Efficient Simulation Design for Risk Management of Large ... · Efficient Simulation Design for...

Efficient Simulation Design for Risk Management of

Large Variable Annuity Portfolios

Mingbin (Ben) Feng, Zhenni Tan, & Jiayi Zhang

Risk Management of Variable Annuities via Simulation

§ VAs: Insurance products that are exposed to market returns & risks§ Market risks exposures are in the form of guaranteed benefits/riders

§ Mortality risks, market risks, & policyholder behaviors§ Interactions among different risks are complicated

§ Multiple riders & customizable features§ Analytical solutions are not sufficient

§ Simulation could be the only reliable method

Ben Feng ([email protected])

Computational Challenge for Simulation

§ Simulation is time consuming§ Complicated individual contract features

§ Large portfolio size and contract diversity

§ Many replications are required for accuracy

§ Predictive analytics can be useful§ Clustering: identify representative contracts

§ Simulation design: more efficient simulation experiments

§ Machine learning/AI: learn & predict without additional simulation



Three-Stages Simulation Design

Compressor

• Given 𝑁-contracts portfolio• Select 𝑛 ≪ 𝑁 “representative” contracts

Simulator

• Given 𝑛 representative contracts • Efficient MC to estimate their values/Greeks

Predictor

• Given 𝑛 representative contracts & the values/Greeks• Predict values for the original 𝑁 contracts & the portfolio

Goal: Estimate the value/Greeks of a VA portfolio comprising 𝑁 contracts• Value of each VA can only be estimated via simulation• Way too many contracts to be simulated individually (𝑁 is too big)

Existing Literature

§ Liu & Tan (2017)§ 11 articles by G. Gan & co-authors in 2013-2017§ Technical reports by T. Colman & co-authors in 2015-2017


Compressor

• Quasi-Monte Carlo (Liu & Tan 2017)• Clustering-based methods (Gan & co-authors)• Moment matching using Johnson’s curve (Colman & co-authors)

Simulator

• Equal number of replications for all contracts• Mostly single-period, some considered nested simulation/multi-periods

Predictor

• Linear approximation (Liu & Tan 2017)• Neural Network/Regression Tree/Random Forest (Colman & co-authors)• Kriging/Radio Basis Functions/Inverse Distance Weighting (Gan & co-authors)

PROBLEM STATEMENT


Stay within the three-step design for illustrationProvide predictive analytical insights & improvements

Demonstrate the effectiveness via improved time & accuracyAchieve improvements with minimal changes to existing design

Selecting Representative ContractsCLUSTERING-BASED COMPRESSOR


The Need for Representative Contracts

§ Large VA portfolio: MC for all 𝑁 contracts is infeasible§ Similar VA contracts: MC for all 𝑁 contracts is wasteful § May be a small & well-chosen set of contracts is sufficient to “represent” the portfolio

§ Idea of clustering:§ Groups similar contracts into clusters (algorithmically)

§ After clustering, the center (centroid) in each cluster “represents” that cluster


§ Given a dissimilarity measure 𝑑 𝑦&, 𝑦( , form 𝑛 ≪ 𝑁 clusters of contracts

§ The goal is to minimize the total in-cluster distance: 𝑆* ≔ 𝑘thcluster, 𝑥* ≔ 𝑘thcentroid

𝑑; == = 𝑑 𝑦&, 𝑥*>

&?@AB∈DE

F

*?@

§ Need to decide: (1) optimal 𝑛, (2) locations of 𝑛 centroids, and (3) memberships of 𝑁 contracts§ Simultaneous optimization is NP-hard

§ Lloyd’s 𝒌-means algorithm is often used in practice (fix 𝑛, iterate (2) & (3) until convergence)

§ For large portfolios (e.g., 𝑁 = 100𝐾 or more) even 𝑘-means could take unacceptably long…then what?


Clustering is good, but how?

§ 400 random points in 0,1 K (e.g., the “portfolio”)§ 8 clusters wanted (e.g., 8 “representative contracts”)§ Euclidean distance (𝑑; denotes total in-cluster distances)§ Benchmarks: overall clustering & simple random sampling


A 2D Illustration


One Existing Proposal: Subset & Clustering§ Divide & conquer: Divide portfolio & centroids into subsets, then perform clustering in each (Gan et al.)

§ Less contracts & less centroids in each subset

§ Redundant centroids § More subsets è higher redundancy

§ Fast selection of representative contracts that have great uniformity property (Liu & Tan, Gan et al.)

§ Requires bounded design space§ In general, uniformity ≠ representativeness§ In some cases, some clusters may have zero memberBen Feng ([email protected])

Other Proposals: Quasi-MC & Latin-Hypercube Sampling

i. Randomly sample (e.g., 50%) of the portfolio (idea: a random sample “represents” the portfolio)ii. Clustering in this sample with all centroids (idea: approximate “overall clustering” as much as possible)


Our Proposal: SRS & Clustering

Subset & Clustering SRS & Clustering

Computational comparison, roughly speaking…


Contracts

Cent

roid

s

Contracts

Cent

roid

s

§ Synthetic VA portfolio: contract attributes

§ Contract attributes are mixed-type: categorical & numerical, continuous & discrete

§ Representative contracts may not be in the original portfolio§ Literature: map centroids to the “nearest neighbor” in the original portfolio

§ But why? How about only necessary rounding?

§ Dissimilarity measure used in previous studies (say 𝑑F&𝑑N num. & cat. variables)

𝑑 𝑥, 𝑦 == 𝑥& − 𝑦& KPQ

&?@+= 𝟏 TBUAB

PV

&?@


Clustering for VAs

Guarantee Type Gender Age Premium GMWB withdrawal rate Maturity

GMDB only, GMDB + GMWB Male, Female 20,21, … , 60 [$10,000, $500,000] 4%, 5%,… , 8% 10,11, … , 25

§ Experiment: compare total in-cluster distances§ 10,000 contracts, 50 desired representative contracts, 10 subsets

§ For fair comparison, randomly select one of the subsets and perform overall clustering

§ Case 1: Independent & Uniform attributes

§ Case 2: Correlated & Non-uniform attributes§ Age skewed to the right (more elderly contracts)

§ Maturity is negatively correlated with age (higher age ⇒ shorter maturity)

§ Withdrawal rate is negatively correlated with maturity (shorter maturity ⇒ higher withdrawal rate)


Clustering for VAs


Clustering for VAs: Results

§ SRS & clustering is more representative than other methods§ SRS & clustering is more resilient to changes in portfolio characteristics§ If subset & clustering was used before, no additional coding effort is required

LESSONS LEARNED


Clustering can help compressing large portfolios

Computational compromise is taken with cautions

SRS & Clustering is representative, resilient, (& fast)

Optimal Allocation of Simulation BudgetMONTE CARLO SIMULATOR


Simulation Budget Allocation: Problem Description

§ Given a fixed simulation budget (say 𝑅 total replications for the whole portfolio)§ How many replications should each contract get?

§ Equal allocation is simple, but we can do (much) better

§ The goal is to estimate the portfolio value accurately, so let’s do just that§ Let 𝜎&K be the (true) per-sample MC variance of the 𝑖-th contract, then the portfolio variance is

𝜎eK ==𝜎&K

𝑟&

F

&?@, where= 𝑟&

F

&?@= 𝑅

§ 𝑟& ≔ no. of replications for contract 𝑖

§ 𝑅 ≔ no. of total replications (budget)


§ For given 𝜎& ’s,

miniB

𝜎eK ==𝜎&K

𝑟&

F

&?@s. t. = 𝑟&

F

&?@= 𝑅

§ Optimal solution (via Lagrange multiplier): 𝑟&∗ ∝ 𝜎&

§ Optimal variance of each contract: mBn

iB∗ ∝ 𝜎&

§ Intuition: simulate more for highly variable contracts§ The plot 𝜎&, 𝑟&∗ should be a straight line

§ But, 𝜎& ’s are generally unknown…then what?§ Product-level knowledge: contract value is a percentage of initial premium

§ Analytical-level knowledge: sample-variances can be estimated, i.e., 𝜎o& ’s


Simulation Budget Allocation: Problem Formulation

Product-level knowledge: Assume $1 VAs have the same sample variance 𝜎pK

§ $𝑃& VA has variance 𝜎&K = 𝑃&K𝜎pK ∝ 𝑃&K, then optimal allocation is 𝑟&∗ ∝ 𝑃&§ If the assumption is true, then

§ For equal allocation, 𝜎&K ∝ 𝑃&K, 𝑃&,mBn

i̅is a quadratic curve

§ For optimal allocation, mBn

iB∗ ∝ 𝑃&, 𝑃&,

mBn

iB∗ is a straight line


Analytical-level knowledge: sample-variances can be estimated, i.e., 𝜎o& ’s

§ Two-stage simulation:§ Stage 1 (pilot run): use a small portion (10%) of the budget to estimate sample standard deviations 𝜎o&§ Stage 2 (full run): allocate the remaining budget so that the overall allocation 𝑟&∗ ∝ 𝜎o& (approx.)


LESSONS LEARNED


Take advantages of data visualization

Optimize the usage of computational resources

Utilize analytical knowledge for better performance

§ 100,000 synthetic contracts (independent & correlated), 25 subsets, same predictor as in Gan (2013)

§ Simulation budget: 10s𝑛 (where 𝑛 is the no. of clusters)§ Baseline: subset & clustering & nearest neighbor + equal allocation§ Clust.Alloc: SRS & clustering & rounding + two-stages optimal allocation

§ Benchmark: ~10u replications for each contract (optimal allocation)§ Portfolio relative error:

PRE ≔∑ �̂�& − ∑ 𝜇&@pp|

&?@@pp|&?@

∑ 𝜇&@pp|&?@

=�̂�e − 𝜇e𝜇e

§ Repeat the whole experiment 100 times to assess accuracy and runtime


What’s the big deal?

Guarantee Type Gender Age Premium GMWB withdrawal rate Maturity

GMDB only, GMDB + GMWB Male, Female 20,21, … , 60 [$10,000, $500,000] 4%, 5%,… , 8% 10,11, … , 25


Accuracy & Runtime

No. of Clusters Contract Attributes Portfolio Absolute Relative ErrorBaseline Clust.Alloc % Improv.

100 Independent 17.18% 2.04% 88%Correlated 15.96% 0.98% 94%

500 Independent 1.95% 0.52% 73%Correlated 1.89% 0.41% 78%

No. of Clusters Experiment Type Compressor Simulator Predictor* TotalClustering NN/Rounding

100 Baseline 2.48 1.98 0.33 100.89 105.68Clust.Alloc 4.35 0.00 0.45 121.32 126.12

500 Baseline 26.72 10.16 1.85 473.40 512.12Clust.Alloc 10.19 0.00 2.38 528.56 541.13

• Average portfolio relative error• Improvements range from 73% - 94%• Improvements are more significant when contract attributes are correlated

• Average runtime (in secs, independent attributes)• New compressor takes longer when the number of clusters is small• New simulator/predictor take longer due to absence of redundant centroids

Comparison of Interpolation MethodsPREDICTORS


§ Given the representative contracts and their estimated values for the representative portfolio

§ How to estimate the quantities for the original portfolio without additional simulation?

§ Comparisons: § Speed: computational complexity and runtime

§ Accuracy: relative errors of predictions to benchmarks

§ Granularity: portfolio-level vs. contract-level predictions

§ A good prediction should be fast, accurate, and preferably has high granularity


What constitutes a good predictor?

§ Size multiple for clusters (Clust.Size):§ Predict contract value by its “nearest representative contract”

§ Just a benchmark, anything worse than this should not be considered

§ Ordinary Least Squares (OLS): Well-studied, standard, fast built-in libraries

§ Kriging (Gaussian Process Modeling):§ Need to solve an 𝑁 + 1 × 𝑁 + 1 system of linear equation for each prediction

§ Exists a shortcut to predict whole portfolio without predicting contract values

§ Predictions for representative contracts exactly equal to their estimated values

§ Stochastic kriging: Similar to kriging, but accounts for Monte Carlo noise§ Predictions for representative contracts can differ from their estimated values


Common Predictors

No. of Clusters Granularity Relative Error (runtime)

KRI.Baseline Clust.Size OLS KRI.Clust.Alloc STO.KRI100 Portfolio 17.18% (9) 2.78% (2) 2.64% (<1) 2.04% (10) 1.75% (N/A)

Contract 961% (101) 92% (2) 294% (<1) 153% (121) 27% (23)500 Portfolio 1.95% (17) 0.85% (10) 1.02% (<1) 0.52% (19) 0.38% (N/A)

Contract 241% (473) 46% (10) 271% (<1) 49% (529) 8% (218)


Accuracy & Runtime

• OLS is not a suitable choice of predictor for this application• Stochastic kriging has the best performance• Does granularity matter to you?

• Example: 100,000 synthetic contracts, 25 subsets, independent attributes

• Portfolio-level relative error ≔ ∑ ~�B�∑ ~B��B��

��B��

∑ ~B��B��

= ~��~�~�

• Contract-level relative error ≔ @@pp|

∑ ~�B�~B~B

@pp|&?@

What new?§ Improved clustering and MC experiment design in

valuation of large VA portfolio§ Over 90% improved accuracy with little overhead

§ Illustration in a three-stages procedure but insights are broadly applicable more generally

§ Comparative study for different predictors§ Proposed a valuable benchmark

What’s next?§ Proofs?

§ Quantify “representativeness” and “resilience”?

§ Tailored design for different estimation tasks§ E.g., “representative contracts” for tail measures?§ E.g., “representative contracts” for Greeks?

§ Synergies among different steps§ E.g., use the fact that mB

iB∗ ∝ 𝜎& in prediction?

§ If some Greeks were estimated in the simulator, use them as sensitivity information in the predictor?

§ Green simulation§ Use previous results to improve current simulation?§ Can we get an accurate answer without simulating?§ Learning from repeated simulation experiments?

Concluding Remarks


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