C redit V alue A djustment by Expected Future Exposure M ethod M u M . Liu, CQF
Agenda • What is CVA and why it must be considered? • Valuation M ethodology – Current Exposure (“ CE” ) – Expected Future Exposure (“EFE”) • M odules needed for “ EFE” – Interest rate simulation model • Spot rate models i.e. Vasicek, Hull-White, etc • Forward rate model i.e. HJM – Probability of default model • Conclusion – Valuation Procedure • Q & A 2
CVA Overview • Company vs. Bank example • Exposure – Expected Positive Exposure (EPE) – Expected Negative Exposure (ENE) • Interpretation of ‘CVA’ • CVA is applied to: – Swaps (interest rate, currency, equity, commodity) – Other OTC private contracts • Bilateral Nature of Swaps • Accounting: IFRS13, FASB ASC 815, and ASC 820. 3
CVA Overview A Typical Exposure Profile for Interest Rate Swaps 4
Valuation M ethodology • CVA can be represented by the following risk-neutral expectation in terms of exposure. B = − Q 0 1 (1 ) CVA R E τ ≤ τ { T } B τ where 1 is the indicator function that can be interpreted as the risk-neutral probability of default when computed within an expectation; R is the recovery rate; is the discount factor; is the exposure at time . The exposure calculation varies depending on which method is used. • This expectation can be rewritten as: ∞ B ∫ = − + δ 0 0 (1 ) ( , ) ( ) CVA R PD t t t E t dt B τ Probability of Default M odel Interest Rate M odel; PCA Spectral Analysis Spot rate models, HJM model, LIBOR market model, etc 5
M odeling Toolkits Things we need for CVA calculation: Principal Component Analysis Spectral Decomposition (Attribution Analysis) An Interest Rate M odel Heath-Jarrow-M orton (HJM ) A PD Bootstrapping Algorithm Iterative algo A Programming Language M atlab 6
Principal Component Analysis (PCA) Forward curve as of M ay 11, 2016. There are more than 30 market instruments that constitute the forward curve. Principal Component Analysis helps us answer the question: which are the most influential tenors? 7
Principal Component Analysis (PCA) PCA is an approach to determine the dominant and independent factors of the evolution of a multivariate system as a whole. 1. Decorrelation: PCA factors are independent and their covariance is zero. 2. Reduced dimensionality 3. Data structure reveal As mentioned before, HJM model is usually implemented with multiple factors. The shape of a yield curve is very complex - in fact, there are many schools of thought about the relationships among the tenors on the yield curve. For instance, does the short-term tenor affect the shape of the long end of the yield curve, or vise versa? The yield curve can move in many different ways, such as parallel shift, concave and convex at different tenors. In determining which tenors have the most significant impact on the shape of the yield curve, we use Principle Component Analysis (PCA) to analyze the relationships among all the tenors based on the forward rates. The tenors that have the most impact on the shape of the yield curve are known as ‘factors’ in PCA analysis. It turns out that the dimensions of the HJM model are determined by the number of factors. It would be the modeler’s choice as to how many factors to model, but typically three factors would be sufficient to model the dynamics of a yield curve. Empirically, the first factor explains 70% of the yield curve variation; first two factors would explain a total of 87% of the variation; and the first three factors would explain about 95% - 97% of the total variation. Thus, modeling more than five factors would be unnecessary. 8
Principal Component Analysis (PCA) The dynamics of yield curve movements are complex. Principal component analysis helps us to identify the key factors that drive the movements. PCA provides independent systematic factors that attribute to the overall movement of the yield curve. = + + + + + (1) (2) (3) ( ) n ... f f e X e X e X e X + V 1 2 3 t t t n To get the components (a.k.s factors), we decompose the covariance matrix obtained from forward rates on all the tenors. ∑ = Α V V ' { } = (1) (2) ( ) n , ..., V e e e λ > λ > λ A is a diagonal matrix with eigenvalues ranked in a descending order ... 1 2 n λ 0 0 1 Α = 0 ... 0 λ 0 0 N • Eigen vectors V are linear combinations of data changes. They form an orthogonal basis. • Eigen values represent dispersion of data around the eigendirection, or variance of curve movement with respect to the orthogonal direction. 9
Principal Component Analysis (PCA) • Top three factors (components) explain 98.6% of the total variation of the yield curve. • Eigenvectors are orthogonal direction of data changes in the direction of minimum variance. (mathematical definition) • Analogy – intuitive interpretation 10
PCA Factors and Volatility τ = λ = i ( , ) , 1,2,3... v t e i N i i The volatility structure for HJ M model is implemented based on this relationship. 11
M odeling Toolkits Things we need for CVA calculation: Principal Component Analysis Spectral Decomposition (Attribution Analysis) An Interest Rate M odel Heath-Jarrow-M orton (HJM ) A PD Bootstrapping Algorithm Iterative algo 12
Heath-J arrow-M orton (HJ M ) M odel • SDEof forward rate under M usiela Parameterization. ∂ τ k k τ ( , ) ∑ f t ∑ ∫ τ = τ + + τ ( , ) ( , ) ( , ) ( , ) df t v t v t s ds dt v t dX ∂ τ i i i 0 = = 1 1 i i • M odels the whole stochastic evolution of the entire yield curve. • Drift rate is a function of volatility. • Non-M arkovian process • If implemented under a tree model, the branches would look very ‘bushy’. • Below is a forward rates dynamics expressed in form of SDE. This SDE is not necessarily arbitrage-free. In α ( ; ) t T order to have an equivalent martingale measure, the functional form of must be chosen in conjunction with the volatility structure. In other words, drift rate will be automatically determined when a volatility structure is specified. = α + σ df t T ( ; ) ( ; ) t T dt ( , ) t T dW t ( ) Challenges: • Drift rate discretization • Volatility structure interpolation • PCA – principal component analysis (involves Eigen value/ Eigen vectors) 13
M PR of Interest Rate Visualization M arket price of risk is, in fact, a stochastic process itself. Source: The M arket Price of Interest-rate Risk: M easuring and M odelling Fear and Greed in the Fixed-income M arkets. Riaz Ahmad and Paul Wilmott. 14
HJ M M odel Derivation (Brief) − ∫ T t f t s ds ( ; ) = Z t T ( ; ) e ∂ = − ∂ f t T ( ; ) log Z t T ( ; ) T Differentiating our forward rate equation (1), we can obtain the following SDE: ∂ = − ∂ ( , ) log ( ; ) df t T Z t T T ∂ ∂ 1 = − − σ − σ 2 r t ( ) ( , ) t T dt ( , ) t T dW t ( ) ∂ ∂ 2 T T Under the risk-neutral measure, the SDEbecomes: ∂ ∂ 1 = σ − − σ 2 Q ( , ) ( , ) ( ) ( , ) ( ) df t T t T r t dt t T dW t ∂ ∂ 2 T T Now applying the chain rule on the drift term, the HJM stochastic differential equation becomes: ∂ ∂ = σ σ − σ Q df t T ( , ) ( , ) t T ( , ) t T dt ( , ) t T dW ( ) t ∂ ∂ T T The term r(t) vanished because we’re differentiating with respect to T . 15
HJ M M odel Derivation (Brief) Let’s introduce a substitution ∂ = − ∂ T σ ( , ) ( , ) v t T t T Integrating both sides: = − ∫ T σ ( , ) t T v t s ds ( , ) t Now our HJM SDE becomes: = − σ + Q ( , ) ( , ) ( , ) ( , ) ( ) df t T t T v t T dt v t T dW t Or, ∫ T = + Q df t T ( , ) v t T ( , ) v t s ds ( , ) v t T dW ( , ) ( ) t t Drift rate structure One of the unique features about HJM model is its non-M arkov feature. The drift rate in HJ M model is highly path-dependent. In a M arkov chain, only the present state of a variable determines the possible future states. M arkov process is a stochastic process without memory. 16
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