ERM-101-12 (Part 1) Measurement and Modeling of Depedencies in Economic Capital
Related Learning Objectives 2b) Evaluate how risks are correlated, and give examples of risks that are positively correlated and risks that are negatively correlated 2c) Analyze and evaluate risk aggregation techniques, including use of correlation, integrated risk distributions and copulas 2f) Analyze the importance of tails of distributions, tail correlations, and low frequency / high severity events 2h) Construct approaches to managing various risks and evaluate how an entity makes decisions about techniques to model, measure and aggregate risks including but not limited to stochastic processes 5b) Define the basic elements and explain the uses of economic capital 5c) Explain the challenges and limits of economic capital calculations and explain how economic capital may differ from external requirements of rating agencies and regulators Key Points of This Reading 1) Understand different correlations (e.g. Pearson correlation, Spearman correlation, and Kendall Tau correlation) 2) Understand different risk aggregation methodologies (e.g. Variance- Covariance matrix, Copulas, and Causal modeling)
Economic Capital Prob. Expected Reserve VaR(95%) Economic Capital Mean 95 th Percentile It is the amount of capital needed to cover losses at a certain risk tolerance level (e.g. VaR(95%))
Risk Aggregation EC (Individual Risks) EC (Total Company) Prob. Expected Reserve VaR(95%) Economic Capital Prob. Expected Reserve Mean VaR(95%) Economic Capital 95 th Percentile Aggregation Method Prob. Expected Reserve VaR(95%) Economic Capital Mean 95 th Percentile Mean 95 th Percentile Prob. Expected Reserve VaR(95%) Economic Capital Mean 95 th Percentile
Solvency II Three Pillars Approach 1. Quantitative requirements 2. Qualitative requirements 3. Supervisory reporting and disclosure 1. Solvency Capital Requirement (SCR) 2. Minimum Capital Requirement (MCR) 1. European Standard Formula 2. Internal Model
Dependency vs. Correlation Dependency There is some link between two random variables, but it does NOT need to follow a linear pattern Correlation There is some link between two random variables, but it does follow a linear pattern Y Y X X
Deficiencies of Using Correlation 1. Correlation is a scalar measure of dependency 2. Possible values of correlation depend on the marginal distribution of the risks 3. Perfectly positively (negatively) dependent risks do not necessarily have a correlation of 1 (-1) 4. A correlation of zero does not imply independence between risks 5. Correlation is not invariant under monotonic transformations 6. Correlation is only defined when the variances of the risks are finite
Three Measures of Correlation Pearson's Rho Spearman's Rho Kendall's Tau ρ ( XY, ) = Cov[ X, Y ] Var( X ) Var( Y ) ρ ( XY, ) = rank Cov[ r( X ), r( Y )] Var( r( X )) Var( r( Y )) τ = S 0.5 nn ( 1) Value Correlation Rank Correlation Rank Correlation It is calculated directly from two data series Changing the value of an individual observation will change the value of Pearson s rho Pearson s rho is attractive as it is widely used and easy to calculate It is calculated from the position (rank) of the variables Changing the value of an individual observation (but not the position of the observation) will not change a rank correlation coefficient Because rank correlation coefficients to not depend on the underlying shape of data series, only the relative position of observations, their results are always valid However, it is only a valid measure of association when the data series on which it is being calculated are jointly elliptical However, whereas Pearson s rho can be used directly in some common multivariate distributions such as the normal and t, the rank correlation coefficients are more usually combined with copulas
Pearson s Rho X and Y are values RN X Y X 2 Y 2 XY (X-x) 2 (Y-y) 2 (X-x)(Y-y) 1 10 20 100 400 200 900 0 0 2 95 25 9025 625 2375 3025 25 275 3 15 10 225 100 150 625 100 250 4 35 15 1225 225 525 25 25 25 5 45 30 2025 900 1350 25 100 50 Sum 200 100 12600 2250 4600 4600 250 600 Cov[ X, Y ] 600 ρ ( XY, ) = = = 0.56 Var( X ) Var( Y ) (4600)(250) r XY n XY i i Xi Yi 5(4600) (200)(100) 2 2 5(12600) ( ) ( ) ( 200) 5(2250) ( 100 ) i i i i = = = 0.56 2 2 2 2 n X X n Y Y
Ranks RN X Y Rank(X) Rank(Y) 1 10 20 1 3 2 95 25 5 4 3 15 10 2 1 4 35 15 3 2 5 45 30 4 5 Sum 200 100 15 15 Ascending Order
Spearman s Rho X and Y are ranks RN X Y X 2 Y 2 XY (X-x) 2 (Y-y) 2 (X-x)(Y-y) (X-Y) 2 1 1 3 1 9 3 4 0 0 4 2 5 4 25 16 20 4 1 2 1 3 2 1 4 1 2 1 4 2 1 4 3 2 9 4 6 0 1 0 1 5 4 5 16 25 20 1 4 2 1 Sum 15 15 55 55 51 10 10 6 8 ρ ( XY, ) = Cov[ X, Y ] 6 = = 0.60 Var( X ) Var( Y ) (10)(10) s r T 2 ( X 1 t Y ) t= t 8 XY, 2 2 = 1 6 = 1 6 = 0.6 TT ( 1) 5 (5 1)
Kendall's Tau Concordant Pair Discordant Pair (X 2,Y 2 ) (X 1,Y 1 ) (X 1,Y 1 ) (X 2,Y 2 ) (X 2 X 1 ) and (Y 2 Y 1 ) Same Sign (X 2 X 1 ) and (Y 2 Y 1 ) Opposite Sign Kendall's Tau = τ = C D 1 nn ( 1) 2
Kendall's Tau RN X Y 1 2 3 4 1 1 3 2 5 4 C (+,+) 3 2 1 D(+,-) C (-,-) 4 3 2 D(+,-) C (-,-) C (+,+) 5 4 5 C (+,+) D(-,+) C (+,+) C (+,+) (X 2 X 1 ) = 5 1 = +4 (Y 2 Y 1 ) = 4 3 = +1 C(+,+) (X 3 X 1 ) = 2 1 = +1 (Y 3 Y 1 ) = 1 3 = -2 D(+,-) Kendall's Tau = C D 7 3 τ = = = 0.5 nn ( 1) 0.5 (5 (5 1)) C D 7 3 τ = = = 0.4 C + D 7+ 3 0.4