OPTION-BASED PRICING AND DEVIATIONS FROM ABSOLUTE PRIORITY IN
CHAPTER 11 BANKRUPTCIES
Ryan A. Oliver
B.A. University of California, Berkeley, 2004
J.D. University of Southern California, Gould School of Law, 2007
THESIS
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF BUSINESS ADMINISTRATION
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
SPRING
2010
OPTION-BASED PRICING AND DEVIATIONS FROM ABSOLUTE PRIORITY IN
CHAPTER 11 BANKRUPTCIES
A Thesis
by
Ryan A. Oliver
Approved by:
____________________________________, Committee Chair
Hao Lin, Ph.D., C.F.A.
____________________________________, Second Reader
Lan Liu, Ph.D.
____________________
Date
ii
Student: Ryan Andrew Oliver
I certify that this student has met the requirements for format contained in the University
format manual, and that this Thesis is suitable for shelving in the Library and credit is to
be awarded for the Thesis.
_____________________________________________
Monica Lam, Ph.D.
Associate Dean for Graduate and External Programs
College of Business Administration
iii
_____________________
Date
Abstract
of
OPTION-BASED PRICING AND DEVIATIONS FROM ABSOLUTE PRIORITY IN
CHAPTER 11 BANKRUPTCIES
by
Ryan A. Oliver
Deviations from absolute priority are simply the result of more accurate valuations of
creditor interests. This paper argues that unsecured claims against a company in Chapter
11 are comprised of call options on the company’s assets. To the extent this is true,
option-pricing formulas provide an explanation for the deviations from absolute priority
in bankruptcy resolutions. These deviations are simply the difference between the
nominal value of a claimant’s interest and the value of that claimant’s interest as
determined by an option-based pricing analysis.
____________________________________, Committee Chair
Hao Lin, Ph.D., C.F.A.
____________________
Date
iv
ACKNOWLEDGEMENTS
The author wishes to thank Dr. Hao Lin, Assistant Professor of Finance in the
College of Business Administration of California State University, Sacramento for his
insight, encouragement, and patience
v
TABLE OF CONTENTS
Page
Acknowledgements………………………………………………………………… v
Chapter
1. INTRODUCTION………………………………………………………… 1
2. ABSOLUTE PRIORITY………………………………………………… 3
3. LITERATURE REVIEW………………………………………………….. 5
4. DEVIATIONS FROM ABSOLUTE PRIORITY ……………………………. 10
5. CONCLUSION………………………………………………………….… 27
References………………………………………………………………………..… 28
Appendix ………………………………………………………………………...… 30
vi
1
Chapter 1.
INTRODUCTION
This paper sets forth a simple, options-based, framework for analyzing and
predicting deviations from absolute priority in Chapter 11 Bankruptcy settlements. The
framework presented herein is inspired by the work of Baird and Bernstein who, in a
2006 Yale Law Journal paper, compellingly argue that uncertainty in valuation,
combined with a priority based bankruptcy regime creates option value for junior
creditors.
Baird and Bernstein posit that this junior creditor option value drives deviations
from absolute priority. They argue that in negotiated settlements, junior creditors are
compensated for the option value they hold as follows.
The uncertainty inherent in valuing a large corporation in financial distress
creates a bargaining dynamic that accounts for many of the puzzling departures from
absolute priority that the standard model cannot explain. “Deviations” from absolute
priority are often nothing of the kind. They are instead the natural product of bargaining
in a system that is committed to respecting priority, but must do so in a world in which
priorities are enforced through a valuation process the outcome of which is uncertain.
This paper accepts the premise set forth by Baird and Bernstein and attempts to
lend further support to their position by setting forth a framework for valuing the option
right junior creditors hold in bankruptcy proceedings. In doing so, this paper also further
explains deviations from absolute priority, the prevalence of which have been observed
and debated by reorganization scholars for decades.
2
An overview of the framework used in this paper to value junior creditor options
and explain deviations from absolute priority is set forth below. First, creditor claims are
recast as combinations of options, relying on the insights of contingent claims analysis.
Next, these individual constituent options are priced using a version of the BlackScholes formula. Then, the financial position of each creditor is determined by
reconstructing each creditor claim pursuant to its option-based value. And, finally, the
results of this procedure are analyzed to explain, and predict deviations from absolute
priority in Chapter 11 bankruptcies.
The results of this pricing procedure show that, within the contours of the
assumptions and simplifications upon which this paper is based, deviations from
absolute priority are driven by junior creditor option value. Thus, to the extent this is
true, factors which increase the value of junior creditor options will also increase the
magnitude of deviations from absolute priority. This paper focuses on the role debtor
asset levels place in determining creditor option value, but a similar analysis could be
conducted by varying any of the inputs to the Black-Scholes formula.
It is the position of this paper that such an option-based analysis of deviations
from absolute priority demonstrates that such deviations are, in practice, simple
recognitions, by the parties, of the true value of bankruptcy creditor claims.
3
Chapter 2
ABSOLUTE PRIORITY
The absolute priority rule is the Chapter 11 Bankruptcy tenet which holds that
senior creditors may insist upon being paid in full before any creditor subordinate to them
receives anything. With respect to unsecured creditors, the Code states that a plan may
be confirmed over the objection of a senior creditor if:
(i) the plan provides that each holder of an interest of such class receive or retain
on account of such interest property of a value, as of the effective date of the plan,
equal to the greatest of the allowed amount of any fixed liquidation preference to
which such holder is entitled, any fixed redemption price to which such holder is
entitled, or the value of such interest; or
(ii) the holder of any interest that is junior to the interests of such class will not
receive or retain under the plan on account of such junior interest any property.1
The absolute priority rule for unsecured creditors holds that each class must
receive the value of its interest (in net present value terms) OR in the alternative; no
subordinate interest may receive anything.
If one assumes that a debtor company’s value is known with certainty and is not
subject to any variation or disagreement among the parties, applying the absolute priority
rule is a simple matter of ranking interests and assets to creditors.
For example, consider a hypothetical company in Chapter 11 with only three
classes of claimants: senior debt-holders with a $500 claim, junior debt-holders with a
$250 claim, and equity-holders who are the residual claimants to the company’s assets.
In this case, company assets are distributed first to the senior creditors until their claim is
satisfied, then to the junior creditors until their claim is satisfied, and finally any residual
1
11 U.S.C.A. § 1129(b)(1).
4
assets are distribute to the equity holders. The table below summarizes the distribution of
interests to creditors when the company is worth varying amounts:
Firm Value
$1,000.00
Position Under Strict Absolute Priority
$ 500.00
Position Under Strict Absolute Priority
$ 250.00
Position Under Strict Absolute Priority
$ 250.00
$750.00
$500.00
Senior Debt
$500.00
$500.00
Junior Debt
$250.00
$
Equity
$
$
-
$250.00
$250.00
$
-
$
-
Distributions Under Absolute Priority Regime
In accordance with absolute priority rules, no subordinate creditor in this example
receives anything until all interests senior to it are paid in full. However, in practice,
deviations from absolute priority are commonplace.
5
Chapter 3.
LITERATURE REVIEW
3.1. Contingent Claim Analysis
This paper posits that deviations from absolute priority can be explained, in part,
by valuing creditor interests as contingent claims on the debtor corporation’s assets. In
doing so, this paper relies on the insights of Contingent Claim Analysis.
A contingent claim is one that pays off only upon the occurrence certain
contingencies; i.e. the value of an underlying asset is greater (less) than a pre-specified
value for a call (put) option. (Epstein 93).
Fisher Black and Myron Scholes, in 1972, first established that an option can be
valued as a function of five variables: the current value of the underlying asset (S), the
variance in value of the underlying asset (  2 ), the strike price of the option (K), the time
to expiration of the option (t), and the riskless interest rate (r). (see appendix for formula).
This Contingent Claim Analysis (“CCA”), has subsequently been refined and expanded
by Black and Scholes (73), Merton (73, 74, 77), and others to show that any asset can be
valued as an option if its value is a function of the value of an underlying asset. Pursuant
to CCA, any corporate liability, including shareholder equity and various types of debt,
can be valued as claims contingent on the value of the underlying corporate assets.
This paper values one such set of corporate liabilities, creditor interests in
bankrupt companies, using a simple version of CCA. This approach is a direct outgrowth
of the arguments set forth in Baird and Bernstein’s 2006 Yale Law Journal Article,
“Absolute Priority, Valuation Uncertainty, and the Reorganization Bargain.”
6
3.2. “Absolute Priority, Valuation Uncertainty, and the Reorganization Bargain”
As Baird and Bernstein write in their Yale Law Journal article, “Absolute Priority,
Valuation Uncertainty, and the Reorganization Bargain,”
Modern Chapter 11 is the equivalent of a provision in a joint venture agreement that calls
for the appointment of an appraiser and uses the number the appraiser sets (or is expected
to set) as the baseline against which to measure the rights of parties. . . . in the
reorganization context, any valuation mechanism that does not involve a transaction that
monetizes the senior investor’s position (through a sale of the business or a buyout of the
position) creates option value in the position of the junior investor.2
The reorganization process creates option value for junior investors through the
interaction of the absolute priority regime, the right to a valuation hearing the Code
grants impaired claimants, and the uncertainty inherent in business valuations.
3.2.1. Absolute Priority
The absolute priority rule is the Chapter 11 Bankruptcy tenet which holds that
senior creditors may insist upon being paid in full before any creditor subordinate to them
receives anything. This absolute priority rule for unsecured creditors requires that each
class must receive the value of its interest (in net present value terms) OR in the
alternative; no subordinate interest may receive anything.
3.2.2. Right to a Valuation Hearing
The conflict between creditor classes is central to the bankruptcy process. The
Code attempts to strike a balance between enforcing senior claimants’ contractual priority
rights and providing some protection for the interests of junior claimants during
2
Douglas G. Baird & Donald S. Bernstein. Absolute Priority, Valuation Uncertainty, and the
Reorganization Bargain. 115 Yale L.J. 1930 (2006).
7
reorganizations. The Code strikes this balance by enforcing absolute priority in contested
plans while allowing any single dissenting class of creditors to protect its position by
rejecting a proposed plan and insisting upon a judicial valuation hearing.
During the judicial valuation hearing, the court finds a total debtor value against
which the claims of creditors may be assessed in light of the absolute priority
requirement. If the court finds that the contested plan is “fair and equitable” and that the
absolute priority requirement is satisfied, the court may “cram-down” a plan over the
objection of dissenting creditors.
Under the Code, any impaired claimant may reject a proposed bankruptcy plan
and insist on this valuation hearing. However, any value found by a court, or by any
other appraiser, is subject to some amount of uncertainty.
3.3.3. Valuation Uncertainty
Business valuation is inherently uncertain as the process relies on predictions
about future performance. Depending on the assumptions made concerning future cash
flows, discount rates, and general economic conditions, the derived value of a business
will vary.
Though an incredible amount of scholarship has been dedicated to the field of
business valuation it is sufficient for the purposes of this paper to note that any valuation
of a distressed business is subject to some amount of variance. Moreover, deriving the
value of a distressed business through an adversarial, litigious, process likely introduces
some additional uncertainty into business valuation.
3.3.4. Interaction among Factors
8
The interaction among absolute priority, the right to a valuation hearing, and
valuation uncertainty creates option value for junior investors. Following the example of
Baird and Bernstein, this paper proposes a hypothetical to illustrate this creation of option
value in the reorganization context. Suppose a debtor company has two creditors: senior
debt with a $500 claim and junior debt with a $250 claim. Both creditors believe that the
company, going forward, is worth exactly $500. Further, both creditors believe that
impartial valuations of the business by bankruptcy courts would have a mean value of
$500 but that any individual valuation could be higher or lower, each with equal
probability.
The bankruptcy code gives any impaired class the right to insist upon judicial
appraisal. This appraisal is subject to some amount of variance and this variance,
combined with a right to a hearing, is a source of value to the junior creditor. Because
the judge has been assumed to be an impartial appraiser, 50% of the time the appraisal
will return a value of more than $500 (“high valuation”) and 50% of the time it will
return a value of less than $500 (“low valuation”). In the former case – the high
valuation – the junior creditor will receive the difference between the appraised value and
$500. In the latter case – the low valuation – the junior creditor will receive nothing.
But, due to the absolute priority regime, the junior creditor will never receive anything
unless the senior creditor has been paid in full. Therefore, the junior creditor faces no
downside risk by insisting upon a valuation hearing, but, conversely, the hearing exposes
the junior creditor to a potential upside based on the possibility of a high valuation.
9
For example, if a plan is proposed whereby the senior creditor will receive $500 –
the face value of its claim and the amount that all parties believe the business to be worth
– the junior creditor has no reason to not reject the plan and ask for a judicial valuation.
Having been offered nothing, the junior creditor has nothing to lose by rejecting the offer
and opting for a judicial appraisal. Half of the time, the judicial valuation will be high –
more than $500 – and the junior creditor will receive that amount minus $500. The other
half of the time the valuation will be low – less than $500 – and the junior creditor will be
no worse off than it was under the original plan.
Faced with a valuation hearing, the payoff profile for the junior creditor appears
as follows:
Payoff Profile for Junior Creditor
Where the X axis represents the value of the company as determined by the bankruptcy
judge, the Y axis represents the junior creditor’s position, and the x represents the amount
of senior debt.
Given this payoff profile, it become apparent that he right to insist on a valuation
hearing has value. The remainder of this paper builds on the work of Baird and Bernstein
10
by proposing a framework for valuing this right using Contingent Claim Analysis,
thereby explaining deviations from absolute priority, under specified conditions.
11
Chapter 4.
ABSOLUTE PRIORITY
This section argues that option pricing explains, in part, deviations from absolute
priority. The argument proceeds as follows. First, senior debt, junior debt, and equity
bankruptcy claims are described in terms of their constituent call option components.
Next, these individual options which make up creditor claims are valued, under numerous
firm asset levels, using the Black Scholes formula. These individual options are then
combined to reconstruct the total financial position of each claimant. This option based
position is then compared to the position of each claimant in a counterfactual regime
which strictly enforced absolute priority. This comparison is made to determine and
analyze deviations from absolute priority and to provide an explanation for the
prevalence of these deviations in bankruptcy agreements.
4.1. Option Components of Bankruptcy Claims
The Black-Scholes-Merton analysis of corporate liabilities as a call options on a
firm’s assets is used in this paper to address bankruptcy liabilities. The absolute priority
regime treats every unsecured interest as a “residual” claimant (to the extent of its claim)
on the assets of the firm after interests senior to it have been paid. Moreover, each of
these claims continues to enjoy limited liability in bankruptcy. Thus, in bankruptcy
proceedings, levels of unsecured debt and equity can be recast as a hierarchy of interests
composed of call options as described in the work of Merton and others.
To simplify this initial description of pricing bankruptcy claims as options, this
paper makes a number of unrealistic, but simplifying, assumptions.
12
Assumptions:
1. All creditor claims are general and unsecured.
2. There are only three classes of creditors: senior debt, junior debt, and equity.
3. The value of the firm is known with certainty and all debt is mature.
4. The standard deviation of the judicial appraisal is known with certainty
5. The length of bankruptcy proceedings is known with certainty.
6. All options expire at, and may only be exercised at, the conclusion of the
bankruptcy proceedings; the call options are European.
7. No dividends are paid during bankruptcy proceedings.
Given these assumptions, creditor claims on companies in bankruptcy proceedings
can be recast and analyzed as combinations of European call options.
Senior Debt:
When a company defaults on its debt, the debt-holders become the owners of the
company. More technically, upon default, senior debt-holders are granted a senior
interest in the assets of the company equal to the outstanding principal and interest on the
debt. This claim is satisfied if a reorganization settlement grants the senior debt-holder
assets equal to this amount. Senior creditors recover dollar for dollar until their claim is
satisfied and then receive nothing beyond that amount. Graphically:
Senior Debt as a Covered Call Position
13
Financially, the senior debt-holders are, in effect, taking a covered call position on
the company’s assets. The senior debt-holders are: (i) long the company’s assets and (ii)
short a European call option on company assets with a strike price equal to the
outstanding senior debt.
Junior Debt:
Junior debt holders are next in line to the company’s assets. After the senior debtholders have been made whole, junior debt-holders are entitled to the asset of the
company in an amount that satisfies this junior claim. Graphically:
Junior Debt as a Bull-Spread Position
Financially, the junior debt holders are, in effect, taking a bull-spread position on
company assets. The junior debt holders are: (i) long a European call option on company
assets with a strike price equal to the outstanding senior debt and (ii) short a European
call option on company assets with a strike price equal to the total amount of outstanding
senior and junior debt.
Equity:
14
Equity holders are still the residual owners of bankrupt companies. After the
senior and junior debt holders’ claims have been satisfied, the equity holders own all of
the remaining assets. Graphically:
Equity as a Call Option
Financially, equity holders are (i) long a European call option on company assets with a
strike price equal to the total amount of outstanding senior and junior debt.
Pricing Options:
Given this view of bankruptcy claims, the financial position of any claimant can
be determined with a straightforward application of the Black-Scholes formula. As set
forth above, six inputs are required to value an option. Here, in this assumed bankruptcy
example, the inputs to the Black Scholes equation are as follows:


S: Current value of the underlying asset. Here, the underlying asset is the
debtor firm and it is assumed to be known and certain. Valuations are
provided for firm values ranging from $1000 to zero, in ten dollar increments.
K: Strike price of the option. Here, the strike price of the option is equal to the
face value of the total amount of debt senior to the claim being valued. Thus
in this case with three classes of claimants, there are two strike prices: (i)
$500, the value of the senior debt and (ii) $750, the combined value of the
senior and junior debt.
15




t: Life to expiration of the option. Here, it is assumed that options expire at
the conclusion of bankruptcy proceedings, which will occur in 6 months (0.5
years).
r = Riskless interest rate corresponding to the life of the option. Here, the
riskless interest rate corresponding to the length of the bankruptcy
proceedings is assumed to be 2%.
 2 : Variance in the ln(value) of the underlying asset. Here, the variance is
driven by the variation in the judicial valuation and assumed to be 25%.
D = dividend yield. Here, D is zero as it has been assumed that cash dividend
payments have been suspended during Chapter 11 proceedings.
With these inputs, the value of senior, junior, and equity claims can be
determined. Figure 6 summarizes the assumptions made in the following example.
Creditor Interests
Claims
Senior
Debt
500
Junior
Debt
250
Equity
Residual
Black Scholes Inputs
Spot Price (Firm Values)
1000
declinig in 10 increments
Strike Prices (Debt Totals)
500 (Sr. Debt)
750 (Sr. Debt + Jr. Debt)
Risk Free Interest Rate
2.00% assumed
Dividend Yield
0% assumed
Standard Deviation of Appraisal
25% assumed
Time (years until end of
proceedings)
0.5 assumed
Summary of Assumptions and Inputs
Figure 6 sets forth the value of options with strike prices at $500 and $750 at firm
values ranging from $1000 to zero. Each of these values is derived using the BlackScholes formula and the assumptions stated above. Calculations and exact values are
attached in the Appendix.
16
34
0
28
0
22
0
16
0
10
0
40
34
0
28
0
22
0
16
0
10
0
40
40
0
46
0
52
0
64
0
58
0
70
0
76
0
88
0
82
0
94
0
600
500
400
300
200
100
0
10
00
Call Value
Call at $500
Firm Value
Call
Call at $750
300
Call Value
250
200
150
100
50
40
0
46
0
52
0
64
0
58
0
70
0
76
0
88
0
82
0
94
0
10
00
0
Firm Value
Call
Expectedly, the value of these calls declines as the firm’s value falls and the value
of the calls do not reach zero until the options are well out of the money.
After determining these option values, one can assess the position of each
claimant by combining the various rights each claimant holds. For example, and
according to the framework set forth above, the senior debtor in this example is
financially (i) long the assets of the firm and (ii) short a European call option with a strike
price equal to the nominal value of the junior debt. Thus, in the case where the firm is
worth $1000: the senior creditor is +1000 and - $504.98 for a total position of 495.02. A
17
summary of each claimant’s position, under a number of different firm values, is
presented below.3
Firm
Value
Senior Debt
Long Firm Short
Option
Call
Based
Position
1000.00
1000.00
900.00
900.00
800.00
800.00
750.00
750.00
700.00
700.00
600.00
500.00
400.00
300.00
250.00
200.00
100.00
10.00
600.00
500.00
400.00
300.00
250.00
200.00
100.00
10.00
504.98
404.99
305.09
255.31
205.96
111.73
-37.58
-4.35
-0.05
0.00
0.00
0.00
0.00
Firm
Value
1000.00
900.00
800.00
750.00
700.00
600.00
500.00
400.00
300.00
250.00
200.00
100.00
10.00
3
See Appendix A for full calculations
Long
Call
500
Junior Debt
Short
Option
Call
Based
750
Position
495.02
504.98
-260.35
244.63
495.01
404.99
-167.59
237.39
494.91
305.09
-87.80
217.29
494.69
255.31
-56.38
198.94
494.04
205.96
-32.33
173.63
488.27
462.42
395.65
299.95
250.00
200.00
100.00
10.00
111.73
37.58
4.35
0.05
0.00
0.00
0.00
0.00
-6.53
-0.47
-0.01
0.00
0.00
0.00
0.00
0.00
105.20
37.11
4.34
0.05
0.00
0.00
0.00
0.00
Equity
Long
Option
Call
Based
500
Position
260.35
167.59
87.80
56.38
32.33
6.53
0.47
0.01
0.00
0.00
0.00
0.00
0.00
260.35
167.59
87.80
56.38
32.33
6.53
0.47
0.01
0.00
0.00
0.00
0.00
0.00
18
Graphically,
Option Based Positions
600
Option Values
500
Senior Debt Option Based
Position
400
Junior Debt Option Based
Position
300
200
Equity Option Based
Position
100
20
90
160
230
300
370
440
510
580
650
720
790
860
930
1000
0
Firm Value
These financial positions are next compared to the positions of claimants under a
regime where absolute priority is strictly enforced. The differences represent deviations
from absolute priority.
Firm
Value
Option
Based
Position
1000
900
800
750
700
600
500
400
300
250
200
100
495.02
495.01
494.91
494.69
494.04
488.27
462.42
395.65
299.95
250.00
200.00
100.00
Senior Debt
Absolute Deviation
Priority
Based
Position
500
500
500
500
500
500
500
400
300
250
200
100
-4.98
-4.99
-5.09
-5.31
-5.96
-11.73
-37.58
-4.35
-0.05
0.00
0.00
0.00
Option
Based
Position
244.63
237.39
217.29
198.94
173.63
105.20
37.11
4.34
0.05
0.00
0.00
0.00
Junior Debt
Absolute Deviation
Priority
Based
Position
250
250
250
250
200
100
0
0
0
0
0
0
-5.37
-12.61
-32.71
-51.06
-26.37
5.20
37.11
4.34
0.05
0.00
0.00
0.00
19
10
10.00
10
0.00
0.00
0
0.00
Equity
Firm
Value
Option
Based
Position
1000
900
800
750
700
600
500
400
300
250
200
100
10
260.35
167.59
87.80
56.38
32.33
6.53
0.47
0.01
0.00
0.00
0.00
0.00
0.00
Deviations from Absolute Priority
Graphically:
Absolute
Priority
Based
Position
250
150
50
0
0
0
0
0
0
0
0
0
0
Deviation
10.35
17.59
37.80
56.38
32.33
6.53
0.47
0.01
0.00
0.00
0.00
0.00
0.00
20
Senior Debt
600
500
Position Value
400
300
200
100
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
0
Firm Value
Option Based Position
Absolute Priority Based Position
Senior Debt
0
10
00
95
0
90
0
85
0
80
0
75
0
70
0
65
0
60
0
55
0
50
0
45
0
40
0
35
0
30
0
25
0
20
0
15
0
10
0
50
0
-5
Deviation
-10
-15
-20
-25
-30
-35
-40
Firm Value
Deviation
When using option-based valuation, the financial position of the senior debtholder
is always worse than it would be under strict absolute priority, i.e. option value is worth
21
less than the face value of the senior debt. Intuitively, this makes sense. In no case can
the senior debtholder recover more than $500; but, in every case there is some chance
that the senior debtholder will recover less. Thus, the expected value of the senior
debtholder’s position is always less than the face value of its claim.
The option-based explanation for this is simple. Senior debtholders are long an asset and
short a call on that same asset at $500. Therefore, at every firm value level, senior
debtholders are giving up a portion of their recovery as determined by the value of the
call which they are short. For example, at $500, senior debtholders are entitled to all of
the $500 of firm value minus the value of a call option at $500. At a firm value of $500,
the difference is $37.58. This point represents the maximum deviation from absolute
priority.
The shape of the deviation graph is a result of the interaction between option
value and absolute priority. The value of the senior creditor’s option based position falls
as the firm value declines. But, the absolute priority based position remains constant
until the face value of the claim is reached and only then begins to fall. Therefore, the
deviation from absolute priority grows steadily until recovery under absolute priority is
determined by available assets. After this point is reached, the value of the absolute
priority position falls in direct proportion to the fall in firm value. During this decline the
deviation shrinks as the senior creditor’s positions under absolute priority and optionbased pricing converge as the firm’s value approaches zero.
Deviation
10
00
95
0
90
0
85
0
80
0
75
0
70
0
65
0
60
0
55
0
50
0
45
0
40
0
35
0
30
0
25
0
20
0
15
0
10
0
-20
-40
-60
Firm Value
Deviation
0
50
Option Based Position
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
Position Value
22
Junior Debt
300
250
200
150
100
50
0
Firm Value
Absolute Priority Based Position
Junior Debt
60
40
20
0
23
Equity
300
250
Position Value
200
150
100
50
50
0
0
100
50
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
0
Firm Value
Option Based Position
Absolute Priority Based Position
Equity
60
Deviation
50
40
30
20
10
10
00
95
0
90
0
85
0
80
0
75
0
70
0
65
0
60
0
55
0
50
0
45
0
40
0
35
0
30
0
25
0
20
0
15
0
10
0
0
Firm Value
Deviation
The position of the equity holder is converse to that of the senior debtholder. At
every firm value, equity holders are in a better position under option-based valuation than
24
they would be under a strict application of absolute priority. As residual owners, equity
claimants hold a call option on the firm in addition to any value they might recover under
absolute priority. Thus, at every firm value, equity holders are better off under optionbased pricing than under a strict application of absolute priority.
The deviations from absolute priority are similarly converse to that of the senior
creditor’s deviations. From a firm value of 0 to a firm value of $750, deviation steadily
grows as the value of the equity option increases – due to the strike price approaching the
spot price – while the position of equity holders under strict absolute priority remains at
zero. Deviation is maximized at $750 because at values above this point equity holders
begin to recover under absolute priority. At values above $750, equity holders begin to
recover under absolute priority. Equity holders’ positions under option based valuation
and absolute priority converge as the firm value continues to rise.
The above graph shows that, at every firm value, deviations among the creditors
average to zero. Value is not created or destroyed depending on the method of valuing
and assigning creditor interests through bankruptcy. All that changes, depending on
method selected, is the way the debtor company is divided.
4.2. Implications of Relaxing Assumptions
In demonstrating the creation of option value, this paper made a number of
simplifying, but unrealistic, assumptions. The mathematical examples and explanations
given in this paper are built upon a number of assumptions which do not hold true in
practice. In comparing bankruptcy claims to publicly traded options, this paper argues
that analogues for each of the six inputs to the Black-Scholes equation for pricing
25
securities could be found for bankruptcy claims. However, the bankruptcy claim inputs
nearly all differ from option inputs in an important respect. The bankruptcy claim inputs
are not known with nearly the same degree of certainty as are those of true options.
This lack of certainty speaks only to the model’s ability to accurately predict the
actual divergences from absolute priority and does not detract from the model’s
usefulness as a partial and theoretical explanation for the prevalence of deviations from
absolute priority in bankruptcy settlements.
The lack of certainty in inputs prevents one from finding a precise option-based
value for the creditors’ position. In any settlement, parties could easily, and in good
faith, reach different conclusions about option value. Depending on the parties’ honest
assessments of: K, the value of the underlying company; σ, the standard deviation of the
underlying company’s value; and T, the time to conclusion of the proceedings, the parties
could come to significantly different conclusions about option value.
Moreover, creditors could use any number of option based analyses to value their
positions. In practice, creditor interests might be better modeled using more sophisticated
tools, such as combinations of capped-style options, but this paper relied on the simplest
option framework possible to simplify the explanation of the creation of option value.
Still, in any case where the parties recognize that a reasonable possibility exists
such that the senior creditors’ claims will not exhaust the debtor’s assets, option value
will be vested in junior creditors. This option value changes the nature of bankruptcy
negotiations. Bankruptcy settlements are largely driven by negotiation among the parties
and option value informs these negotiations. Sophisticated creditors are certainly aware
26
of the dynamics that create option value and will bargain to capture that value. Nothing
in the bankruptcy code calls for a determination of option value. Instead, the parties’
individual, private, assessment of option value will, in part, determine their bargaining
positions. As a bankruptcy plan’s treatment of a creditor’s interest approaches that
creditor’s appraisal of its position, the creditor becomes more likely to approve the plan.
27
Chapter 5.
CONCLUSION
Deviations from absolute priority are simply the result of more accurate
valuations of creditor interests. It is not the face value of claims that are important in
negotiated settlements. Rather, it is their true value; the value best determined through
option pricing that drives settlements.
This paper argues that unsecured claims against a company in Chapter 11 are
comprised of call options on the company’s assets. To the extent this is true, optionpricing formulas provide a quantitative explanation for the deviations from absolute
priority in bankruptcy resolutions.4 These deviations are the simply the difference
between the nominal value of a claimant’s interest and the value of that claimant’s
interest as determined by option pricing tools, especially the Black-Scholes formula.
The Black-Scholes formula reflects the role that asset values, claim values, time,
variance, and interest rates play in determining creditor claim values. It is the position of
this paper that it is these factors, not mere inefficiencies in the bankruptcy process itself,
which drive bankruptcy settlements. Deviations from absolute priority are a consequence
of the call option value junior creditors hold. In negotiated settlements, senior creditors,
in effect, purchase these options from the junior creditors, allowing subordinate creditors
to recover at the expense of senior interests.
To garner the approval of junior creditors, and to avoid the expense and
uncertainty of a judicial appraisal, Chapter 11 plan drafters will offer junior creditors
The structure of this section is inspired by the format of “Pricing the Securities of Companies of Chapter
11: An Options Approach” by William H. White.
4
28
some value. Junior creditors will tend to reject any plan where the value they are offered
is lower than the value – determined through option pricing – that the junior creditor
expects to receive from a judicial valuation. Over time, equilibrium should be reached
between the senior creditor’s right to insist on absolute priority and the junior creditor’s
right to call for a judicial valuation.
Financially, this equilibrium point should occur where the bankruptcy plan
compensates each creditor according to the value it holds. Thus, a junior creditor who is
out of the money, in terms of nominal value, will tend to receive something under the
negotiated plan whenever enough valuation uncertainty exists to give the junior creditor
some chance of recovering under a judicial valuation. Conversely, a senior creditor will
tend to voluntarily surrender its right to insist on priority in distributions whenever
enough valuation uncertainty exists such that the senior creditor’s expected value – based
on option value – is less than the nominal value of it claim. Senior creditors will make
this concession in order to avoid the expense and uncertainty of litigation.
Assuming that negotiation involves only minimal transaction costs, all parties
should prefer to avoid the costs of litigation and instead negotiate a settlement that
reflects their true financial positions. These positions are most accurately determined
using option pricing, not by reference to nominal debt values. This process theoretically
explains the prevalence of deviations from absolute priority.
To explain deviations from absolute priority in terms of valuation uncertainty,
previous scholars have pointed to informational asymmetries between classes of
claimants (note) and deficiencies in the bankruptcy process which hinder the enforcement
29
of senior creditors’ rights (note). While these arguments certainly contribute to the
debate, it is unlikely – given the sophistication and expertise of reorganization
professionals – that these afore-cited inefficiencies can account for the routine divergence
between absolute priority in theory and bankruptcy settlement in practice. This paper,
and the scholarship it rests upon, suggests that it is the option-like structure of bankruptcy
claims themselves that drives, in large part, the empirically observed deviations from
absolute priority.
30
REFERENCES
Baird, Douglas G. and Donald S. Bernstein. (2006), Absolute Priority, Valuation
Uncertainty, and the Reorganization Bargain, Yale Law Journal 115, 1930.
Black and Scholes. (1973), The Pricing of Options and Corporate Liabilities, Journal of
Political Economy (May/June) pg. 7, 637.
Bebchuk, Lucian A. (1988), A New Approach to Corporate Reorganizations, Harvard
Law Review 101,775.
Bonbright, James C. and Milton M. Bergerman. (1928) Two Rival Theories of Priority
Rights of Secured Holders in a Corporate Reorganization. Columbia Law Review 28,
127.
Charitou, Andreas and Lenos Trigeorgis (Draft 2000), Option-Based Bankruptcy
Prediction. Social Science Research Network Electronic Paper Collection:
http://papers.ssrn.com/paper.taf?abstract_id=248709.
Eberhart, Allan C. et. Al., Security Pricing and Deviations from the Absolute Priority
Rule in Bankruptcy Proceedings. 45 J. Fin. 1457 (1990).
Epstein, David G., Steve H. Nickles, & James J. White. Bankruptcy pg 2. West
Publishing Co., St. Paul, Minn., 1993.
Gilson Stuart C., Edith S. Hotchkiss, Richard S. Ruback. (2000). Valuation of Bankrupt
Firms. The Review of Financial Studies 13, 1, 43.
Hillegeist, Stephen A., Elizabeth K. Keating, Donald P. Cram, and Kyle G. Lundstedt.
(2004), Assessing the Probability of Bankruptcy, Review of Accounting Studies 9, 1.
In re All Media Properties, Inc. 5 B.R. 126, 142 n. 5 (Bankr. S.D.Tex.1980), order aff’d,
646 F.2d 193 (5th Cir.1981).
In re AOV industries, Inc., 792 F.2d 1140 (D.C. Cir. 1986).
Kennedy. (1979), The Commencement of a Case Under the New Bankruptcy Law, Wash
& Lee Law Review, 36, 977.
Klee, All You Ever Wanted to Know about Cramdown Under the New Bankruptcy Code,
53 Am. Bankr. L.J. 133.
McDonald, R. (2002). Derivatives Markets. First Edition.
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Merton, R.C. (1973), Theory of Rational Option Pricing, Bell Journal of Economics and
Management Science 4, pp. 141-183.
Merton, R.C. (1974), On the Pricing of Corporate Debt: The Risk Structure of Interest
Rates, Journal of Finance 29, 449.
Merton, R.C. (1977), On the Pricing of Contingent Claims and the Modigliani-Miller
Theorem, Journal of Financial Economics 5, 241.
Schwartz, Alan. (1998), A Contract theory Approach to Business Bankruptcy. Yale Law
Journal 107, 1807.
Vassalou , M. and Y. Xing (2004). Default Risk in Equity Returns, Journal of Finance.
2, 831.
White, William H. (1990). Pricing the Securities of Companies in Chapter 11: An
Options Approach. NYU Stern School of Business.
http://www.williamhwhite.biz/images/WHW_Pricing_Ch_11_Secs__An_Options_Appr.pdf
WORKS CONSULTED
Arzac, Enrique R. (2008). Valuation for Mergers, Buyouts, and Restructurings. John
Wiley & Sons, Inc. New York.
Baker, H. Kent. (2003), Understanding Financial Management: A Practical Guide.
Blackwell Publishing., Malden MA.
Brealey, Richard A., Stewart C. Myers, and Franklin Allen. (2006), Principles of
Corporate Finance. McGraw-Hill Irwin. New York.
Damodaran, Aswath. (2002), Investment Valuation: Tools and Techniques for
Determining the Value of Any Asset. John Wiley & Sons, Inc. New York.
Nickles, Steve H. and David G. Epstein. (2006). Bankruptcy & Related Law. Thompson
West. St. Paul, MN.
32
APPENDIX
Senior Debt
Firm Long Short
Value Firm Call
1000 1000
-505
990 990
-495
980 980
-485
970 970
-475
960 960
-465
950 950
-455
940 940
-445
930 930
-435
920 920
-425
910 910
-415
900 900
-405
890 890
-395
880 880
-385
870 870
-375
860 860
-365
850 850
-355
840 840
-345
830 830
-335
820 820
-325
810 810 -315.1
800 800 -305.1
790 790 -295.1
780 780 -285.2
770 770 -275.2
760 760 -265.2
750 750 -255.3
740 740 -245.4
730 730 -235.5
720 720 -225.6
710 710 -215.8
700 700
-206
Option Based
Position
495.0239391
495.0236685
495.0233238
495.0228851
495.0223271
495.0216179
495.0207173
495.0195748
495.0181265
495.0162925
495.0139725
495.0110409
495.0073407
495.0026762
494.9968034
494.9894196
494.980149
494.9685269
494.9539799
494.935802
494.9131265
494.8848921
494.8498037
494.8062854
494.752426
494.6859162
494.6039758
494.5032715
494.3798229
494.2288992
494.0449038
Absolute
Priority
Based
Position
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
Deviation
-4.976061
-4.976332
-4.976676
-4.977115
-4.977673
-4.978382
-4.979283
-4.980425
-4.981874
-4.983707
-4.986027
-4.988959
-4.992659
-4.997324
-5.003197
-5.01058
-5.019851
-5.031473
-5.04602
-5.064198
-5.086874
-5.115108
-5.150196
-5.193715
-5.247574
-5.314084
-5.396024
-5.496729
-5.620177
-5.771101
-5.955096
33
690
680
670
660
650
640
630
620
610
600
590
580
570
560
550
540
530
520
510
500
490
480
470
460
450
440
430
420
410
400
390
380
370
360
350
340
330
320
310
690 -196.2
680 -186.4
670 -176.8
660 -167.2
650 -157.6
640 -148.2
630 -138.9
620 -129.7
610 -120.6
600 -111.7
590
-103
580 -94.53
570 -86.28
560
-78.3
550 -70.61
540 -63.25
530 -56.24
520 -49.61
510 -43.38
500 -37.58
490 -32.23
480 -27.34
470 -22.92
460 -18.97
450 -15.48
440 -12.44
430
-9.84
420 -7.645
410 -5.827
400
-4.35
390 -3.175
380 -2.262
370 -1.569
360 -1.058
350 -0.691
340 -0.436
330 -0.265
320 -0.155
310 -0.087
493.8212499
493.5502271
493.2228616
492.8287746
492.3560412
491.7910575
491.1184241
490.3208526
489.3791091
488.2720048
486.9764477
485.4675693
483.7189407
481.7028884
479.3909219
476.7542765
473.7645723
470.39458
466.6190774
462.4157681
457.7662241
452.6568039
447.0794882
441.0325706
434.5211394
427.5572918
420.1600344
412.3548417
404.1728704
395.6498575
386.8247609
377.7382305
368.4310204
358.9424633
349.3091267
339.5637528
329.7345508
319.844873
309.9132572
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
500
490
480
470
460
450
440
430
420
410
400
390
380
370
360
350
340
330
320
310
-6.17875
-6.449773
-6.777138
-7.171225
-7.643959
-8.208942
-8.881576
-9.679147
-10.62089
-11.728
-13.02355
-14.53243
-16.28106
-18.29711
-20.60908
-23.24572
-26.23543
-29.60542
-33.38092
-37.58423
-32.23378
-27.3432
-22.92051
-18.96743
-15.47886
-12.44271
-9.839966
-7.645158
-5.82713
-4.350142
-3.175239
-2.261769
-1.56898
-1.057537
-0.690873
-0.436247
-0.265449
-0.155127
-0.086743
34
300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
-0.046
-0.023
-0.011
-0.005
-0.002
-8E-04
-3E-04
-9E-05
-3E-05
-7E-06
-1E-06
-3E-07
-5E-08
-6E-09
-6E-10
-5E-11
-3E-12
-1E-13
-3E-15
-4E-17
-3E-19
-1E-21
-1E-24
-3E-28
-9E-33
-2E-38
-5E-46
-8E-57
-6E-74
-2E-108
299.9537813
289.9766424
279.9888619
269.9950178
259.9979232
249.9991993
239.9997169
229.9999091
219.9999738
209.9999933
199.9999985
189.9999997
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
-0.046219
-0.023358
-0.011138
-0.004982
-0.002077
-0.000801
-0.000283
-9.09E-05
-2.62E-05
-6.69E-06
-1.49E-06
-2.85E-07
-4.56E-08
-5.97E-09
-6.19E-10
-4.9E-11
-2.81E-12
-1.14E-13
0
0
0
0
0
0
0
0
0
0
0
0
35
Junior Debt
Firm Long Short Option Based
Value Call
Call
Position
500
750
1000
990
980
970
960
950
940
930
920
910
900
890
880
870
860
850
840
830
820
810
800
790
780
770
760
750
740
730
720
710
700
690
680
670
660
650
640
630
620
610
600
505
495
485
475
465
455
445
435
425
415
405
395
385
375
365
355
345
335
325
315.1
305.1
295.1
285.2
275.2
265.2
255.3
245.4
235.5
225.6
215.8
206
196.2
186.4
176.8
167.2
157.6
148.2
138.9
129.7
120.6
111.7
-260.4
-250.8
-241.2
-231.7
-222.3
-213
-203.7
-194.5
-185.4
-176.5
-167.6
-158.9
-150.3
-141.8
-133.5
-125.4
-117.4
-109.7
-102.2
-94.87
-87.8
-80.97
-74.41
-68.11
-62.1
-56.38
-50.95
-45.83
-41.01
-36.51
-32.33
-28.45
-24.89
-21.63
-18.66
-15.99
-13.6
-11.47
-9.59
-7.948
-6.525
244.6252637
244.2194935
243.7614182
243.2451436
242.6642592
242.0118179
241.2803202
240.4617041
239.5473396
238.5280322
237.3940347
236.1350687
234.7403585
233.1986778
231.4984105
229.6276282
227.5741836
225.3258232
222.8703178
220.1956127
217.2899975
214.1422947
210.7420663
207.0798374
203.1473346
198.937736
194.4459301
189.6687793
184.605383
179.2573359
173.6289745
167.727606
161.5637133
155.1511276
148.507163
141.6527042
134.6122405
127.4138385
120.0890466
112.6727228
105.2027815
Absolute
Priority
Based
Position
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
250
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
Deviation
-5.374736
-5.780507
-6.238582
-6.754856
-7.335741
-7.988182
-8.71968
-9.538296
-10.45266
-11.47197
-12.60597
-13.86493
-15.25964
-16.80132
-18.50159
-20.37237
-22.42582
-24.67418
-27.12968
-29.80439
-32.71
-35.85771
-39.25793
-42.92016
-46.85267
-51.06226
-45.55407
-40.33122
-35.39462
-30.74266
-26.37103
-22.27239
-18.43629
-14.84887
-11.49284
-8.347296
-5.38776
-2.586161
0.0890466
2.6727228
5.2027815
690
680
670
660
650
640
630
620
610
600
590
580
570
560
550
540
530
520
510
500
490
480
470
460
450
440
430
420
410
400
390
380
370
360
350
340
330
320
310
300
290
280
270
260
28.4511
24.8861
21.626
18.6641
15.9913
13.5967
11.4677
9.5901
7.94817
6.52521
5.3037
4.26559
3.39265
2.66679
2.07033
1.58631
1.19868
0.89256
0.65437
0.47192
0.33445
0.23269
0.15875
0.10607
0.06933
0.04426
0.02756
0.01671
0.00985
0.00563
0.00312
0.00167
0.00086
0.00042
0.0002
9.1E-05
3.9E-05
1.6E-05
6.2E-06
2.2E-06
7.5E-07
2.4E-07
6.8E-08
1.8E-08
28.451144
24.88606
21.626011
18.664062
15.991255
13.596702
11.467737
9.5901009
7.9481681
6.5252137
5.3037009
4.2655899
3.3926542
2.6667938
2.0703349
1.5863051
1.1986756
0.8925606
0.6543708
0.4719151
0.3344518
0.2326905
0.1587493
0.1060742
0.0693281
0.0442591
0.0275573
0.0167071
0.0098453
0.0056285
0.0031151
0.0016653
0.0008578
0.0004246
0.0002013
9.117E-05
3.928E-05
1.604E-05
6.18E-06
2.235E-06
7.55E-07
2.366E-07
6.836E-08
1.806E-08
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
28.45114
24.88606
21.62601
18.66406
15.99125
13.5967
11.46774
9.590101
7.948168
6.525214
5.303701
4.26559
3.392654
2.666794
2.070335
1.586305
1.198676
0.892561
0.654371
0.471915
0.334452
0.23269
0.158749
0.106074
0.069328
0.044259
0.027557
0.016707
0.009845
0.005628
0.003115
0.001665
0.000858
0.000425
0.000201
9.12E-05
3.93E-05
1.6E-05
6.18E-06
2.24E-06
7.55E-07
2.37E-07
6.84E-08
1.81E-08
36
37
Firm
Value
1000
990
980
970
960
950
940
930
920
910
900
890
880
870
860
850
840
830
820
810
800
790
780
770
760
750
740
730
720
710
700
Long
Call 500
260.351
250.757
241.215
231.732
222.313
212.967
203.699
194.519
185.435
176.456
167.592
158.854
150.252
141.799
133.505
125.383
117.446
109.706
102.176
94.8686
87.7969
80.9728
74.4081
68.1139
62.1002
56.3763
50.9501
45.8279
41.0148
36.5138
32.3261
Equity
Option
Absolute
Based
Priority
Position
Based
Position
260.3508
250
250.75684
240
241.21526
230
231.73197
220
222.31341
210
212.96656
200
203.69896
190
194.51872
180
185.43453
170
176.45568
160
167.59199
150
158.85389
140
150.2523
130
141.79865
120
133.50479
110
125.38295
100
117.44567
90
109.70565
80
102.1757
70
94.868585
60
87.796876
50
80.972813
40
74.40813
30
68.113877
20
62.100239
10
56.376348
0
50.950094
0
45.827949
0
41.014794
0
36.513765
0
32.326122
0
Deviation
10.3508
10.75684
11.21526
11.73197
12.31341
12.96656
13.69896
14.51872
15.43453
16.45568
17.59199
18.85389
20.2523
21.79865
23.50479
25.38295
27.44567
29.70565
32.1757
34.86859
37.79688
40.97281
44.40813
48.11388
52.10024
56.37635
50.95009
45.82795
41.01479
36.51376
32.32612
38
240
230
220
210
200
190
180
170
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
3E-04
9E-05
3E-05
7E-06
1E-06
3E-07
5E-08
6E-09
6E-10
5E-11
3E-12
1E-13
3E-15
4E-17
3E-19
1E-21
1E-24
3E-28
9E-33
2E-38
5E-46
8E-57
6E-74
#####
-9E-10
-2E-10
-3E-11
-4E-12
-5E-13
-5E-14
-4E-15
-3E-16
-1E-17
-5E-19
-1E-20
-2E-22
-2E-24
-8E-27
-2E-29
-1E-32
-3E-36
-2E-40
-8E-46
-1E-52
-2E-61
-9E-74
-3E-93
#####
0.000283058
9.08678E-05
2.61891E-05
6.68795E-06
1.49025E-06
2.84549E-07
4.55702E-08
5.96611E-09
6.1907E-10
4.90239E-11
2.82753E-12
1.12028E-13
2.83026E-15
4.14008E-17
3.08614E-19
9.85809E-22
1.05895E-24
2.69057E-28
9.46698E-33
1.94092E-38
5.06476E-46
8.24501E-57
5.9242E-74
1.5995E-108
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0002831
9.087E-05
2.619E-05
6.688E-06
1.49E-06
2.845E-07
4.557E-08
5.966E-09
6.191E-10
4.902E-11
2.828E-12
1.12E-13
2.83E-15
4.14E-17
3.086E-19
9.858E-22
1.059E-24
2.691E-28
9.467E-33
1.941E-38
5.065E-46
8.245E-57
5.924E-74
1.6E-108