1、Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,1,Credit Risk Management,Elements of Financial Risk Management Chapter 12 Peter Christoffersen,Overview,Credit risk can be defined as the risk of loss due to a counterpartys failure to honor an obligation in part or in
2、 full Credit risk can take several forms For banks credit risk arises fundamentally through its lending activities Nonbank corporations that provide short-term credit to their debtors face credit risk as well Investors who hold a portfolio of corporate bonds or distressed equities need to get a hand
3、le on the default probability of the assets in their portfolio,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,2,Overview,Default risk, a key element of credit risk, introduces an important source of nonnormality into the portfolio Credit risk can also arise in the
4、form of counterparty default in a derivatives transaction The chapter is structured as follows: Section 2 provides a few stylized facts on corporate defaults Section 3 develops a model for understanding the effect on corporate debt and equity values of corporate default.,Elements of Financial Risk M
5、anagement Second Edition 2012 by Peter Christoffersen,3,Overview,Default risk will have an important effect on how corporate debt is priced, but default risk will also impact the equity price. The model will help us understand which factors drive corporate default risk Section 4 builds on single-fir
6、m model from Section 3 to develop a portfolio model of default risk. The model and its extensions provide a framework for computing credit Value-at-Risk Section 5 discusses further issues in credit risk including recovery rates, measuring credit quality through ratings, and measuring default risk us
7、ing credit default swaps,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,4,Figure 12.1: Exposure to Counterparty Default Risk,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,5,Brief History of Corporate Defaults,Credit rating agenci
8、es such as Moodys and Standard & Poors maintain databases of corporate defaults through time In Moodys definition corporate default is triggered by one of three events: a missed or delayed interest or principal payment a bankruptcy filing a distressed exchange where old debt is exchanged for new deb
9、t that represents a smaller obligation for the borrower,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,6,Table 12.1: Largest Moodys-Rated Defaults,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,7,Figure 12.2: Annual Average Corpor
10、ate Default Rates for Speculative Grade Firms, 1983-2010,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,8,Figure 12.2: Annual Average Corporate Default Rates for Speculative Grade Firms, 1983-2010,Elements of Financial Risk Management Second Edition 2012 by Peter C
11、hristoffersen,9,Modeling Corporate Default,The Merton model of corporate default provides important insights into the valuation of equity and debt when the probability of default is nontrivial The model also helps us understand which factors affect the default probability Consider the situation wher
12、e we are exposed to the risk that a particular firm defaults This risk could arise from the fact that we own stock in the firm, or it could be that we have lent the firm cash, or it could be because the firm is a counterparty in a derivative transaction with us,Elements of Financial Risk Management
13、Second Edition 2012 by Peter Christoffersen,10,Modeling Corporate Default,We would like to use observed stock price on the firm to assess the probability of the firm defaulting Assume that the balance sheet of the company in question is of a particularly simple form The firm is financed with debt an
14、d equity and all the debt expires at time t+T The face value of the debt is D and it is fixed. The future asset value of the firm, At+T , is uncertain,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,11,Equity is a Call Option on the Assets of the Firm,At time t+T wh
15、en the companys debt comes due the firm will continue to operate if At+T D but the firms debt holders will declare the firm bankrupt if At+T D and the firm will go into default The stock holders of the firm are the residual claimants on the firm and to the stock holders the firm is therefore worth,E
16、lements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,12,when the debt comes due This is exactly the payoff function of a call option with strike D that matures on day t+T,Equity is a Call Option on the Assets of the Firm,Figure 12.4 shows the value of firm equity Et+T as
17、a function of the asset value At+T at maturity of the debt when the face value of debt D is $50 The equity holder of a company can therefore be viewed as holding a call option on the asset value of the firm It is important to note that in the case of stock options the stock price was the risky varia
18、ble In the present model of default, the asset value of the firm is the risky variable but the risky equity value can be derived as an option on the risky asset value,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,13,Figure 12.4: Equity Value as Function of Asset V
19、alue when Face Value of Debt is $50,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,14,Equity is a Call Option on the Assets of the Firm,The BSM formula can be used to value the equity in the firm in the Merton model Assuming that asset volatility, sA, and the risk-
20、free rate, rf , are constant, and assuming that the log asset value is normally distributed we get the current value of the equity to be,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,15,where,Equity is a Call Option on the Assets of the Firm,Note that the risk-fre
21、e rate, rf , is not the rate earned on the companys debt; it is instead the rate earned on risk-free debt that can be obtained from the price of a government bond Investors who are long options are long volatility The Merton model therefore provides the additional insight that equity holders are lon
22、g asset volatility The option value is particularly large when the option is at-the-money; that is, when the asset value is close to the face value of debt,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,16,Equity is a Call Option on the Assets of the Firm,In this c
23、ase if the manager holds equity he or she has an incentive to increase the asset value volatility (perhaps by taking on more risky projects) so as to increase the option value of equityThis action is not in the interest of the debt holders as we shall see now,Elements of Financial Risk Management Se
24、cond Edition 2012 by Peter Christoffersen,17,Corporate Debt is a Put Option Sold,The simple accounting identity states that the asset value must equal the sum of debt and equity at any point in time and so we have,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,18,w
25、here we have used the option payoff on equity described earlier. We use Dt+T to denote the market value of the debt at time t+T. Solving for the value of company debt, we get,Figure 12.5: Market Value of Debt as a Function of Asset Value when Face Value of Debt is $50,Elements of Financial Risk Mana
26、gement Second Edition 2012 by Peter Christoffersen,19,Corporate Debt is a Put Option Sold,Figure 12.5 shows the payoff to the debt holder of the firm as a function of the asset value At+T when the face value of debt D is $50Comparing Figure 12.5 with the option payoffs we see that the debt holders l
27、ook as if they have sold a put option although the out-of-the-money payoff has been lifted from 0 to $50 on the vertical axis corresponding to the face value of debt in this example,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,20,Corporate Debt is a Put Option So
28、ld,Figure 12.5 suggests that we can rewrite the debt holder payoff as,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,21,which shows that the holder of company debt can be viewed as being long a risk-free bond with face value D and short a put option on the asset va
29、lue of the company, At+T , with a strike value of D We can therefore use the model to value corporate debt; for example, corporate bonds,Corporate Debt is a Put Option Sold,Using the put option formula from Chapter 10 the value today of the corporate debt with face value D is,Elements of Financial R
30、isk Management Second Edition 2012 by Peter Christoffersen,22,where d is again defined by,The debt holder is short a put option and so is short asset volatility If the manager takes actions that increase the asset volatility of the firm, then the debt holders suffer because the put option becomes mo
31、re valuable,Implementing the Model,Stock return volatility needs to be estimated for the BSM model to be implemented In order to implement the Merton model we need values for sA and At, which are not directly observableIn practice, if the stock of the firm is publicly traded then we do observe the n
32、umber of shares outstanding and we also observe the stock price, and we therefore do observe Et,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,23,where NS is the number of shares outstanding,Implementing the Model,From the call option relationship earlier we know t
33、hat Et is related to sA and At via the equation,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,24,This gives us one equation in two unknowns We need another equation The preceding equation for Et implies a dynamic for the stock price that can be used to derive the
34、following relationship between the equity and asset volatilities:,where s is the stock price volatility,Implementing the Model,The stock price volatility can be estimated from historical returns or implied from stock option pricesWe therefore now have two equations in two unknowns, At and sA The two
35、 equations are nonlinear and so must be solved numerically using, for example, Solver in Excel Note that a crucially powerful feature of the Merton model is that we can use it to price corporate debt on firms even without observing the asset value as long as the stock price is available,Elements of
36、Financial Risk Management Second Edition 2012 by Peter Christoffersen,25,The Risk-Neutral Probability of Default,The risk-neutral probability of default in the Merton model corresponds to the probability that the put option is exercised It is simply,Elements of Financial Risk Management Second Editi
37、on 2012 by Peter Christoffersen,26,Note that this probability of default is constructed from risk neutral distribution of asset values and so it may well be different from the actual physical probability The physical default probability could be derived in the model but would require an estimate of
38、the physical growth rate of firm assets.,The Risk-Neutral Probability of Default,Default risk is also sometimes measured in terms of distance to default, which is defined as,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,27,The interpretation of dd is that it is th
39、e number of standard deviations the asset value must move down for the firm to default As expected, distance to default is increasing in the asset value and decreasing in the face value of debt,The Risk-Neutral Probability of Default,The distance to default is also decreasing in the asset volatility
40、Note that the probability of default is,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,28,The probability of default is therefore increasing in asset volatility,Portfolio Credit Risk,The Merton model gives powerful intuition about corporate default and debt pricing
41、 It enables us to link the debt value to equity price and volatility, which in the case of public companies can be observed or estimated While much can be learned from the Merton model, we have several motivations for going further First, we are interested in studying the portfolio implications of c
42、redit risk Default is a highly nonlinear event and furthermore default is correlated across firms and so credit risk is likely to impose limits on the benefits to diversification,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,29,Portfolio Credit Risk,Second, certai
43、n credit derivatives, such as collateralized debt obligations (CDOs), depend on the correlation of defaults that we therefore need to model Third, for privately held companies we may not have information necessary to implement the Merton model Fourth, even if we have the information needed, for a po
44、rtfolio of many loans, the implementation of Mertons model for each loan would be cumbersome To keep things relatively simple, we will assume a single factor model similar to the market index model For simplicity, we will also assume normal distribution,Elements of Financial Risk Management Second E
45、dition 2012 by Peter Christoffersen,30,Portfolio Credit Risk,We will assume a multivariate version of Mertons model in which the asset value of firm i is log normally distributed,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,31,where zi,t+T is a standard normal va
46、riable As before, the probability of default for firm i is,where,Portfolio Credit Risk,We will assume further that the unconditional probability of default on any one loan is PD This implies that the distance to default is now,Elements of Financial Risk Management Second Edition 2012 by Peter Christ
47、offersen,32,for all firms A firm defaults when the asset value shock zi is less than -ddi or equivalently less than -1(PD) We will assume that the horizon of interest, T, is one year so that T = 1 and T is therefore left out of the formulas in this section For ease of notation, time subscript, t is
48、suppressed,Factor Structure,The relationship between asset values across firms will be crucial for measuring portfolio credit riskAssume that the correlation between any firm i and any other firm j is , which does not depend on i nor jThis equi-correlation assumption implies a factor structure on th
49、e n asset values We have,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,33,where the common factor F and the idiosyncratic are independent standard normal variables,Factor Structure,Note that the zis will be correlated with each other with coefficient because they are all correlated with the common factor F with coefficient Using the factor structure we can solve for in terms of zi and F as,Elements of Financial Risk Management Second Edition 2012 by Peter Christoffersen,34,
