13.1 Adjusted Present Value

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Problem. A firm has the opportunity to do a one-shot project. It requires a date 0 initial outlay for new

investment of $250,000. During the initial five-years, it will generate the following before-tax cash flows:

date 1 = $120,000, date 2 = $140,000, date 3 = $180,000, date 4 = $130,000, date 5 = $80,000, and

$40,000 each year thereafter. The project’s tax rate is 40.0%, it’s unlevered cost of capital is 10.0%, and

the riskfree rate (= cost of debt) is 3.0%. The company has precommitted to a particular quantity of debt

on the following dates to support this project: date 0 = $150,000, date 1 = $130,000, date 2 = $110,000,

date 3 = $90,000, date 4 = $70,000, and $40,000 each year thereafter. What is the project’s NPV as

calculated using the APV method? What is the present value of future cash flows to both debt and equity?

Other modules in this chapter will analyze the same problem using the FTE and WACC methods and

verify that all three methods yield the same results.

Solution Strategy. Value this two-stage project under the APV method by calculating the NPV of the

unlevered investments plus the present value of the debt tax shield. Then use the project NPV result to

calculate the PV of Future Cash Flows for each date in the future. This will be used by the other modules

in this chapter to calculate the amount of Equity and Cost of Equity Capital for each date.

FIGURE 13.1 Spreadsheet for Three Valuation Methods Using The APV Method.

How To Build This Spreadsheet Model.

1. Inputs. Enter the overall project inputs in the range B4:B7, the Before-Tax Cash flows in the

range C12:H12, and the Debt amounts in the range B22:G22. Lock in the first seven rows as

titles by selecting cell A8 and clicking on Window | Freeze Panes.

2. Taxes and After-Tax Cash Flow. Taxes paid is the (Before-tax Cash Flow) * (Tax Rate). Enter

=C12*$B$5 in the cell C13 and copy it across. The $ sign on $B$5 lock in the absolute cell

reference for the Tax Rate. After-tax Cash Flow is Before-tax Cash Flow less Taxes. Enter

=C12-C13 in cell C14 and copy it across.

3. Present Value of Future Cash Flows. Using the unlevered cost of capital, discount an infinite

series of constant cash flows using the infinite annuity formula: (After-tax Cash Flow) /

(Unlevered Cost of Capital). Enter =H14/$B$6 in cell G16. Discount the explicitly forecast

horizon cash flows using a recursive, one-period-at-a-time approach: PV of Future Cash Flow (t)

= [After-tax Cash Flow (t+1) + PV of Future Cash Flow (t+1)] / (1 + Unlevered Cost of Capital).

Enter =(G14+G16)/(1+$B$6) in cell F16 and copy it leftwards to the range B16:E16.

4. New Investment and NPV of Unlevered Investment. To get the NPV of the Unlevered

Investment, subtract the Initial Outlay for New Investment. Enter =-$B$4 in cell B17 and

=B16+B17 in cell B18.

5. Tax Shield. We adopt the convention that interest to be paid on date t+1 based on debt which

issued on date t. A tax shield is defined as the quantity of taxes avoided by deducing interest

expense. It is calculated as Tax Shield (t+1) = Debt (t) * (Riskfree Rate) * (Tax Rate). Enter

=B22*$B$7*$B$5 in cell C23 and copy it across.

6. PV of Future Tax Shield. Using the riskfree rate, discount an infinite series of constant tax

shields using the infinite annuity formula: (Tax Shield) / (Riskfree Rate). Enter =H23/$B$7 in

cell G25. Discount the explicitly forecast horizon tax shields using a recursive, one-period-at-atime

approach: PV of Future Tax Shield (t) = [Current Date Tax Shield (t+1) + PV of Future Tax

Shield (t+1)] / (1 + Riskfree Rate). Enter =(G23+G25)/(1+$B$7) in cell F25 and copy it

leftwards to the range B25:E25.

7. NPV of the Project and PV of Future Cash Flows. The NPV of the Project under APV is the

sum of: (A.) NPV of the Unlevered Investment and (B.) PV of Debt Tax Shield. Enter

=B18+B25 in cell B26. The PV of Future Cash Flows under APV is the sum of the PV of Future

Cash Flows and PV of Debt Tax Shield. Enter =B16+B25 in cell B27 and copy it across.

We see that the NPV of the Project under APV is $221.48 and the PV of Future Cash Flows under APV

starts at $471.48 and declines to $260.00. As a special case, the same spreadsheet model can be used for a

single-stage, infinite horizon project. This is implemented by holding the Cash Flows and Debt amounts

constant over the Explicit Forecast Horizon (the first stage) and identical to the Cash Flows and Debt

amounts over the Infinite Horizon (the second stage).