Discounted Cash Flow


Discounted Cash Flow

The concept of DCF for projects often governs whether or not a project will be selected to go forward, or whether or not a change in scope requiring new resources will be approved. Those are two good reasons why project managers should be familiar with the concept. The controller's office usually makes the DCF calculations for the project manager. However, the project manager often influences the risk factors that go into determining the discount rate. Thus, the project manager is not entirely a passive partner in the DCF deliberations.

The Discount Rate

The discount rate, sometimes called an interest rate, but also called the cost of capital rate or factor, is the discount applied to future cash flows to account for the risk that those flows may not happen as planned or to create "equivalence" with another investment opportunity. [4] Discounting gives us a present-time feel for the value of future flows, having taken risk and alternate opportunities into account.

Many things can be considered in determining the discount rate: real interest that could be earned on an alternate opportunity, the risk of inflation, the risk that the market will not be accepting of the project deliverables, the risk that customers will not have the capacity to pay in the future because of some mishap in the economy, the risk that the project will not be completed on time in order to exploit a market opportunity, or that the project will consume more capital than planned.

It is not uncommon for different discount rates to be applied to different projects in the same company because of the different risk factors faced by each project. Ordinarily, to make discounting practical, all projects of similar type are given the same discount factor, such as all pharmaceutical projects discounted at one rate and all real estate projects discounted at another rate.

Net Present Value and Net Future Value

You may recognize that discounting is the inverse of the familiar idea of compounding. Compounding begins with a present value and forecasts a future value based on an interest rate or capital factor rate. Discounting begins with a forecasted future value to which a discount factor is applied to obtain a present value of the amount. In the same project, the compounding factor and the discount rate are the same rate. Therefore, it is relatively easy to work from the present to the future or from the future back to the present.

Most of us who have had a savings or investment account are familiar with the compounding formula, so let's begin with compounding as a lead-in to discounting. Let's use "k" as the compounding (or discounting factor) and compute the values at the end of the compounding period:

Future value (FV) = Present value (PV) * (1 + k)N

where N is the number of periods to be compounded. N takes on values from N = 0 to some value "m".

To take a simple numerical example, if "k" = 8% and "N" = 3, then we can calculate the future value:

FV = PV * (1 + 0.08)3

FV = PV * 1.2597

Thus, if we had an opportunity to invest $10,000 for three years, we have an expectation that $12,597 will be returned to us at the end of three years, given that the compounding factor is 8%. The return on our investment is the net of the investment and the total return: $12,597 - $10,000 = $2,597.

If we had other opportunities to evaluate, the opportunity costs would be the difference in returns between each opportunity. The factor 1.2597 is called the future value factor. Future value factors are distinguished by always being a number equal to or larger than 1. Future value factors are easily calculated in ExcelŽ using the "power" function or they can be found in future value tables in finance books. Table 5-6 is an abridged collection of future value factors.

Table 5-6: Future Value Factors

Discount Rate

Year 0

Year 1

Year 2

Year 3

Year 4

5%

1

1.05

1.1025

1.157625

1.215506

7%

1

1.07

1.1449

1.225043

1.310796

9%

1

1.09

1.1881

1.295029

1.411582

These factors are easily calculated using a spreadsheet. The formula in any particular cell is: Factor = (1 + discount rate)N where N is the number of the year, 0 to 4.

In Excel, the "power" function can be used to calculate the Factor equation. There are two arguments in "power": the first is (1 + discount rate), and the second is N.

Now, about discounting: discounting begins in the future with forecasted flows; the DCF factor "k" is used to find the present value. We solve for present value using the future value formula given above:

PV = FV/(1 + k)N

where N is the number of periods to be compounded.

The present value factors are the inverse of the future value factors. For instance, for m = 3, the present value factor using 8% is 1/1.2597, or 0.7938. All present value factors will be numbers less than or equal to 1. Table 5-7 is an abridged example of present values.

Table 5-7: Present Value Factors

Discount Rate

Year 0

Year 1

Year 2

Year 3

Year 4

5%

1

0.952381

0.907029

0.863838

0.822702

7%

1

0.934579

0.873439

0.816298

0.762895

9%

1

0.917431

0.84168

0.772183

0.708425

These factors are easily calculated using a spreadsheet. The formula in any particular cell is: Factor = 1/(1 + discount rate)N where N is the number of the year, 0 to 4.

In Excel, the "power" function can be used to calculate the denominator in the Factor equation. There are two arguments in "power": the first is (1 + discount rate), and the second is N.

Using the 8% factors and m = 3, suppose we had a forecast for future flows due to a present investment of $10,000 as follows:

N = 0, $10,000 present-time investment outflow

and future forecasted inflows:

N = 1, $3,000, k = 1.08, 1/k = 0.9259

N = 2, $6,000, k2 = 1.082, 1/k2 = 0.8573

N = 3, $7,000, k3 = 1.083, 1/k3 = 0.7938

The present value of these flows given a discount of 8% is given by:

PV(inflows) = $3,000 * 0.9259 + $6,000 * 0.8573 + $7,000 * 0.7938 = $13,478

NPV of return = -$10,000 + $13,478 = $3,478 at the end of three periods.

Figure 5-3 provides a graphical presentation of the cash flows. It is usually simpler to calculate the NPV directly by entering all "outflows" as negative numbers and all "inflows" as positive numbers. In the figure, outflows are arrows pointing down and the inflows are arrows pointing up. With this convention, the above would be written as:

NPV = PV (all inflows) - PV (all outflows)

NPV = PV (all inflows - all outflows)

click to expand
Figure 5-3: NPV Example.

At 8% discount rate:

NPV

=

-$10,000 + $3,000 * 0.9259 + $6,000 * 0.8573 + $7,000 * 0.7938

 

=

$3,478

NPV is our first risk-adjusted measure of project financial performance. If the outflows are the cash expended on the project and the inflows are the cash received because of the project, then in the case illustrated above it makes sense to do the project because the NPV $0. If the NPV is negative, then more money is going out of the business than is coming into the business on a risk-adjusted basis; it makes no financial sense to do projects of negative NPV:

Decision policy: Only projects with NPV $0 will be selected for execution.

Internal Rate of Return

Now what about this question: Given that the future flows are forecasted, as they were in the example we have just studied, what is the discount factor that makes the NPV exactly $0? We are interested in that factor, which we will call the IRR (internal rate of return), because any discount factor that might be imposed by the controller that is higher than the IRR would cause the NPV to be negative and the project would not be approved.

IRR = Discount rate for which NPV = 0

We cannot solve for the IRR directly. The IRR can only be solved iteratively. For instance, for the project just discussed, using the present value factor tables, we find that at a discount of 24%, the NPV is slightly negative. At 23%, the NPV is slightly positive, meaning the IRR is between 23 and 24%. IRR can be solved in ExcelŽ using the IRR function that solves iterative equations. The exact solution is 23.598%.

In some companies, the IRR is called the "hurdle rate." No project can be approved with a discount above the hurdle rate. The IRR is the upper bound of the discount factor for a positive NPV.

Decision policy: The project is acceptable if k < IRR.

When k = IRR, the project is usually not accepted.

Benefit/Cost Ratio

The benefit/cost ratio (BCR) is a figure of merit that is often used to evaluate projects. Project managers may be asked to participate in calculating the BCR by evaluating the risks in the project that affect cash flows. The BCR is defined as:

BCR = (PV of all cash inflows)/(PV of all cash outflows)

Defined as above, if BCR > 1, then the project has greater inflow than outflow, the NPV will be positive, and, all other things being equal, the project should be on a list of good candidates:

Decision policy: The project is acceptable if BCR > 1.

Break-Even Point

Using the DCFs, the break-even point is when the cumulative project DCFs go from cash negative to cash positive. In equation form, the break-even point occurs when:

Σ PV of discounted cash outflows = Σ PV of discounted cash inflows

Economic Value Add

EVA is a risk-adjusted quantitative measure of project performance. Unlike the previous measures, EVA is focused on earnings. [5] The idea of EVA is that a project should earn more in profits than the cost of the project's capital employed. Remember that capital employed is the liability or equity (resources "owned" by creditors and owners) that finances the expenses of the project. If the project were less profitable than the cost paid for its capital employed, then the creditors and owners would do better to employ their capital elsewhere. Such a competitive rationale should be a component of any rational decision policy.

When working with EVA we use the same risk factors and notation as already developed. The cost of capital factor (CCF) is the discount rate, "k":

The cost of capital employed ($CCE) = k * $Capital employed

Thus, if a project absorbs $100,000 in capital for a year, and the risk factor, CCF, is 8%, then the CCE is $8,000. The project's earnings must exceed $8,000 or else more is being spent on capital than is being earned on the same capital. A special note: the time frame for comparison of earnings and CCE should be identical or else each is subject to different risks. It is common practice to discount all figures to present time for a common comparison.

The first step in looking at EVA is to get a handle on earnings. Earnings are profits and they are portrayed on the P&L. Typically, the P&L would show:

(Revenues - Cash expenses - Noncash expenses) * (1 - Tax rate) = Earnings after tax (EAT)

Immediately we see that noncash expenses save real cash outlays on taxes by subtracting from taxable revenue. For our purposes, we will assume the noncash expense is depreciation of capital assets and that the revenue is from the deliverables of the project. [8]

If we then sum up the EAT for each period and compare the summed EAT to the CCE, we have the EVA for the project.

EVA = EAT - k * CCE

To put EVA in a project context, consider Table 5-8. We assume a $500,000 capital investment in the project and a five-year straight-line depreciation of $100,000 per year. [9] Our capital investment on the balance sheet is relieved ("relieved" is accounting terminology for "reduced") by a depreciation expense each year; each year the capital employed in the project goes down accordingly, and the CCE is less each year as well. Each year we multiply the investment balance on the balance sheet by the cost of capital factor, k, to calculate the cost of capital for that year. Discounting by 1/(1 + k)N each year provides the present value of the cost of capital employed (PV CCE).

Table 5-8: Depreciation in EVA Example

Present

1

2

3

4

5

Total

 

$0.00

$100,000.00

$100,000.00

$100,000.00

$100,000.00

$100,000.00

$500,000.00

Depreciation

$500,000.00

$400,000.00

$300,000.00

$200,000.00

$100,000.00

$0.00

 

Capital employed

 

0.08

0.08

0.08

0.08

0.08

 

Discount rate, "k"

 

$40,000.00

$32,000.00

$24,000.00

$16,000.00

$8,000.00

$120,000.00

CCE

 

$37,037.04

$27,434.84

$19,051.97

$11,760.48

$5,444.67

$100,729.00

PV of CCE

Present = Period 0.

Capital employed = nondepreciated value of asset remaining.

PV ot CCE = present value of the cost of capital employed.

Now in Table 5-9 we calculate the EVA. For purposes of this example, let's assume the project business case proposes an earnings figure, before noncash additions, of $50,000 per year. That $50,000 is the earnings figure before any risk adjustments. To find the EVA, we simply make the risk adjustments by finding the present value of the EAT and subtracting the PV CCE, and that provides the PV EVA.

Table 5-9: EVA of Project

Present

1

2

3

4

5

Total

 
 

$50,000

$50,000

$50,000

$50,000

$50,000

$250,000

Business case EAT

 

$46,296.30

$42,866.94

$39,691.61

$36,751.49

$34,029.16

$199,635.50

PV of EAT

$0.00

$37,037.04

$27,434.84

$19,051.97

$11,760.48

$5,444.67

$100,729.00

PV of CCE

 

$9,259.26

$15,432.10

$20,639.64

$24,991.01

$28,584.49

$98,906.50

PV of EVA

 

0.08

0.08

0.08

0.08

0.08

 

Discount rate, "k"

PV of EAT = present value of earnings after tax.

PV of CCE = present value of the cost of capital employed.

PV of EVA = present value of economic value add.

We see in this example that the EVA is comfortably positive, so this project earns more than it costs to employ the capital.

Economic Value Add and Net Present Value Equivalence

Fortunately for project managers, NPV and EVA are exactly equivalent. NPV is computationally much more straightforward, so EVA-NPV equivalence is a very big convenience indeed. Let's see how this equivalence works in the example project. Table 5-10 shows the calculations.

Table 5-10: EVA-NPV of Project

Present

1

2

3

4

5

Total

 

EVA of Project

 

$50,000.00

$50,000.00

$50,000.00

$50,000.00

$50,000.00

$250,000.00

Business case EAT

 

$46,296.30

$42,866.94

$39,691.61

$36,751.49

$34,029.16

$199,635.50

PV of EAT

$0.00

$37,037.04

$27,434.84

$19,051.97

$11,760.48

$5,444.67

$100,729.00

PV of CCE

 

$9,259.26

$15,432.10

$20,639.64

$24,991.01

$28,584.49

$98,906.50

PV of EVA

 

0.08

0.08

0.08

0.08

0.08

 

Discount rate, "k"

PV of EAT = present value of earnings after tax.

PV of CCE = present value of the cost of capital employed.

PV of EVA = present value of economic value add.

NPV of Project

 

$50,000.00

$50,000.00

$50,000.00

$50,000.00

$50,000.00

$250,000.00

Business case EAT

$0.00

$100,000.00

$100,000.00

$100,000.00

$100,000.00

$100,000.00

$500,000.00

Depreciation

 

$150,000.00

$150,000.00

$150,000.00

$150,000.00

$150,000.00

 

NCF

 

0.08

0.08

0.08

0.08

0.08

 

Discount rate, "k"

 

$138,888.88

$128,600.82

$119,074.84

$110,254.48

$102,087.48

$598,906.50

PV of NCF

      

-$500,000.00

PV of investment

      

$98,906.50

NPV

PV of NCF = present value of net cash flow.

First, we must reorient ourselves to cash flow rather than earnings. Tom Pike's ditty — "Cash is a fact but profit is an opinion" — jumps to mind. To show equivalence between NPV and EVA, we must find the cash flow from cash earnings. Remember that we define cash earnings as net cash flow (NCF). Also recall the definition previously given of net cash flow: the after-tax earnings with the noncash expenses on the expense statement added back in:

NCF = EAT + Noncash expense

NCF = EAT + Depreciation

From this point the calculations are straightforward: add the EAT and the depreciation together to obtain NCF, calculate NCF present value, and the result is the present value of the cash inflows from earnings. Then subtract the present value of the outflows. The result is the NPV. We see that, to the penny, the EVA and the NPV are the same, though the calculations for NPV are usually much simpler since cash is much easier to measure and track than EAT.

NPV (NCF from operations) = EVA (EAT)

[4]Higgins, Robert C., Analysis for Financial Management, Irwin McGraw-Hill, Boston, MA, 1998, chap. 1, p. 238.

[5]P.T. Finegan first wrote about EVA in the publication Corporate Cashflow. [6] Shawn Tully [7] made EVA popular in a Fortune magazine article after Finegan wrote about EVA.

[5]Finegan, P.T., Financial incentives resolve the shareholder-value puzzle, Corporate Cashflow, pp. 27–32, October 1989.

[5]Tully, S., The real key to creating wealth, Fortune, pp. 38–50, September 1993.

[8]If the project does not generate revenue, but generates cost savings instead, the cost savings can be plugged in since they also create increased earnings.

[9]Depreciation is not always uniformly the same figure each year. There are "accelerated" depreciation methods that take more expense in the early years and less in the later years. The controller will make the decision about which depreciation formula to follow.