Solving a problem using CAPM and capital structure theory

I am given the following problem:

A firm currently owns assets worth $$4$$ milllion dollars that have a beta of 1. The risk-free interest rate is 10% and the market risk premium is 8%. Suppose the firm has the opportunity to invest in a project that will earn a 13% rate of return for certain into the indefinite future. The cost of the project is $$1$$ million dollars. Should the firm make the investment?

How do I solve it using CAPM or capital structure theory?

Is it correct to say that the firm should not invest because, using CAPM, the required rate of return should be:

$$r_i = r_f + \beta_ * (E[R_{Mkt}] - r_f) = 18\%$$

@chsk gives the correct answer if the firm has to raise funds using the equity market and the cost of those funds is the same as that for the firm. However, I would recommend that this firm do the project.

As indicated in the project, the risk free rate of return is 10%, and this project "...will earn a 13% rate of return for certain into the indefinite future." So this is a risk-less project with a return above that of the risk free rate. So, if you can borrow at the risk free rate, such a project will be profitable. That should be clear by introspection, but here are some numbers, even assuming that there are no cash flows during construction: $$\mathrm{PV} = \-1,000,000 + \sum_{t\ge 1} \frac{ \130,000 }{ 1.1^t } \approx \295,000$$ and if cash flows appear during the initial period the value is about \\$425,000.

In general, the internal cost of capital should be tied to project risk and not overall firm cost of capital. Consider that we could start a new firm that does only the new project. What is the appropriate cost of capital to lend to this new firm? Well, by assumption, the new project has no risk. Therefore, the appropriate cost of funds should be the risk free rate (10% in this problem).

In the real world this might be tricky. You might have synergies which complicate considering the project as a stand alone firm. You might not actually be able to find a investor that will provide funds to you based on project risk. You might have other frictions which put you in a pecking order project financing setting. But, safely in the confines of a study problem, these frictions are not of first order importance.

The way that I would answer this problem is as follows:

The appropriate cost of funds for this project should be the risk free rate because there is not uncertainty to the project's returns. Therefore, the NPV of this project should be positive (show math to that effect). However, there are a number of reasons (sketch some of these out) why it might be difficult to raise funds for a risk-less project at the risk free rate. It is likely that an upper bound on market financing on this project would be the firm's average cost of capital, which is 18%. At an 18% discount rate this is not an economical project (show math to that effect). This firm should review its financing choices to determine their marginal cost of capital for this project. The internal rate of return on this project is 13% (show math to that effect), so as long as the firm's cost of funds is $$<13$$% then they should fund this project.

Assuming that the project has the same beta as the firm's existing assets -- and assuming I'm remembering what I learned in my introductory CorpFin class correctly --, you are indeed right. The present value of future cash flows is, in general,

$$\mathrm{PV} = \sum_{t=1}^T \frac{ \mathbb{E}[ \tilde{c}_t ] }{ (1 + \mu_t)^t }$$

where

$$\mu_t = r_f + (\mu_M - r_f) \cdot \beta_t$$

and

$$\mathbb{E}[ \tilde{c}_t ] = \int_{-\infty}^\infty f(c_t) \, c_t \, dc_t$$

Here, $$T = \infty$$, and for all $$t$$, we have $$\beta_t = 1$$, $$\mu_t = 0.18$$, and $$\mathbb{E}(\tilde{c}_t) = 130.000$$. The present value of the investment's future cash flows, then, is

$$\mathrm{PV} = \sum_{t\ge 1} \frac{ 130.000 }{ 1.18^t } \approx 722.222$$

which is less than one million, and the investment should not be undertaken.

Using a hurdle rate for determining whether the investment should be undertaken is also possible, but I generally find the present value-based approach cleaner. It shows you how much you'd gain (or, in this case, lose) on the investment, and unlike IRR-based calculations it also works for investments where future cash flows change sign more than once (imagine, say, a mining operation where you have to pay for clean-up after resources have been mined).