# Mm Stock Beta Equity Return

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Value of company

Bonds     Warrants     Bonds + warrants
Solvay Business School                         Risky debt                                      A.Farber
Introduction

Risky debt valuation - Binomial model
Professor André Farber
Revised May 2003

This worksheet illustrates the valuation of a risky zero-coupon based on a three-step binomial
version of Merton's model.

An all-equity firm wants to raise some amount of external financing (the proceed of the issue)
to finance an investment project. It considers issuing a 3-year zero-coupon. You are asked to
analyze the term of the issue using a binomial model with 1 step per year.

You are given the following data:
Market data: risk-free interest rate , market risk premium
Company data: number of shares outstanding, current stock price, standard deviation of
returns (volatility), beta asset
Zero-coupon: Face value, Maturity (3 years), Issue price

- to calculate the market values of the equity and of the debt following the issue;
- to compute the expected return on equity and the expected return on debt
Solvay Business School    Risky debt    A.Farber
Introduction

the issue)
Solvay Business School                          Risky debt                                           Page 4/11
Data and questions

Risky debt valuation using the binomial model
Professor André Farber

Basic data:
Number shares                           500
Price per share                          25
Volatility                             40%
Beta Asset                              1.2
Riskless rate                            4%
Zero-coupon
Face value of issue                  12,000
Maturity                                 3
Proceed                              10,000

To analyze this issue, use a binomial model with 1 step per year.

1. Build a three-step binomial tree for the value of company

We will first calculate the market values of debt and equity.
Remember that to achieve this, we apply the risk-neutral valuation principle.
We assume that we are in a risk-neutral world. This is done by using risk-neutral probabilities.

2. Calculate the value of the debt and of the equity at maturity
3. Based on the risk-neutral valuation principle, compute the following:
- the market value of the debt
- the market value of the equity
- the market value of the option to default
4. Is the issue properly priced? Who gains or looses from any mispricing?

The next questions analyze expected returns on the equity and on the debt.
To do so, it is useful to move back to the "real" world and to calculate the "true" probabilities.
These true probabilities are obtained in two steps.
Step 1: Based on the CAPM, compute the expected return on equity of the unlevered firm (ra)
Step 2: Compute the true probability (q) as the solution of: q*u + (1-q) * d = 1+ra

5. Calculate the expected return on the equity of the levered firm.
Two differents methods leading to the same result:
a) To use the CAPM, proceed as follow:
- calculate the delta of equity
- calculate the beta of equity for the levered firm
Solvay Business School                          Risky debt                                         Page 5/11
Data and questions
- use CAPM equation to get the expected return on equity
b) Direct calculation based on true probabilities:
- calculate the realized returns in the up and down cases;
- calculate the expected return using the true probabilities.

6. Calculate the expected return on the debt of the levered firm.
Once again, two different methods lead to the same result:
a) View the risky debt as a portfolio composed of risk-free debt and a (short) put option
Step 1: calculate the expected return on the put option
- calculate the delta of the put (using the put-call parity relationship or based on the binomial tree)
-calculate the beta of the put
-use the CAPM equation to get the expected return on the put option
Step 2: calculate the expected return on the risky debt as a weighted average of returns on
the risk-free debt and on the put option
b) Direct calculation based on true probabilities:
- calculate the realized returns in the up and down cases;
- calculate the expected return using the true probabilities.

7. The final step in our analysis is to calculate the weighted average cost of capital.
This a way to check whether previous results are correct.
MM 1958 applies in this model: the WACC is independent of the capital structure.
Solvay Business School       Risky debt       Page 6/11
Data and questions
Solvay Business School       Risky debt       Page 7/11
Data and questions

binomial tree)
Solvay Business School                                 Risky debt                                                 Page 8/11
Solution
Risky debt valuation
DATA                                                       SOLUTION (summary - see below for details)
Number shares                       500                    Initial value of equity 12,500
Price per share                   25.00                    Value of company        22,500
Volatility                      40.00%                     Equity                  12,605
Beta Asset                         1.20                    Debt                     9,895
Market risk premium                 5%                     Yield to maturity       6.64%
Riskless rate                    4.00%
Zero-coupon                                                Interest rate on issue     6.27%
Face value of issue              12,000                    Expected return on equity 13.86%
Maturity                              3                    Expected return on debt    5.08%
Proceed                          10,000                    WACC                      10.00%

Parameters of binomial model (1 step per year)
u=       1.4918
d=      0.6703
pneutral =     0.450

Time               0             1              2            3            Risk neutral probabilities
1. VALUE OF COMPANY                                                            # paths     Proba/path    Probability
74,703              1          9.11%          9.11%
50,075
33,566                        33,566              3         11.14%          33.41%
22,500                       22,500
15,082                        15,082              3         13.61%          40.84%
10,110
6,777              1         16.64%          16.64%

Values at maturity
VALUE OF ZERO-COUPON BOND                                                Riskless debt           - Put = Risky debt     Equity
12,000         12,000                0        12,000     62,703
11,538
11,095                        12,000         12,000                0        12,000     21,566
9,895                       11,538
9,634                       12,000         12,000                0        12,000      3,082
8,776
6,777         12,000           5,223          6,777         0

Risk-neutral expected future values                                            12,000             869         11,131     14,178
Discount factor (using the risk-free interest rate)                             0.889           0.889          0.889      0.889
3. Market values                                                               10,668             773          9,895     12,605

2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011
Solvay Business School                        Risky debt                                      Page 9/11
Solution
Notes:
Market value of the company                                       22,500
Value before issue                                  12,500
+ New issue                                           10,000
Given the term of the issue of the zero coupon
Market value of equity                                12,605
Market value of debt                                   9,895
Yield to maturity (this is not an expected return)    6.64%

The value of the option to bankrupt is the value of the put option
Market value of riskless debt                          10,668
Market value of risky debt                              9,895
Value of the put option                                   773

4. Note: the bond issue in overvalued
Issue price                                           10,000
Market value                                           9,895
Difference                                               105
The mispricing has an impact on the market value of equity:
Market value of equity after issue                    12,605           =
Initial market value of equity                        12,500
Difference                                               105

5. Expected return on equity
Beta asset                                                1.20
Risk premium on the market portfolio                   5.00%
Expected return on unlevered firm (CAPM)              10.00%
Proba of up movement (true proba) q                     0.523 > risk-neutral p = .450
At time 1:
V          E                 D
Up           33,566     22,471            11,095
Down         15,082      5,448             9,634

Delta of equity (viewed as a call option)               0.92 Reminder : delta = (Cu-Cd)/(uS-dS)
Omega of equity = delta *V/E                            1.64
Beta equity = Omega * Beta asset                        1.97
Expected return on equity                           13.86% using CAPM
Other calculation leading to same result:
Return on equity if up                               78.28%
Return on equity if down                            -56.78%
Expected return on equity                           13.86%
Note: the expected return on equity will vary over time as the value of the company changes.

2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011
Solvay Business School                     Risky debt                                     Page 10/11
Solution
6. Expected return on debt

At time 1:                        Riskless
V      debt            Put
Up            33,566   11,095               0
Down          15,082   11,095          1,461
Delta put                                            -0.08 =delta call - 1 or (Pu-Pd)/(uS-dS)
Omega of the put option = Delta*V/Put                -2.30
Beta of the put option = Omega * Beta asset          -2.76
Expected return on the put option                 -9.81%
Check
Return on put if up                            -100.00% per period
Return on put if down                            89.09% per period
Expected return                                  -9.81%
Beta debt                                            0.22
Expected return on the risky debt                 5.08%
Details of calculation
Market     Fraction Expected Beta
Value      invested return
Riskless debt               10,668       1.078    4.00%        0.00
Put                           -773     -0.078    -9.81%       -2.76
Risky debt                   9,895       1.000    5.08%        0.22

Direct calculation.
Current value of risky debt               9,895
At time 1:                        V        Debt Return
Up               33,566      11,095    12.12%
Down             15,082       9,634     -2.64%
Expected return on the risky debt                   5.08%
Note: the expected return on the risky debt varies over time
7. Weighted average cost of capital
Market     Expected
value     return
Company                                             22,500              100.00%
Equity                                              12,605    13.86%     56.02%
Debt                                                  9,895    5.08%     43.98%
Cost of debt

Weighted average cost of capital                  10.00%

Reminder : by CAPM using beta asset              10.00%
This had to be expected since we are in a MM world

2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011
Solvay Business School                        Risky debt                                        Page 11/11
Solution

Here we illustrate the evolution over time of the beta equity, cost of equity, beta debt, cost of debt
0           1           2
Beta equity
1.56
1.79
1.97                    2.46
2.58
3.38
Expected return on equity
11.80%
12.96%
13.86%                   16.32%
16.91%
20.88%
Beta debt
0.00
0.00
0.22                    0.00
0.42
0.87
Expected return on risky debt
4.00%
4.00%
5.08%                    4.00%
6.09%
8.35%

2fab2ba1-cdb5-49cf-a152-becffa58048d.xls 4/14/2011

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