This document summarizes an application of Bayesian analysis to forecast insurance loss payments. It begins with an overview of Bayesian methodology and credibility theory. It then presents a case study using a Bayesian hierarchical model to forecast loss reserves for workers' compensation claims data from 10 large insurance companies. Key steps included specifying the probability model, performing Markov chain Monte Carlo simulations for computation and inference, and checking model fit. Results showed the Bayesian model provided greater reserve estimates than traditional methods and accounted for uncertainty in long-term predictions.
MAT 510 Exceptional Education - snaptutorial.comDavisMurphyB11
For more classes visit
www.snaptutorial.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
MAT 510 Exceptional Education - snaptutorial.comDavisMurphyB11
For more classes visit
www.snaptutorial.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
For more course tutorials visit
www.newtonhelp.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
The Statistical Thinking Strategy has significant commonality with the scientific method. Which of the following statistical thinking principles is NOT generally associated with the scientific method?
Question 3
Mat 510 Enhance teaching / snaptutorial.comBaileya19
For more classes visit
www.snaptutorial.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
The Statistical Thinking Strategy has significant commonality with the scientific method. Which of the
Automatic algorithms for time series forecastingRob Hyndman
Many applications require a large number of time series to be forecast completely automatically. For example, manufacturing companies often require weekly forecasts of demand for thousands of products at dozens of locations in order to plan distribution and maintain suitable inventory stocks. In these circumstances, it is not feasible for time series models to be developed for each series by an experienced analyst. Instead, an automatic forecasting algorithm is required.
In addition to providing automatic forecasts when required, these algorithms also provide high quality benchmarks that can be used when developing more specific and specialized forecasting models.
I will describe some algorithms for automatically forecasting univariate time series that have been developed over the last 20 years. The role of forecasting competitions in comparing the forecast accuracy of these algorithms will also be discussed.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
In this presentation I gave in the 3rd Edition: Advanced Model Validation Conference, I first introduced the regulatory expectation on model risk aggregation and the general industry practices, and then discussed the typical qualitative approach with key enhancement opportunities highlighted.
Statistics Assignment Help from the Statistics Assignment Experts. Statistics assignment help is the most common assignment that students are mostly demand for. Statistics is the branch of mathematics, comprises the collection, summarizing, analysis, interpretation, and presentation of data.
A detailed roadmap through the Analyze phase of the DMAIC methodology that navigates the user through the various tools and concepts for leading a Six Sigma project.
International Food Policy Research Institute (IFPRI) organized a three days Training Workshop on ‘Monitoring and Evaluation Methods’ on 10-12 March 2014 in New Delhi, India. The workshop is part of an IFAD grant to IFPRI to partner in the Monitoring and Evaluation component of the ongoing projects in the region. The three day workshop is intended to be a collaborative affair between project directors, M & E leaders and M & E experts. As part of the workshop, detailed interaction will take place on the evaluation routines involving sampling, questionnaire development, data collection and management techniques and production of an evaluation report. The workshop is designed to better understand the M & E needs of various projects that are at different stages of implementation. Both the generic issues involved in M & E programs as well as project specific needs will be addressed in the workshop. The objective of the workshop is to come up with a work plan for M & E domains in the IFAD projects and determine the possibilities of collaboration between IFPRI and project leaders.
Quantitative Analysis For Management 13th Edition Render Test BankJescieer
Full download : http://alibabadownload.com/product/quantitative-analysis-for-management-13th-edition-render-test-bank/ Quantitative Analysis For Management 13th Edition Render Test Bank
Business Bankruptcy Prediction Based on Survival Analysis Approachijcsit
This study sampled companies listed on Taiwan Stock Exchange that examined financial distress between
2003 and 2009. It uses the survival analysis to find the main indicators which can explain the business
bankruptcy in Taiwan. This paper uses the Cox Proportional Hazard Model to assess the usefulness of
traditional financial ratios and market variables as predictors of the probability of business failure to a
given time. This paper presents empirical results of a study regarding 12 financial ratios as predictors of
business failure in Taiwan. It showed that it does not need many ratios to be able to anticipate potential
business bankruptcy. The financial distress probability model is constructed using Profitability, Leverage,
Efficiency and Valuation ratio variables. In the proposed steps of business failure prediction model, it used
detail SAS procedure. The study proves that the accuracies of classification of the mode in overall accuracy
of classification are 87.93%.
Lawrence D. Schall (Professor of Finance and Business Economics, University of Washington)
Gary L. Sundem (Associate Professor of Accounting, University of Washington)
William R. Geijsbeek, Jr. (Finance Manager, The Boeing Company)
For more course tutorials visit
www.newtonhelp.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
The Statistical Thinking Strategy has significant commonality with the scientific method. Which of the following statistical thinking principles is NOT generally associated with the scientific method?
Question 3
Mat 510 Enhance teaching / snaptutorial.comBaileya19
For more classes visit
www.snaptutorial.com
MAT 510 Final Exam Set 1
Question 1
Improvement is needed for an organization to survive because:
Question 2
The Statistical Thinking Strategy has significant commonality with the scientific method. Which of the
Automatic algorithms for time series forecastingRob Hyndman
Many applications require a large number of time series to be forecast completely automatically. For example, manufacturing companies often require weekly forecasts of demand for thousands of products at dozens of locations in order to plan distribution and maintain suitable inventory stocks. In these circumstances, it is not feasible for time series models to be developed for each series by an experienced analyst. Instead, an automatic forecasting algorithm is required.
In addition to providing automatic forecasts when required, these algorithms also provide high quality benchmarks that can be used when developing more specific and specialized forecasting models.
I will describe some algorithms for automatically forecasting univariate time series that have been developed over the last 20 years. The role of forecasting competitions in comparing the forecast accuracy of these algorithms will also be discussed.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
In this presentation I gave in the 3rd Edition: Advanced Model Validation Conference, I first introduced the regulatory expectation on model risk aggregation and the general industry practices, and then discussed the typical qualitative approach with key enhancement opportunities highlighted.
Statistics Assignment Help from the Statistics Assignment Experts. Statistics assignment help is the most common assignment that students are mostly demand for. Statistics is the branch of mathematics, comprises the collection, summarizing, analysis, interpretation, and presentation of data.
A detailed roadmap through the Analyze phase of the DMAIC methodology that navigates the user through the various tools and concepts for leading a Six Sigma project.
International Food Policy Research Institute (IFPRI) organized a three days Training Workshop on ‘Monitoring and Evaluation Methods’ on 10-12 March 2014 in New Delhi, India. The workshop is part of an IFAD grant to IFPRI to partner in the Monitoring and Evaluation component of the ongoing projects in the region. The three day workshop is intended to be a collaborative affair between project directors, M & E leaders and M & E experts. As part of the workshop, detailed interaction will take place on the evaluation routines involving sampling, questionnaire development, data collection and management techniques and production of an evaluation report. The workshop is designed to better understand the M & E needs of various projects that are at different stages of implementation. Both the generic issues involved in M & E programs as well as project specific needs will be addressed in the workshop. The objective of the workshop is to come up with a work plan for M & E domains in the IFAD projects and determine the possibilities of collaboration between IFPRI and project leaders.
Quantitative Analysis For Management 13th Edition Render Test BankJescieer
Full download : http://alibabadownload.com/product/quantitative-analysis-for-management-13th-edition-render-test-bank/ Quantitative Analysis For Management 13th Edition Render Test Bank
Business Bankruptcy Prediction Based on Survival Analysis Approachijcsit
This study sampled companies listed on Taiwan Stock Exchange that examined financial distress between
2003 and 2009. It uses the survival analysis to find the main indicators which can explain the business
bankruptcy in Taiwan. This paper uses the Cox Proportional Hazard Model to assess the usefulness of
traditional financial ratios and market variables as predictors of the probability of business failure to a
given time. This paper presents empirical results of a study regarding 12 financial ratios as predictors of
business failure in Taiwan. It showed that it does not need many ratios to be able to anticipate potential
business bankruptcy. The financial distress probability model is constructed using Profitability, Leverage,
Efficiency and Valuation ratio variables. In the proposed steps of business failure prediction model, it used
detail SAS procedure. The study proves that the accuracies of classification of the mode in overall accuracy
of classification are 87.93%.
Lawrence D. Schall (Professor of Finance and Business Economics, University of Washington)
Gary L. Sundem (Associate Professor of Accounting, University of Washington)
William R. Geijsbeek, Jr. (Finance Manager, The Boeing Company)
With the FASB’s current expected credit loss (CECL) model due to be released before the end of the year, there are many changes that banks and credit unions should plan for. These slides accompany a webinar, and cover a summary of the expected loss model as proposed, the do's and don'ts for bankers as they prepare, and ways that CECL will impact the ALLL calculation. View the corresponding webinar recording here: http://web.sageworks.com/cecl-prep-webinar/
With the current expected credit loss (CECL) model for the Allowance on the horizon, bankers will be asked to create future-looking methodologies that adjust for reasonable and supportable forecasts. Without adequate modeling experience, that can be a challenge for community banks and credit unions.
Watch the full webinar here: http://web.sageworks.com/forward-looking-alll-adjustments/
The data set used in this project is available in the Kaggle and contains nineteen columns (independent variables) that indicate the characteristics of the clients of a fictional telecommunications corporation. The Churn column (response variable) indicates whether the customer departed within the last month or not. The class No includes the clients that did not leave the company last month, while the class YES contains the clients that decided to terminate their relations with the company. The objective of the analysis is to obtain the relation between the customer’s characteristics and the churn.
We review basic reserving methodologies for reserving general insurance like lag analysis and chain ladder. We then move forward to consider multiple stochastic loss reserving models in detail and show how they uncover more insights than basic reserving models.
BA is used to gain insights that inform business decisions and can be used to automate and optimize business processes. Data-driven companies treat their data as a corporate asset and leverage it for a competitive advantage. Successful business analytics depends on data quality, skilled analysts who understand the technologies and the business, and an organizational commitment to data-driven decision-making.
Business analytics examples
Business analytics techniques break down into two main areas. The first is basic business intelligence. This involves examining historical data to get a sense of how a business department, team or staff member performed over a particular time. This is a mature practice that most enterprises are fairly accomplished at using.
Predictive Analysis - Using Insight-informed Data to Determine Factors Drivin...ThinkInnovation
Primary Goals
1. To determine what factors are driving the lead conversion process.
2. To Identify which leads are more likely to convert to paid customers.
Data Description
3. Dataset consists of 4613 rows and 15 columns.
Modelling Strategies
4. Plan
4.1 Perform Dummy Encoding
4.2 List Variables for Modeling
4.3 Identify metric of interest to judge model's performance
5. Build
5.1 Build Logistic Regression Model (Preliminary Model)
5.2 Observe the metrics of the model
6. Improve
6.1 Identify the significant variables
6.2 Rebuild model
6.3 Observe the metrics of the models
7. Decide
7.1 Compare the results of Logistic Regression model (Base model) and Decision Tree Model
7.2 Conclude on best model for this project
8. Recommend
8.1 Determine factors driving the lead conversion process
8.2 Recommend what that may help to identify which leads are more likely to convert to paying customers
Author: Anthony Mok
Date: 16 Nov 2023
Email: xxiaohao@yahoo.com
Forecasting operational risk losses for CCAR poses a number of challenges, not least of which is uncertainty about the best methodology for modeling the relationship between a bank’s future losses and the macroeconomic environment.
MLX 2018 - Marcos López de Prado, Lawrence Berkeley National Laboratory Comp...Mehdi Merai Ph.D.(c)
Presented by: Marcos López de Prado, Lawrence Berkeley National Laboratory Computational Research Division
MLX FinTech Conference II, Toronto, May 2018.
More info at: https://www.machinelearningx.net
High level overview of Predictive Analytics techniques - Decision Trees, Regressions, Time Series Forecasting, Exponential Smoothing, etc.
Was put together to train friends and mentees. Based on personal learnings/research and no proprietary info, etc. and no claims on 100% accuracy. Also every institution/organization/team uses it own steps/methodologies, so please use the one relevant for you and this only for training purposes.
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
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Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
If you are looking for a pi coin investor. Then look no further because I have the right one he is a pi vendor (he buy and resell to whales in China). I met him on a crypto conference and ever since I and my friends have sold more than 10k pi coins to him And he bought all and still want more. I will drop his telegram handle below just send him a message.
@Pi_vendor_247
Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins with.
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
Latino Buying Power - May 2024 Presentation for Latino CaucusDanay Escanaverino
Unlock the potential of Latino Buying Power with this in-depth SlideShare presentation. Explore how the Latino consumer market is transforming the American economy, driven by their significant buying power, entrepreneurial contributions, and growing influence across various sectors.
**Key Sections Covered:**
1. **Economic Impact:** Understand the profound economic impact of Latino consumers on the U.S. economy. Discover how their increasing purchasing power is fueling growth in key industries and contributing to national economic prosperity.
2. **Buying Power:** Dive into detailed analyses of Latino buying power, including its growth trends, key drivers, and projections for the future. Learn how this influential group’s spending habits are shaping market dynamics and creating opportunities for businesses.
3. **Entrepreneurial Contributions:** Explore the entrepreneurial spirit within the Latino community. Examine how Latino-owned businesses are thriving and contributing to job creation, innovation, and economic diversification.
4. **Workforce Statistics:** Gain insights into the role of Latino workers in the American labor market. Review statistics on employment rates, occupational distribution, and the economic contributions of Latino professionals across various industries.
5. **Media Consumption:** Understand the media consumption habits of Latino audiences. Discover their preferences for digital platforms, television, radio, and social media. Learn how these consumption patterns are influencing advertising strategies and media content.
6. **Education:** Examine the educational achievements and challenges within the Latino community. Review statistics on enrollment, graduation rates, and fields of study. Understand the implications of education on economic mobility and workforce readiness.
7. **Home Ownership:** Explore trends in Latino home ownership. Understand the factors driving home buying decisions, the challenges faced by Latino homeowners, and the impact of home ownership on community stability and economic growth.
This SlideShare provides valuable insights for marketers, business owners, policymakers, and anyone interested in the economic influence of the Latino community. By understanding the various facets of Latino buying power, you can effectively engage with this dynamic and growing market segment.
Equip yourself with the knowledge to leverage Latino buying power, tap into their entrepreneurial spirit, and connect with their unique cultural and consumer preferences. Drive your business success by embracing the economic potential of Latino consumers.
**Keywords:** Latino buying power, economic impact, entrepreneurial contributions, workforce statistics, media consumption, education, home ownership, Latino market, Hispanic buying power, Latino purchasing power.
how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
#pi network
#pi coins
#money
4. 4
Bayesian methodology
• Fundamental question is
• With the posterior distribution of the parameters, the distribution of any
quantities of interest can be obtained
• The key is the Bayes’ rule:
Given data and a specified model, what is the distribution of the parameters?
Posterior distribution is proportional to data distribution * prior distribution
5. 5
Application to the actuarial field
• Most of you are Bayesian!
– Bornhuetter-Ferguson type reserving to regulate data or account for
information not in data with prior knowledge of the average loss ratio
– Credibility. Bühlmann and Gisler (2005) said
“Credibility theory belongs mathematically to the area of Bayesian statistics
[and it] is motivated by questions arising in insurance practice.”
• So, when you are talking about these, you are thinking in a Bayesian world
• But……Few of you are doing Bayesian analysis!
• Now, there is an opportunity to be a real Bayesian!
6. 6
More on credibility
• Credibility theory refers to
• We should not retain the word just for the actuarial credibility formulas
• Recall that the actuarial credibility theory started asking the following question:
“any procedure that uses information (‘borrows strength’) from samples from different, but
related, populations.” –- Klugman (1987)
Given a group of policyholders with some common risk factor and past claims experience,
what is the Bayes’ premium to be charged for each policyholder?
Bayes’ Premium
Bayesian Analysis
Credibility to
borrow information
Bülmann formulas …… Hierarchical model
7. 7
Credibility example
• Credibility formulas are only linear approximations to overcome computational difficulties:
– No closed form except for some simple models and distributions
– Hard to estimate the population parameters
• Now, there is no reason to linearly approximate Bayesian methods as advances in statistical
computation in the past several decades have enabled more complex and realistic models to be
constructed and estimated
• Consider the following example in Workers’ Comp (see Scollnik 2001):
• Question: What’s the expected count for year 5, given the observed claim history?
• Let’s do the Bayesian hierarchical analysis and then compare it with other estimates
Year
Group 1 Group 2 Group 3
Payroll # Claims Payroll # Claims Payroll # Claims
1 280 9 260 6
2 320 7 275 4 145 8
3 265 6 240 2 120 3
4 340 13 265 8 105 4
8. 8
Visualization of the
hierarchies
)exposure(~claims# kikik Pois θ×
),(log~log 2
0 σθθ Nk
• Intuitively, we assume claim count to be a
Poisson distribution
• Credibility view assumes that each group has a
different claim rate per exposure µk, but each µk
arises from the same distribution, say
• We call µ0 and ¾ hyper-parameters
• If µ0 is estimated using all the data, so will each
µk. Thus, the estimation of one group will borrow
information from other groups, and will be pooled
toward the overall mean
• Assign non-informative (flat) priors so that ¾ and
µ0are estimated from the data, e.g.
)100,0(~);100,0(~ 2
0 NU θσ
Data
Group
11. 11
Case study in loss reserving
Part II
Paper: A Bayesian Nonlinear Model for Forecasting Insurance Loss
Payments.
Yanwei (Wayne) Zhang, CNA Insurance Company
Vanja Dukic, Applied Math, University of Colorado-Boulder
James Guszcza, Deloitte Consulting LLP
Paper is available at http://
www.actuaryzhang.com/publication/bayesianNonlinear.pdf.
13. 13
Challenges
• Challenges in current loss reserving practices:
– Most stochastic models need to be supplemented by a tail factor, but the
corresponding uncertainty is hard to be accounted for
– Inference at an arbitrary point is hard to obtain, e.g., 3 months or 9 months
– Too many parameters! Parsimony is the basic principle of statistics
– Treat accident year, development lag, or both independently
– Focus on one triangle, lack a method to blend industry data
– Usually rely on post-model selection using judgment:
• Input of point estimate is almost meaningless, but large leverage
• Extra uncertainty is not accounted for
14. 14
Benefits of the Bayesian hierarchical model to be built
• Allows input of external information and expert opinion
• Blending of information across accident years and across companies
• Extrapolates development beyond the range of observed data
• Estimates at any time point can be made
• Uncertainty of extrapolation is directly included
• Full distribution is available, not just standard error
• Prediction of a new accident year can be achieved
• Minimizes the risk of underestimating the uncertainty in traditional models
• Estimation of company-level and accident-year-level variations
15. 15
Steps in the Bayesian
analysis
• Steps in a Bayesian analysis
– Setting up the probability model
• Specify the full distribution of data and the priors
• Prior distribution could be either informative or non-informative, but need to result in a
proper joint density
– Computation and inference
• Usually need to use sampling method to simulate values from the posterior distribution
– Model checking
• Residual plot
• Out-of-Sample validation
• Sensitivity analysis of prior distribution
16. 16
Visualization of data
• Workers’ Comp Schedule P data (1988-1997) from 10 large companies
• Use only 9 years’ data, put the 10th
year as hold-out validation set
18. 18
Probability model
• We use the Log-Normal distribution to reflect the skewness and ensure that cumulative
losses are positive
• We use a nonlinear mean structure with the log-logistic growth curve: t!
/(t!
+µ!
)
• We use an auto-correlated process along the development for forecasting
• We build a multi-level structure to allow the expected ultimate loss ratios to vary by
accident year and company:
– In one company, loss ratios from different years follow the same distribution with a
mean of company-level loss ratio
– Different company-level average loss ratios follow the same distribution with a
mean of the industry-level loss ratio
• Growth curve is assumed to be the same within one company, but vary across
companies, arising from the same industry average growth curve
• Assign non-informative priors to complete model specification
Expected cumulative loss = premium * expected loss ratio * expected emergence
20. 20
Computation and model estimation
• Such a specification does not result in a closed-form posterior distribution
• Must resort to sampling method to simulate the distribution
• We use Markov Chain Monte Carlo [MCMC] algorithms
– The software WinBUGS implements the MCMC method
– Always need to check the convergence of the MCMC algorithm
• Trajectory plot
• Density plot
• Autocorrelation plot
24. 24
Estimation of loss ratios
• Industry average loss ratio is 0.693 [0.644, 0.748]
• Variations across company is about twice as large as those across accident years
25. 25
Loss reserve estimation results
• Autocorrelation is about 0.479 [0.445, 0.511]
• Industrial average emergence percentage at 108 months is about 93.5%
• Bayesian reserves projected to ultimate are greater than the GLM estimates
projected to 108 months, by a factor about 1.4.
Company
Estimate at ultimate Estimate at the end of the 9th year
Bayesian Bayesian GLM-ODP
Reserve Pred Err 50% Interval Reserve Pred Err Reserve Pred Err
1 260.98 46.84 (230.80,292.54) 170.33 25.98 155.99 10.90
2 173.13 22.00 (159.37,188.60) 136.20 15.13 139.63 7.11
3 216.19 13.95 (206.70,224.83) 151.82 9.01 130.71 4.53
4 81.95 7.39 (77.17,87.14) 63.28 4.80 54.69 3.46
5 44.60 6.69 (40.33,49.21) 37.95 5.14 33.56 2.12
6 48.86 5.27 (45.48,52.41) 38.31 3.97 37.00 2.05
7 34.45 2.19 (33.03,35.90) 26.21 1.49 25.11 0.91
8 22.91 2.06 (21.62,24.32) 16.46 1.37 16.83 0.72
9 30.66 5.62 (27.11,34.42) 22.58 3.22 18.39 1.52
10 19.88 1.35 (18.94,20.80) 15.47 0.91 17.71 0.68
28. 28
Out-of-Sample test
• We use only 9 years of data to train the model, and validate on the 10th
year
– Note that this is the cash flow of the coming calendar year
• Policies written in the past
• Policies to be written in the coming year (need an estimated premium)
• For 4 companies, we also have observed data for the bottom right part
• The coverage rates of the 50% and 95% intervals in the two validation sets are
• The model performs fairly well overall, but long-term prediction is a little under
expectation
50% Interval 95% Interval
Set 1 57% 95%
Set 2 40% 81%
29. 29
Sensitivity analysis
• Change the prior distribution of the industry-level loss ratio to more
realistic distributions
• 6 scenarios: Gamma distribution with mean 0.5, 0.7 and 0.9, variance 0.1
and 0.2, respectively
30. 30
Discussion of the model
• The model used in this analysis provides solutions to many existing challenges
• The model can be further improved:
– Inflation can be readily included with an appropriate model
– Prior information can be incorporated on the accident-year or company level
– Build in more hierarchies: states, lines of business, etc…
– Include triangles that have more loss history to stabilize extrapolation
• For future research:
– Which curve to choose? (model risk)
– Correlation among lines? (Copula)
31. 31
Summary
• Introduced Bayesian hierarchical model as a full probability model that allows
pooling of information and inputs of expert opinion
• Illustrated application of the Bayesian model in insurance with a case study of
forecasting loss payments in loss reserving using data from multiple companies
• The application of Bayesian model in insurance is intuitive and promising. I hope
more people will start exploiting it and applying it to their work.
• You may download this presentation, the paper and code from my website:
http://www.actuaryzhang.com/publication/publication.html ;
Or contact me at: actuaryzhang@uchicago.edu
32. 32
Reference
• Bülmann and Gisler (2005). A Course in Credibility Theory and its Applications.
• Clark D. R. (2003). LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood
Approach. Available at http://www.casact.org/pubs/forum/03orum/03041.pdf.
• Guszcza J. (2008). Hierarchical Growth Curve Models for Loss Reserving. Available at
http://www.casact.org/pubs/forum/08orum/7Guszcza.pdf.
• Klugman S (1987). Credibility for Classification Ratemaking via The Hierarchical Normal
Linear Model.Available at http://www.casact.org/pubs/proceed/proceed87/87272.pdf.
• Scollnik D. P. M. (2001). Actuarial Modeling with MCMC and BUGS, North American
Actuarial Journal 5(2): 96-124.
• Zhang et al. (2010). A Bayesian Nonlinear Model for Forecasting Insurance Loss Payments.
Available at http://www.actuaryzhang.com/publication/bayesianNonlinear.pdf.
35. 35
WinBUGS
• BUGS (Bayesian inference Using Gibbs Sampling ) was developed by the MRC Biostatistics
Unit, and it has a number of versions. WinBUGS is one of them.
• We can work directly in WinBUGS, but better to submit batch run from other software
– R: package R2WinBUGS
– SAS: macro %WINBUGSIO
– Excel: add-in BugsXLA
• R is most handy when working with WinBUGS, but we will focus on Excel here
• The excel add-in BugsXLA is developed by Phil Woodward, and provides a great user
interface to work with WinBUGS
• It allows the specification of typical Bayesian hierarchical models, but enhancement is needed
to fit more complicated and customized models
• I will illustrate this using the simple Workers’ Comp Frequency model
36. 36
BugsXLA
• Download and install WinBUGS at http://www.mrc-bsu.cam.ac.uk/bugs/
• Download and install the Excel add-in BugsXLA at http://www.axrf86.dsl.pipex.com/
• Put the data into long format
37. 37
BugsXLA
• Click the “Bayesian analysis” button
• Specify input data
• Specify categorical variables
• In the new window, move the variable
“Group” to the “FACTORS” column
• Can specify the levels and the ordering
with “Edit Factor Levels”
38. 38
BugsXLA
• Specify Poisson Distribution
for the response variable “Claims”
• Want to use identity link, but the
only option is “log”
• But for this simple example, we can
just re-parameterize the model
• Put “Payroll” as offset
• Put “Group” as random effect
• We are done specifying the model.
Now, click “MCMC Options” to
customize simulations
39. 39
BugsXLA
• Burn-in: number of simulations to discard
from the beginning
• Samples: number of samples to draw
• Thin: sample every kth
simulations
• Chains: number of chains
• Import Stats: summary statistics for the
parameters and simulations
• Import Sample: the simulated outcomes
for each parameter
40. 40
BugsXLA
• After clicking “OK” in the “Bayesian analysis” dialog,
a “Prior Distribution” dialog pops up
• Change the distribution here so that the group
effect is Normally distributed, with a large variance,
say, the standard deviation is uniform on (0,100)
• Click “Run WinBUGS”
• Then,
41. 41
BugsXLA
• Simulation results are imported
– Estimation summary
– Model checks
– Simulated outcomes
• Calculate the mean for each group
• Plot the result