Stochastic Models. Credit Risk. Walt Pohl. May 16, Department of Business Administration
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1 Stochastic Models Credit Risk Walt Pohl Universität Zürich Department of Business Administration May 16, 2013
2 Default From the point of view of a lender, debt pays a fixed amount at predictable times, unless the borrower defaults: fails to pay the correct amount at the correct time. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
3 Default From the point of view of a lender, debt pays a fixed amount at predictable times, unless the borrower defaults: fails to pay the correct amount at the correct time. This is the primary risk to the loan, known as credit risk. Determining the credit risk is the key element to determining the debt s interest rate. (If there s no risk of default, then it should be the risk-free rate.) Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
4 Default and Collateral Most debt is backed by collateral if the borrower defaults, the creditor can seize the collateral. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
5 Default and Collateral Most debt is backed by collateral if the borrower defaults, the creditor can seize the collateral. There are intermediate outcomes short of total default (and seizure, if the loan is collateralized): Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
6 Default and Collateral Most debt is backed by collateral if the borrower defaults, the creditor can seize the collateral. There are intermediate outcomes short of total default (and seizure, if the loan is collateralized): Borrowers may temporarily suspend payments, only to resume them later. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
7 Default and Collateral Most debt is backed by collateral if the borrower defaults, the creditor can seize the collateral. There are intermediate outcomes short of total default (and seizure, if the loan is collateralized): Borrowers may temporarily suspend payments, only to resume them later. Lenders and borrowers may agree to a restructuring. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
8 Default and Collateral Most debt is backed by collateral if the borrower defaults, the creditor can seize the collateral. There are intermediate outcomes short of total default (and seizure, if the loan is collateralized): Borrowers may temporarily suspend payments, only to resume them later. Lenders and borrowers may agree to a restructuring. We will assume default is a purely binary decision, though. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
9 Pricing Credit Risk Pricing credit risk entails answering certain questions: Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
10 Pricing Credit Risk Pricing credit risk entails answering certain questions: How likely is default? Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
11 Pricing Credit Risk Pricing credit risk entails answering certain questions: How likely is default? How much of an interest premium does the creditor demand for bearing credit risk? Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
12 Pricing Credit Risk Pricing credit risk entails answering certain questions: How likely is default? How much of an interest premium does the creditor demand for bearing credit risk? How hard it is to diversify credit risk? How correlated are defaults? Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
13 Pricing Credit Risk, cont d The obvious way to answer the first and third question is by using historical data, but in many instances there is not enough data. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
14 Pricing Credit Risk, cont d The obvious way to answer the first and third question is by using historical data, but in many instances there is not enough data. Practitioners hit upon an alternative use market price data. This gives you a forward-looking measure of how the market prices credit risk. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
15 Corporate Bonds Bonds sold by a corporation usually have the entire firm as collateral if the firm defaults, the shareholders are wiped out, and the bondholders get the firm s assets. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
16 Corporate Bonds Bonds sold by a corporation usually have the entire firm as collateral if the firm defaults, the shareholders are wiped out, and the bondholders get the firm s assets. The shareholders will want to maintain control of the firm as long as its assets are worth more than its debt. If the value of the firm s assets drops too much, the shareholders allow default. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
17 Merton Model This leads to the Merton model for default. The value of a firm s assets, V t follow a geometric Brownian motion, dv = µvdt + σvdw. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
18 Merton Model This leads to the Merton model for default. The value of a firm s assets, V t follow a geometric Brownian motion, dv = µvdt + σvdw. The firm s stock then behaves like an option if the value of the firm s assets drops too low, then the value of the stock goes to zero, and the firm defaults. The observed stock price is the value of this option. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
19 Merton Model, cont d Standard option theory allows you to estimate the parameters of this process, and calculuate the probability of default based on the firm s stock price. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
20 Merton Model, cont d Standard option theory allows you to estimate the parameters of this process, and calculuate the probability of default based on the firm s stock price. This is one example of how market data can be used to calculate the probability of default. This is somewhat special, though for mortgages, housing prices can t be observed continuously, and do not serve as a good predictor of default. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
21 Enter The Li Model The Li model is a standard model to assess the risk other kinds of credit porfolios by market price data. It helped create a large market in pools of personal and commercial debt. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
22 Enter The Li Model The Li model is a standard model to assess the risk other kinds of credit porfolios by market price data. It helped create a large market in pools of personal and commercial debt. It acheived public notoriety when it starred in Felix Salmon s article Recipe for Disaster: The Formula That Killed Wall Street (Wired, ). Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
23 Li Model Basics In a credit portfolio, not only the individual chances of default matter, but how these defaults are correlated. Determining correlations from data is particularly hard, since you need enough data to compute the correlation of each asset with the other. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
24 Li Model Basics In a credit portfolio, not only the individual chances of default matter, but how these defaults are correlated. Determining correlations from data is particularly hard, since you need enough data to compute the correlation of each asset with the other. The Li model splits the credit portfolio problem into two pieces: Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
25 Li Model Basics In a credit portfolio, not only the individual chances of default matter, but how these defaults are correlated. Determining correlations from data is particularly hard, since you need enough data to compute the correlation of each asset with the other. The Li model splits the credit portfolio problem into two pieces: The probability of default before time T, which is treated as a random variable, with a hazard rate. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
26 Li Model Basics In a credit portfolio, not only the individual chances of default matter, but how these defaults are correlated. Determining correlations from data is particularly hard, since you need enough data to compute the correlation of each asset with the other. The Li model splits the credit portfolio problem into two pieces: The probability of default before time T, which is treated as a random variable, with a hazard rate. The dependence of default times, which are modeled by a Gaussian copula. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
27 Credit Derivatives Up until the crash, there was a booming market in over-the-counter credit derivatives contracts whose payoffs were complicated functions of debt. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
28 Credit Derivatives Up until the crash, there was a booming market in over-the-counter credit derivatives contracts whose payoffs were complicated functions of debt. For example, debt pools were divided into tranches. When interest payments arrived, senior tranches would get paid before junior tranches. So default would affect junior tranches before they affected senior tranches. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
29 Credit Derivatives Up until the crash, there was a booming market in over-the-counter credit derivatives contracts whose payoffs were complicated functions of debt. For example, debt pools were divided into tranches. When interest payments arrived, senior tranches would get paid before junior tranches. So default would affect junior tranches before they affected senior tranches. Middle tranches was itself pooled, and divided into tranches again. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
30 Credit Derivatives Up until the crash, there was a booming market in over-the-counter credit derivatives contracts whose payoffs were complicated functions of debt. For example, debt pools were divided into tranches. When interest payments arrived, senior tranches would get paid before junior tranches. So default would affect junior tranches before they affected senior tranches. Middle tranches was itself pooled, and divided into tranches again. To sell these different kinds of debt, practitioners needed a model to work out their prices. This is what the Li model allowed. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
31 Li Model In Practice For simplicity, suppose the portfolio only has two assets in it, and that the hazard rate is a constant. Then we only need three numbers to fit the model: Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
32 Li Model In Practice For simplicity, suppose the portfolio only has two assets in it, and that the hazard rate is a constant. Then we only need three numbers to fit the model: The hazard rate for each asset. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
33 Li Model In Practice For simplicity, suppose the portfolio only has two assets in it, and that the hazard rate is a constant. Then we only need three numbers to fit the model: The hazard rate for each asset. The correlation for the Gaussian copula. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
34 Li Model In Practice For simplicity, suppose the portfolio only has two assets in it, and that the hazard rate is a constant. Then we only need three numbers to fit the model: The hazard rate for each asset. The correlation for the Gaussian copula. As soon as we know the price of three credit derivatives on this portfolio, we know the whole model. Then we can price any other derivative. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
35 Li Model In Practice, cont d Of course, people developed more sophisticated models, but they all had this feature that market prices could be used to predict other market prices. They frequently relied on the Gaussian copula, since it provided a parsimonous model of dependence. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
36 Li Model In Practice, cont d Of course, people developed more sophisticated models, but they all had this feature that market prices could be used to predict other market prices. They frequently relied on the Gaussian copula, since it provided a parsimonous model of dependence. This was in a era where default rates were unusually low, making market prices unusually low. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
37 What Happened? The Gaussian copula has a very subtle property that makes it somewhat special the main effect of the dependence is in the middle. As you get further and further into the tails of the copula (quantiles near 0% or 100%), the dependence disappears. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
38 What Happened? The Gaussian copula has a very subtle property that makes it somewhat special the main effect of the dependence is in the middle. As you get further and further into the tails of the copula (quantiles near 0% or 100%), the dependence disappears. So not only were prices calculated in an era where default rates were unusually low, the model also underestimated the risk of defaults being bunched together. The prices of credit derivatives were wrong along two dimensions both the risk of individual default, and the chances of defaults happening together were misspecified. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
39 What Happened?, cont d Banks that were holding these mispriced assets were suddenly exposed to large losses in their portfolios. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
40 What Happened?, cont d Banks that were holding these mispriced assets were suddenly exposed to large losses in their portfolios. Banks are very interconnected banks borrow heavily from each other to make up short-term needs. So banks had lent money to other banks under the assumption that they were good credit risks, when they were not. This is how the crisis rippled through the system, imperiling the entire global banking system. Walt Pohl (UZH QBA) Stochastic Models May 16, / 15
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