Circular File

This is an embellishment of the Circular File example of Section 18.3 in (BM). The objective here is to do what is hinted at in footnote 21 on page 517, i. e., calculate the change in market values that undertaking the investment opportunity in the middle of page 517 entails, using the tools of Chapter 20. The relevant tools are risk neutral valuation (recall that we must be able to compute market values of assets in this fashion if there is no arbitrage opportunities), which we have learned because we did Chapter 20 before we did Chapter 18. The exposition in Section 18.3 is thus enhanced by removing any doubt that the authors are waving their hands in these calculations.

To keep things simple, I will restrict the cash flow possibilities to the two states in the investment opportunity. This will generate numbers for the changes in market values of Circular File’s debt and equity that are different than those in (BM), but the qualitative features are preserved. The essence of the restriction is this. There are three market values given in the example that can be used to determine the three unknowns we need and they do so uniquely, i. e., they exhaust the degrees of freedom of this restriction to two cash flow states. To generate the exact changes in the text, one would need some more degrees of freedom, e. g., more than two cash flow states.

So we take as data of the example the market value balance sheet at the top of page 517, and the investment opportunity in the middle of page 517. We wish to determine the cash flows and the risk neutral probability of the high cash flow state that generate the market values in this data. Let denote this risk neutral probability and recall that the risk free rate in this example is 5%. The market value of the cash flow in the investment opportunity is 8 since it costs 10 and its NPV = -2. This is enough to determine since we must have that

,(1)

if there is to be no arbitrage opportunity. Solving (1) yields = 0.07.

Denote the cash flows from assets in place (those that yield the market values at the top of page 517) by and , where the subscripts denote the high and low cash flow states. For these to generate the market value of 25 for the debt claim, it must be that

,(2)

since for this problem to be interesting it must be that in the high cash flow state Circular File is solvent and creditors get only their promised payment of 50, but in the low cash flow state, Circular File is bankrupt and creditors get everything, i. e., they get . Plugging into (2) the value of = 0.07, and solving for this low cash flow yields . Note that this number is less than 50 so Circular will indeed be bankrupt in the low cash flow state.

To determine the value of we can use one of the two remaining market values in the data for the example, either the total market value of Circular File or the market value of Circular File’s equity. The equity is the easiest. If there is no arbitrage, we must have that

,(3)

since in the low cash flow state, creditors get all of , but in the high cash flow state they only get their promised payment of 50. Plugging into (3) the value of = 0.07 and solving gives .

So now Circular File’ market value balance sheet at the top of page 517 can be generated from the prospective cash flows of 125 if the high cash flow state occurs and 22.75/.93 if the low cash flow state occurs. To check this out, we should have that

which you should verify by plugging in the values we have computed for , and .

What happens if Circular File undertakes the investment opportunity in the middle of page 517? The first thing to note is that it must pay 10 to do this now. The text says that it has 10 in cash. This must be 10 of net working capital, but this is then 10(1.05) = 10.50 in terms of the future cash flows and . So with the investment opportunity undertaken, in the high cash flow state Circular File’s cash flow will be 125 – 10.50 + 120 = 234.50 and in the low cash flow state its cash flow will be 22.75/.93 – 10.50 + 0 = 12.985/.93. We can then compute the value of Circular File’s assets, its debt and its equity using risk neutral valuation. The value of its assets must be 28 since the project has NPV = -2. The risk neutral calculation is

= = = 28.

The value of Circular File’s debt is

= = = 15.70.

The value of Circular File’s equity is therefore 28 – 15.70 = 12.30. The risk neutral calculation is

= = = 12.30.

Again, note that the value of Circular File’s debt and equity are not 20 and 8, respectively, as in the balance sheet at the bottom of page 517. Note however that the value of Circular File’s debt is less than it was before the project was undertaken, by 25 – 15.70 = 9.30. Note also that the value of Circular File’s equity is more than it was before the project was undertaken by 12.30 – 5 = 7.30. So we have that , and , where = -10 + 8 = -2, = -9.30, and = 7.30. Firm value falls by the amount of the project’s negative NPV, but the value of creditors’ claim falls by more, so the value of equity increases. This is the essence of the incentive problem of risk shifting or asset substitution.

The incentive problem of under investment, a problem pointed out first by Stew Myers, is the one described on page 518. There Circular File is as described at the top of page 517 but it cannot use its net working capital to undertake a project that has positive NPV. It cost 10 and is worth 15. Assume that Circular File’s assets in place are a reflection of possible future cash flows of and , with the risk neutral probability of of = 0.07. Then we know that = 125 and = 22.75/.93.

You can then calculate the possible project cash flows that generate the value of Circular File’s debt of 33 and equity of 12, as given in the market value balance sheet in the middle of page 518. Let and denote the project cash flows in the high and low cash flow states, respectively. The equations that determine these numbers are as follows.

,(4)

.(5)

Solving (4) and (5) give the project cash flow in the high cash flow state is = 105 and the cash flow in the low cash flow state is = 8.4/.93. You check that the value of this project cash flow is 15.

The values in the market value balance sheet in the middle of page 518 are the values if Circular File’s shareholders contribute the cost of 10 to undertake the project. But if they do this, they would be trading equity worth 5 and cash of 10 for equity worth 12, a net loss of 3. In essence the project adds value of 5 but it increases the value of Circular File’s debt by 8, the project’s NPV and 3 of shareholders’ wealth. Circular File’s shareholders would never go for this. They would, therefore, forego this positive NPV project.

This is the essence of the under investment problem. With debt in its capital structure, a firm’s shareholders may find it in their interest to forego value-increasing (positive NPV) investments because too much of the value increase goes to creditors.