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 p 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 p since we must have that
8
p 120   1  p  0 
,
1.05
(1)
if there is to be no arbitrage opportunity. Solving (1) yields p = 0.07.
Denote the cash flows from assets in place (those that yield the market values at
the top of page 517) by ch and cl , 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
25 
p (50)  (1  p )cl
,
1.05
(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 cl .
Plugging into (2) the value of p = 0.07, and solving for this low cash flow yields
cl  22.75 .93 . 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 ch 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
5
p  ch  50   1  p  0 
,
1.05
(3)
since in the low cash flow state, creditors get all of cl , but in the high cash flow state
they only get their promised payment of 50. Plugging into (3) the value of p = 0.07 and
solving gives ch  125 .
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
30 
pch  1  p  cl
,
1.05
which you should verify by plugging in the values we have computed for p , ch and cl .
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 cl and ch . 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
.07  234.50   .93 12.985 / .93 16.415  12.985
29.40
=
=
= 28.
1.05
1.05
1.05
The value of Circular File’s debt is
.07  50   .93 12.985 / .93
3.50  12.985 16.485
=
=
= 15.70.
1.05
1.05
1.05
The value of Circular File’s equity is therefore 28 – 15.70 = 12.30. The risk neutral
calculation is
.07  234.5  50   .93  0 
.07 184.5  12.915
=
=
= 12.30.
1.05
1.05
1.05
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 V  D  E , and
V  D  E , where V = -10 + 8 = -2, D = -9.30, and E = 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 cl and ch , with the risk neutral probability of
ch of p = 0.07. Then we know that ch = 125 and cl = 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 wh and wl denote the project cash flows in the high and low
cash flow states, respectively. The equations that determine these numbers are as follows.
33 
12 
.07  50   .93   22.75 .93  wl 
1.05
.07 125  wh  50   .93  0 
.
1.05
,
(4)
(5)
Solving (4) and (5) give the project cash flow in the high cash flow state is wh = 105 and
the cash flow in the low cash flow state is wl = 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.