Stochastic Model Efficiency Applications: Cluster-Distance Sampling to Tackle Sampling Errors and Bias Yvonne C. Chueh, PhD, ASA Paul H. Johnson, Jr., PhD Joint work between the University of Illinois at Urbana-Champaign (UIUC) and Central Washington University (CWU) Funded by The Actuarial Foundation Y. Chueh and P. Johnson Jr (IME 2013) July 2013 1 / 20
Introduction and Purpose Introduction Practitioners/researchers are challenged to make credible inferential statements about the population distribution of important economic variables These distributions often involve a large number of policyholders and economic scenarios Well known challenge of running a stochastic asset/liability model is the long run-time Successful projection of these population distributions is important for actuaries Pricing, reserving, budgeting risk capital Y. Chueh and P. Johnson Jr (IME 2013) July 2013 2 / 20
Introduction and Purpose Introduction (Continued) To analyze the population distrbution of economic outcomes, model efficiency approaches are often utilized Model Efficiency : Mathematical approaches that reduce the number of economic scenarios required to achieve a given level of precision in stochastic actuarial modeling (Rosner 2011) Model efficiency approaches include (Rosner 2011): Transfer Scenario Order Importance Sampling Curve Fitting Representative Scenarios Y. Chueh and P. Johnson Jr (IME 2013) July 2013 3 / 20
Introduction and Purpose Purpose We discuss the development of CSTEP , a high performance computation software tool that utilizes representative scenarios to reduce sampling errors arising from small stochastic economic scenario samples, especially at the tails Overview of CSTEP Description of method of Representative Scenarios Original definition of distance between two economic scenarios employed in CSTEP New (economic) definition of distance between two economic scenarios employed in CSTEP Brief comparison of original and economic distance definitions using statutory ending surplus data from a real block of general annuities, provided by Ed Cowman, FSA Y. Chueh and P. Johnson Jr (IME 2013) July 2013 4 / 20
Representative Scenarios CSTEP CSTEP: Cluster Sampling for Tail Estimation of Probability (Chueh and Johnson 2012, Johnson et al. 2013) Upgrade from SALMS (Stochastic Asset Liability Model Sampling) used since 2003 CSTEP is open source, high performance computation software Universe capacity: 8,388,608 scenarios with up to 4500 time periods each Flexible sample size, reversible and reusable sampling Rate sampling (interest rate, equity return, index) Y. Chueh and P. Johnson Jr (IME 2013) July 2013 5 / 20
Representative Scenarios Representative Scenarios Consider a population of N rate paths Editable distance formulas similar to Euclidean distance are used to select n representative (pivot) scenarios where n <<< N (Chueh 2002) Choose an arbitrary path out of the N paths and call it Pivot 1 Calculate the distances of the remaining N - 1 paths to Pivot 1, the path with the largest distance to Pivot 1 is Pivot 2 Assign each of remaining N - 2 paths to the closest of Pivots 1 and 2, forming two disjoint paths Calculate the distances of the remaining N - 2 paths to Pivots 1 and 2, the path with the largest distance to Pivots 1 and 2 is Pivot 3 Assign each of remaining N - 3 paths to the closest of Pivots 1, 2, and 3, forming three disjoint paths ... repeat until n Pivots A probability is then assigned to each representative scenario Y. Chueh and P. Johnson Jr (IME 2013) July 2013 6 / 20
Scenario Distance Formulas Scenario Distance Formulas In order to employ the method of representative scenarios, we need to be able to calculate the distance between two scenarios and tie that distance to the model output The original version of CSTEP employed a theorem of high-dimensional continuity (Continuity Theorem 1) The new version of CSTEP employs a modified theorem of high-dimensional continuity that improves the tail fit for volative economic scenarios and equity-based insurance guarantees (Continuity Theorem 2) Y. Chueh and P. Johnson Jr (IME 2013) July 2013 7 / 20
Scenario Distance Formulas Continuity Theorem 1 Consider two n-period rate scenarios: x = ( r 1 , r 2 , ..., r n ) and s = ( r ′ 1 , r ′ 2 , ..., r ′ n ) Let CF t denote the net cash flow at the end of period t under scenario x ; CF ′ t is similarly defined under scenario s Let f ( x ) = � n � t 1 + r k and f ( s ) = � n � t 1 1 t = 1 CF t t = 1 CF ′ k = 1 t k = 1 1 + r ′ k Let the distance between x and s : �� n t = 1 [ � t 1 + r k − � t 1 1 k ] 2 d X ( x , s ) = k = 1 k = 1 1 + r ′ Let the distance between f ( x ) and f ( s ) : d Y ( f ( x ) , f ( s )) = | � n � t � t 1 1 t = 1 [ CF t 1 + r k − CF ′ k ] | k = 1 t k = 1 1 + r ′ Y. Chueh and P. Johnson Jr (IME 2013) July 2013 8 / 20
Scenario Distance Formulas Continuity Theorem 1 (Continued) Given an arbitrary risk scenario s ∈ X , ∀ ǫ > 0 ∃ δ = ǫ 2 √ nM ∋ ∀ scenario vectors x ∈ X : d X ( x , s ) ≤ δ is a sufficient condition ∋ d Y ( f ( x ) , f ( s )) ≤ ǫ M = max t ( | CF t | , | CF ′ t | ) The above illustrates uniform continuity Y. Chueh and P. Johnson Jr (IME 2013) July 2013 9 / 20
Scenario Distance Formulas Continuity Theorem 1: Proof [ d Y ( f ( x ) , f ( s ))] 2 = | � n � t � t 1 1 k ] | 2 1 + r k − CF ′ t = 1 [ CF t k = 1 t k = 1 1 + r ′ ≤ ( 2 M ) 2 | � n t = 1 [ � t 1 + r k − � t 1 1 k ] | 2 k = 1 k = 1 1 + r ′ ≤ n ( 2 M ) 2 � n t = 1 [ � t 1 + r k − � t 1 1 k ] 2 k = 1 k = 1 1 + r ′ = n ( 2 M ) 2 [ d X ( x , s )] 2 ≤ n ( 2 M ) 2 δ 2 = ǫ 2 Then: d X ( x , s ) ≤ δ ensures that d Y ( f ( x ) , f ( s )) ≤ ǫ Y. Chueh and P. Johnson Jr (IME 2013) July 2013 10 / 20
Scenario Distance Formulas Original CSTEP Distance Formulas Let p denote a pivot scenario Significance Method: �� n t = 1 [ � t 1 1 + r k ] 2 d X ( x , p ) = k = 1 Relative Present Value (RPV) Method: �� n t = 1 [ � t 1 + r k − � t 1 1 k ] 2 d X ( x , p ) = 1 + r p k = 1 k = 1 Y. Chueh and P. Johnson Jr (IME 2013) July 2013 11 / 20
Scenario Distance Formulas Continuity Theorem 2 Let C t ≈ CF t and C ′ t ≈ CF ′ t (determined by historical experience for a similar block of business or a sample(s) of the population distribution) Let the distance between x and s : �� n � t 1 � t 1 k ] 2 d ∗ X ( x , s ) = t = 1 [ C t 1 + r k − C ′ k = 1 t k = 1 1 + r ′ Given an arbitrary risk scenario s ∈ X , ∀ ǫ > 0 ∃ δ = ǫ > 0 ∋ ∀ scenario vectors x ∈ X : d ∗ X ( x , s ) ≤ δ is a sufficient condition ∋ d Y ( f ( x ) , f ( s )) ≤ ǫ Y. Chueh and P. Johnson Jr (IME 2013) July 2013 12 / 20
Scenario Distance Formulas Continuity Theorem 2: Proof [ d Y ( f ( x ) , f ( s ))] 2 = | � n � t � t 1 1 k ] | 2 t = 1 [ CF t 1 + r k − CF ′ k = 1 t k = 1 1 + r ′ ≈ | � n � t � t 1 1 k ] | 2 t = 1 [ C t 1 + r k − C ′ t k = 1 k = 1 1 + r ′ X ( x , s )] 2 = [ d ∗ ≤ δ 2 = ǫ 2 Then: d ∗ X ( x , s ) ≤ δ ensures that d Y ( f ( x ) , f ( s )) ≤ ǫ Y. Chueh and P. Johnson Jr (IME 2013) July 2013 13 / 20
Scenario Distance Formulas New (Economic) CSTEP Distance Formulas Let p denote a pivot scenario Economic Significance Method: �� n � t 1 1 + r k ] 2 d ∗ X ( x , p ) = t = 1 [ C t k = 1 Economic Present Value (EPV) Method: �� n � t � t 1 1 k ] 2 d ∗ X ( x , p ) = t = 1 [ C t 1 + r k − C ′ k = 1 k = 1 t 1 + r ′ Y. Chueh and P. Johnson Jr (IME 2013) July 2013 14 / 20
Comparison of RPV and EPV Methods Cowman Data Statutory ending surplus data was the stochastic model output from a real block of general annuities with annual cash flows using proprietary stochastic scenario generator 10,000 stochastic economic interest rate scenarios were considered, where each scenario is a random path of 20 one-year US treasury yield rates x = ( i 1 , i 2 , , i 20 ) Compare the percentage differences of tail metrics under the RPV and EPV methods with 100 scenarios each, to corresponding tail metrics for the full-run distribution of 10,000 scenarios For EPV method, the average statutory ending surplus of an RPV sample of 100 scenarios at each time divided by the average statutory ending surplus at the end of first year was an approximation for C t Y. Chueh and P. Johnson Jr (IME 2013) July 2013 15 / 20
Comparison of RPV and EPV Methods Percentage Difference (VaR): 100 Scenarios, RPV vs EPV 10.0% 5.0% 0.0% 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 RPV -5.0% 2.pdf EPV -10.0% -15.0% -20.0% Y. Chueh and P. Johnson Jr (IME 2013) July 2013 16 / 20
Comparison of RPV and EPV Methods Percentage Difference (CTE): 100 Scenarios, RPV vs EPV 2.00% 0.00% 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 -2.00% -4.00% -6.00% RPV -8.00% 3.pdf EPV -10.00% -12.00% -14.00% -16.00% -18.00% Y. Chueh and P. Johnson Jr (IME 2013) July 2013 17 / 20
Applications and Next Steps Applications and Next Steps CSTEP can be used with another software application: AMOOF (Chueh and Curtis 2004, Johnson et al. 2013) We have validated the new CSTEP distance formulas using the Cowman data We are conducting a study of new CSTEP distance formulas in conjunction with Milliman with the intent to publish our findings in a peer-reviewed actuarial journal Y. Chueh and P. Johnson Jr (IME 2013) July 2013 18 / 20
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