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Technical presentation on an implementation of the Brennan-Schwartz Corporate Bond Model.
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Brennan-Schwartz
Corporate Bond Model
Douglas Cohen
10/14/2008
• 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
• 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
• 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)
• 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
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
• 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
• 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
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