Credit Risk Cheat Sheet, Cheat Sheet of Credit and Risk Management

Download Credit Risk Cheat Sheet and more Cheat Sheet Credit and Risk Management in PDF only on Docsity! Credit Risk Cheat Sheet Credit Risk Cheat Sheet Abstract This short document lists the main formulas, concepts and definitions of the class. Framed definitions starting with É are the key concepts of the class that must be known. Æ are important information to keep in mind as general knowledge. Î refers to traps and points of attention. BOND VALUATION Lecture 1 É Price of a bond. The price of a bond serving fixed coupons C (fixed rate bond) in (t1, ..., tn), which maturity is T and nominal N is: B̄A(0, T ) = n ∑ i=1 C (1+ rAi ) ti + N (1+ rAT ) T where: rAi = ri + s A with ri , the risk-free rate in i, and s A, the so-called "Z-spread" or more com- monly, the spread of the counterparty A. Î What is a risk-free rate?. the risk-free rate is usually considered as the Constant Maturity Swap (CMS) price, for different maturities. For ex- ample, in Europe, the 10 years, risk-free rate, would be the CMS 10y that exchanges a fixed rate with EURIBOR 3M (ticker BBG being EUSA10Y). É Price of a bond – continuous rate and coupon. Considering a contin- uous coupon, the formula for a bond of nominal 1, is: B̄A(0, T ) = 1+ (c − rA) 1− e−r AT rA (only if we consider ri and s A as constant) Î A chicken and egg problem. Note that this is a chicken and egg problem: the spread is extracted from the price and the price is deduced from the spread. In practice, the market, by buying and selling bonds, agrees to a price from which one can extract a spread to price new bonds or credit derivatives. É The implied probability of default. The no-arbitrage assumption gives us: Q(τ > T | τ > t) = B̄A(t, T ) B(t, T ) where τ is the rv of the time of default. And thus, given the value of risky bonds and risk-free bonds (B(t, T )): 1− PD =Q(τ > T | τ > t) = e−s A(T−t) Æ Bootstrapping the spreads. Bootstrapping is an iterative process that aims at extracting zero-coupon rates from bonds with coupons. Suppose firm A has n bonds (B̄A1 , B̄ A 2 , ..., B̄ A n) of nominal 1, with a coupon rate c. • from B̄A1 , we know that:B̄ A 1 = 1+c (1+rA1 ) 1 and thus that r A 1 = 1+c BA1 − 1; • from B̄A2 , we know that:B̄ A 2 = c (1+rA1 ) 1 + 1+c (1+rA2 ) 2 , from which, know- ing rA1 , we can extract the value of r A 2 ; • going on like this, we can extract the value of (rA1 , r A 2 , ..., r A n ). Of course this technique can be applied to risky or non-risky bonds. É Reduced form models. Reduced-form models consist in modeling the conditional law of the random time of default: τ < t + dt | τ > t The most applied reduced-form models assumes that the probability of de- fault is the product of the infinitesimal time step with a so-called default intensity (considered constant here): Q(τ < t + dt | τ > t) = λ× dt which is equivalent to say that: the random time of default follows an ex- ponential law of rate λ. Thus, the survival probability of the studied counterparty is: Q(τ > t) = exp (−λt) which is consistent with the no-arbitrage formula introduced earlier. Æ Hazard rate modeling. • Constant: Time homogeneous Poisson Process; • Deterministic: Time deterministic inhomogeneous Poisson Pro- cess; • Stochastic: Time-varying and stochastic Poisson Process as the Cox, Ingersoll, Ross (CIR) model. É Implied probability of default taking into account recovery. Let R be the recovery rate, the implied probability of default taking into account recovery is: PD= 1− B̄(0,T )B(0,T ) 1− R 1 Credit Risk Cheat Sheet É The Expected Loss (EL). The Expected Loss (EL) on a credit exposure can be split in three parts: • PD: the Probability of Default (see above); • LGD: the Loss Given Default is equal to 1−R, where R is the recovery rate, that is the proportion of the exposure that the lender retrieves; • EAD: the Exposure At Default, that can be fixed (for bullet bonds for examplea) or not (exposure of derivatives towards a counterparty for example). Resulting in: EL= E(EAD× 1{τ<M} × LGD) = ︸︷︷︸ assump. E(EAD)×E(1{τ<M}) ︸ ︷︷ ︸ PD ×E(LGD) aBonds which notional is reimbursed at the maturity PRICING CDS Lecture 1 The value of the fixed leg of a CDS is: Fixed(0, T ) = s(0, T ) 1− e−(r+λ)T r +λ The value of the floating leg of a CDS is: Floating(0, T ) = (1− R) λ λ+ r 1− e−(λ+r)T
É The spread of a CDS. The spread of a CDS is: s = λ(1− R) É The sensitivity of a CDS. The sensitivity of the MtM is the risky dura- tion, DV: DV(t, T,λ) = 1− e−(r+λ)(T−t) r +λ É Valuation of a CDS. Let s0, be the spread in t = 0, and st , the spread today, in t. The Present Value of the protection seller is: PV (s0, st ) = DV(0, t,λ)(s0 − st ) Æ Other CDS-like products. • CMCDS: Constant Maturity Credit Default Swaps are like CDS ex- cept that the premium paid by the protection buyer is calculated every 3 months (or 6 months) based on the spread of the refer- ence entity at that time. • ABSCDS: Asset-Backed Security Credit Default Swaps are CDS whose reference is not an entity, but an ABS (see below). TRANSITION MATRICES Lecture 2 Æ Ratings. Ratings are an evaluation of the credit risk of a debtor, performed by credit rating agencies. Moody’s S&P Fitch Rating description Aaa AAA AAA Prime Aa1 AA+ AA+ High gradeAa2 AA AA Aa3 AA- AA- A1 A+ A+ Upper medium gradeA2 A A A3 A- A- Baa1 BBB+ BBB+ Lower medium gradeBaa2 BBB BBB Baa3 BBB- BBB- Ba1 BB+ BB+ Non-investment grade / SpeculativeBa2 BB BB Ba3 BB- BB- B1 B+ B+ Highly speculativeB2 B B B3 B- B- Caa1 CCC CCC Substantial risksCaa2 CCC CCC Caa3 CCC CCC- Ca CC CC Extremely speculative C C C Default imminent C RD DDD In default É Transition matrices. In credit risk, a transition matrix, Mt,t+1 = (mi j)i j , is a matrix where: mi j = P(Gradet+1 = j | Gradet = i) É The generator of an homogeneous Markov chain. The generator for a Markov chain (Mt,t+n)n is the matrix Q so that: ∀(t, T ), Mt,T = exp ((T − t)Q) with exp(A) = ∑ n≥0 An n! Would such a matrix exist, we have: Q = ∑ n>0 (−1)n−1 (Mt,t+1 − I)n n Î Markovian property. The existence of such a generator matrix is based on the assumption of the Markov propriety that states (in the dis- crete case) that: P(Gradet+1 = j | Gradet = i) = P(Gradet+1 = j | Gradet = i, Gradet−1 = it−1, ...,Gradet−h = it−h) which is not observed in practice. STATISTICAL TOOLS TO ASSESS CREDIT RISK Lecture 2 É Logistic regression. The logistic regression model can be defined the following ways: p(X ) = P(Y = 1|X ) = eβ0+β1X1+...+βp Xp 1+ eβ0+β1X1+...+βp Xp or equivalently ln  p(X ) 1− p(X ) ‹ = β0 + β1X1 + . . .+ βpXp where X = X1, . . . , Xp
Æ Pros and Cons of the logistic regression. • Pros: no close-formula but easy to fit with MLE, can easily be interpreted thanks to odds ratios. • Cons: cannot capture non-linear relationships between the vari- ables. 2 Credit Risk Cheat Sheet RETURN AND VALUE CREATION FOR A BANK Lecture 5 É Return On Equity (ROE). The Return On Equity of a Business Line is: ROE= Net Income of the BL Regulatory Capital allocated to the BL É Risk Adjusted Return of Capital (RAROC). The Risk Adjusted Return On Capital of a Business Line is: RAROC= Net Income of the BL−Average loss of the BL Economic Capital allocated to the BL É Weight Average Cost of Capital (WACC). The Weighted Average Cost of Capital is: WACC= (r + k1)T1 + (r + k2)T2 + (r + kd )D where, T1/r1, T2/r2 and D/rD is the proportion / the cost (spread) of Tier 1 capital, Tier 2 Capital and debt in the liabilities of the bank and r the risk-free rate. Î WACC and hurdle rate.. The WACC is the hurdle rate of bank activi- ties, that is, the minimum return necessary to create value. É Economic Value Added (EVA). The Economic Value Added for a bank is: EVA = Net Income of the bank − Average loss of the bank − WACC ∗ Liabilities Another definition of the Economic Value Added for a bank is: EVA = Net Income of the bank − Average loss of the bank − k ∗ Economic Capital where k is the cost of capital, that is: k = (r + k1)T1 + (r + k2)T2. É Risk Adjusted Returns On Risk Adjusted Capital (RARORAC). The Risk Adjusted Return On Risk Adjusted Capital for a bank is: RARORAC= RAROC− k Another definition of the RARORAC is: RARORAC= RAROC− k× Allocated Economic Capital Used Economic Capital É Cost of capital – A CAPM approach. The Capital Asset Pricing Model states that the average return of the stock i, E(ri), follows: E(ri) = βi(E(rM )− r f ) + r f where βi = ρi,M σi σM , rM is the return of the whole stock market, σM is the volatility of the whole stock market, σi is the volatility of the stock i, ρi,M is the correlation between the returns of the market and the ones of i, r f is the risk-free rate. É Cost of capital – The Gordon-Shapiro Approach. The Gordon-Shapiro approach states that the valuation of a firm, P, is a function of the expected growth g of its cash-flows (Dt ) and the expected return k of the sharehold- ers: P = ∞ ∑ t=1 Dt (1+ k)t = D1 k− g with Dt = Dt−1 × (1+ g) Æ What is the cost of equity?. Shares, contrary to bounds, do not specify the future returns. Nonetheless, their expected return, necessary to compute the WACC, can be extracted from the market using the CAPM approach (E(ri)) or the Gordon Shapiro approach (k). COUNTERPARTY RISK Lecture 6 É Counterparty risk – Definition. The counterparty risk is defined as the risk that the counterparty to a transaction defaults before the final set- tlement of the transaction’s cash-flows. It can be a bilateral risk as the exposure may vary with the market conditions. Î What drive the counterparty risk?. Counterparty risk is driven by (i) OTC contract’s market value risk drivers, (ii) the counterparty credit spread, and (iii) the correlation between the underlying and the proba- bility of default of the counterparty (Wrong Way Risk – WWR – and Right Way Risk – RWR). Æ Use of counterparty risks measurement. • Pricing: to take into this risk when pricing a derivative (Counter- party Value Adjustment – CVA); • Risk management: for internal and regulatory purposes (SIMM, KCVA, PFE, etc. – see below). É Exposures – EE, PFE, MPFE, EPE, EEPE. V (t) denotes the market value of a derivative, at time t. Counterparty exposure is equal to E(t) = V (t)+ =max(0, V (t)). Expected Exposure (EE). EE(t) = EQ(E(t)) Potential Future Exposure (PFE). PFEα(t) = qα(E(h)) Maximum Potential Future Exposure (MPFE). MPFE(t) =max h<s (E(h)) Effective Positive Exposure (EPE). EPE(t) = 1 t ∫ t 0 EE(s)ds Expected Effective Positive Exposure (EEPE). EEPE(t) = 1 t ∫ t 0 max h<s (EE(h))ds PFE is then considered as the EAD in the regulatory formula used to com- pute RWA. É CVA – Credit Value Adjustment. CVA= (1− R) ∫ T 0 EQ(E(s))dSC (s) Where SC is the survival function of the counterparty. Æ Two ways to mitigate counterparty risk: • Netting: in presence of multiple trades with a counterparty, net- ting agreements allow, in the event of default of one of the coun- terparties, to aggregate the transactions before settling claims (E(t) = (ΣV (i))+ 6= Σ(V (i))+). • Collateralizing: collateral is a property or other assets that a counterparty offers as a way for the counterpary to secure the 5 Credit Risk Cheat Sheet exposure. Î Other counterparty risk metrics. There are several other counter- party risk metrics such as: • SIMM: SIMM stands for Standard Initial Margin Model and is used to compute the initial margin of non-cleared derivatives; • KCVA is a VaR estimate of the CVA required by Basel III that im- pose a capital charge. Æ IRC and CRM, credit market risk metrics: IRC and CRM are credit market risk metrics that are not counterparty risk metrics as they mea- sure the credit risk due to the potential default(s) of the underlying refer- ence(s) on derivative products. With the VaR and the stressed VaR, these measure are part of the market RWA. É IRC – Incremental Risk Charge The IRC is a risk metrics that captures risk due to adverse rating migrations on vanilla credit securities such as bonds and CDS on corporates and sovereigns within the trading book. É CRM – Comprehensive Risk Measure • The CRM is a risk metrics that, as the IRC, captures the risks due to adverse rating migrations on credit securities. It applies to credit cor- relation portfolios (CDO, CLO, CBO, etc.) within the trading book. • The CRM also captures risks due to credit spread, recovery rates and base correlations variations. 6