Slides for Risk Management Credit Risk

Transcription

1 Slides for Risk Management Credit Risk Groll Seminar für Finanzökonometrie Prof. Mittnik, PhD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 1 / 97

2 1 Introduction to default risk 2 Additional risk components 3 Estimating default probabilities Credit Ratings Based on asset value models 4 Credit portfolio risk: default correlation Effects in defaults only mode 5 Estimating default correlations Based on asset value models Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 2 / 97

3 Definition Introduction to default risk Definition Credit risk predominantly comprises the risk of losses arising from an inability of a counterparty of a financial contract to fulfill promised payments. The case of a counterparty failing to meet its financial obligations is called default. in a broader context credit risk also is perceived as entailing credit spread risk downgrade risk Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 3 / 97

4 Default risk Introduction to default risk each counterparty is assumed to have an inherent probability of default, which can not be directly observed in order to assess the risk arising from this possible event of default, the following quantitative characteristics have to be estimated for each counterparty: probability of default (PD): the probability of a default event exposure at default (EAD): the amount that still has to be repayed by the borrower at the time of his default (could be random) loss given default (LGD): the fraction of the still outstanding obligations that the borrower is not able to repay, given that he defaults Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 4 / 97

5 Default risk Introduction to default risk the loss arising from a counterparty is given by L = EAD }{{ LGD} 1 }{{} D amount of money default indicator where D denotes the event of default with associated probability P (D) = PD assuming independence between the amount of loss EAD LGD and the occurrence of default 1 D, the expected loss (EL) can be calculated as E [L] = E [EAD LGD 1 D ] = E [EAD LGD] E [1 D ] = E [EAD LGD] PD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 5 / 97

6 Introduction to default risk Simplifications argumentation against independence: times of financial turmoil both increase likelihood of default events and amounts of losses, since remaining firm assets usually achieve less liquidation value because of low market prices caused by low demand due to bad market conditions: LGD and 1 D should not be independent assumption: EAD is not a random variable this does not hold for financial contracts with varying underlying exposure like in the case of market-driven instruments such as swaps and forwards Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 6 / 97

7 Example Introduction to default risk A bank has granted a loan with nominal value 100 to an industrial company. The company in turn has used the borrowed money to bring up enough capital for an investment in a new product line. However, the success of the new product line and hence the size of the future profits are uncertain. Hence, the ability of the company to repay the loan depends on the success of the investment project, which has the following distribution of profits: profit probability 1% 2% 3% 4% 90% Given EAD = 100, assess the risk arising to the bank by calculation of 1.PD 2.EL 3.LGD 4.VaR Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 7 / 97

8 Introduction to default risk Example: loss distribution given repayment C for the investment, the associated loss to the bank is given by L = max {EAD C, 0} Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 8 / 97

9 Example: PD Introduction to default risk PD = 3% + 2% + 1% = 6% Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 9 / 97

10 Example: EL Introduction to default risk EL = 94% 0 + 3% % % 90 = 2.7 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 10 / 97

11 Introduction to default risk Example: LGD since default is given, outcomes without loss are excluded Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 11 / 97

12 Introduction to default risk Example: LGD probabilities of events have to be scaled up in order to sum up to probability 1 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 12 / 97

13 Introduction to default risk Example: LGD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 13 / 97

14 Introduction to default risk Example: LGD LGD = E [L L > 0] = P (L = 20) P (L > 0) = = P (L = 60) P (L > 0) P (L = 90) P (L > 0) 90 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 14 / 97

15 Introduction to default risk Example: VaR cumulative distribution of losses loss L probability P (L l) 94% 97% 99% 100% Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 15 / 97

16 Introduction to default risk Example: VaR getting VaR 0.95 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 16 / 97

17 Present value Additional risk components given certain payments c 1,..., c n at future points in time t + 1,... t + n, the present value V t of the future payments is calculated by discounting: c 1 V t = (1 + r + c 2 t+1) (1 + r t+1) (1 + r + + c n t+2) (1 + r t+1) (1 + r t+n) c 1 = (1 + R + c 2 t+1) (1 + R t+2) c n (1 + R t+n) n, where r t+i denotes the (discrete) interest rate between (t + i 1) and (t + i), and R t+n denotes the (discrete) annualized interest rate for periods t + 1 to t + n Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 17 / 97

18 Yield curve Additional risk components based on observable prices for investment opportunities that are generally considered riskless (government bonds of industrial countries), one can extract prevailing market interest rates in a yield curve given the prevailing interest rates, the present value of any new stream of cash flows can be calculated Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 18 / 97

19 Additional risk components Valuation under uncertainty leaving the world of riskless and guaranteed future payments: cash flow c i only occurs in case of state D i, with probability p i := P (D i occurs) what about the present value of the stream of uncertain future cash flows? regarding the problem as a sum of uncertain present values, n i=1 1 Di V (i) t = n c i 1 Di (1 + R t+i ) i, i=1 the problem can be reduced to a lottery in the present the expected payoff of this lottery can be calculated straightforward: [ n ] n n E 1 Di V (i) t = P (D i occurs) V (i) t = p i V (i) t i=1 i=1 however: the present value that people assign to the lottery over a number of uncertain values does not equal its expectation Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 19 / 97 i=1

20 Risk aversion Additional risk components if one asks people to participate in a game where they win 1 in case of heads at a coin toss, but lose 1 in case of tail, it would not be difficult to find people that are willing to play however, increasing the bet to instead of 1, people in general would not be willing to participate in the game, unless they get an adjustment to their favor the adjustment demanded as a compensation for the risk involved in the game is called risk premium, depending both on the odds of the game, as well as on the amount of money involved in case of adverse outcomes Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 20 / 97

21 Additional risk components Valuation under uncertainty transfering the perception of risk aversion to the valuation of uncertain payoffs requires that each term has to be devaluated additionally to discounting n c i V t = f (c i, p i ) (1 + R t+i ) i, i=1 which usually is written as an additional spread s i on the riskless interest rate: n c i V t = (1 + R t+i + s i ) i i=1 according to arbitrage theory compensation for risk has to take place in a consistent manner across the market (risk neutral martingal measure) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 21 / 97

22 Additional risk components Additional risk components besides the risk of a default of the borrower, holding a bond (or loan) also entails the risk of a depreciation, which comes into play if it shall be sold prior to maturity and default there are three components in the valuation of a bond that can cause depreciations: interest rate risk: depreciations caused by increases of the discount interest rate downgrade risk: the probability of default of the borrower increases leading to a higher demanded risk compensation when reselling the bond spread risk: changes in the prevailing risk aversion of investors that lead to higher demanded risk compensations besides unchanged probability of default Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 22 / 97

23 Additional risk components Zero rate curves in reality, neither exact default probabilities nor exact risk premiums can be individually observed in the market however, given the bonds of several firms with approximately equal default probabilities, an associated yield curve can be inferred on the basis of the observable bond prices hence, even without knowledge about exact probabilities of default or associated risk premiums individually, the value of any new bond with comparable risk characteristics can be calculated Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 23 / 97

24 TED spread Additional risk components TED spread (Treasury Bill Eurodollar Difference): difference between interest rates on three-months interbank loans and three-months U.S. government debt ("T-bills") LIBOR (London Interbank Offered Rate): interest rates at which banks borrow unsecured funds of each other in the London money market while three-months U.S. government debt is largely considered riskless, overnight lending in the interbank money market involves default risk TED spread measures additional risk premium demanded in interbank money markets example: T-bill rates of 3.40% and LIBOR rates of 3.70% lead to TED spread of 30 (denoted in bps) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 24 / 97

25 TED spread Additional risk components two possible explanations for increasing TED spread loss in confidence of the credit worthiness of banks: default probabilities have increased compensation demanded for any given portion of risk has increased: risk aversion in the market has increased Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 25 / 97

26 Additional risk components Example: revaluation A bank holds a corporate bond with principal 100, annual coupon payments of 6 and maturity 3 years in its portfolio. According to the internal rating system of the bank, the issuer of the bond can be classified as reliable borrower. However, since the bank is planning to resell the bond within the next months, the riskmanagement division shall assess the loss associated with a decreasing credit quality of the borrower and an associated downgrade to the rating category unreliable. Yields for both internal rating categories are given by maturity in years reliable 2% 2.8% 3.2% unreliable 3.4% 4.8% 5.4% Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 26 / 97

27 Additional risk components Example: revaluation future cash flows associated with the bond in case of no default: period t = 1 t = 2 t = 3 cash flow calculating the current value of the bond: P reliable t = 6 (1.02) + 6 (1.028) (1.032) 3 = calculating value in case of downgrade: P unreliable t = 6 (1.034) + 6 (1.048) (1.054) 3 = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 27 / 97

28 Estimating default probabilities Main risk components default risk: counterparty actually failing to meet its financial obligations realizing only in case of default, when debt is still not resold to third party determining factor: default probability downgrade risk: losses induced by decreasing credit quality realizing even without actual default in case of debt reselling determining factor: credit quality changes due to fluctuations of default probabilities problem: default probabilities are not observable, and fluctuations in default probabilities hence all the less Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 28 / 97

29 Estimating default probabilities Estimation methodologies estimation methodologies differ, depending on the publicly available market information about the borrowing firm: without listed stocks, traded debt (bonds) or available credit rating: analysis based on fundamental values and quantitative business ratios derived from financial statement (e.g. Altmann s Z-score) available credit rating: derive default probability from historic default rates of equally rated firms traded bonds: derived from credit spreads listed stocks: derived from asset value model using observed equity prices (Merton model) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 29 / 97

30 Estimating default probabilities Rating definitions Moody s Credit Ratings Long-Term Corporate Obligation Ratings: Aaa Obligations rated Aaa are judged to be of the highest quality, with minimal credit risk. Aa Obligations rated Aa are judged to be of high quality and are subject to very low credit risk. A Obligations rated A are considered upper-medium grade and are subject to low credit risk. Baa Obligations rated Baa are subject to moderate credit risk. They are considered medium-grade and as such may possess certain speculative characteristics. Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 30 / 97

31 Estimating default probabilities Rating definitions Moody s Credit Ratings Ba Obligations rated Ba are judged to have speculative elements and are subject to substandtial credit risk. B Obligations rated B are considered speculative and are subject to high credit risk. Caa Obligations rated Caa are judged to be of poor standing and are subject to very high credit risk. Ca Obligations rated Ca are highly speculative and are likely in, or very near, default, with some prospect of recovery of principal and interest. C Obligations rated C are the lowest rated class of bonds and are typically in default, with little prospect for recovery of principal or interest. Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 31 / 97

32 Estimating default probabilities Distribution over classes Credit Ratings Distribution of European Bond Issuers by Whole Letter Rating 2009 Rating Category percentage Aaa 4.2 Aa 20.8 A 31.8 Baa 22.3 Ba 7.4 B 9.5 Caa-C 4.0 Investment-Grade 79.2 Speculative-Grade 20.8 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 32 / 97

33 Transition matrix Estimating default probabilities Credit Ratings One-Year-Average Ratings Transition for Europe initial rating year end Aaa Aa A Baa Ba B Caa-C Defaults Aaa Aa A Baa Ba B Caa-C Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 33 / 97

34 Recovery Rates Estimating default probabilities Credit Ratings seniority of the bond determines its recovery rate in case of default revaluation: provide a credit spread corresponding to each category as well Europe Sr. Secured Loans 55.5 Sr. Unsecured Loans 43.0 Sr. Secured Bonds 38.7 Sr. Unsecured Bonds 24.5 Sr. Subordinated Bonds 34.3 Subordinated Bonds 25.4 Jr. Subordinated Bonds n.a. standard deviation is missing Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 34 / 97

35 Introduction Estimating default probabilities Based on asset value models idea: use information incorporated in equity prices in order to estimate default probabilities first appearance in On the pricing of corporate debt: The risk structure of interest rates (Merton, 1974) asset value models: default occurs when value of firm s assets is below the nominal debt value two of most widely used credit risk models are based on asset value model interpretation: KMV-Model: estimation of default probabilities from adjusted asset value model CreditMetrics: estimation of default correlations based on asset value model Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 35 / 97

36 Estimating default probabilities Asset-value - default connection Based on asset value models with given liabilities the occurrence of default depends on the evolution of the firm s asset prices Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 36 / 97

37 Default probabilities Estimating default probabilities Based on asset value models with known distribution of asset prices at the end of the time horizon the default probability equals the fraction of asset price paths with values below the debt level Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 37 / 97

38 Problem Estimating default probabilities Based on asset value models although assets are listed in financial statements, market values for assets are usually not completely existing example: market prices of physical capital (machines, real estate, property) are not existing, simply because these assets are kept by the firm and are not traded based on the sparsity of information about asset values, estimating asset value dynamics seems impossible even exact numbers for debt may not be known: hidden financial obligations due to commitments, subsidiaries or contingent debt levels (swaps) solution: try to overcome asset value problems through incorporation of market data for equity Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 38 / 97

39 Equity as option Estimating default probabilities Based on asset value models assets A t financed by equity E t and debt D t A t = E t + D t share holders as the owners of the firm have the right to liquidate the firm at any time: paying off the debt and taking over the remaining assets two scenarios: A T < D t : total value of assets are below financial obligations - no assets left for equity holders A T D t : after repaying debt equity holders are left with net profit of A T D T payoff to equity holders is given by max (A T D T, 0) = (A T D T ) + equal to call option on asset values A T with strike price D T observable equity prices are the prices that investors are willing to pay for this payoff in T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 39 / 97

40 Payoff call option Estimating default probabilities Based on asset value models goal: assume functional form for dynamics of asset values and adjust parameters of dynamics to equity prices observable in the market Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 40 / 97

41 Brownian motion Estimating default probabilities Based on asset value models Definition A real-valued stochastic process (B t ) 0 t is called Brownian motion, if 1 B 0 = 0 2 the function t B t (ω) is continuous P-a.s. 3 the increments B t B s are independent with distribution B t B s N (0, t s) for any 0 s < t. Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 41 / 97

42 Brownian motion Estimating default probabilities Based on asset value models changing volatility of Brownian motion: because of V (σb t ) = σ 2 V (B t ) = σ 2 t the increments of the rescaled Brownian motion (σb t ) 0 t< are distributed according to σ (B t B s ) N ( 0, σ 2 (t s) ) example: for σ = 0.4 the rescaled Brownian motion has variance σ 2 = = 0.16 per time period Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 42 / 97

43 Brownian motion Estimating default probabilities Based on asset value models discrete approximation of Brownian motion Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 43 / 97

44 Brownian motion Estimating default probabilities Based on asset value models simulated paths with finer resolution can be obtained by using n normally distributed random variables with variance 1 n : for X i N (0, 1) the scaled random variable σx i = 1 n X i has variance ( ) V n 1 X i = 1 n V (X i) = 1 n, so that ( n ) ( n ) V σx i = σ 2 V X i i=1 = 1 n i=1 n V (X i ) i=1 = 1 n n 1 = 1 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 44 / 97

45 Estimating default probabilities Deterministic dynamics Based on asset value models consider deterministic function as degenerated special case of stochastic processes A t = A 0 exp (rt) value changes of process per time evolving da t dt = (A t ) = d (A 0 exp (rt)) dt da t = ra t dt = A 0 exp (rt) r = A t r intuitive interpretation: for any given value of the process, how does this process value change, given that we look at process for time dt example: given world population 7 billion, how does population number change in one year Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 45 / 97

46 this dynamics is often used to model the evoluation of money on a bank account with constant rate of interest: given current account surplus of 1000, how does amount of money change within next year? Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 46 / 97 Exponential growth Estimating default probabilities Based on asset value models dynamics of A t on logarithmic scale d (log A t ) dt = (log A t ) = 1 A t A t = 1 A t A t r = r

47 Stochastic dynamics Estimating default probabilities Based on asset value models goal: process with random changes, with size of changes depending on current value da t = A t σdb t justification: absolute size of stock price changes depends on current stock price solution of stochastic differential equation: A t = A 0 exp ( 12 ) σ2 t + σb t proof: because of erratic behaviour of Brownian motion, approximation to changes of process must incorporate derivatives of higher order Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 47 / 97

48 Stochastic dynamics Estimating default probabilities Based on asset value models application of Ito s formula df (I t ) = F (I t ) di t F (I t ) d I t to A t = A 0 exp ( 12 ) σ2 t + σb t = A 0 exp (I t ) = F (I t ) leads to da t = exp ( 12 ) σ2 t + σb t ( 12 ) σ2 dt + σdb t + 12 ( exp 12 ) σ2 t + σb t σ 2 d B t = A t ( 12 ) σ2 dt + σdb t Atσ2 dt = A t σdb t Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 48 / 97

49 Stochastic dynamics Estimating default probabilities Based on asset value models on a logarithmic scale, with (log x) = 1 x : ds t = d (log A t ) = 1 da t 1 A t 2 1 d A t A 2 t = 1 A t σdb t 1 A t 2 = σdb t 1 2 σ2 dt 1 A 2 t σ 2 dt A 2 t Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 49 / 97

50 Stochastic dynamics Estimating default probabilities Based on asset value models because of convexity of exp, a symmetric erratic behaviour on the logarithmic scale would lead to positive drift for real world asset prices hence, dynamics of logarithmic world show negative drift when expectation of real world is zero Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 50 / 97

51 Estimating default probabilities Geometric Brownian motion Based on asset value models incorporate drift term in real world dynamics da t = A t µdt + A t σdb t solution: A t = A 0 exp ((µ 12 ) ) σ2 t + σb t dynamics on logarithmic scale ds t = d (log A t ) = 1 A t da t d A t A 2 t = µdt + σdb t 1 2 σ2 dt = (µ 12 ) σ2 dt + σdb t Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 51 / 97

52 Estimating default probabilities Geometric Brownian motion Based on asset value models goal: asset value expectation shall follow exponential growth, with random fluctuation attached first guess: introduce random fluctuations by attachment of Brownian motion in log-world Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 52 / 97

53 Estimating default probabilities Geometric Brownian motion Based on asset value models convexity increases upward deviations Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 53 / 97

54 Estimating default probabilities Geometric Brownian motion Based on asset value models expectation in real world is greater than expectation in log-world hence: to get µ in real world, adjustment 1 2 σ2 is needed in log-world Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 54 / 97

55 Option pricing Estimating default probabilities Based on asset value models key assumptions: asset value process follows a geometric Brownian motion arbitrage-free world price of C T = (A T D T ) + is less than E [C T ], since risk averse investors want compensation for risk according to arbitrage theory, the arbitrage-free price C 0 of C T is given by the expectation of the discounted payoff C T under a risk-neutral equivalent measure Q : C 0 = E Q [ CT e rt hence, the equity value E t of a firm is related to the asset value process A t and the debt value D t by [ (AT ) ] D + T E 0 = E Q e rt ] Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 55 / 97

56 Dynamics under Q Estimating default probabilities Based on asset value models changing to the risk-neutral measure Q nullifies excess returns above the risk-free interest rate: da t = A t rdt + A t σdb t with solution A t = A 0 exp ((r 12 ) ) σ2 t + σb t additionally discounting leads to dynamics with expected return equal to zero: ( ) ( ) At At d e rt = e rt σdb t Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 56 / 97

57 Estimating default probabilities Black-Scholes formula Based on asset value models denoting the discounted asset value process (A t /e rt ) with  t, the solution to the dynamics under the risk-neutral measure is (  t =  0 exp σb t 1 ) 2 σ2 t denoting the discounted value of the debt by ˆD T, the value of the call option of the assets  t with strike price ˆD T is given by [ ) ] + C 0 = E (ÂT ˆD T ˆ ) = (ÂT ˆD T 1 {  T > ˆD T} dω = Ω ˆ (x ˆD T ) 1 {x> ˆD T} df  T (x) hence, calculation of the expectation requires the cumulative distribution function and the probability density function of the random variable  T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 57 / 97

58 Estimating default probabilities Cumulative distribution function Based on asset value models using knowledge of logarithmic asset value process ) FÂT (x) = P (ÂT x ( ) = P log  T log x with log  T = log A σ2 T + σb T and log  T N ( log A 0 1 ) 2 σ2 T, σ 2 T log  T log A σ2 T σ T N (0, 1) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 58 / 97

59 Estimating default probabilities Cumulative distribution function plugging in: ) FÂT (x) = P (ÂT x ( ) = P log  T log x ( log ÂT log A = P σ2 T σ T ( ) log x log A = Φ σ2 T σ T ( ) = Φ log x A σ2 T σ T := Φ ( h (x)), with h (x) = Based on asset value models log x log A σ2 T σ T ( ) log x 1 A0 2 σ2 T σ T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 59 / 97 )

60 Estimating default probabilities Probability density function Based on asset value models density function is derivative of cdf: because of ( h (x)) = fât (x) = F Â T (x) = (Φ ( h (x))) = Φ ( h (x)) ( h (x)) ( 1 = φ ( h (x)) σ T ( 1 = φ ( h (x)) xσ T ( ) log x A σ2 T σ T = 1 x A 0 ), ) 1 A 0 ( ) log x A 0 σ T + 0 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 60 / 97

61 Estimating default probabilities Based on asset value models using density function we get ˆ Â T dfât = {ÂT ˆD T} = = = ˆ ˆD T ˆ ˆ ˆ ˆ ( ) = A 0 ( ) xf (x) dx ( 1 x φ ( h (x)) ˆD T xσ T φ ( h (x)) ˆD T σ T dx φ (h (x)) ˆD T σ T dx φ (g (x)) xσ T dx ˆD T = A 0 [ Φ (g (x))] ˆD ( ( )) T = A 0 Φ g ˆD T, ) dx with g (x) = h (x) + σ T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 61 / 97

62 Estimating default probabilities Based on asset value models ( ) holds because of ( φ (g (x)) = φ h (x) + σ ) T ( = 1 h (x) + σ T exp 2π 2 so that ( = 1 exp h (x)2 2π 2 = 1 exp ( h (x)2 2π 2 ( ) = φ (h (x)) exp log x A 0 σ T ( ( )) x = φ (h (x)) exp log = φ (h (x)) x A 0, ) 2 ) 2h (x) σ T + σ 2 T 2 ) ( exp h (x) σ T σ2 T 2 A 0 + σ2 T 2 φ (h (x)) = φ (g (x)) A0 x σ T σ2 T 2 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 62 / 97 )

63 Estimating default probabilities Based on asset value models ( ) holds because of furthermore, ( Φ (g (x))) = φ (g (x)) (g (x)) = φ (g (x)) ( h (x) + σ ) T = φ (g (x)) (h (x)) ( ) 1 = φ (g (x)) xσ T g ( ) = h ( ) + σ T ( ) log A 0 = σ 1 T 2 σ T + σ T = σ T σ T = Φ (g ( )) = 0 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 63 / 97

64 Estimating default probabilities Black-Scholes formula Based on asset value models E [C] = = ˆ = = ˆ ˆ (ÂT ˆD T ) + dfât (x) (ÂT ˆD T ) 1 {  T > ˆD T} df  T (x) {ÂT > ˆD T} ˆ ˆD T = A 0 Φ = v  T dfât (x) ˆD T ˆ{ÂT > ˆD T} df  T (x) )  T dfât (x) ˆD T P (ÂT > ˆD T ( ( )) g ˆD T ( A 0, ˆD T, T, σ ( ( )) ˆD T Φ h ˆD T ) with ( ) x log A σ2 T h (x) = σ, g (x) = h (x) + σ T T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 64 / 97

65 Summary Estimating default probabilities Based on asset value models assumptions made by Black-Scholes world: asset prices follow geometric Brownian motion trading appears in continuous time no transaction prices borrowing and lending at risk-free rate is possible at arbitrarily high amounts Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 65 / 97

66 Summary Estimating default probabilities Based on asset value models given these assumptions, the probability of default is P (Y = 1) = P (A T D T ) = P (log A T log D T ) ( = P log A 0 + (r 12 ) ) σ2 T + σb T log D T ( = P σb T log D T log A 0 (r 12 ) ) σ2 T ( B T = P log D T log A 0 ( r 1 2 σ2) ) T T σ T ( log DT log A 0 ( r 1 2 σ2) ) T = Φ σ T Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 66 / 97

67 Summary Estimating default probabilities Based on asset value models according to option pricing theory, the observable equity prices E 0 is a function of the non-observable parameters A 0 and σ ( ) E 0 = v A 0, ˆD T, T, σ given we take a first estimate of σ, the only unknown parameter in the equation is the asset price locally inverting the option price formula hence gives an estimate of A 0 repeating this procedures for a series of points in time with known equity prices gives an estimated time series of asset prices given the time series of asset prices, parameters σ and µ can be estimated, and can be used as a second guess input in the locally inverted option price formula repeating this procedure gives estimated values of σ and µ hence, PD can be estimated Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 67 / 97

68 Credit portfolio risk: default correlation Defaults as binomials Effects in defaults only mode given two binomial random variables X 1 and X 2 with { 1 with probability p X i B (1; p), i.e., X i = 0 with probability (1 p) the event 1 is interpreted as default, with default probability P (X 1 = 1) = P (X 2 = 1) = p joint distribution for case of independence X X 2 0 p (1 p) (1 p) 2 (1 p) 1 p 2 p (1 p) p p (1 p) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 68 / 97

69 Credit portfolio risk: default correlation Default correlation Effects in defaults only mode notation X X 2 0 β γ (1 p) 1 α β p p (1 p) calculate covariance default correlation ϱ X1,X 2 = Cov (X 1, X 2 ) = E [X 1 X 2 ] E [X 1 ] E [X 2 ] = P (X 1 = 1, X 2 = 1) p 2 = α p 2 α p 1 p 2 p1 (p 1 1) p 1 =p 2 = α p2 p 2 (p 2 1) p (p 1) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 69 / 97

70 Credit portfolio risk: default correlation Example: independence Effects in defaults only mode given default probabilities p 1 = p 2 = 0.1, joint probabilities in case of independency are given by X X covariance and correlation are given by Cov (X 1, X 2 ) = α p 2 = = 0 ϱ X1,X 2 = Cov (X 1, X 2 ) p (1 p) = 0 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 70 / 97

71 Credit portfolio risk: default correlation Example: dependence Effects in defaults only mode the same individual default probabilities p 1 = p 2 = 0.1 also could lead to the following joint distribution X X associated default correlation and covariance are Cov (X 1, X 2 ) = = 0.07 ϱ X1,X 2 = Cov (X 1, X 2 ) p (1 p) = = note: the probability of joint defaults is 8 times higher in the second case Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 71 / 97

72 Credit portfolio risk: default correlation Example: graphical representation Effects in defaults only mode expressing probabilities as pillars above the events {"default", "no default"} Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 72 / 97

73 Credit portfolio risk: default correlation Effects in defaults only mode Example: interpretation as asset value model Goal: find joint default probability thinking in terms of asset value model: for p = 0.1, default happens in 10% worst possible asset path realizations already known: for any given nominal debt value D T the probability of default depends on the distribution of the underlying asset value by p = P (A T < D T ) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 73 / 97

74 Credit portfolio risk: default correlation Effects in defaults only mode Example: interpretation as asset value model the occurrence of joint defaults can be interpreted as both asset values lying below their respective debt level joint default in asset value model: joint asset distribution F required joint default probability: PD = F (D 1, D 2 ) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 74 / 97

75 Credit portfolio risk: default correlation Effects in defaults only mode Example: interpretation as asset value model joint default probabilities can be equivalently described in terms of quantiles: PD = P (A 1 < D 1, A 2 < D 2 ) = F (D 1, D 2 ) = C (F 1 (D 1 ), F 2 (D 2 )) = C (PD 1, PD 2 ) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 75 / 97

76 Credit portfolio risk: default correlation Effects in defaults only mode Example: interpretation as asset value model hence, given that PD i is already known: the exact marginal distribution of assets does not provide additional information joint default probabilities depend on asset pair copula only marginal asset distributions need not be modelled joint default distributions can be described by copula densities Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 76 / 97

77 Credit portfolio risk: default correlation Example: associated copula density Effects in defaults only mode given joint distribution, density heights of an associated copula are calculated according to both default one default no default h 1,1 = P (X 1 = 1, X 2 = 1) P (X 1 = 1) P (X 2 = 1) = = 8 h 1,0 = h 0,1 = P (X 1 = 1, X 2 = 0) P (X 1 = 1) P (X 2 = 0) = = h 0,0 = P (X 1 = 0, X 2 = 0) P (X 1 = 0) P (X 2 = 0) = = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 77 / 97

78 Credit portfolio risk: default correlation Effects in defaults only mode Example: representation as copula density given associated copula density, probability of joint default can be calculated as probability mass in respective interval: P (X 1 = 1, X 2 = 1) = width X1 width X2 heights = = 0.08 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 78 / 97

79 Credit portfolio risk: default correlation Example: maximum dependence Effects in defaults only mode with given individual default probabilities p 1 = p 2 = p the maximum possible dependency occurs when defaults always appear together: X X associated default correlation and covariance are Cov max (X 1, X 2 ) = = 0.09 ϱ max = = 1 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 79 / 97

80 Credit portfolio risk: default correlation Example: negative correlation Effects in defaults only mode with given individual default probabilities p 1 = p 2 = p the maximum diversification effects occur, when defaults never appear together: X X associated default correlation and covariance are Cov min (X 1, X 2 ) = = 0.01 ϱ min = = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 80 / 97

81 Credit portfolio risk: default correlation Graphical representation Effects in defaults only mode Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 81 / 97

82 Credit portfolio risk: default correlation Effects in defaults only mode Graphically derivation of joint default distribution for given copula density, partition unit cube according to individual default probabilities since exact asset path realizations are not necessary, individual intervals can be represented with uniform distribution of mean height Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 82 / 97

83 Credit portfolio risk: default correlation Analytical derivation Effects in defaults only mode given copula C, the probability mass in rectangle [a 1, b 1 ] [a 2, b 2 ] is given by copula h-volume: P (a 1 < A 1 b 1, a 2 < A 2 b 2 ) = =P (A 1 b 1, A 2 b 2 ) P (A 1 b 1, A 2 a 2 ) P (A 1 a 1, A 2 b 2 ) + P (A 1 a 1, A 2 a 2 ) =C (b 1, b 2 ) C (b 1, a 2 ) C (a 1, b 2 ) + C (a 1, a 2 ) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 83 / 97

84 Credit portfolio risk: default correlation h-volume derivation Effects in defaults only mode graphical representation of probability mass in [a 1, b 1 ] [a 2, b 2 ] Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 84 / 97

85 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula let individual default probabilities be given by p 1 = p 2 = 0.05 calculate joint default probability with Gaussian copula with parameter ρ = 0.57 P (A , A ) = C Gau (0.05, 0.05; ρ = 0.57) calculate probability of one default = P (A , A 2 > 0.05) = P (A 1 > 0.05, A ) = p 1 α = = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 85 / 97

86 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula calculate probability of no default P (A 1 > 0.05, A 2 > 0.05) = 1 2 β α = = X X Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 86 / 97

87 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula calculate joint default probability with Clayton copula with parameter α = 1.3 P (A , A ) = C Clay (0.05, 0.05; α = 1.3) = ( u α 1 + u α 2 1 ) 1/α = ( ) 1/1.3 = calculate probability of one default P (A , A 2 > 0.05) = P (A 1 > 0.05, A ) = = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 87 / 97

88 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula calculate probability of no default P (A 1 > 0.05, A 2 > 0.05) = 1 2 β α = = X X Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 88 / 97

89 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula assuming that underlying asset paths follow individual standard normal distributions, both models exhibit nearly equal correlations while the left model is usually used to derive joint default probabilities in practice, the right model exhibits substantially higher joint default probabilities (0.3 instead of 0.15) moreover, since both models exhibit equal correlations, practitioners might even not be aware of the risk due to possibly wrong model specifications Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 89 / 97

90 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula measuring model differences default correlations portfolio defaults ϱ Gau X 1X 2 = α p = = p (1 p) ϱ Clay X 1X 2 = α p = p (1 p) = Gaussian Clayton number defaults probability cumulative default value-at-risk: VaR0.98 Gau = 1, VaRClay 0.98 = 2 Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 90 / 97

91 Credit portfolio risk: default correlation Effects in defaults only mode Example: Gaussian vs. Clayton copula without copulas, would such joint default probabilities ever be imaginable? calculating Gaussian copula parameter ρ, leading to equally large joint default probability of 0.03 with individual default probabilities of 0.05 requires solution of C Gau ρ (0.05, 0.05)! = 0.03 trial and error approximation leads to ρ = 0.88 : C Gau 0.88 (0.05, 0.05) = practitioners could deem such a high correlation between assets as very unlikely! Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 91 / 97

92 Factor model Estimating default correlations Based on asset value models Goal: determine asset correlation asset returns are assumed to be a composite of influences from individual country and industry factors (X i ) 1 i n and a firm specific idiosyncratic component ɛ, ɛ N (0, 1), and factors and idiosyncratic component are assumed to be uncorrelated: for the case of two factors we get Cov (X i, ɛ) = 0 r = a 1 X 1 + a 2 X 2 + bɛ the associated asset return variance is given by V (r) = a 2 1V (X 1 ) + a 2 2V (X 2 ) + 2a 1 a 2 Cov (X 1, X 2 ) + b 2 V (ɛ) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 92 / 97

93 Estimating default correlations Variance decomposition Based on asset value models the asset return variance can be decomposed into a part arising from common market fluctuations and a part arising from idiosyncratic components variance of non-idiosyncratic part: V (a 1 X 1 + a 2 X 2 ) = a 2 1V (X 1 ) + a 2 2V (X 2 ) + 2a 1 a 2 Cov (X 1, X 2 ) = a 2 1σ 2 X 1 + a 2 2σ 2 X 2 + 2a 1 a 2 ρ X1,X 2 σ X1 σ X2 instead of explicitly specifying coefficient b, it suffices to know fraction c of variance explained by non-idiosyncratic components in order to calculate overall variance V (r) = V (a 1X 1 + a 2 X 2 ) c Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 93 / 97

94 Estimating default correlations Correlations from factor models Based on asset value models given factor models of two assets asset covariance is given by r = a 1 X 1 + a 2 X 2 + bɛ, r = a 1 X 3 + a 2 X 4 + bε, Cov (r, r) =a 1 a 1 Cov (X 1, X 3 ) + a 1 a 2 Cov (X 1, X 4 ) + a 2 a 1 Cov (X 2, X 3 ) + a 2 a 2 Cov (X 2, X 4 ), hence, asset correlation can be calculated according to the standard formula by Cov (r, r) ρ r,r = σ r σ r Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 94 / 97

95 Example Estimating default correlations Based on asset value models The company ABC is associated with country-specific risks of countries Germany and Spain. According to the financial analyst in charge, the cash-flows of the company are produced in both countries at a ratio of 3 to 1, and the fraction of the overall variance explained by country specific factors is 0.4. Determine the coefficients of the factor model, when the volatility of the German index is σ G = 1.4, the volatility of the Spain index is σ S = 1.2 and their correlation is given by ρ G,S = 0.3. model specification: r = 0.75X G X S + bɛ, ɛ N (0, 1) Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 95 / 97

96 Example Estimating default correlations Based on asset value models variance induced by country specific factors V (0.75X G X S ) = V (X G ) V (X S ) Cov (X G, X S ) = ( ) = overall variance: 0.4 V (r) = V (r) = = Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 96 / 97

97 Example Estimating default correlations Based on asset value models Remarks: overall variance in factor model depends on scaling: instead of coefficients 0.75 and 0.25 we also could have taken values 3 and 1, or any other multiple, leading to different asset variances covariance between assets depends on scaling, too however, through focussing on correlations, units of measurement become normalized: scaling effects drop out for both assets Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik, PhD 97 / 97