Brennan Schwartz Corporate Bond Model

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Brennan-Schwartz Corporate Bond Model Douglas Cohen 10/14/2008

description

Technical presentation on an implementation of the Brennan-Schwartz Corporate Bond Model.

Transcript of Brennan Schwartz Corporate Bond Model

Page 1: Brennan Schwartz Corporate Bond Model

Brennan-Schwartz

Corporate Bond Model

Douglas Cohen

10/14/2008

Page 2: Brennan Schwartz Corporate Bond Model

• Firm Value Process• dV = μ V dt + σv V dZV

• V is the value of the firm’s assets• μ r , short-term risk-free rate• dt ≈ Δt, where Δt is for a one month period

• σv is asset volatility• KMV, Pete’s Expected Asset Volatility, etc

• dZV is a wiener process weighted by σv

• dZV = N(0,1) (Δt)1/2

• Change in firm value equals risk-neutral growth component plus (minus) the random drift component

Page 3: Brennan Schwartz Corporate Bond Model

• Default Events– Balance Sheet Default

• Asset to liability ratio falls below 0.8

– Cash Flow Default• Annual interest expense to liability ratio falls below

0.10• Asset to liability ratio falls below 1.0

– Corporate Bond Valuation• If an issuer defaults before bond maturity

– Bond holder receives recovery rate times par value

• Else– Bond holder receives all scheduled payments

Page 4: Brennan Schwartz Corporate Bond Model

• Interest Expense & Dividends• dV(t) = dAssetValue(t) - Outflows(t)

• Outflows(t) = Interest Expense + Dividends• Rolling Debt Feature

• Total debt = fixed rate debt + floating debt• Company begins with a quantity of fixed rate debt that

has maturities from 1 year to 7 year• KMV liability data

• As debt matures, fixed rate debt becomes floating debt• The total quantity of debt held by the firm never changes

throughout the simulation• Other structural models vary the quantity of debt by

using a mean-reverting target leverage ratio process• Companies increase (decrease) debt holdings when

asset value increases (decreases)

Page 5: Brennan Schwartz Corporate Bond Model

• Interest Rates• Fixed rate

• Taken from Citi proxy bond data• Used for all the debt at onset of the simulation• By year 8, all the debt becomes floating

• Floating • Short-rate + Spread• Spread is a function of A/L ratio• Short-rate is the short-term risk-free rate produced

by the two-factor interest rate model

• Dividends• Planning to include this feature • dV = (μ – ς) V dt + σv V dZV

• ς is the Dividend Payout Rate• Deducted from the growth rate, as opposed to

interest rate expense that is deducted from the change in asset value

Page 6: Brennan Schwartz Corporate Bond Model

Refinance Spread as a function of A/L Ratio

0

200

400

600

800

1000

1200

0 1 2 3 4 5 6

Assets/Liabilities

Sp

read

s

Page 7: Brennan Schwartz Corporate Bond Model

• Two-Factor Interest Rate Model • drt = (θ + u - αrt-1) dt + σrdZr

• Mean-reverting process, i.e. u

• du = - βudt + σudZu

• Takes forward rates into account, i.e. θ

• Wiener process weighted by interest rate volatility, i.e. σrdZr

• Provides a source of long-term volatility in the model

• Contributes to asset value process, interest expense and discounting of cash flows

• High (Low) short-term rates provide large (small) risk-free asset value growth

• High (Low) short-term rates provide large (small) monthly interest rate expense

• High (Low) short-term rates provide large (small) discount factors for bond’s scheduled cash flows

• Dominating effects vary from bond to bond

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• Interest rate and asset value correlation• dZV = ρrv dZr + (1 – ρrv)1/2 dZ

• ρrv = dZV dZr

• Allows the model to correlate interest rate volatility with asset value volatility

• Outside studies have found that local sensitivities to this parameter have little significance• Preliminary results seem to confirm this

conjecture

Page 9: Brennan Schwartz Corporate Bond Model

Example A/L Ratio over Time

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

9.000

10.000

05/28/05 10/10/06 02/22/08 07/06/09 11/18/10 04/01/12 08/14/13 12/27/14 05/10/16

Date

A/L

Rat

io

Receive All Cash

Incur Principal Loss

Initial A/L Ratio = 1.911

Initial Issuer Interest Expense = $373,567,482

LGD