E4731: IEOR Credit Risk and Credit Derivatives
Lecture 04: Structural models of default.
Notes originally written by Prof. Rama Cont
Instructor: Xuedong He
Spring, 2012
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Capital structure of a firm
The structural approach to default modeling was initiated by [Black and Scholes, 1973] and [Merton, 1973] . The starting point of this approach is the capital structure of the firm. A firm can raise capital either by issuing equity (stocks) or by issuing debt (bonds). The assets of the firm are divided into debt and equity.
Vt : total assets of the firm St : equity Dt : value of debt Equity + Debt = St + Dt = Vt .
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Lehman Brothers stock price in 2008
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Enron stock price 2000-2001
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Winstar Communication before default in April 2001
Moody's KMV]
[source:
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Default defined in terms of capital structure
Default happens when assets are not sufficient to pay back debt. Debt: modeled as zero-coupon bond with nominal L and maturity T Default at t = T if firm is unable to pay back debt: VT < L. Default only happens at maturity T : = T 1VT <L + 1VT L Payoff to bondholders: DT = min(VT , L) = L - max(L - VT , 0) Shareholders get what is left after debt is paid back: ST = max(VT - L, 0)
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Default defined in terms of capital structure (Cont'd)
The above setting is the capital structure in Merton (1974) model Equity behaves like a call option on Vt with maturity T and strike L Bond behaves like a portfolio of a default-free bond of nominal L and a short position in put option on Vt with maturity T and strike L. Risk-neutral pricing (assuming the short rates are independent of the asset value) St = B(t, T )EQ [max(VT - L, 0) | Ft ] Dt = B(t, T )EQ [L - max(L - VT , 0) | Ft ] = Vt - St .
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Merton (1974) model
[Merton, 1974] assumes
constant short rate (interest rate) r, so B(t, T ) = exp[-r(T - t)]. Black Scholes dynamics for asset value (under Q): dVt = rVt dt + Vt dWt .
Therefore, St = CBS (t, Vt , T, L, ) Dt = Vt - CBS (t, Vt , T, L, ) = LB(t, T ) - PBS (t, Vt , T, L, ) where CBS (t, Vt , T, L, ) and PBS (t, Vt , T, L, ) are call price and put price in standard Black-Scholes formulae.
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Black Scholes formula
If the interest rate is r, then a call option with strike K and maturity T is valued at CBS (t, Vt , T, K, ) where CBS (t, V, T, K, ) = V N (d1 ) - KB(t, T )N (d2 )
2 V ln B(t,T )K + (T - t) ln V + (r + )(T - t) 2 2 = d1 = K T -t T -t 2
V ln B(t,T )K - (T - t) 2 d2 = d1 - T - t = T -t u 1 z2 N (u) = exp(- )dz 2 2 -
2
By put-call parity, PBS (t, V, T, K, ) = KB(t, T )N (-d2 ) - V N (-d1 )
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Sensitivity to parameters
The value of the risky debt is increasing in V increasing in L decreasing in r decreasing in time-to-maturity decreasing in volatility
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Credit spread
Define the continuously compounded yield y(t, T ) of the risk bond by L 1 ln Dt = exp[-y(t, T )(T - t)]L, i.e., y(t, T ) = T - t Dt y(t, T ) is a promised yield Define the continuously compounded credit spread by s(t, T ) := y(t, T ) - r In Merton (1974) model, -1 s(t, T ) = ln [N (d2 ) + N (-d1 )/ t ] T -t where LB(t, T ) t := V (t) is nominal debt ratio. Thus, the credit spread is a function of nominal debt ratio
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70
60
50 Credit spread (bps)
40
30
20
10
0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. Asset volatility is fixed at 20%. The nominal debt ratio t is fixed at 0.5.
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300
250
Credit spread (bps)
200
150
100
50
0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. Asset volatility is fixed at 20%. The nominal debt ratio t is fixed at 0.85.
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2000
Credit spread (bps)
1500
1000
500
0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. Asset volatility is fixed at 20%. The nominal debt ratio t is fixed at 1.1.
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14
12
10 Credit spread (bps)
8
6
4
2
0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. The face value of the debt is 100. Asset volatility is fixed at 20% and the riskless interest rate is equal to 5%. The asset value Vt is fixed at 200.
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40 35 30 Credit spread (bps) 25 20 15 10 5 0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. The face value of the debt is 100. Asset volatility is fixed at 20% and the riskless interest rate is equal to 5%. The asset value Vt is fixed at 150.
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2000
Credit spread (bps)
1500
1000
500
0
0
2
4 6 Time to maturity
8
10
Figure: Credit spreads as a function of time to maturity in a Merton model. The face value of the debt is 100. Asset volatility is fixed at 20% and the riskless interest rate is equal to 5%. The asset value Vt is fixed at 90.
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Term structure of term credit spreads
In the Merton model the term structure of term credit spreads can be increasing: for good quality credit decreasing: for distressed firms increase to a maximum and then decrease: leveraged case
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250
200
Credit spread (bps)
150
100
50
0 0.1
0.15
0.2
0.25
0.3 Volatility
0.35
0.4
0.45
0.5
Figure: Credit spreads as a function of volatility in a Merton model. The time to maturity is fixed at 1yr and the riskless interest rate is equal to 5%. The face value of the debt is 100 and the asset value is 200.
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20 18 16 Credit spread (bps) 14 12 10 8 6 4
0
2
4 6 Interest Rate (%)
8
10
Figure: Credit spreads as a function of interest rate in a Merton model. The time to maturity is fixed at 1yr and the volatility is equal to 20%. The face value of the debt is 100 and the asset value is 150.
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400 350
Value of asset, equity, and bond
3000 Asset Equity Bond
2500
300 250 200 150 100 500 50 0 0
Credit spread (bps)
2000
1500
1000
0
1
2 Time
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5
0
1
2 Time
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Figure: Simulated credit spread. The maturity is 5yr and the volatility is equal to 20%. The interest rate is 5%. The face value of the debt is 100 and the initial asset value is 100.
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Behavior of short term credit spreads
In the Merton model the credit spread has a peculiar behavior for short maturities: s(t, T ) = Vt -1 ln N (d2 ) + N (-d1 ) , T -t LB(t, T ) as t T
with Vt to be fixed, i.e., Vt V If L < V , then s(t, T ) 0, If L > V , then s(t, T ) , as t T as t T
Even if the value of the asset follows a general diffusion process, we still have the same behavior of short term credit spreads
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Recovery rate
In the Merton model the recovery rate of debtholders is random: defined in terms of nominal value it is R= VT L
The (risk-neutral) expected recovery rate EQ (R) is given by EQ VT Vt N (-d1 ) EQ [VT 1VT <L |Ft ] VT < L, Ft = = L LN (-d2 ) LB(t, T ) N (-d2 )
The value of the risky debt can be written as Dt = LB(t, T ) 1 - Q(default) 1 - EQ (R) Question: what is the expected recovery rate under the objective probability?
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Calibration to equity values
To use the model, we need calibrate its parameters r is easy to calibrate to market data Maturity T and face value L are not easy to handle, because the real capital structure is far more complicated than assumed in the model. At the moment, let us accept this simple model and assume we are able to extract T and L from the balance sheet. Volatility is not observed and thus need to be calibrated Unlike option pricing, the state process here value of asset V is not observed either. Therefore, both volatility and current value of asset V need to be calibrated to observed data
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Calibration to equity values (Cont'd)
Stock value is observed, so St = CBS (t, Vt , T, L, ) is one equation used in calibration Second equation is not easy. There are two possible ways. Method 1 use stock volatility. Applying It^ formula, we have the equity volatility o S (t) = BS (t, Vt ) Vt St
Use standard method to estimate the stock volatility, then we have two equations for two unknowns Drawback: the equity volatility is not constant, so the estimation might not be justified
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Calibration to equity values (Cont'd)
Method 2 Use historical data of stock price and St = CBS (t, Vt , T, L, ) to update by iteration Suppose we have a sequence of stock price St0 , St1 , . . . , StN . Suppose in the n-th step, we have an estimate of the volatility of asset value: n . In the n + 1-th step:
1 Use n and the stock prices to invert the Black-Scholes formula, deriving a sequence of implied asset value Vtn+1 , Vtn+1 , . . . , Vtn+1 0 1 N Use Vtn+1 , Vtn+1 , . . . , Vtn+1 to estimate , assuming Vt is a 0 1 N geometric Brownian motion. Denote by n+1 the estimate Update n by n+1
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Estimate Volatility
Assume St follows the geometric Brownian motion with , volatility i.e., dSt = St dt + St dWt As a result, ln St is a drifted Brownian motion, i.e., ln St = ln S0 + - 2 2 t + Wt
As a result, ln St has independent increments and is normally distributed, i.e., 2 - (t2 - t1 ), 2 (t2 - t1 ) ln St2 - ln St1 N 2
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Estimate Volatility (Cont'd)
Suppose we observe a series of samples Si at time ti , i = 0, 1, . . . , n with ti - ti-1 = t, i = 1, . . . , n Let Xi := ln Sti - ln Sti-1 , i = 1, . . . , n. Then, Xi 's are i.i.d and normally distributed with mean - variance 2 t. Thus, the sample variance 2 := 1 (n - 1)t
n 2 2
t and
(Xi - X)2
i=1 1 n n i=1 Xi
is an unbiased estimate of 2 , where X := sample mean.
is the
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Calibration to credit spreads
Alternatively we can try to pick parameters to match observed the credit spread(s) or risk-neutral default probabilities stripped from credit default swaps to Q(VT < L) = N (-d2 ). Some practical difficulties: Given the strange behavior of Merton spreads at short maturities, matching actual spreads can require unrealistic values of volatility.
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Merton model: summary
Structural models of default link default probabilities and recovery rates to the firms capital structure define default as an endogenous event link the value of risky bonds to the value of a stock provide a way to explain default risk in terms of fundamentals of the firm define a unified framework for valuation of equity and credit-linked instruments and hybrids
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Merton model: drawbacks
Short term credit spreads are too low. Default is assumed to be a predictable event, given asset value. What about Enron, Tyco, WorldCom, Parmalat,..? Default can only occur at one date (maturity of debt). Capital structure is unrealistically simple
Only one debt No coupon payment
Calibration and computation are not easy in general Therefore, the market standard is to use another approach: reduced-form approach. We will study this approach in next few lectures.
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Merton model: extensions
1 Default barrier models
Default time := inf {t [0, T ] | Vt C} where C is an exogenously given default boundary When default happens, the bondholders take over the company. The value then is C1 ( ). For example, C1 ( ) = LR, where R is recovery rate. If there is no default before T , the bondholders receive min(VT , L) at maturity. The payoff of the bond is similar to that of a barrier option. Thus, the machinery of option pricing can be borrowed If the asset value process Vt is a diffusion, then we still have the same behavior of spreads at short maturities as in Merton model.
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Merton model: extensions (Cont'd)
2 Unexpected default
jumps in asset value: d ln Vt = rdt + dWt + hd(Nt - t) where Nt is a Poisson process with intensity Unknown default barrier: := inf {t [0, T ] | Vt C} where the default barrier C is an unknown random variable. Both these two models can generate nonzero short term credit spreads, because the default time is unpredictable now.
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Merton model: extensions (Cont'd)
3 Including coupons. Consider a coupon bond with one coupon L1 has to be paid at date t1 and face value L has to be paid at T . For t > t1 , if the firm is still alive and the assets are worth Vt , we can value the bond using standard Merton model Dt = Vt - CBS (t, Vt , T, L, ) At t1 , the firm has to pay the coupon, otherwise it defaults. The situation here is complicated and it critically depends on the assumptions we make on what equity oweners, who control the firm, are allowed to do with the firm's assets.
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Merton model: extensions (Cont'd)
3.1 First, assume that equity owners are not allowed to use the firm's assets to pay debt. Then they have to finance the debt payment either by paying "out of their own pockets" or by issuing new equity to finance the coupon payment. These two options actually make no difference. (Here we assume the option to issue new debt is not considered.) Paying the coupon will leave them with equity worth CBS (t1 , Vt1 , T, L, ). Therefore, the shareholders will pay the coupon only if L1 < CBS (t1 , Vt1 , T, L, ). Otherwise, they will default. This gives the equity value and bond value immediately before the coupon payment date t1 . For any t < t1 , risk-neutral pricing can be applied by considering t1 to be the terminal date.
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Merton model: extensions (Cont'd)
3.2 Second, assume asset sales are allowed. We consider what happens at t1 . If assets are worth more than L1 , it is never optimal to default, since this leaves them with 0. In this case, the shareholders have three options: selling assets, paying "out of their own pockets", and issuing new equity to finance the coupon payment. It can be shown that the shareholders will choose to sell assets. The intuition is that the first option is an expense to both shareholders and bondholders and the second and third option is an expense to shareholders only. If assets are worth less than L1 , then shareholders will default. Then, we have the corresponding equity value and bond value at t1 Anything before t1 can be valued by risk-neutral pricing.
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Merton model: extensions (Cont'd)
The assumptions we make on asset sales are critical for our valuation Even if there is only one coupon, the valuation is not easy. The payoff is similar to that of a compounded option. With multiple coupon payments, the valuation is even much more complicated
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Default Probability
A practical use of structural model: KMV model In Merton model, Risk-neutral default probability is Q( = T ) = Q(VT < L) = N - ln(V0 /L) + (r - 1 2 )T 2 T
We are interested in objective default probability, i.e., under true probability measure P. Assume under P, asset price follows 1 Vt = V0 exp ( - 2 )t + Wt , 2 Then the default probability under P is P( = T ) = P(VT < L) = N - ln(V0 /L) + ( - 1 2 )T 2 T
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t 0.
Default Probability (Cont'd)
We could define ln V0 - ln L + ( - 1 2 )T 2 T as a measure for "distance to default" (DDMerton ). If we can compute DDMerton , then we can map it to default probability via some (decreasing) function f (). In Merton model, the function is N (-) The is essentially the philosophy behind Moody's KMV model
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Moody's KMV
The measure for distance to default actually used in Moody's KMV is DD = V -D V
where V is the current value of asset, is the annual volatility of value of asset, and D is default point Then, DD is mapped to the probability of subsequent default over a horizon of 1 year, i.e., Annual probability of default = f (DD) for some function f . The computed probability of default is called expected default frequency (EDF) in Moody's KMV.
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Expected Default Frequency
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Distance to default
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Moody's KMV (Cont'd)
The measure used in Moody's KMV does not include the drift of the assets Two reasons
Though the drift affects the default probability greatly for medium to long maturities, the effect is relatively small in short term, e.g., 1 year. It seems that the volatility is the main determinant in short term. It is very hard to estimate the drift accurately, so adding it to the model adds almost no information
The default point used in Moody's KMV is D = L1 + L2 2
where L1 is the liabilities of the firm in short term, and L2 is the liabilities in long term.
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Moody's KMV (Cont'd)
If we know the mapping f (), we can compute the EDF of a specific firm as follows: 1 Extract information from the balance sheet and compute the default point D 2 Observe the stock price from the market and estimate the current value of asset V and its volatility (Method 2) 3 Compute DD 4 Map DD to EDF via f () Of course, we need estimate f () using historical data 1 Compute DD for each firm in the data 2 Group DDs into small intervals. For instance, one interval could be [0.15, 0.25] 3 Use the default frequency within each bucket as the estimate of EDF. For instance, the default frequency of the firms whose DDs fall into [0.15, 0.25] could be regarded as EDF for DD = 0.2, i.e., f (0.2) 4 Produce an empirical curve of f ()
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Further reading
For structural approach, one can refer to Chapter 2, [Lando, 2004] For Moody's KMV model, one can refer to [Crosbie and Bohn, 2003]
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References
Black, F. and Scholes, M. (1973). The pricing of options and coporate liabilities. Journal of Political Economy, 81(3):637654. Crosbie, P. and Bohn, J. (2003). Modeling Default Risk. Lando, D. (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press, Princeton. Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1):141183. Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2):449470.
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