# Introduction to Bankruptcy by tue52347

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```									                                                                    RISKY DEBT VALUATION

RISKY DEBT AND THE WACC USING EXCEL
A. Farber
This version: January 2006

This workbook contains several worksheets to illustrate the choice of a capital structure

Riskless debt                    This worksheet illustrates the problem when debt is riskless.
Can be used to illustrate MM 58 and MM 63
State prices                     In this worksheet, we identify options implicit in risky debt in a simple setting.
Valuation is achieved using state prices.
Binomial 1 period                This worksheet uses the binomial model to value a bond with 1 year to maturity
Binomial 4 periods               In this worksheet, we generalize the binomial option model to several periods
Merton                           Here use the Merton model (a binomial model with a large number to time steps)
Leland                           The Merton model does not take into account interest tax shields and costs of bankruptcy.
The Leland model is the first in a recent family to incorporate these dimensions.

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                     Introduction                                              3/1/2011
RISKY DEBT VALUATION

Capital structure - Introduction - Certainty - Perpetuities

Debt                                    5,000 (risk free)                  EBIT               1,500
EBIT                                     1,500 (perpetuity)                Interest             200     ==>      Debt =    5,000
Tax rate                                  34%                              Taxes                442
rD                                          4% (=risk free rate)           Net Income           858     ==>    Equity =     6,600
Beta Asset                                 1.00                                                                     V=     11,600
Mkt risk premium                            6%
rA                                     10.00%

Value                 Exp.Ret.      Beta                       Value                Exp.Ret.   Beta
VU                      9,900.00       85.3%     10.00%        1.00        Equity       6,600.00       56.9%   13.00%     1.50
V(TaxShield)             1,700.00       14.7%     4.00%        0.00        Debt         5,000.00       43.1%    4.00%     0.00
V                       11,600.00      100.0%     9.12%        0.85                     11,600.00     100.0%    9.12%     0.85

EBIAT =EBIT(1-Tc)                         990
WACC                                    8.53%
V                                       11600

This worksheet illustrates the standard approach found in textbooks.
The assumptions underlying the calculations are:
(1) EBIT and debt are constant perpetuities
(2) debt is riskless. Therefore, the cost of debt is equal to the risk-free rate.

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                  Riskless debt                                              3/1/2011
RISKY DEBT VALUATION

The worksheet can be used to illustate both Modigliani Miller 1958 (no taxes) and Modigliani Miller 1963 (corporate taxes)
To illustrate MM58, set the tax rate equal to 0.
Key points to notice:
(1) the value of the company does not change
(2) the beta of equity and the cost of equity increase
(3) the WACC does not change
To illustrate MM63, set the tax rate to a positive rate.
Key points to notice:
(1) the value of the company increases by the value of the tax shield.
(2) the value of the tax shield is equal to TaxRate * Debt
(3) the beta of equity and the cost of equity increase
(4) the WACC decreases
(5) the value of the company can be calculated as:
V = VU + VTS
V = Equity + Debt
V = EBIAT/WACC

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                      Riskless debt                                                      3/1/2011
RISKY DEBT VALUATION

Risky debt - Introduction using state prices
1 period - No taxes
Now                Prosperity Depression               Valuing options: Strike =               70.0
Probabilities                                   0.535      0.465                              Now         Prosperity Depression
Gov Bond                       100.0             105.0     105.0                  Call           34.9        79.2       0.0
V unlevered                    100.0             149.2     67.0                   Put             1.5         0.0       3.0
State prices                                     0.44       0.51
Risk-free rate                    5.0%                                            Put call parity
Mkt risk premium                 6.00%                                            Stock           100.0    149.2       67.0       VU
Put              1.5      0.0        3.0
Debt face value                          70                                                       101.5    149.2       70.0
Now           Prosperity Depression                    Call            34.9     79.2        0.0        Equity
Debt                                65.1    70.0      67.0                        RFBond          66.7     70.0        70.0       Risk-free debt
Equity                              34.9    79.2       0.0                                        101.5    149.2       70.0
100.0
Beta       Exp.Ret.                                         Beta        Exp.Ret.
VU                                 100.0        100%       1.00       11.00%      Equity         34.9        34.9%     2.76        21.58%
Debt            65.1       65.1%     0.06        5.34%
V                                  100.0        100%       1.00       11.00%                     100.0      100.0%     1.00        11.00%

YTM of debt                        7.5%

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                      State prices                                                       3/1/2011
RISKY DEBT VALUATION

Risky debt - 1 period binomial model

VU                   100                  dt                 1
Volatility              40.00%                   u              1.492
Beta asset                  1.00                  d              0.670
Risk-free rate                     5%                  RNProba p      0.462
Mkt risk premium                     6%                                Now               Up     Down
Debt Face value                        70              V             100               149     67
Maturity                          1 (years)      Equity        34.9              79       0
Debt          65.1              70      67
vu            0.44               1       0
vd            0.51               0       1

Value                        Beta      Exp.Ret.                                         Beta     Exp.Ret.
VU                  100.00          100.0%        1.00      11.00%          Equity    34.85       34.9%      2.77      21.59%
Debt       65.15      65.1%      0.06      5.33%
V                   100.00          100.0%        1.00      11.00%                    100.00     100.0%                11.00%

Yield to maturity                     7.45%                              Face value                  70.0
Spread (bp)                             245                              Default probability      53.78% (risk neutral)
Loss|Default                 3.0
Risk-free debt                         66.67                             Expected loss|Default        1.6
Value of put option                     1.52                             Exp Recovery Rate        51.50%
Value of risky debt        65.15 =(FaceValue - Default probability * Loss|D

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                    Binomial 1 period                                                  3/1/2011
RISKY DEBT VALUATION

This worksheet uses a 1-period binomial option pricing model to value risky debt.
The time interval (dt) is 1 year.
The value of the company can either go up or down.
Up State Down State
VU         u*VU        d*VU
u and d are gross returns (=1 + return) in both states.
u and d are related to the volatility of the company:
u=exp(Volatility)        d=1/u
Valuation is achieved using risk-neutral valuation (see Ross Westerfield Jaffe 7th ed. p. 633)
The risk-neutral probability of an up (p) is:
p = (1+rf-d)/(u-d)
Note that this is equivalent to:
p*u+(1-p)*d=1+rf
Using p, the expected (gross) return is equal to the (gross) risk-free rate.
The 1-period risk-neutral valuation formula is:
f = [p*fu + (1-p)*fd]/(1+rf)
f is the current value of any security, fu and fd are the values in the two future states.
[p*fu + (1-p)*fd] is the expected value in a risk neutral world.
f is obtained by discounting this risk-neutral expected value at the risk-free interest rate.
Future values for the equity and the debt are calculated in cells G9:H10.
The equity is a call option on the company: Su = Max(0,Vu-F) and Sd = Max(0,Vd-F)
where Su and Sd are the value of equity in the two states.
For the debt: Bu=Min(Vu,F) and Bd = Min(Vd,F)

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                         Binomial 1 period                                      3/1/2011
RISKY DEBT VALUATION

where Bu and Bd are the values of debt in the two states.
Current values using risk neutral valuation appear in cells (F9:F10)

RISK NEUTRAL VALUATION AND STATE PRICE COMPARED
In this setting, the state price valuation formula is:
f = vu*fu + vd*fd
where vu and vd are state prices.
Comparing the 1-period risk-neutral valuation formula with the state price valuation formula leads to:
vu = p/(1+rf)
vd = (1-p)/(1+rf)
(see cells F11:F12 - state prices are the prices of digital options)

BETA AND EXPECTED RETURN CALCULATIONS
The beta of the equity (J15) is calculated using the following formula:
BetaEquity = BetaAsset * DeltaEquity * (V/Equity)
with
DeltaEquity = (Su-Sd)/(Vu-Vd)
(This is the standard formula for the delta of an option in the binomial model)
Similarly, the beta of the debt (J16) is:
BetaDebt = BetaAsset * DeltaDebt * (V/Debt)
with

Conclusions (same as in the State prices worksheet)
(1) MM 1958 holds - V = VU

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                           Binomial 1 period                                 3/1/2011
RISKY DEBT VALUATION

(2) WACC is constant
(3) Risky debt = Risk-free debt - Put option
The put option is the price that shareholders pay to have the right to default.

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                              Binomial 1 period                           3/1/2011
RISKY DEBT VALUATION

- Default probability * Loss|Default)/(1+rf)

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls        Binomial 1 period   3/1/2011
RISKY DEBT VALUATION

Risky debt - Binomial 4 periods

VU                                    100             dt                  0.50                    Here I illustrate the valuation of a 2-year bond
Volatility                        40.00%              u                  1.327                    using a binomial model with 2 steps per year.
Beta asset                           1.00             d                 0.754                     The calculation are longer but the underlying logic is the sam
Risk-free rate                         5%             rF/period         2.47%
Mkt risk premium                       6%             RNProba p         0.473
Debt Face value                        70
Maturity (year)                         2             Valuation: see below

Value                   Beta      Exp.Ret.                                                 Beta      Exp.Ret.
VU                               100.00     100.0%       1.00       11.00%               Equity     42.47       42.5%        2.02      17.11%
Debt       57.53       57.5%        0.25      6.49%
V                                100.00     100.0%       1.00       11.00%                          100.00     100.0%                  11.00%

Yield to maturity                            10.31%                                    Face value                   70.0
Spread (bp)                                     531                                    Default probability       35.43%
Loss if no recovery          70.0
Risk-free debt                                63.49                                    Expected recovery            63.4
Value of put option                            5.96                                    Expecte Loss|Default         18.6
Exp Recovery Rate         90.61%
Value of risky debt          46.7

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                    Binomial 4 periods                                                            3/1/2011
RISKY DEBT VALUATION

Risky debt Merton Model
Input:
Value of firm                              100                         Delta - N(d1)                      0.8299
Dividend yield                          0.00%                          RN Proba NoDefault N(d2)           0.3779
Volatility                             40.00%
Beta asset                                   1
Mkt risk premium                        5.00%
Face value of debt                          70
Maturity (year)                             10
Interest rate                           5.00%

Value                  Beta       Exp.Ret.                                   Beta      Exp.Ret.
VU                                      100       100.0%      1.00      10.00%         Equity    66.9     66.9%    1.24      11.20%
Debt      33.1     33.1%    0.51       7.57%
V                                       100       100.0%      1.00      10.00%                   100     100.0%    1.00      10.00%

Yield to maturity            7.50%               Face value                           70
Spread (bp)                    250               Default probability             67.10%
Loss if no recovery                  70
Riskless debt                48.52               Expected recovery                37.65
Put                          15.47               Exp.Loss|Default                 32.35
33.05               Exp Recovery Rate               53.78%
Value of risky debt               31.29

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                  Merton                                                    3/1/2011
RISKY DEBT VALUATION

Risky debt - Leland (1994)

Data
Value of unlevered firm                 100                  Level of bankruptcy               VB              13.68
Volatility            30.00%                  PV of \$1 if bankruptcy            pB              0.110
Beta asset                1.00                                                    pB'             0.001
Coupon                   2.00
Risk free rate                 5.00%
Mkt risk premium                  5.00%
Bankruptcy cost                    50%
Corporate tax rate                   35%

Value                  Beta      Exp.Ret                    Value             Beta   Exp.Ret
VU                       100.0     89.5%        1.00      10.00%      Equity         75.4    67.5%     1.31   11.54%
VTS                        12.5    11.2%        0.14       5.68%      Debt           36.4    32.5%     0.11   5.56%
- VBC                       0.8    -0.7%        -1.11     -0.56%
V                         111.7   100.0%        0.92       9.59%      V              111.7   100.0%    0.92   9.59%

Leverage ratio D/VL                   0.33
Yield on debt                       5.50%
Yield spread (bp)                       50

Riskless debt                        40.0
Put option                            3.6

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                                      Leland                                              3/1/2011
RISKY DEBT VALUATION

To EMF 2006 participants: this model illustrates the trade-off theory of the capital structure.
This theory is explained in RWJ 7th ed. Chap 16
The model is complex but you can use this spreadsheet to get a feeling of how the data change the results.

The Leland model is an extension of the Merton model that incorporate tax shields and costs of bankruptcy.
Debt is a perpetuity paying a constant coupon.
Bankruptcy takes place if the value of the assets drops below some level VB (calculated in cell I4)
Bankruptcy costs are expressed as a fraction of the value if bankruptcy takes place.
The value of the levered firm is equal to:
the value of the unlevered firm
+ the present value of the tax shield (VTS)
- the present value of bankruptcy costs

a4cce477-5a0b-4c85-877c-90cc7a6dd40a.xls                          Leland                                               3/1/2011

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