Circular File
This is an embellishment of the Circular File
example of Section 18.3 in (BMA). The
objective here is to do what is hinted at in
footnote 19 on page 483, i. e., calculate the
change in market values that undertaking the
investment opportunity at the top of page 483
entails, using the tools of Chapter 21. 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 21 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 (BMA), 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 once
the risk free rate is specified, 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 of Circular File on
page 482, and the investment opportunity at the
top of page 483. 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 assume 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
p 120   1  p  0 
,
8
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 on page 482)
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
p  ch  50   1  p  0 
,
5
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’s market value balance
sheet on page 482 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
pch  1  p  cl
,
30 
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 at the top of page 483?
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
1.05
=
16.415  12.985 29.40
=
= 28.
1.05
1.05
The value of Circular File’s debt is
.07  50   .93 12.985 /.93
3.50  12.985
=
=
1.05
1.05
16.485
= 15.70.
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
=
=
1.05
1.05
1.05
= 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 in the middle of page 483.
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 at the bottom of page 483 and the
top of page 484. There Circular File is as
described as on page 482, 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 at the top of page
484. 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 
.07  50   .93   22.75 .93  wl 
1.05
.07 125  wh  50   .93  0 
.
12 
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
at the top of page 484 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 of 5 plus 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. In the language of real
options, they would not exercise their options to
invest in an optimal way.