# Some hints for Problem Set 10

The classic capital budgeting model (such as you learned in Finance 3310) implicitly assumes that the firm has unlimited liability and faces linear taxes. When these assumptions hold, then the net present value (NPV) of a project is calculated by estimating expected values of future incremental after-tax cash flows and discounting them at an appropriate risk-adjusted discount rate. However, we showed during yesterday’s class meeting how limited liability and nonlinear taxes imply that the net present value of a project depends upon the manner in which incremental after-tax cash flows interact with cash flows from existing assets. Consequently, the after-tax value of equity is equal to the difference between the pre-tax value of equity and the value of the government’s tax claim (both of which we model as call options on the firm’s assets). Furthermore, project NPV corresponds to the difference in after-tax value of equity (assuming the project is undertaken), minus the after-tax value of equity (assuming the project is not undertaken).

Problem Set 10 provides an opportunity to apply these concepts.  Here are some hints for parts A through E of Problem Set 10 :

1. In part A, apply the option pricing framework to determine the pre-tax value of equity (V(E), where V(E) = V(Max(0,F-B)), the value of debt (V(D), where V(D) = V(B – Max(0,B-F)), and the value of taxes (V(T), where V(T) = $\tau$V(Max(0,F-TS)), assuming that this investment is not undertaken.  Helpful hint: we performed these same calculations in class yesterday for the problem described on pp. 17-18 of the http://fin4335.garven.com/fall2017/risk_costly_chapter7.pdf teaching note.
2. In order to determine whether the project should be undertaken, in part B you need to  after-tax equity value (i.e., V(E) – V(T)) which obtains under the assumption that the investment is undertaken. Once you obtain that result, the net present value (NPV) of the project is the difference between the after-tax value of equity (V(E) – V(T)) in part A (which you have already calculated) and the after-tax value of equity which obtains if the investment is undertaken. The decision to invest or not to invest depends upon whether the NPV of the investment is positive (in which case you undertake the project) or negative (in which case you do not undertake the project).
3. An investment tax credit (ITC) is quite literally a check sent by the U.S. Department of the Treasury to the company; thus, the NPV when there is an ITC is equal to the NPV that you calculated in part B plus the value of the ITC.  At that point, whether you invest or don’t invest depends upon whether NPV is positive or negative (as in part B).  The ITC in this case increases project NPV by \$1 million.
4. In order to answer part D, you need to redo the calculation described in the first paragraph above using a 20% tax rate rather than a 35% tax rate.
5. In order to answer part E, you can figure out the tax rate at which the firm is indifferent about making the investment by trial-and-error, or better yet, adapt the Tax Options spreadsheet located on the lectures notes page to the parameters upon problem set 10 is based and use Solver.