gradient which is delta-y/delta-x , gives delta-x , which is subtracted from R ( substitute x) to give another R' that is closer to the IRR, when NPV = 0 ( or the x value where y is zero, or where the function crosses the x-axis ). Using R' as the next R, this is repeated until a good approximation of IRR is achieved. The gradient is always recalculated at the start of each iteration , using the derivative function.
So if npv = CF$_0$ / R$^0$ + CF$_1$ / R$^1$ + .. + CF$_n$ / R$^n$,
or npv = CF$_0$ + CF$_1$ x R$^{-1}$ + CF$_2$ x R$^{-2}$ + .. + CF$_n$ x R$^{-n}$,
then d(npv)/dR = 0 + - CF$_1$x R $^{-2}$ + - 2 CF$_2$x R $^{-3}$ ... + -n x CF$_n$ x R$^{-n-1}$.
See
* REDIRECT Computation instead of Table Lookup