├── .gitignore ├── LifeCon1 ├── .Rbuildignore ├── .gitignore ├── .travis.yml ├── Bibliography │ ├── Bibliography.Rmd │ ├── LDAReference.bib │ ├── LossDataAnalyticsReference.bib │ └── packages.bib ├── Chapters │ ├── EmergingCosts29Dec2011B_utf8.Rmd │ ├── InterestRateRisk05Nov2012.Rmd │ ├── JointLife31May2017.Rmd │ ├── MultiStateLecture24October2014.Rmd │ ├── MultipleDecrementsLecture07Oct2012.Rmd │ ├── PensionLecture06Nov2014.Rmd │ ├── PolicyValues.Rmd │ └── UnivLifeLecture14Nov2012.Rmd ├── Data │ ├── CLAIMLEVEL.csv │ └── PropertyFundInsample.csv ├── Figures │ ├── IRR_BinaryDF.eps │ ├── IRR_DiscreteRVDF.eps │ ├── IRR_InverseDF.eps │ ├── IRR_MixedDF.eps │ ├── JL_ActMathCh3Figure4.png │ ├── JL_ActMathCh3Figure6.png │ ├── JL_JointLifeTimeline.pdf │ ├── MDM_MultipleDecrement.pdf │ ├── MDM_RetirementSystemTurnoverL.pdf │ ├── MDM_RetirementSystemTurnoverR.pdf │ ├── MSM_HealthyDisabledDead.pdf │ ├── MSM_JointLifeStatus.pdf │ ├── MSM_LifeDeathOtherDeath.pdf │ ├── MSM_MultipleDecrement.pdf │ ├── MSM_TwoState.pdf │ ├── PC_RetirementSystemTurnoverL.pdf │ ├── PC_RetirementSystemTurnoverR.pdf │ ├── PV_F1WholeLife.eps │ ├── PV_F2WholeLifeDistn.eps │ ├── PV_FWholeLifeTimeLine.eps │ ├── UL_Fig9DHWSN43TypeAULProfitTest.pdf │ ├── UL_Figs6,7DHWSN42TypeBULProfitTest.pdf │ ├── UL_Figs6,7DHWSN42TypeBULProfitTestB.pdf │ ├── UL_LifeCycleExample.pdf │ ├── UL_UniversalLifeFigure1.png │ ├── UL_VaryingBenefitAmounts.pdf │ └── Unused │ │ ├── ActMathCh3Figure5.pdf │ │ └── JointLifeStatus.pdf ├── LifeCon1Version1.0.Rproj ├── ShowHide.js ├── _bookdown.yml ├── _build.sh ├── _deploy.sh ├── _output.yml ├── disqus.html ├── index.Rmd ├── preamble.tex ├── style.css └── toc.css ├── LifeCon2 ├── .Rbuildignore ├── .gitignore ├── .travis.yml ├── Bibliography │ ├── Bibliography.Rmd │ ├── LDAReference.bib │ ├── LossDataAnalyticsReference.bib │ └── packages.bib ├── Chapters │ ├── 01-SurvivalModelsLifeTables.Rmd │ ├── 02-Selection.Rmd │ ├── 03-InsuranceBenefitsI (OLD).Rmd │ ├── 03-InsuranceBenefitsI.Rmd │ ├── 04-InsuranceBenefitsII.Rmd │ ├── 05-AnnuitiesI.Rmd │ ├── 06-AnnuitiesII.Rmd │ ├── 07-PremiumCalculationI.Rmd │ ├── 08-PremiumCalculationII.Rmd │ ├── 09-ReservesI.Rmd │ ├── 10-ReservesII.Rmd │ ├── 11-MultipleStateModels.Rmd │ ├── 12-MultipleDecrementsI.Rmd │ ├── 13-MultipleDecrementsII.Rmd │ ├── 14-MultipleLivesI.Rmd │ ├── 15-MultipleLivesII.Rmd │ ├── 16-OtherTopics.Rmd │ └── 18-MAFAppendix.Rmd ├── Data │ ├── CLAIMLEVEL.csv │ └── PropertyFundInsample.csv ├── Figures │ └── Appendix_MSM.pdf ├── LICENSE ├── LifeCon2.epub ├── LifeCon2Version1.0.Rproj ├── ShowHide.js ├── _bookdown.yml ├── _build.sh ├── _deploy.sh ├── _output.yml ├── disqus.html ├── index.Rmd ├── lifecon.sty ├── mathspec.sty ├── preamble.tex ├── style.css ├── temp │ ├── ActMathAnalytics.pdf │ ├── ActMathAnalytics.synctex.gz │ ├── ActMathAnalytics.tex │ └── lifecon.sty └── toc.css ├── LifeCon3 ├── .Rbuildignore ├── .gitignore ├── .travis.yml ├── Bibliography │ ├── Bibliography.Rmd │ ├── LDAReference.bib │ ├── LossDataAnalyticsReference.bib │ └── packages.bib ├── Chapters │ ├── 01-SurvivalModelsLifeTables.Rmd │ └── 04-NetPremiums.Rmd ├── Figures │ ├── Ch1_01_ex1.1.16.JPG │ ├── Ch1_02_ex_1.1.17.JPG │ ├── Ch1_03_1.2.1.JPG │ ├── Ch1_04_1.2.3a.JPG │ ├── Ch1_05_1.2.3b.JPG │ ├── Ch1_06_ex_1.2.1a.JPG │ ├── Ch1_07_ex_1.2.1b.JPG │ ├── Ch1_08_ex_1.2.3.JPG │ ├── Ch1_09_ex_1.3.2.JPG │ ├── Ch1_10_1.4.2.JPG │ ├── Ch1_11_ex_1.4.4.JPG │ ├── Ch1_12_ex_1.4.5.JPG │ └── Ch4_01_4.2.JPG ├── LifeCon3Version1.0.Rproj ├── ShowHide.js ├── _bookdown.yml ├── _build.sh ├── _deploy.sh ├── _output.yml ├── disqus.html ├── index.Rmd ├── preamble.tex ├── style.css └── toc.css └── README.md /.gitignore: -------------------------------------------------------------------------------- 1 | .Rproj.user 2 | .Rhistory 3 | .RData 4 | .Ruserdata 5 | -------------------------------------------------------------------------------- /LifeCon1/.Rbuildignore: -------------------------------------------------------------------------------- 1 | ^.*\.Rproj$ 2 | ^\.Rproj\.user$ 3 | -------------------------------------------------------------------------------- /LifeCon1/.gitignore: -------------------------------------------------------------------------------- 1 | .Rproj.user 2 | .Rhistory 3 | .RData 4 | _publish.R 5 | _book 6 | _bookdown_files 7 | rsconnect 8 | -------------------------------------------------------------------------------- /LifeCon1/.travis.yml: -------------------------------------------------------------------------------- 1 | language: r 2 | cache: packages 3 | 4 | env: 5 | global: 6 | - secure: "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" 7 | 8 | before_script: 9 | - chmod +x ./_build.sh 10 | - chmod +x ./_deploy.sh 11 | 12 | script: 13 | - ./_build.sh 14 | - ./_deploy.sh 15 | -------------------------------------------------------------------------------- /LifeCon1/Bibliography/Bibliography.Rmd: -------------------------------------------------------------------------------- 1 | `r if (knitr:::is_html_output()) '# Bibliography {-}'` 2 | -------------------------------------------------------------------------------- /LifeCon1/Bibliography/LDAReference.bib: -------------------------------------------------------------------------------- 1 | @Book{xie2015, 2 | title = {Dynamic Documents with {R} and knitr}, 3 | author = {Yihui Xie}, 4 | publisher = {Chapman and Hall/CRC}, 5 | address = {Boca Raton, Florida}, 6 | year = {2015}, 7 | edition = {2nd}, 8 | note = {ISBN 978-1498716963}, 9 | url = {http://yihui.name/knitr/}, 10 | } 11 | 12 | @InProceedings{III2015, 13 | Title = {International Insurance Fact Book}, 14 | Author = {{Insurance~Information~Institute}}, 15 | Year = {2015}, 16 | Note = {[Retrieved on May 10, 2016]}, 17 | Publisher = {Insurance Information Institute}, 18 | Url = {http://www.iii.org/sites/default/files/docs/pdf/international_insurance_factbook_2015.pdf} 19 | } 20 | 21 | @Book{bowers1986actuarial, 22 | Title = {Actuarial Mathematics}, 23 | Author = {Bowers, Newton L. and Gerber, Hans U. and Hickman, James C. and Jones, Donald A. and Nesbitt, Cecil J.}, 24 | Publisher = {Society of Actuaries Itasca, Ill.}, 25 | Year = {1986} 26 | } 27 | 28 | @Book{dickson2013actuarial, 29 | Title = {Actuarial Mathematics for Life Contingent Risks}, 30 | Author = {Dickson, David C. M. and Hardy, Mary and Waters, Howard R.}, 31 | Publisher = {Cambridge University Press}, 32 | Year = {2013} 33 | } 34 | 35 | @InProceedings{survey2013, 36 | Title = {2013 Insurance Predictive Modeling Survey}, 37 | Author = {Earnix}, 38 | Year = {2013}, 39 | Note = {[Retrieved on July 7, 2014]}, 40 | Publisher = {Earnix and Insurance Services Office, Inc.}, 41 | Url = {http://earnix.com/2013-insurance-predictive-modeling-survey/3594/} 42 | } 43 | 44 | @Article{bailey1960, 45 | Title = {Two studies in automobile ratemaking}, 46 | Author = {Bailey, Robert A. and LeRoy, J. Simon}, 47 | Journal = {Proceedings of the Casualty Actuarial Society Casualty Actuarial Society}, 48 | Year = {1960}, 49 | Number = {I}, 50 | Volume = {XLVII} 51 | } 52 | 53 | @InProceedings{SASsurvey, 54 | Title = {Building believers: How to expand the use of predictive analytics in claims}, 55 | Author = {Gorman, Mark and Swenson, Stephen}, 56 | Year = {2013}, 57 | Note = {[Retrieved on August 17, 2014]}, 58 | Publisher = {SAS}, 59 | Url = {http://www.sas.com/resources/whitepaper/wp_59831.pdf} 60 | } 61 | 62 | @inproceedings{venter1983transformed, 63 | title={Transformed beta and gamma distributions and aggregate losses}, 64 | author={Venter, Gary}, 65 | booktitle={Proceedings of the Casualty Actuarial Society}, 66 | volume={70}, 67 | number={133 \& 134}, 68 | pages={289--308}, 69 | year={1983} 70 | } 71 | 72 | @article{mcdonald1984some, 73 | title={Some generalized functions for the size distribution of income}, 74 | author={McDonald, James B}, 75 | journal={Econometrica: journal of the Econometric Society}, 76 | pages={647--663}, 77 | year={1984}, 78 | publisher={JSTOR} 79 | } 80 | -------------------------------------------------------------------------------- /LifeCon1/Bibliography/LossDataAnalyticsReference.bib: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ewfrees/LifeCon/66e09a97ef093c46778c3d5724f196abdea845f0/LifeCon1/Bibliography/LossDataAnalyticsReference.bib -------------------------------------------------------------------------------- /LifeCon1/Bibliography/packages.bib: -------------------------------------------------------------------------------- 1 | @Manual{R-base, 2 | title = {R: A Language and Environment for Statistical Computing}, 3 | author = {{R Core Team}}, 4 | organization = {R Foundation for Statistical Computing}, 5 | address = {Vienna, Austria}, 6 | year = {2016}, 7 | url = {https://www.R-project.org/}, 8 | } 9 | @Manual{R-bookdown, 10 | title = {bookdown: Authoring Books and Technical Documents with R Markdown}, 11 | author = {Yihui Xie}, 12 | year = {2016}, 13 | note = {R package version 0.3}, 14 | url = {https://CRAN.R-project.org/package=bookdown}, 15 | } 16 | @Manual{R-knitr, 17 | title = {knitr: A General-Purpose Package for Dynamic Report Generation in R}, 18 | author = {Yihui Xie}, 19 | year = {2016}, 20 | note = {R package version 1.15.1}, 21 | url = {https://CRAN.R-project.org/package=knitr}, 22 | } 23 | @Manual{R-rmarkdown, 24 | title = {rmarkdown: Dynamic Documents for R}, 25 | author = {JJ Allaire and Joe Cheng and Yihui Xie and Jonathan McPherson and Winston Chang and Jeff Allen and Hadley Wickham and Aron Atkins and Rob Hyndman}, 26 | year = {2016}, 27 | note = {R package version 1.2}, 28 | url = {https://CRAN.R-project.org/package=rmarkdown}, 29 | } 30 | -------------------------------------------------------------------------------- /LifeCon1/Chapters/EmergingCosts29Dec2011B_utf8.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Emerging Costs" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | # Emerging Costs 12 | ## Cash Values {#S:CashValue} 13 | 14 | Policy values quantify the obligations, or liabilities, of a life 15 | insurance company. 16 | 17 | - They are used by managers and investors to determine the financial 18 | position of the company. 19 | 20 | - They are also used by managers and regulators to determine the 21 | likelihood that a company will continue as an ongoing operation to 22 | fulfill promises made to policyholders. 23 | 24 | Policy values are also used to determine the allocation of investment 25 | assets needed to fulfill company obligations from issuing policies - 26 | this is the topic of Section \[S:AssetShares\]. This section is devoted 27 | to the application of policy values to determine a policy's value to the 28 | insured upon cancelation of the policy, its *cash value*. 29 | 30 | An insurance policy provides a combination of protection against 31 | unforeseen risks (the insurance element) as well as a savings plan (the 32 | investment component). When a policyholder decides to terminate the 33 | contract prematurely (before called for in the plan design), in many 34 | cases the policyholder will receive an amount known as a cash value, 35 | *withdrawal benefit*, or *nonforfeiture benefit*. You can think of the 36 | cash value as the policy value minus a surrender charge, that is, 37 | $$\begin{aligned} 38 | _k CV = ~ _k V -~ _k SC.\end{aligned}$$ Because of the difficulty of 39 | collecting additional funds from a withdrawing policyholder, 40 | $_k CV \geq 0.$ 41 | 42 | Determining the cash value (or, equivalently the surrender charge) is 43 | not straightforward. One needs to think about fairness to the continuing 44 | policyholders as well as fairness to those that terminate the contract. 45 | One viewpoint is that those that surrender their policy have not 46 | fulfilled their contract and therefore are not entitled to any benefits. 47 | Another viewpoint is that those that surrender their policy should 48 | receive all of their premiums minus a charge for the insurance 49 | protection received. 50 | 51 | It is also important to recognize the timing and incidence of expenses 52 | because most insurance contracts are costly to underwrite and incur 53 | substantial expenses in the early years. In Section \[S:DAExpenses\], we 54 | discuss an alternative reserving basis that acknowledges high early 55 | expenses. 56 | 57 | ## Asset Shares {#S:AssetShares} 58 | 59 | The “asset share” of a policy in force is the share of the insurer’s 60 | assets attributable to that policy. Here, expenses, interest, mortality 61 | and so forth are calculated based on the insurer’s experience for 62 | similar policies over the period. 63 | 64 | Asset share calculations are similar to those used for policy values 65 | although the purposes of the two concepts are quite different. The 66 | policy value represents an amount the insurer *needs to have*, the asset 67 | share represents the amount that the insurer *actually does have*. 68 | 69 | Asset shares are typically calculated recursively, such as using 70 | relationship similar to those for policy values, e.g., $$\begin{aligned} 71 | (~_k AS + G_k - e_k)(1+i_k) &= & 72 | q_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1}\right) + q_{[x]+k}^{(w)} ~_{k+1} CV + p_{[x]+k}^{(\tau)} ~_{k+1} AS\end{aligned}$$ 73 | where 74 | 75 | - at the beginning of the year, $~_k AS$ is the asset share, $G_k$ is 76 | the contract premium, $e_k$ are annual expenses per contract 77 | 78 | - at death, $q_{[x]+k}^{(d)}$ is the probability of death, $b_{k+1}$ 79 | is the insurance benefit, and $E_k$ are settlement expenses per 80 | contract 81 | 82 | - at withdrawal, $q_{[x]+k}^{(w)}$ is the probability of withdrawal, 83 | $~_{k+1} CV$ is the cash value, 84 | 85 | - at survival to the end of the year, 86 | $p_{[x]+k}^{(\tau)} = 1 - (q_{[x]+k}^{(d)}+q_{[x]+k}^{(w)})$ is the 87 | probability of survivorship and $~_{k+1} AS$ is the asset share. 88 | 89 | Asset shares define the notion of how quickly a policy builds policy 90 | values (assets) and hence profits. 91 | 92 | The main advantage of this recursive approach is that we can examine the 93 | incidence (timing) of profits. 94 | 95 | The recursive relation can be expanded to include other ancillary 96 | benefits, taxes, more complex expense structures, and so forth. 97 | 98 | Asset shares are primarily used to set premiums by defining a profit 99 | goal. For example, 100 | 101 | - PV(profit) = $x$% PV (premium) 102 | 103 | - A return on investment = $x$% 104 | 105 | - The product breaks even $n$ years or sooner. 106 | 107 | Asset shares can easily be used for flexible (variable) plans. 108 | 109 | They can also be used to determine reserves, dividend schedules, 110 | withdrawal benefits or so-called “model offices” (combining asset shares 111 | over several plans, age at issue, new versus old business, and so on). 112 | 113 | **Example (SoA \# 336).** 114 | 115 | For a fully discrete insurance of 1000 on ($x$), you are given: 116 | 117 | - $~_4 AS = 396.63$ is the asset share at the end of year 4. 118 | 119 | - $~_5 AS = 694.50$ is the asset share at the end of year 5. 120 | 121 | - $G = 281.77$ is the gross premium. 122 | 123 | - $~_5 CV = 572.12$ is the cash value at the end of year 5. 124 | 125 | - $c_4 = 0.05$ is the fraction of the gross premium paid at time 4 for 126 | expenses. 127 | 128 | - $e_4 = 7.0$ is the amount of per policy expenses paid at time 4. 129 | 130 | - $q_{x+4}^{(1)} = 0.09$ is the probability of decrement by death. 131 | 132 | - $q_{x+4}^{(2)} = 0.26$ is the probability of decrement by 133 | withdrawal. 134 | 135 | Calculate $i$. 136 | 137 | **Solution.** 138 | 139 | At the beginning of the year, the asset share plus net income available 140 | is $$\begin{aligned} 141 | ~_4 AS + G(1-e_4) - e_4 = 396.63 +281.77(1-0.05) - 7 = 657.3115 .\end{aligned}$$ 142 | At the end of the year, funds must be sufficient to pay those who 143 | survive, die and withdraw: $$\begin{aligned} 144 | & & p_{x+4}^{(\tau)} ~_5 AS + 1000 q_{x+4}^{(1)} + ~_5 CV q_{x+4}^{(2)} \\ 145 | &~~~~~~~~=& (1 - 0.09 - 0.26) (694.50) + 1000 (0.09) + (572.12) (0.26) = 690.1762 .\end{aligned}$$ 146 | Using the relation $657.3115 (1+i) =690.1762$, we get 147 | $i = 4.9999\% \approx 5\%$. 148 | 149 | **Example (SoA \# 242).** 150 | 151 | For a fully discrete whole life insurance of 10,000 on ($x$), you are 152 | given: 153 | 154 | - $~_{10} AS =1600$ is the asset share at the end of year 10. 155 | 156 | - $G = 200$ is the gross premium. 157 | 158 | - $~_{11} CV =1700$ is the cash value at the end of year 11. 159 | 160 | - $c_{10} = 0.04$ is the fraction of gross premium paid at time 10 161 | for expenses. 162 | 163 | - $e_{10} ƒ= 70$ is the amount of per policy expense paid at time 10. 164 | 165 | - Death and withdrawal are the only decrements. 166 | 167 | - $q_{x+10}^{(d)} = 0.02$ 168 | 169 | - $q_{x+10}^{(w)} = 0.18$ 170 | 171 | - $i = 0.05$ 172 | 173 | Calculate $~_{11} AS$, the asset share at the end of year 11. 174 | 175 | *Solution.* 176 | 177 | At the beginning of the year, the asset share plus net income available 178 | is $$\begin{aligned} 179 | ~_{10} AS + G(1-c_{10}) - e_{10} = 1600 + 200(1-0.04) - 70 = 1722.\end{aligned}$$ 180 | At the end of the year, funds must be sufficient to pay those who 181 | survive, die and withdraw: $$\begin{aligned} 182 | & & p_{x+4}^{(\tau)} ~_{11} AS + 10000 q_{x+10}^{(d)} + ~_{11} CV q_{x+10}^{(w)} \\ 183 | &~~~~~~~~=& (1 - 0.02 - 0.18) ~_{11} AS + 10000 (0.02) + (1700) (0.18) = 0.8 ~_{11} AS +506.\end{aligned}$$ 184 | Thus, with $i = 0.05$, we have $(1.05) 1722 = 0.8 ~_{11} AS +506$, or 185 | $~_{11} AS = 1627.625$. 186 | 187 | ## Emerging Costs {#S:Expenses} 188 | 189 | Under “cash accounting,” a person or firm counts income when it is 190 | received and claims deductions when money is paid. Annual cash flows may 191 | consist of income items: premium income, interest and dividends on 192 | assets, maturity proceeds of assets, and reinsurance recoveries in 193 | respect of claims. The outflows may consist of: claims settle or amounts 194 | paid on account, reinsurance premiums, expenses, tax, and dividends. In 195 | cash accounting, the profit, or net income, is simply the excess of 196 | money received over money paid. Following cash makes life easy - one 197 | simply has to look at the checkbook to see when money is coming in and 198 | when it goes out. Projecting annual cash flows as they “emerge,” or 199 | emerging cash flows, is an important task. 200 | 201 | However, firms cannot simply follow cash to determine profitability. 202 | Consider the case of a savings bank that receives deposits from their 203 | customers without making any payments. This money received is not a 204 | “profit” to the bank - rather the bank receives the deposit as money 205 | coming in but also has an obligation to return the money when demanded 206 | by the customer. That is, the bank’s obligations have increased with the 207 | customer deposit. 208 | 209 | Under “accrual accounting,” revenues and costs are recognized when they 210 | are incurred, not necessarily when related payments are received or 211 | made. In many businesses, the timing of incurral is easy to identify. 212 | Typically, a company records revenue as incurred when a product or 213 | service is shipped or billed to a customer, not when payments are 214 | actually made (they could be much later if you are buying on a payment 215 | plan). To assess profitability, a company’s costs need to be matched to 216 | the revenue. 217 | 218 | Matching costs to revenue is difficult in life insurance because of the 219 | potential time lag between premiums (one type of revenue) and benefits. 220 | To understand the cost of a life insurance product, let us begin by 221 | considering a net basis, ignoring expenses from doing business as well 222 | as any profit motivation. 223 | 224 | Suppose that we have 1,000 identical policyholders age $x$ who purchase 225 | a policy with a generic premium payment schedule {$P_h$} and benefit 226 | schedule {$b_h$}. At some future time point, say $h$ years later, we 227 | expect there to be $1000 ~_h p_x$ policyholders alive with each policy 228 | having value $_h V$. Thus, the company’s total obligation is 229 | $1000 ~_h p_x ~_h V$. One year later, the company’s obligation will be 230 | $1000 v ~_{h+1} p_x ~_{h+1} V$. The increase in this obligation is a 231 | cost to the company. Using the recursive reserve formula, this cost may 232 | be expressed as $$\begin{aligned} 233 | 1000 ~v _{h+1} p_x ~_{h+1} V &-& 1000 ~_h p_x ~_h V \\ &= 234 | 1000 ~_h p_x \left( v p_{x+h}~_{h+1} V - ~_h V \right) \\ 235 | &= 1000 ~_h p_x \left( v p_{x+h}~_{h+1} V - \left\{ v q_{x+h} 236 | b_{h+1} - P_h +v p_{x+h}~_{h+1} V 237 | \right\}\right) \\ 238 | &= 1000 ~_h p_x \left( P_h - v q_{x+h} b_{h+1} \right) ,\end{aligned}$$ 239 | with the relation $~_{h+1} p_x = ~_h p_x ~p_{x+h}$. Thus, the cost is 240 | simply the expected cash inflow, premiums in excess of benefits. This is 241 | anticipated as we are working on a net basis which implies zero profits. 242 | 243 | ## Annual Profits {#S:AnnProfit} 244 | 245 | It is important to monitor the actual experience of a block of business. 246 | To this end, we use carats, or “hats,” to denote actual values. We use 247 | 248 | - $\hat{i}_k$ is the actual rate of return from invested assets, 249 | 250 | - $\hat{e}_k$ is the actual annual expenses per contract, 251 | 252 | - at death, $\hat{q}_{[x]+k}^{(d)}$ is the realized fraction of 253 | deaths, 254 | 255 | - at withdrawal, $\hat{q}_{[x]+k}^{(w)}$ is the realized fraction of 256 | withdrawals, and 257 | 258 | - at survival to the end of the year, 259 | $\hat{p}_{[x]+k}^{(\tau)} = 1 - (\hat{q}_{[x]+k}^{(d)}+\hat{q}_{[x]+k}^{(w)})$ 260 | is realized fraction that survived. 261 | 262 | With these quantities, the profit during the year (at time $k+1$) is 263 | $$\begin{aligned} 264 | Profit_{k+1} &= & 265 | \left( _k AS + G_k -\hat{e}_k\right) (1+ \hat{i}_k) - 266 | \hat{q}_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1}\right) - \hat{q}_{[x]+k}^{(w)} ~_{k+1} CV - \hat{p}_{[x]+k}^{(\tau)} ~_{k+1} AS .\end{aligned}$$ 267 | For this illustration, we have that assumed premiums $G_k$ and 268 | settlement expenses $E_k$ are the same as actual values. We also assume 269 | that $ _k AS$ represents the terminal (year-end) company obligation 270 | (that does not depend on experience). Recall the recursive asset share 271 | calculation $$\begin{aligned} 272 | (~_k AS + G_k - e_k)(1+i_k) = 273 | q_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1} - ~_{k+1} AS \right) + q_{[x]+k}^{(w)} ( _{k+1} CV - ~_{k+1} AS) + ~_{k+1} AS\end{aligned}$$ 274 | Substituting for $_{k+1} AS$, we may write the profit during the year is 275 | $$\begin{aligned} 276 | Profit_{k+1} &= & 277 | \left( _k AS + G_k -\hat{e}_k\right) (1+ \hat{i}_k) - 278 | \hat{q}_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1} - ~_{k+1} AS\right) \\ 279 | &~~~~~~~~-& \hat{q}_{[x]+k}^{(w)} (~_{k+1} CV -~_{k+1} AS) - ~_{k+1} AS \\ 280 | &= & 281 | \left( _k AS + G_k \right) (\hat{i}_k - i_k) \\ 282 | &~~~~~~~~+& e_k (1+ \hat{i}_k) - \hat{e}_k (1+ i_k) \\ 283 | &~~~~~~~~+& \left(b_{k+1} + E_{k+1} - ~_{k+1} AS\right) (q_{[x]+k}^{(d)} - \hat{q}_{[x]+k}^{(d)} ) \\ 284 | &~~~~~~~~+& \left( _{k+1} CV - ~_{k+1} AS\right) (q_{[x]+k}^{(w)} - \hat{q}_{[x]+k}^{(w)} ) .\end{aligned}$$ 285 | This is one way to decompose the profit into identifiable components, 286 | known as the *analysis of surplus*. Here, we interpret these portions of 287 | the profit: 288 | 289 | - $\left( _k AS + G_k \right) (\hat{i}_k - i_k)$ - due to favorable 290 | investment experience, 291 | 292 | - $e_k (1+ \hat{i}_k) - \hat{e}_k (1+ i_k)$ - due to favorable 293 | expense experience, 294 | 295 | - $\left(b_{k+1} + E_{k+1} - ~_{k+1} AS\right) (q_{[x]+k}^{(d)} - \hat{q}_{[x]+k}^{(d)} )$ - 296 | due to favorable mortality experience, and 297 | 298 | - $\left( _{k+1} CV - ~_{k+1} AS\right) (q_{[x]+k}^{(w)} - \hat{q}_{[x]+k}^{(w)} )$ - 299 | due to favorable withdrawal experience. 300 | 301 | ## Profit Testing {#S:ProfitTest} 302 | 303 | In the prior section, we defined profits based on actual experience. It 304 | is also helpful to define a profit at the plan design stage, before 305 | experience is realized. Instead of using actual experience, we now 306 | remove the carats and employ a *profit test basis* which is the set of 307 | assumptions used to examine the incidence and timing of profits. 308 | 309 | The profit during the year (at time $k+1$) is $$\begin{aligned} 310 | Pr_{k+1} &= & 311 | \left( _k AS + G_k -e_k\right) (1+ i_k) - 312 | q_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1}\right) - q_{[x]+k}^{(w)} ~_{k+1} CV - p_{[x]+k}^{(\tau)} ~_{k+1} AS .\end{aligned}$$ 313 | When summarizing profits, it can be helpful to remind ourselves that 314 | profits are only available for those policies in force at the beginning 315 | of the year. Thus, we might define a term such as 316 | $\Pi_{k+1} = ~_k p_{[x]}^{(\tau)} Pr_{k+1}$ for profits discounted for 317 | survivorship. The term *profit signature* is used for the vector of 318 | discounted profits 319 | $\boldsymbol{\Pi} = \left(\Pi_0, \Pi_1 \ldots\right)^{\prime}$. 320 | 321 | Profits depend on all the assumptions, including assumed interest. To 322 | summarize profits, one measure used is the *internal rate of return* 323 | (*IRR*), defined to be the solution of the equation $$\begin{aligned} 324 | \sum_k \left(\frac{1}{1+j} \right)^k \Pi_k = \sum_k \left(\frac{1}{1+j} \right)^k ~_{k-1} p_{[x]}^{(\tau)} Pr_k = 0,\end{aligned}$$ 325 | where profits are calculated using interest rate $i_k \equiv j$. The 326 | *IRR* is the solution of a nonlinear equation and so may not exist or 327 | may have multiple solutions. For the multiple solution problem, we can 328 | use the *hurdle rate*, defined to be the smallest *IRR* so that the 329 | contract is deemed adequately profitable. 330 | 331 | We can summarize profits using an assumed discount rate, $r$. Define the 332 | *net present value*, also known as the *expected present value of future 333 | profits*, to be $$\begin{aligned} 334 | NPV = \sum_k \left(\frac{1}{1+r} \right)^k \Pi_k = \sum_k \left(\frac{1}{1+r} \right)^k ~_{k-1} p_{[x]}^{(\tau)} Pr_k,\end{aligned}$$ 335 | where profits are calculated using interest rate $i_k \equiv r$. 336 | 337 | Another profit measure is the *discounted payback period*, the first 338 | time that the sum of discounted profits is non-negative. Here, the 339 | discounting is for (1) survival and (2) for interest, using risk 340 | discount rate $r$. That is, the discounted payback period is the 341 | smallest value of $m$ such that $$\begin{aligned} 342 | \sum_{k=0}^m \left(\frac{1}{1+r} \right)^k \Pi_k \ge 0.\end{aligned}$$ 343 | 344 | ## Deferred Acquisition Expenses and Modified Premium Reserves {#S:DAExpenses} 345 | 346 | In Chapter 7, we introduced the random future net and gross loss 347 | variables. Taking expectations, we have the corresponding policy values, 348 | 349 | $~_t V^n =$ EPV future benefits - EPV future net premiums 350 | 351 | and 352 | 353 | $~_t V^g =$ EPV future benefits + EPV future expenses - EPV future gross 354 | premiums. 355 | 356 | We interpret the difference, EPV future gross premiums - EPV future net 357 | premiums, to be the expected present value (EPV) of future expense 358 | loadings. Thus, we may define the EPV of expenses as 359 | 360 | $~_t V^e =~_t V^g - ~_t V^n$ = EPV future expenses - EPV future expense 361 | loadings. 362 | 363 | If we further assume that $~_0 V^g = ~_0 V^n =0 $ as is customary under 364 | the equivalence principle, then $~_0 V^e =0$. As we have seen in 365 | Section \[S:ExpenseAugment\], it is common for expenses to be high 366 | during early policy durations relative to later durations. It is also 367 | common for expense loading to be flat, meaning that in general one can 368 | anticipate a negative expense value in early durations. A negative 369 | expense value is referred to as a *deferred acquisition cost*, or DAC. 370 | 371 | Insurers use policy values for reserves, or liabilities, held against 372 | future obligations. At least at early durations, an insurer is required 373 | to hold more capital under a net premium basis than a gross premium 374 | basis. While it is not common to use gross premium basis in many 375 | jurisdictions, in the US the net premium valuation basis has been 376 | preferred, in part because it restricts an insurer’s ability to 377 | manipulate reserves by making accounting adjustments. 378 | 379 | Changes in reserves are expenses to a company and using a larger net 380 | premium basis can mean large expenses that may penalize smaller, 381 | fast-growing companies. Thus, a number of modified premium reserve bases 382 | have been introduced over the years, the most commonly used one being 383 | the 384 | 385 | **Full Preliminary Term Approach.** Let $P^n$ be the valuation net 386 | premium that we assume level for now. Further, define, for valuation 387 | purposes, 388 | 389 | - $_1 P_{[x]}^n = b_1 v q_{[x]}$ a modified first year cost of 390 | insurance and 391 | 392 | - $P^{FPT}$ a modified premium for subsequent years. 393 | 394 | The idea is the $P^n - ~ _1 P_{[x]}^n$ is the additional first year 395 | expense allowance. Thus, we have 396 | $0 \leq ~_1 P_{[x]}^n \leq P^n \leq P^{FPT}$. 397 | 398 | Let $h$ be the number of premium-payment periods and $j$ ($\leq h$) be 399 | the number of years that $P^{FPT}>P^n$. Then, $$\begin{aligned} 400 | _1 P_{[x]}^n + P^{FPT} a_{x:\overline{j-1|}} = P^n \ddot{a}_{x:\overline{j|}} .\end{aligned}$$ 401 | This is equivalent to considering splitting the policy into two 402 | components, a 1-year term and a separate contract issued to the same 403 | life one year later, if the life survives. 404 | 405 | With this $V_k^{FPT} = V_k^n, k\ge j$, that is, the full preliminary 406 | term reserve equals the net premium reserve when the premiums are the 407 | same. For $1 \le k < j$, we have $$\begin{aligned} 408 | V_k^{FPT} = \textrm{EPV Future Benefits} - P^{FPT} \ddot{a}_{[x]+k:\overline{j-k|}} - P^n ~_{j-k|h-j} \ddot{a}_{[x]+k} .\end{aligned}$$ 409 | Thus, for $1 \le k < j$ we have $V_k^{FPT} < V_k^n$ and 410 | $$\begin{aligned} 411 | V_k^n - V_k^{FPT}& =& \left\{\textrm{EPV Future Benefits} - P^{FPT} \ddot{a}_{[x]+k:\overline{k-j|}} - P^n ~_{j-k|h-j} \ddot{a}_{[x]+k} \right\}\\ 412 | & ~~~ -& \left\{\textrm{EPV Future Benefits} - P^n \ddot{a}_{[x]+k:\overline{k-j|}} - P^n ~_{j-k|h-j} \ddot{a}_{[x]+k} \right\} \\ 413 | & = & (P^{FPT}- P^n) \ddot{a}_{[x]+k:\overline{k-j|}} .\end{aligned}$$ 414 | We can think of the modified reserve $V_k^{FPT}$ as the net premium 415 | reserve minus a reserve for the unamortized excess premium. 416 | 417 | **Special Case 1. Ordinary Whole Life.** Here, take $j = \omega - x$ and 418 | $_1 P_{[x]}^n = v _1 q_{[x]} = A_{[x]:\overline{1|}}^1$. Thus, 419 | $A_{[x]:\overline{1|}}^1+ P^{FPT} a_x =P_x \ddot{a}_x$ and 420 | $$\begin{aligned} 421 | P^{FPT} &= \frac{P_x \ddot{a}_x - A_{[x]:\overline{1|}}^1}{a_x} = \frac{A_x - A_{[x]:\overline{1|}}^1}{a_x} \\ 422 | &= \frac{_1 E_x A_{x+1}}{ _1 E_x \ddot{a}_{x+1}} = \frac{A_{x+1}}{\ddot{a}_{x+1}} = P_{x+1} .\end{aligned}$$ 423 | 424 | **Special Case 2. $n$-pay, $m$-year Endowment.** Here, take $j = n$. 425 | Then, use the same logic in Special Case 1 to check that 426 | $P^{FPT} = _{n-1} P_{x+1:\overline{m-1|}}.$ Further, $$\begin{aligned} 427 | V_k^{FPT} &= A_{x+k:\overline{m-k|}} - _{n-1} P_{x+1:\overline{m-1|}} \ddot{a}_{x+k:\overline{m-k|}} 428 | = ~_{k-1}^{n-1} V_{x+1:\overline{m-1|}},\end{aligned}$$ that is, a 429 | reserve for a life age ($x+1$) at duration $k-1$ for an $n-1$-pay, 430 | $m-1$-year endowment policy. 431 | -------------------------------------------------------------------------------- /LifeCon1/Chapters/PensionLecture06Nov2014.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Pension Contingencies" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | # Pension Contingencies 12 | ## Introduction 13 | 14 | We think of a pension plan as a type of deferred compensation scheme - 15 | an arrangement made by an employer wherein benefits are accrued during a 16 | working lifetime of an employee and are paid during retirement. 17 | 18 | Several *ancillary* (i.e., auxiliary, assisting) benefits are typically 19 | associated with such a scheme: 20 | 21 | - survivor’s pension 22 | 23 | - lump sum death benefit 24 | 25 | - disability benefit 26 | 27 | - withdrawal benefit. 28 | 29 | The main benefit is a life annuity payable at retirement. 30 | 31 | There are two major categories of pension plans 32 | 33 | - *Defined Contribution Plans*. This type of plan specifies how much 34 | the employer (and employee) will contribute into the plan. 35 | Contributions are accumulated until withdrawal and then are 36 | typically converted into a life annuity. 37 | 38 | - *Defined Benefit Plans*. This type of plan specifies the level of 39 | benefit promised by a plan. Although the plan may call for some 40 | employee contributions, it is typically the responsibility of the 41 | employer, as plan sponsor, to make sure that the plans has 42 | sufficient funds to provide the promised benefits. 43 | 44 | When designing the plan, participants look to a desired *replacement 45 | ratio*, defined to be $$\begin{aligned} 46 | R = \frac{\text{pension income in the year after retirement}}{\text{salary in the year before retirement}} ,\end{aligned}$$ 47 | e.g., 50-70%. 48 | 49 | Pension contingencies is similar to life contingencies but there are 50 | pragmatic (and theoretical) differences: 51 | 52 | - Pension plans deal with smaller numbers of people. Hence, the 53 | uncertainty takes on a greater role. 54 | 55 | - Unlike life insurance, there is no real “underwriting” concept in 56 | pension work. Thus, pension actuaries have less control over many 57 | variables that go into projections, e.g., salary scales in benefit 58 | formulas, demographic variables such as number of new entrants, 59 | age/service composition of the workforce. 60 | 61 | - There is a different attitude towards “conservatism.” In pricing, 62 | the life actuary errs on the side of the company in setting 63 | assumptions and uses dividends/bonuses to address equity 64 | considerations. However, for the pension actuary, an error on the 65 | low side can cause plan ruination while an error on the high side 66 | means that the current profit of an employer is considerably 67 | reduced. 68 | 69 | - Because the employer is a major contributor to costs of a plan, we 70 | tend to take a “macro” in lieu of a “micro” viewpoint. That is, a 71 | plan’s overall ability to meet objectives may outweigh equity 72 | concerns for each individual participant. 73 | 74 | Employers sponsor plans for a number of reasons, including: 75 | 76 | - competitions for new employees; 77 | 78 | - to facilitate turnover of older employees by ensuring that they can 79 | retire 80 | 81 | - retention - to provide incentive for employees to stay with the 82 | employer 83 | 84 | - pressure from trade unions 85 | 86 | - tax efficient method of remunerating employees 87 | 88 | - responsibility to employees who have contributed to the company. 89 | 90 | ## Demographic Assumptions 91 | 92 | We assume we have available a (primary) multidecrement table with $T$ is 93 | the time to termination of employment. The cause of termination may 94 | include withdrawal, death in service, retirement for disability, or 95 | retirement for age-service. Recall our example discussed earlier: 96 | 97 | **Special Case 2. Pension Plan Termination and Retirement Model.** In 98 | this special case, we might let 99 | 100 | - 0 means Active 101 | 102 | - 1 means Retired 103 | 104 | - 2 means Disabled 105 | 106 | - 3 means Death 107 | 108 | - 4 means Termination 109 | 110 | - 5 means Withdraw for Other Reasons 111 | 112 | This model is relevant for certain retirement systems where benefit 113 | amounts vary by the reason for leaving employment. 114 | 115 | Source: 2003 *Society of Actuaries Pension Plan Turnover Study*. 116 | 117 | http://www.soa.org/research/experience-study/ 118 | pension/research-2003-soa-pension-plan-turnover-study.aspx 119 | 120 | ![Aggregate Turnover – Termination and Retirement. Termination estimated 121 | probabilities are smoothed via Whittaker-Henderson Type B whereas 122 | retirement estimated probabilities are unsmoothed. Estimated 123 | probabilities are in 124 | percent.](Figures\PC_RetirementSystemTurnoverL.pdf){width=".5\textwidth"} 125 | 126 | ![Aggregate Turnover – Death, Disability and Other. All estimated 127 | probabilities are smoothed. The vertical scale indicates that 128 | probabilities of turnover from these sources are much smaller than 129 | retirement and termination. Estimated probabilities are in 130 | percent.](Figures\PC_RetirementSystemTurnoverR.pdf){width=".5\textwidth"} 131 | 132 | We may also have secondary decrement tables that summarize quantities 133 | such as $\bar{a}_x^r$ - a life annuity for a retiree aged $x$. This life 134 | annuity may follow a different mortality pattern than those in service 135 | and, indeed, may use a different interest rate for discounting present 136 | values. 137 | 138 | ## Interest and Salary Assumptions 139 | 140 | Many plans, such as multiple employer plans, provide pension benefits in 141 | terms of \$*K* per month per year of service. So, for example, if an 142 | individual who has worked for 30 years receives $K=10$ \$, then that 143 | person receives $10 \times 30 = 300$ \$ per month at retirement. 144 | 145 | Further, it is also common for plans to provide a percent of salary at 146 | retirement. To this end, we may project salaries using 147 | 148 | $$\begin{aligned} 149 | (ES)_{x+k} = (AS)_x \frac{s_{x+k}}{s_x},\end{aligned}$$ 150 | 151 | where 152 | 153 | - $(ES)_{x+k}$ is the estimated or projected salary at age $x+k$, 154 | 155 | - $(AS)_x$ is the actual salary at age $x$, and 156 | 157 | - $s_x$ is the salary scale at age $x$. 158 | 159 | The salary scale $s_x$ may come from an integer age table or be based on 160 | a continuous mathematical function. For example, it is common to use 161 | $s_y = 1.04^y$ to represent a 4% salary escalation assumption. 162 | 163 | **Salary Rate**.\ 164 | Let $\bar{s}_y$ be the annual rate of salary at age $y$ so that the 165 | salary received from age $y$ to $y+1$ is $$\begin{aligned} 166 | s_y = \int_0^1 \bar{s}_{y+t} dt ,\end{aligned}$$ Thus, 167 | $$\begin{aligned} 168 | \frac{s_y}{s_x} &= \frac{\int_0^1 \bar{s}_{y+t} dt}{\int_0^1 \bar{s}_{x+t} dt} \\ 169 | &= \frac{\text{salary received from age y to y+1 }}{\text{salary received from age x to x+1 }} ,\end{aligned}$$ 170 | Now, suppose that $\bar{s}_y = 1.04^{y-20}$. Then $$\begin{aligned} 171 | s_y &= \int_0^1 \bar{s}_{y+t} dt = \int_0^1 1.04^{y-20+t} dt \\ 172 | &= 1.04^{y-20} \int_0^1 1.04^t dt =1.04^{y-20} \times \text{constant}\end{aligned}$$ 173 | Thus, $$\begin{aligned} 174 | \frac{s_y}{s_x} &= \frac{1.04^{y-20} \times \text{constant}}{1.04^{x-20} \times \text{constant}} \\ 175 | &= \frac{1.04^{y-20}}{1.04^{x-20} } =\frac{\bar{s}_y}{\bar{s}_x} \end{aligned}$$ 176 | Using the rate or the salary scale gives the same result. 177 | 178 | **AMLCR Example 10.2.** 179 | 180 | The pension benefit final average salary is the average salary three 181 | years before retirement. Assume a salary scale $s_y = 1.04^y$. Members’ 182 | salaries are increased each year, six months before the valuation date. 183 | 184 | - a\) A member aged 35 received \$75,000 in the year before the valuation 185 | date. Calculate the projected final average salary assuming retirement 186 | at age 65. 187 | 188 | - b\) A member aged 55 received was paid salary at a rate of \$100,000 per 189 | year before at that time. Calculate the projected final average salary 190 | assuming retirement at age 65. 191 | 192 | *Solution:* a) Received salary of 75,000 from age 34 to 35, so 193 | $(AS)_{34}=75000$. Thus, $$\begin{aligned} 194 | \text{Pred 3 Year Avg Salary}&= \frac{(ES)_{62}+(ES)_{63}+(ES)_{64}}{3} \\ 195 | &= (AS)_{34} \frac{s_{62}+s_{63}+s_{64}}{3 s_{34}} \\ 196 | &= (75000) \frac{1.04^{62}+1.04^{63}+1.04^{64}}{3 (1.04^{34})} \\ 197 | &= (75000) \frac{1.04^{28}+1.04^{29}+1.04^{30}}{3 } = 234,018.80.\end{aligned}$$ 198 | 199 | b\) Using mid-year assumption, the actual salary from 54.5 to 55.5 is 200 | 100,000. Thus, we have $$\begin{aligned} 201 | \text{Pred 3 Year Avg Salary} &= \frac{(ES)_{62}+(ES)_{63}+(ES)_{64}}{3} \\ 202 | &= (AS)_{54.5} \frac{s_{62}+s_{63}+s_{64}}{3 s_{54.5}} \\ 203 | &= (100000) \frac{1.04^{62}+1.04^{63}+1.04^{64}}{3 (1.04^{54.5})} = 139,638.80.\end{aligned}$$ 204 | 205 | $\Box$ 206 | 207 | ## Defined Contribution Plans 208 | 209 | In a defined contribution plan, the actuarial present value is the 210 | accumulation under interest of contributions made by or for the 211 | participants. The benefit is an annuity that can be purchased by such 212 | accumulation. The accumulated valued is typically available upon death 213 | and, under certain conditions, upon withdrawal before retirement. 214 | 215 | One task is to set the contribution rate to meet the target replacement 216 | rate for a “model” employee. 217 | 218 | **Actuarial Mathematics Example 11.5.2.** The contribution rate, which 219 | is to be applied as a proportion of salary, is calculated for a new 220 | participant at age 30. Assume that contributions are made at mid-year 221 | and use the UDD assumption for active multiple decrements. 222 | 223 | - Assume that there are no withdrawal benefits for the first 5 years 224 | but after that the accumulated contributions are vested, that is, 225 | become the property of the withdrawing participant, and will be 226 | applied toward an annuity to start no earlier than age 60. 227 | 228 | - Upon death after the end of the 5-year vesting period but before 229 | retirement income has commenced, the accumulated contributions are 230 | paid out. 231 | 232 | - For age-retirement at age 65, the benefit is a 10-year certain and 233 | life annuity with a benefit rate of 50% of the average salary over 234 | the 5 years between ages 60 and 65. 235 | 236 | Find the contribution level to provide for this benefit. 237 | 238 | *Solution.* 239 | 240 | The expected present value of benefits is $$\begin{aligned} 241 | APVB = ~_5 p_{30}^{(\tau)} ~v^{35} (0.5) ~(ES)_{65} ~\bar{a}_{\overline{65:\overline{10|}}}^r\end{aligned}$$ 242 | where $~\bar{a}_{\overline{65:\overline{10|}}}^r$ is the APV of a 243 | 10-year certain and life annuity to a retiree and $$\begin{aligned} 244 | (ES)_{65} = \frac{s_{60}+s_{61}+s_{62}+s_{63}+s_{64}}{5 s_{30}}\end{aligned}$$ 245 | is the projected (estimated) salary at age 65. Note that because 246 | contributions are as a percentage of salary, we may take the actual 247 | salary to equal 1. Thus, 248 | $(0.5) ~(ES)_{65} ~\bar{a}_{\overline{65:\overline{10|}}}^r$ is the 249 | present value of retirement benefit at age 65. We discount this for 250 | interest using $v^{35}$. Because those withdrawing after 5 years receive 251 | their full benefit, we need only consider the fraction that survive 5 252 | years, $~_5 p_{30}^{(\tau)}$. 253 | 254 | The expected present value of contributions for the first five years is 255 | $$\begin{aligned} 256 | APVC_1 &= 257 | c \sum_{k=0}^4 v^{k+1/2} ~ \frac{s_{30+k}}{s_{30}} ~_{k+1/2} p_{30}^{(\tau)} \\ 258 | &= 259 | c \sum_{k=0}^4 v^{k+1/2} ~ \frac{s_{30+k}}{s_{30}} ~_k p_{30}^{(\tau)} \left(1-0.5 q_{30+k}^{(\tau)} \right) ,\end{aligned}$$ 260 | using the UDD assumption. The expected present value of contributions 261 | for the next 30 years is $$\begin{aligned} 262 | APVC_2 = 263 | c ~_5 p_{30}^{(\tau)} \sum_{k=5}^{34} v^{k+1/2}~ \frac{s_{30+k}}{s_{30}} .\end{aligned}$$ 264 | Thus, we determine the contribution rate $c$ through equating 265 | $APVC_1 +APVC_2 = APVB$. 266 | 267 | $\Box$ 268 | 269 | ## Valuation of Benefits 270 | 271 | ### Final Salary Plans 272 | 273 | $$\begin{aligned} 274 | Benefit = (Yrs Of Service)\times(Final Average Salary)\times (Accrual Rate)\end{aligned}$$ 275 | 276 | Accrual Rate - $\alpha$, typically 1-2%, Wisconsin Retirement System is 277 | 1.6%. 278 | 279 | Entry age $x$, current age $y$, Retirement age 60. You could decompose 280 | this as: $$\begin{aligned} 281 | (60-x) (FAS) (AR) = (y-x) (FAS) (AR) +(60-y) (FAS) (AR)\end{aligned}$$ 282 | think of $(y-x) (FAS) (AR)$ as from past service, the accrued benefit. 283 | This is the part that the plan has agreed to fund based on past service 284 | and so is a liability of the plan (for valuation purposes). 285 | 286 | **Withdrawal Pension** Similarly, $(y-x) (FAS) (AR)$ is the amount that 287 | a person is due when he/she leaves the plan, except that $FAS$ is 288 | replaced by actual salary. 289 | 290 | This is sometimes adjusted by cost of living (inflation) increases 291 | (mandatory in the UK). 292 | 293 | - Projected Unit Credit - salaries are projected to exit date 294 | 295 | - Traditional (Current) Unit Credit - salaries are not projected 296 | 297 | ### Career Average Earnings Plans 298 | 299 | $$\begin{aligned} 300 | Benefit = (Yrs Of Service)\times(Career Average Salary)\times(Accrual Rate)\end{aligned}$$ 301 | 302 | *Career average revalued plan*, inflation adjustment is made before 303 | averaging salaries. 304 | -------------------------------------------------------------------------------- /LifeCon1/Chapters/UnivLifeLecture14Nov2012.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Universal Life" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | # Universal Life 12 | ## Introduction to Universal Life 13 | 14 | A **Universal Life** (UL) contract combines life insurance with an 15 | investment product in a transparent, flexible format. The policyholder 16 | may vary the amount and timing of premiums, within some constraints. 17 | Here is how it works: 18 | 19 | - The **premium** is deposited into an account, which is used to 20 | determine the death and survival benefits. 21 | 22 | - The insurer credits **interest** to the account at regular intervals 23 | (typically monthly). 24 | 25 | - The account, made up of the premiums and credited interest, is 26 | subject to deductions for management **expenses** and for the **cost 27 | of life insurance** coverage. 28 | 29 | - The account balance or **account value** (AV) is the balance of 30 | funds in the account. 31 | 32 | The account value represents the insurer’s liability, analogous to the 33 | reserve under a traditional contract. 34 | 35 | - Universal Life (UL) policies are sold as “permanent” life contracts 36 | (with tax advantages) 37 | 38 | - The AV represents the underlying value of the contract, and is used 39 | to determined the cash value 40 | 41 | With the UL, the policyholder’s money is not associated with specific 42 | assets, they are held in “notional” accounts. Credited interest declared 43 | need not reflect actual earnings 44 | 45 | In contrast, the **Variable Universal Life** (VUL), monies are held in 46 | separate accounts. Interest credited is directly associated with the 47 | yields on these funds. This is studied in Chapter 12 of DHW (but not by 48 | us). 49 | 50 | ![image](Figures\UL_LifeCycleExample.pdf){width="120.00000%"} 51 | 52 | **Origins of Universal Life** 53 | 54 | Universal Life (UL) account values are based on an accumulation 55 | (recursive) process - known for well over 100 years. The idea behind UL 56 | apparently came up in the 1960’s, was made public in the 1970’s, and 57 | took off as a product in the 1980’s. Here are some references from Black 58 | and Skipper on *Life Insurance*, 59 | 60 | - George R. Dinney, “Universal Life,” *The Actuary*, Vol. XV, 61 | supplement (September 1981). p. 1; 62 | 63 | - Timothy Lynch, ”Universal Life Insurance: A Primer,” *The Journal of 64 | the American Society of Chartered Life Underwriters*, Vol. XXXVI 65 | (July 1982) 66 | 67 | According to Black and Skipper, the UL concept was at first not welcomed 68 | by most persons in the North American life insurance business. It was 69 | perceived as a threat to the orderly development of the industry, and as 70 | not being in consumers’ or agents’ best interests. 71 | 72 | Universal life policy sales in the United States had a meteoric rise 73 | from an effective zero market share of new sales in 1979 to over 38 74 | percent in 1985, its peak year. Since then, its share has declined to 75 | around a quarter of new individual life premiums-still a major 76 | proportion. 77 | 78 | ## Design Features of a UL Policy 79 | 80 | We now review in more detail the key design features of a UL policy. 81 | 82 | - **Death Benefit:** On the policyholder’s death the total benefit 83 | paid is the account value of the policy, plus an **additional death 84 | benefit** (ADB). 85 | 86 | There are two types of death benefit: 87 | 88 | - **Type A** offers a *level* total death benefit $\equiv$ the 89 | **Face Amount** of the policy. Here, Face Amount = AV + ADB. As 90 | the account value (AV) increases, the ADB decreases. 91 | 92 | - **Type B** offers a *level ADB*. The amount paid on death would 93 | be the AV plus the level ADB selected by the policyholder. The 94 | policyholder may have the option to adjust the ADB to allow for 95 | inflation. 96 | 97 | - Corridor factor = $\frac{\text{AV + ADB}}{\text{AV}}$. 98 | 99 | - The ADB is required to be significant, e.g., around 2.5 at age 40, 100 | decreasing to 1.05 at age 90. 101 | 102 | ![image](Figures\UL_VaryingBenefitAmounts.pdf){width="120.00000%"} 103 | 104 | - **Premiums:** These may be subject to some minimum level, but 105 | otherwise are highly flexible. 106 | 107 | - **Expense Charges:** These are deducted from the account value. The 108 | rates will be variable at the insurer’s discretion, subject to a 109 | maximum specified in the original contract. 110 | 111 | - **Credited Interest:** Usually the credited interest rate will be 112 | decided at the insurer’s discretion, but it may be based on a 113 | published exogenous rate, such as yields on government bonds. A 114 | minimum guaranteed annual credited interest rate will be specified 115 | in the policy document. 116 | 117 | 118 | 119 | - **Cost of Insurance:** Each year the UL account value is subject to 120 | a charge to cover the cost of the selected death benefit cover. 121 | 122 | - The charge is called the Cost of Insurance, or CoI. 123 | 124 | - Usually, the CoI is calculated using an estimate (perhaps 125 | conservative) of the mortality rate for that period, so that, as 126 | the policyholder ages, the mortality charge (per \$1 of ADB) 127 | increases. 128 | 129 | - The CoI is then the single premium for a 1-year term insurance 130 | with sum insured equal to the ADB. 131 | 132 | - **Surrender Charge:** If the policyholder chooses to surrender the 133 | policy early, the surrender value paid will be the policyholder’s 134 | account balance reduced by a surrender charge. 135 | 136 | - The main purpose of the charge is to ensure that the insurer 137 | receives enough to pay its acquisition expenses. 138 | 139 | - The total cash available to the policyholder on surrender is the 140 | account value minus the surrender charge (or zero if greater), 141 | and is referred to as the **Cash Value** of the contract at each 142 | duration. 143 | 144 | 145 | 146 | - **Secondary Guarantees:** A common feature is a no lapse guarantee 147 | 148 | - Coverage continues even if the account value declines to zero, 149 | provided that the policyholder pays a pre-specified minimum 150 | premium 151 | 152 | - This guarantee comes into play if expense and mortality charges 153 | exceed the minimum premium 154 | 155 | - **Policy Loans:** Option: policyholder takes out a loan using the 156 | cash value as collateral. 157 | 158 | **Example: DHW SN 4.1: Type B UL Account Values** 159 | 160 | - A universal life policy is sold to a 45 year old man. The initial 161 | premium is \$2250 and the ADB is a fixed \$100,000. The policy 162 | charges are: $$\begin{aligned} 163 | \text{Cost of Insurance:}&&\text{120\% of the mortality of the Standard Select Mortality Model.}\\ 164 | &&\text{5\% per year interest.}\\ 165 | \text{Expense Charges:}&&\text{\$48 + 1\% of premium at the start of 166 | each year.}\end{aligned}$$ 167 | 168 | - Surrender penalties at each year end are the lesser of the full 169 | account value and the following surrender penalty schedule: 170 | 171 | ------------------- ------ ------ ------ ------ ------ ------- 172 | Year of surrender 1 2 3-4 5-7 8-10 $>$10 173 | Penalty 4500 4100 3500 2500 1200 0 174 | ------------------- ------ ------ ------ ------ ------ ------- 175 | 176 | - Assume (i) the policy remains in force for 20-years, (ii) interest 177 | is credited to the account at 5% per year, (iii) the policyholder 178 | pays the full premium of \$2250 each year, (iv) all cash flows occur 179 | at policy anniversaries. 180 | 181 | - Project the account value and the cash value at each year end for 182 | the 20-year projected term. 183 | 184 | 185 | 186 | - Each year, the insurer deducts from the account value the expense 187 | charge and the cost of insurance (which is the price for a 1-year 188 | term insurance with sum insured equal to the ADB), and adds to the 189 | account value any new premiums paid, and the credited interest for 190 | the year. 191 | 192 | - The first year is: 193 | 194 | ------------------- ------------------------------------------------------ 195 | Premium 2250 196 | Expense Charge $48 + 0.01\times 2250 = 70.50$ 197 | CoI $100,000\times 1.2 \times 0.0006592\times v = 75.34$ 198 | Interest Credited $0.05\times (2250-70.50-75.34) = 105.21$ 199 | Account Value $2250-70.50-75.34+105.21=2209.37$ 200 | Cash Value max$(2209.37-4500,0) = 0$ 201 | ------------------- ------------------------------------------------------ 202 | 203 | - The second year is:\ 204 | 205 | ------------------- ------------------------------------------------------ 206 | Premium 2250 207 | Expense Charge $48 + 0.01\times 2250 = 70.50$ 208 | CoI $100,000\times 1.2 \times 0.0007973\times v = 91.13$ 209 | Interest Credited $0.05\times (2209.37+2250-70.50-91.13) = 214.89$ 210 | Account Value $2209.37+2250-70.50-91.13+214.89=4512.63$ 211 | Cash Value max$(4512.63-4100,0) = 412.63$ 212 | ------------------- ------------------------------------------------------ 213 | 214 | ![image](Figures\UL_UniversalLifeFigure1.png){width=".9\textwidth"} 215 | 216 | ## Recursive Formulas for Universal Life 217 | 218 | ### Review of Asset Shares and Emerging Profits 219 | 220 | Recall our recursive asset share formula $$\begin{aligned} 221 | (~_k AS + G_k - e_k)(1+i_k) &= & 222 | q_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1}\right) \\ 223 | &~~~~+& q_{[x]+k}^{(w)} ~_{k+1} CV + p_{[x]+k}^{(\tau)} ~_{k+1} AS\end{aligned}$$ 224 | 225 | Asset shares are calculated during the year (using experience or another 226 | basis). Then, the profit during the year (at time $k+1$) is 227 | $$\begin{aligned} 228 | Pr_{k+1} &= & \left( _k AS + G_k -e_k\right) (1+ i_k) - 229 | q_{[x]+k}^{(d)} \left(b_{k+1} + E_{k+1}\right) \\ 230 | &~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV - p_{[x]+k}^{(\tau)} ~_{k+1} AS 231 | .\end{aligned}$$ 232 | 233 | Define $\Pi_{k+1} = ~_k p_{[x]}^{(\tau)} Pr_{k+1}$ to be profits 234 | discounted for survivorship. 235 | 236 | To summarize profits, we might use the net present value 237 | $$\begin{aligned} 238 | NPV = \sum_k \left(\frac{1}{1+r} \right)^k \Pi_k\end{aligned}$$ or 239 | another summary measure. 240 | 241 | ### Universal Life with Type B Death Benefit 242 | 243 | We can now apply these same basic principles to account value formulas. 244 | 245 | Use $AV$ for account value instead of AS, use $FA$ for face amount 246 | instead of benefit, and ignore lapsation. From the asset share formula, 247 | $$\begin{aligned} 248 | (~_k AV + G_k - e_k)(1+i_k^c) &= & 249 | q_{[x]+k} \left(FA_{k+1} + E_{k+1}\right) + p_{[x]+k} ~_{k+1} AV \\ 250 | &= q_{[x]+k} \left(FA_{k+1} + E_{k+1} - ~_{k+1} AV \right) +~_{k+1} 251 | AV\end{aligned}$$ 252 | 253 | With only one decrement (death), we have dropped the cause notation. 254 | Further, use $i_k^c$ for the interest credited. Now, we will define the 255 | Cost of Insurance to be $$\begin{aligned} 256 | CoI_{k} =v_q q_{[x]+k} \left(FA_{k+1} + E_{k+1} - ~_{k+1} AV 257 | \right)\end{aligned}$$ where $v_q$ is a discount factor. In the prior 258 | example, we used 5% for both the interest credited and the CoI discount 259 | factor. 260 | 261 | With discount factor $v_q$, the Cost of Insurance is $$\begin{aligned} 262 | CoI_{k} =v_q q_{[x]+k} \left(FA_{k+1} + E_{k+1} - ~_{k+1} AV 263 | \right) .\end{aligned}$$ With this, we define the account value 264 | $$\begin{aligned} 265 | ~_{k+1} AV = (~_k AV + G_k - e_k -CoI_k )(1+i_k^c)\end{aligned}$$ 266 | 267 | For Type B UL, the additional death benefit is $$\begin{aligned} 268 | ADB_{k+1} = FA_{k+1} - ~_{k+1} AV = \text{constant}\end{aligned}$$ so 269 | the recursion as presented is easy to calculate. Not so for Type A. We 270 | can use account values (with a possibly different interest rate) to 271 | determine the profit during the year (at time $k+1$) $$\begin{aligned} 272 | Pr_{k+1} &= & \left( _k AV + G_k -e_k\right) (1+ i_k) - 273 | q_{[x]+k}^{(d)} \left(FA_{k+1} + E_{k+1}\right) \\ 274 | &~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV - p_{[x]+k}^{(\tau)} ~_{k+1} AV 275 | .\end{aligned}$$ 276 | 277 | **Example: DHW SN 4.2: Type B UL Profit Test** 278 | 279 | - A universal life policy is sold to a 45 year old man, policy in 280 | force for 20 years. Premium is \$2250 for six years and the ADB is a 281 | fixed \$100,000. 282 | 283 | - 100% of the mortality of the Standard Select Mortality Model, 284 | 285 | - interest on all funds is 7 %, interest credited is 5% 286 | 287 | - Incurred expenses are \$2000 at inception, \$45 plus 1% of premium 288 | at renewal, \$50 on surrender (even if no cash value is paid), \$100 289 | on death. 290 | 291 | - Surrender (withdrawal) rates are:\ 292 | 293 | ------ ------ ------ ------ ------ ------- ------ 294 | Year 1 2-5 6-10 11 12-19 20 295 | Rate 0.05 0.02 0.03 0.10 0.15 1.00 296 | ------ ------ ------ ------ ------ ------- ------ 297 | 298 | - Project the profits at each year end for the 20 years. 299 | 300 | 301 | 302 | - The expense at policy initiation is 2,000, so $Pr_0 =-2000$. 303 | $$\begin{aligned} 304 | Pr_{k+1} &= & \left( _k AV + G_k -e_k\right) (1+ i_k) - 305 | q_{[x]+k}^{(d)} \left(FA_{k+1} + E_{k+1}\right) \\ 306 | &~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV - p_{[x]+k}^{(\tau)} ~_{k+1} AV 307 | .\end{aligned}$$ 308 | 309 | - The first year is:\ 310 | 311 | -------------------------------- ------------------------------------------------------ 312 | Account Value brought forward 0 313 | Premium 2250 314 | Expenses 0 (all accounted for in $Pr_0$) 315 | Interest Earned 0.07 $\times$ 2250 = 157.50 316 | Expected Death Costs 0.0006592 $\times$ (100 000 + 2209.37 + 100) = 67.44 317 | Expected Surrender Costs 0.999341 $\times$ 0.05 $\times$ (0 + 50) = 2.50 318 | Expected Cost of AV 319 |   for continuing policyholders 0.999341 $\times$ 0.95 $\times$ 2209.37 = 2097.52 320 | $Pr_1$ 2250 + 157.50 - 67.44 - 2.50 - 2097.52 = 240.04 321 | -------------------------------- ------------------------------------------------------ 322 | 323 | - $q^{(d)}=q^{\prime (d)}=0.0006592$, 324 | $q^{(w)}=(1-q^{\prime (d)})q^{\prime (w)}=0.999341 \times 0.05$, 325 | and $p^{(\tau)}=(1-q^{\prime (d)})(1-q^{\prime (w)})=(0.999341) 326 | (0.95)$ 327 | 328 | 329 | 330 | - $$\begin{aligned} 331 | Pr_{k+1} &= & \left( _k AV + G_k -e_k\right) (1+ i_k) - 332 | q_{[x]+k}^{(d)} \left(FA_{k+1} + E_{k+1}\right) \\ 333 | &~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV - p_{[x]+k}^{(\tau)} ~_{k+1} AV 334 | .\end{aligned}$$ 335 | 336 | - The second year is:\ 337 | 338 | -------------------------------- ------------------------------------------------------- 339 | Account Value brought forward 2209.37 340 | Premium 2250 341 | Expenses 45 + 0.01 $\times$ 2250 = 67.50 342 | Interest Earned 0.07 $\times$ (2209.37 + 2250 - 67.50) = 307.43 343 | Expected Death Costs 0.0007973 $\times$ (100 000 + 4512.63 + 100) = 83.41 344 | Expected Surrender Costs 0.9992027 $\times$ 0.02 $\times$ (412.63 + 50) = 9.25 345 | Expected Cost of AV 346 |   for continuing policyholders 0.9992027 $\times$ 0.98 $\times$ 4512.63 = 4418.85 347 | $Pr_2$ 2209.37 + 2250 - 67.50 + 307.43 - 83.41 348 |         - 9.25 - 4418.85 =187.79 349 | -------------------------------- ------------------------------------------------------- 350 | 351 | ![image](Figures\UL_Figs6,7DHWSN42TypeBULProfitTest.pdf){width=".9\textwidth"} 352 | 353 | ![image](Figures\UL_Figs6,7DHWSN42TypeBULProfitTestB.pdf){width=".9\textwidth"} 354 | 355 | ## Type A Universal Life 356 | 357 | We now modify our definition of the Cost of Insurance to accommodate 358 | Type A policies and the “corridor factor.” To simplify matters, drop 359 | settlement expenses and define the unmodified version to be 360 | $$\begin{aligned} 361 | CoI_{k}^f =v_q q_{[x]+k} \left(FA_{k+1} - ~_{k+1} AV ^f \right)\end{aligned}$$ 362 | and the associated account value $$\begin{aligned} 363 | ~_{k+1} AV^f = (~_k AV + G_k - e_k -CoI_k ^f )(1+i_k^c)\end{aligned}$$ 364 | Recall that we can think of the corridor factor as 365 | $\frac{\text{AV + ADB}}{\text{AV}}=\frac{\text{ADB}}{\text{AV}}+1$. So, 366 | now suppose that the corridor factor $\gamma_{k+1}$ is (exogenously) 367 | given. Define a modified cost of insurance $$\begin{aligned} 368 | CoI_{k}^c &=v_q q_{[x]+k} (\gamma_{k+1} -1) \times ~_{k+1} AV ^c\end{aligned}$$ 369 | and the associated account value $$\begin{aligned} 370 | ~_{k+1} AV^c = (~_k AV + G_k - e_k -CoI_k ^c )(1+i_k^c)\end{aligned}$$ 371 | 372 | The account value is thus $$\begin{aligned} 373 | ~_{k+1} AV = \min \left(~_{k+1} AV^f , ~_{k+1} AV ^c \right) .\end{aligned}$$ 374 | With $$\begin{aligned} 375 | ~_{k+1} AV^f &= (~_k AV + G_k - e_k -CoI_k ^f )(1+i_k^c)\\ 376 | ~_{k+1} AV^c &= (~_k AV + G_k - e_k -CoI_k ^c )(1+i_k^c)\end{aligned}$$ 377 | we can write this recursively as $$\begin{aligned} 378 | ~_{k+1} AV = (~_k AV + G_k - e_k -CoI_k )(1+i_k^c) ,\end{aligned}$$ 379 | where $$\begin{aligned} 380 | CoI_k = \max \left(CoI_k ^f , CoI_k ^c \right)\end{aligned}$$ 381 | 382 | To use the recursive formula for the account value, we need to calculate 383 | the cost of insurance. To this end, we start with $$\begin{aligned} 384 | CoI_{k}^f &=v_q q_{[x]+k} \left(FA_{k+1} - ~_{k+1} AV ^f \right) \\ 385 | ~_{k+1} AV^f &= (~_k AV + G_k - e_k -CoI_k ^f )(1+i_k^c)\end{aligned}$$ 386 | We can write $$\begin{aligned} 387 | CoI_{k}^f = v_q q_{[x]+k} \left(FA_{k+1} - (~_k AV + G_k - e_k 388 | -CoI_k ^f )(1+i_k^c) \right)\end{aligned}$$ so $$\begin{aligned} 389 | CoI_{k}^f (1-v_q q_{[x]+k}(1+i_k^c)) = v_q q_{[x]+k} \left(FA_{k+1} 390 | - (~_k AV + G_k - e_k )(1+i_k^c) \right)\end{aligned}$$ which yields 391 | $$\begin{aligned} 392 | CoI_{k}^f &=\frac{v_q q_{[x]+k} \left(FA_{k+1} - (~_k AV + G_k - 393 | e_k )(1+i_k^c) \right)}{1-v_q q_{[x]+k}(1+i_k^c)}\end{aligned}$$ 394 | 395 | Similarly, using $$\begin{aligned} 396 | CoI_{k}^c &= v_q q_{[x]+k} (\gamma_{k+1} -1) \times ~_{k+1} AV ^c \\ 397 | ~_{k+1} AV^c &= (~_k AV + G_k - e_k -CoI_k ^c )(1+i_k^c)\end{aligned}$$ 398 | We can write $$\begin{aligned} 399 | CoI_{k}^c &= v_q q_{[x]+k} (\gamma_{k+1} -1) (~_k AV + G_k - e_k 400 | -CoI_k ^c )(1+i_k^c)\end{aligned}$$ so $$\begin{aligned} 401 | CoI_{k}^c (1+v_q q_{[x]+k} (\gamma_{k+1} -1)(1+i_k^c)) = v_q 402 | q_{[x]+k} (\gamma_{k+1} -1) (~_k AV + G_k - e_k )(1+i_k^c)\end{aligned}$$ 403 | which yields $$\begin{aligned} 404 | CoI_{k}^c &=\frac{v_q q_{[x]+k} (\gamma_{k+1} -1) (~_k AV + G_k - 405 | e_k )(1+i_k^c)}{1+v_q q_{[x]+k}(\gamma_{k+1} -1)(1+i_k^c)}\end{aligned}$$ 406 | 407 | **Example: DHW SN 4.3: Type A UL Profit Test** Consider the following UL 408 | policy issued to a life age 45: 409 | 410 | - Face Amount \$100 000 and the policyholder surrenders the contract 411 | after 20 years. 412 | 413 | - Assume level premiums of \$3 500 are paid annually in advance 414 | 415 | - Assume that the credited interest rate is 4% per year. 416 | 417 | - Death Benefit Type A with corridor factors ($\gamma_k$)\ 418 | 419 | ------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ -- -- 420 | $k$ 1 2 3 4 5 6 7 8 9 10 421 | $\gamma_k$ 2.15 2.09 2.03 1.97 1.91 1.85 1.78 1.71 1.64 1.57 422 | $k$ 11 12 13 14 15 16 17 18 19 20 423 | $\gamma_k$ 1.50 1.46 1.42 1.38 1.34 1.30 1.28 1.26 1.24 1.24 424 | ------------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ -- -- 425 | 426 | - CoI based on: 120% of SSSM, 4% interest; the CoI is calculated 427 | assuming the fund earns 4% interest during the year. 428 | 429 | - Expense Charge: 20% of first premium + \$200, 3% of subsequent 430 | premiums. 431 | 432 | - Surrender Penalties: 433 | 434 | --------------------- ---------- ------ ------ ----- --------- 435 | Year 1 2 3-4 5-7 $\ge 8$ 436 | Surrender Penalties \$ 2 500 2100 1200 600 0 437 | --------------------- ---------- ------ ------ ----- --------- 438 | 439 | We wish to: 440 | 441 | - \(a) Project the account and cash values for this policy. 442 | 443 | - \(b) Profit test the contract using the basis below. Use annual steps, 444 | and determine the NPV and DPP using a risk discount rate of 10% per 445 | year. 446 | 447 | *Solution:* I will only go through the details for part (a). The study 448 | note provides information about the solution to part (b). 449 | 450 | To begin, we first note that the CoI discount rate equals the credited 451 | interest. This means $$\begin{aligned} 452 | v_q (1+i_k^c)=1\end{aligned}$$ which simplifies our calculations. 453 | 454 | For the First Year ($k=0$), we have 455 | 456 | ---------------- ------------------------- 457 | Premium 3500 458 | Expense Charge 200 + (0.20) 3500 = 900 459 | ---------------- ------------------------- 460 | 461 | The CoI assuming the death benefit is Face Amount - Account Value 462 | $$\begin{aligned} 463 | CoI_0^f &=\frac{q_{[45]} \left(v_q FA_1 - (~_0 AV + G_0 - e_0 ) \right)}{1-q_{[45]}} \\ 464 | &= \frac{(1.2) q_{[45]} (v_{.04}100 000 - (3500-900))} {1 - (1.2) 465 | q_{[45]}} = 74.07\end{aligned}$$ CoI assuming the death benefit is based 466 | on the corridor factor $$\begin{aligned} 467 | CoI_0^c &=\frac{q_{[45]} (\gamma_1 -1) (~_0 AV + G_0 - e_0 )}{1+(1.2 q_{[45]})(\gamma_1 -1)} \\ 468 | &= \frac{1.2(0.0006592) (2.15 -1) (3500-900 )}{1+ 469 | 1.2(0.0006592)(2.15 -1)}=2.36\end{aligned}$$ 470 | 471 | --------------- ---------------------------------------- 472 | $CoI_0$ $\max(CoI_0^f, CoI_0^c) = 74.07$ 473 | Account Value (3500 - 900 - 74.07) (1.04) = 2 626.97 474 | Cash Value max(2626.97 - 2500, 0) = 126.97 475 | --------------- ---------------------------------------- 476 | 477 | Example: DHW SN 4.3: Type A UL Profit Test 478 | 479 | ![image](Figures\UL_Fig9DHWSN43TypeAULProfitTest.pdf){width=".9\textwidth"} 480 | -------------------------------------------------------------------------------- /LifeCon1/Figures/IRR_BinaryDF.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-3.0 EPSF-3.0 2 | %%DocumentNeededResources: font Helvetica 3 | %%+ font Helvetica-Bold 4 | %%+ font Helvetica-Oblique 5 | %%+ font Helvetica-BoldOblique 6 | %%+ font Symbol 7 | %%Title: R Graphics Output 8 | %%Creator: R Software 9 | %%Pages: (atend) 10 | %%BoundingBox: 0 0 576 360 11 | %%EndComments 12 | %%BeginProlog 13 | /bp { gs gs } def 14 | % begin .ps.prolog 15 | /gs { gsave } def 16 | /gr { grestore } def 17 | /ep 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"bookdown::render_book('index.Rmd', 'bookdown::gitbook')" 4 | -------------------------------------------------------------------------------- /LifeCon1/_deploy.sh: -------------------------------------------------------------------------------- 1 | #!/bin/sh 2 | 3 | set -e 4 | 5 | [ -z "${GITHUB_PAT}" ] && exit 0 6 | [ "${TRAVIS_BRANCH}" != "master" ] && exit 0 7 | 8 | git config --global user.email "xie@yihui.name" 9 | git config --global user.name "Yihui Xie" 10 | 11 | git clone -b gh-pages https://${GITHUB_PAT}@github.com/${TRAVIS_REPO_SLUG}.git book-output 12 | cd book-output 13 | cp -r ../_book/* ./ 14 | git add --all * 15 | git commit -m"Update the book" || true 16 | git push origin gh-pages 17 | -------------------------------------------------------------------------------- /LifeCon1/_output.yml: -------------------------------------------------------------------------------- 1 | bookdown::pdf_book: 2 | includes: 3 | in_header: preamble.tex 4 | toc: TRUE 5 | toc_depth: '3' 6 | latex_engine: xelatex 7 | citation_package: natbib 8 | keep_tex: yes 9 | bookdown::epub_book: default 10 | bookdown::gitbook: 11 | css: style.css 12 | includes: 13 | in_header: ShowHide.js 14 | config: 15 | toc: 16 | before: | 17 |
  • Loss Data Analytics
    • 18 | after: | 19 |
  • Published with bookdown
    • 20 | edit: https://github.com/rstudio/bookdown-demo/edit/master/%s 21 | -------------------------------------------------------------------------------- /LifeCon1/disqus.html: -------------------------------------------------------------------------------- 1 |
    2 | 20 | -------------------------------------------------------------------------------- /LifeCon1/index.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Actuarial Mathematics Analytics" 3 | author: "An open text authored by the Actuarial Community" 4 | date: "`r Sys.Date()`" 5 | site: bookdown::bookdown_site 6 | output: 7 | bookdown::gitbook: 8 | includes: 9 | after_body: disqus.html 10 | documentclass: book 11 | bibliography: [Bibliography/LDAReference.bib, Bibliography/packages.bib] 12 | biblio-style: apalike 13 | link-citations: yes 14 | #github-repo: rstudio/bookdown-demo 15 | description: "Actuarial Mathematics Analytics is an interactive, online, freely available text. 16 | - The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. 17 | - A subset of the book will be available in pdf format for low-cost printing. 18 | - The online text will be available in multiple languages to promote access to a worldwide audience." 19 | --- 20 | 21 | # Preface {-} 22 | 23 | ####Book Description {-} 24 | 25 | **Actuarial Mathematics Analytics** is an interactive, online, freely available text. 26 | 27 | - The online version contains many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote *deeper learning*. 28 | 29 | - A subset of the book is available for *offline reading* in pdf and EPUB formats. 30 | 31 | - The online text will be available in multiple languages to promote access to a *worldwide audience*. 32 | 33 | 34 | 35 | ####What will success look like? {-} 36 | 37 | The online text will be freely available to a worldwide audience. The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. Moreover, a subset of the book will be available in pdf format for low-cost printing. The online text will be available in multiple languages to promote access to a worldwide audience. 38 | 39 | ####How will the text be used? {-} 40 | 41 | This book will be useful in actuarial curricula worldwide. It will cover the loss data learning objectives of the major actuarial organizations. Thus, it will be suitable for classroom use at universities as well as for use by independent learners seeking to pass professional actuarial examinations. Moreover, the text will also be useful for the continuing professional development of actuaries and other professionals in insurance and related financial risk management industries. 42 | 43 | ####Why is this good for the profession? {-} 44 | 45 | An online text is a type of open educational resource (OER). One important benefit of an OER is that it equalizes access to knowledge, thus permitting a broader community to learn about the actuarial profession. Moreover, it has the capacity to engage viewers through active learning that deepens the learning process, producing analysts more capable of solid actuarial work. 46 | Why is this good for students and teachers and others involved in the learning process? 47 | 48 | Cost is often cited as an important factor for students and teachers in textbook selection (see a recent post on the [$400 textbook](https://www.aei.org/publication/the-new-era-of-the-400-college-textbook-which-is-part-of-the-unsustainable-higher-education-bubble/)). Students will also appreciate the ability to “carry the book around” on their mobile devices. 49 | 50 | 51 | ####Why loss data analytics? {-} 52 | 53 | Although the intent is that this type of resource will eventually permeate throughout the actuarial curriculum, one has to start somewhere. Given the dramatic changes in the way that actuaries treat data, loss data seems like a natural place to start. The idea behind the name *loss data analytics* is to integrate classical loss data models from applied probability with modern analytic tools. In particular, we seek to recognize that big data (including social media and usage based insurance) are here and high speed computation s readily available. 54 | 55 | 56 | ####Project Goal {-} 57 | 58 | The project goal is to have the actuarial community author our textbooks in a collaborative fashion. 59 | 60 | To get involved, please visit our 61 | [Loss Data Analytics Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/). 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | -------------------------------------------------------------------------------- /LifeCon1/preamble.tex: -------------------------------------------------------------------------------- 1 | \usepackage{booktabs} 2 | \setcounter{secnumdepth}{2} 3 | -------------------------------------------------------------------------------- /LifeCon1/style.css: -------------------------------------------------------------------------------- 1 | p.caption { 2 | color: #777; 3 | margin-top: 10px; 4 | } 5 | p code { 6 | white-space: inherit; 7 | } 8 | pre { 9 | word-break: normal; 10 | word-wrap: normal; 11 | } 12 | pre code { 13 | white-space: inherit; 14 | } 15 | -------------------------------------------------------------------------------- /LifeCon1/toc.css: -------------------------------------------------------------------------------- 1 | #TOC ul, 2 | #TOC li, 3 | #TOC span, 4 | #TOC a { 5 | margin: 0; 6 | padding: 0; 7 | position: relative; 8 | } 9 | #TOC { 10 | line-height: 1; 11 | border-radius: 5px 5px 0 0; 12 | background: #141414; 13 | background: linear-gradient(to bottom, #333333 0%, #141414 100%); 14 | border-bottom: 2px solid #0fa1e0; 15 | width: auto; 16 | } 17 | #TOC:after, 18 | #TOC ul:after { 19 | content: ''; 20 | display: block; 21 | clear: both; 22 | } 23 | #TOC a { 24 | background: #141414; 25 | background: linear-gradient(to bottom, #333333 0%, #141414 100%); 26 | color: #ffffff; 27 | display: block; 28 | padding: 19px 20px; 29 | text-decoration: none; 30 | text-shadow: none; 31 | } 32 | #TOC ul { 33 | list-style: none; 34 | } 35 | #TOC > ul > li { 36 | display: inline-block; 37 | float: left; 38 | margin: 0; 39 | } 40 | #TOC > ul > li > a { 41 | color: #ffffff; 42 | } 43 | #TOC > ul > li:hover:after { 44 | content: ''; 45 | display: block; 46 | width: 0; 47 | height: 0; 48 | position: absolute; 49 | left: 50%; 50 | bottom: 0; 51 | border-left: 10px solid transparent; 52 | border-right: 10px solid transparent; 53 | border-bottom: 10px solid #0fa1e0; 54 | margin-left: -10px; 55 | } 56 | #TOC > ul > li:first-child > a { 57 | border-radius: 5px 0 0 0; 58 | } 59 | #TOC.align-right > ul > li:first-child > a, 60 | #TOC.align-center > ul > li:first-child > a { 61 | border-radius: 0; 62 | } 63 | #TOC.align-right > ul > li:last-child > a { 64 | border-radius: 0 5px 0 0; 65 | } 66 | #TOC > ul > li.active > a, 67 | #TOC > ul > li:hover > a { 68 | color: #ffffff; 69 | box-shadow: inset 0 0 3px #000000; 70 | background: #070707; 71 | background: linear-gradient(to bottom, #262626 0%, #070707 100%); 72 | } 73 | #TOC .has-sub { 74 | z-index: 1; 75 | } 76 | #TOC .has-sub:hover > ul { 77 | display: block; 78 | } 79 | #TOC .has-sub ul { 80 | display: none; 81 | position: absolute; 82 | width: 200px; 83 | top: 100%; 84 | left: 0; 85 | } 86 | #TOC .has-sub ul li a { 87 | background: #0fa1e0; 88 | border-bottom: 1px dotted #31b7f1; 89 | filter: none; 90 | display: block; 91 | line-height: 120%; 92 | padding: 10px; 93 | color: #ffffff; 94 | } 95 | #TOC .has-sub ul li:hover a { 96 | background: #0c7fb0; 97 | } 98 | #TOC ul ul li:hover > a { 99 | color: #ffffff; 100 | } 101 | #TOC .has-sub .has-sub:hover > ul { 102 | display: block; 103 | } 104 | #TOC .has-sub .has-sub ul { 105 | display: none; 106 | position: absolute; 107 | left: 100%; 108 | top: 0; 109 | } 110 | #TOC .has-sub .has-sub ul li a { 111 | background: #0c7fb0; 112 | border-bottom: 1px dotted #31b7f1; 113 | } 114 | #TOC .has-sub .has-sub ul li a:hover { 115 | background: #0a6d98; 116 | } 117 | #TOC ul ul li.last > a, 118 | #TOC ul ul li:last-child > a, 119 | #TOC ul ul ul li.last > a, 120 | #TOC ul ul ul li:last-child > a, 121 | #TOC .has-sub ul li:last-child > a, 122 | #TOC .has-sub ul li.last > a { 123 | border-bottom: 0; 124 | } 125 | #TOC ul { 126 | font-size: 1.2rem; 127 | } 128 | -------------------------------------------------------------------------------- /LifeCon2/.Rbuildignore: -------------------------------------------------------------------------------- 1 | ^.*\.Rproj$ 2 | ^\.Rproj\.user$ 3 | -------------------------------------------------------------------------------- /LifeCon2/.gitignore: -------------------------------------------------------------------------------- 1 | .Rproj.user 2 | .Rhistory 3 | .RData 4 | _publish.R 5 | _book 6 | _bookdown_files 7 | rsconnect 8 | -------------------------------------------------------------------------------- /LifeCon2/.travis.yml: -------------------------------------------------------------------------------- 1 | language: r 2 | cache: packages 3 | 4 | env: 5 | global: 6 | - secure: "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" 7 | 8 | before_script: 9 | - chmod +x ./_build.sh 10 | - chmod +x ./_deploy.sh 11 | 12 | script: 13 | - ./_build.sh 14 | - ./_deploy.sh 15 | -------------------------------------------------------------------------------- /LifeCon2/Bibliography/Bibliography.Rmd: -------------------------------------------------------------------------------- 1 | `r if (knitr:::is_html_output()) '# Bibliography {-}'` 2 | -------------------------------------------------------------------------------- /LifeCon2/Bibliography/LDAReference.bib: -------------------------------------------------------------------------------- 1 | @Book{xie2015, 2 | title = {Dynamic Documents with {R} and knitr}, 3 | author = {Yihui Xie}, 4 | publisher = {Chapman and Hall/CRC}, 5 | address = {Boca Raton, Florida}, 6 | year = {2015}, 7 | edition = {2nd}, 8 | note = {ISBN 978-1498716963}, 9 | url = {http://yihui.name/knitr/}, 10 | } 11 | 12 | @InProceedings{III2015, 13 | Title = {International Insurance Fact Book}, 14 | Author = {{Insurance~Information~Institute}}, 15 | Year = {2015}, 16 | Note = {[Retrieved on May 10, 2016]}, 17 | Publisher = {Insurance Information Institute}, 18 | Url = {http://www.iii.org/sites/default/files/docs/pdf/international_insurance_factbook_2015.pdf} 19 | } 20 | 21 | @Book{bowers1986actuarial, 22 | Title = {Actuarial Mathematics}, 23 | Author = {Bowers, Newton L. and Gerber, Hans U. and Hickman, James C. and Jones, Donald A. and Nesbitt, Cecil J.}, 24 | Publisher = {Society of Actuaries Itasca, Ill.}, 25 | Year = {1986} 26 | } 27 | 28 | @Book{dickson2013actuarial, 29 | Title = {Actuarial Mathematics for Life Contingent Risks}, 30 | Author = {Dickson, David C. M. and Hardy, Mary and Waters, Howard R.}, 31 | Publisher = {Cambridge University Press}, 32 | Year = {2013} 33 | } 34 | 35 | @InProceedings{survey2013, 36 | Title = {2013 Insurance Predictive Modeling Survey}, 37 | Author = {Earnix}, 38 | Year = {2013}, 39 | Note = {[Retrieved on July 7, 2014]}, 40 | Publisher = {Earnix and Insurance Services Office, Inc.}, 41 | Url = {http://earnix.com/2013-insurance-predictive-modeling-survey/3594/} 42 | } 43 | 44 | @Article{bailey1960, 45 | Title = {Two studies in automobile ratemaking}, 46 | Author = {Bailey, Robert A. and LeRoy, J. Simon}, 47 | Journal = {Proceedings of the Casualty Actuarial Society Casualty Actuarial Society}, 48 | Year = {1960}, 49 | Number = {I}, 50 | Volume = {XLVII} 51 | } 52 | 53 | @InProceedings{SASsurvey, 54 | Title = {Building believers: How to expand the use of predictive analytics in claims}, 55 | Author = {Gorman, Mark and Swenson, Stephen}, 56 | Year = {2013}, 57 | Note = {[Retrieved on August 17, 2014]}, 58 | Publisher = {SAS}, 59 | Url = {http://www.sas.com/resources/whitepaper/wp_59831.pdf} 60 | } 61 | 62 | @inproceedings{venter1983transformed, 63 | title={Transformed beta and gamma distributions and aggregate losses}, 64 | author={Venter, Gary}, 65 | booktitle={Proceedings of the Casualty Actuarial Society}, 66 | volume={70}, 67 | number={133 \& 134}, 68 | pages={289--308}, 69 | year={1983} 70 | } 71 | 72 | @article{mcdonald1984some, 73 | title={Some generalized functions for the size distribution of income}, 74 | author={McDonald, James B}, 75 | journal={Econometrica: journal of the Econometric Society}, 76 | pages={647--663}, 77 | year={1984}, 78 | publisher={JSTOR} 79 | } 80 | -------------------------------------------------------------------------------- /LifeCon2/Bibliography/LossDataAnalyticsReference.bib: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ewfrees/LifeCon/66e09a97ef093c46778c3d5724f196abdea845f0/LifeCon2/Bibliography/LossDataAnalyticsReference.bib -------------------------------------------------------------------------------- /LifeCon2/Bibliography/packages.bib: -------------------------------------------------------------------------------- 1 | @Manual{R-base, 2 | title = {R: A Language and Environment for Statistical Computing}, 3 | author = {{R Core Team}}, 4 | organization = {R Foundation for Statistical Computing}, 5 | address = {Vienna, Austria}, 6 | year = {2016}, 7 | url = {https://www.R-project.org/}, 8 | } 9 | @Manual{R-bookdown, 10 | title = {bookdown: Authoring Books and Technical Documents with R Markdown}, 11 | author = {Yihui Xie}, 12 | year = {2016}, 13 | note = {R package version 0.3}, 14 | url = {https://CRAN.R-project.org/package=bookdown}, 15 | } 16 | @Manual{R-knitr, 17 | title = {knitr: A General-Purpose Package for Dynamic Report Generation in R}, 18 | author = {Yihui Xie}, 19 | year = {2016}, 20 | note = {R package version 1.15.1}, 21 | url = {https://CRAN.R-project.org/package=knitr}, 22 | } 23 | @Manual{R-rmarkdown, 24 | title = {rmarkdown: Dynamic Documents for R}, 25 | author = {JJ Allaire and Joe Cheng and Yihui Xie and Jonathan McPherson and Winston Chang and Jeff Allen and Hadley Wickham and Aron Atkins and Rob Hyndman}, 26 | year = {2016}, 27 | note = {R package version 1.2}, 28 | url = {https://CRAN.R-project.org/package=rmarkdown}, 29 | } 30 | -------------------------------------------------------------------------------- /LifeCon2/Chapters/02-Selection.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Selection" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | 12 | # Selection 13 | 14 | ## Key Concepts 15 | 16 | For a life table based on an insured population, one must consider for 17 | each individual both (i) the age of policy issue and (ii) the time that 18 | has elapsed since policy issue. This is because the insurer typically 19 | underwrites individuals that purchase a policy. Through the 20 | underwriting, the insurer learns additional information about the 21 | individual’s survival distribution that the insurer would not know for a 22 | life randomly drawn from the general population. This additional 23 | information must be accounted for in the calculation of various 24 | quantities such as survival probabilities for the individual and the 25 | value of the individual’s policy. 26 | 27 | - Consider an individual now aged x + t who purchased a policy at 28 | age x. We say that the individual was **selected**, or **select**, 29 | at age x (and time $t$ = 0). 30 | 31 | - The additional information gained from underwriting the above 32 | individual, obtained by surveys and/or a medical examination, is 33 | assumed to apply for a certain number of years after policy issue 34 | called the **select period**. 35 | 36 | - Say the select period is $d$ years. For $t$ $<$ $d$, one accounts 37 | for the initial selection of the above individual at age x; the 38 | individual’s current age would be written as \[x\] + t (the select 39 | brackets \[ \] denote the initial age of selection). For $t$ $\ge$ 40 | $d$, one no longer accounts for the initial selection of the 41 | individual at age x, and the individual’s age would be written 42 | simply as x + t (with no select brackets \[ \]). 43 | 44 | - An individual has **select mortality** for ages/times within the 45 | select period that differs from the mortality of the general 46 | population. An individual has **ultimate mortality** for ages/times 47 | beyond the select period where their mortality is assumed to be the 48 | same as a life from the general population. 49 | 50 | - A life table that accounts for both select and ultimate mortality is 51 | called a **select-and-ultimate life table**. 52 | 53 | - A life table that ignores selection completely is called an 54 | **aggregate life table**. 55 | 56 | - The previous formulas for the quantities considered so far, such as 57 | survival probabilities, are still valid in the event of selection. 58 | One simply has to use information from the select part of the 59 | select-and-ultimate life table for ages/times within the select 60 | period. 61 | 62 | - For example, with a select period of 3 years, ${}_{2}p_{[x]}$ = 63 | $(p_{[x]})$$(p_{[x] + 1})$ and ${}_{5}p_{[x]}$ = $(p_{[x]})$$(p_{[x] 64 | + 1)}$$(p_{[x] + 2})$$(p_{x + 3})$$(p_{x + 4})$. The p’s with select 65 | brackets would come from the select part of the select-and-ultimate 66 | life table, and the p’s without select brackets would come from the 67 | ultimate part. 68 | 69 | - An illustrative select-and-ultimate table is the 70 | 71 | **Standard Select and Ultimate Survival Model**. This table is 72 | provided in Appendix D of Dickson et al. For brevity, I will refer 73 | to this table as the **Standard Select Survival Model**. 74 | 75 | ## Exercises 76 | 77 | 2.1. Mortality follows the select-and-ultimate life table: 78 | 79 | $\begin{array}{r|r|r|r|r} 80 | \hline 81 | \mathbf{x} & \mathbf{l_{[x]}} & \mathbf{l_{[x] + 1}} & \mathbf{l_{x + 2}} & \mathbf{x + 2}\\ 82 | \hline 83 | 30 & 9,906 & 9,904 & 9,901 & 32 \\ 84 | 31 & 9,902 & 9,900 & 9,897 & 33 \\ 85 | 32 & 9,898 & 9,896 & 9,892 & 34 \\ 86 | 33 & 9,894 & 9,891 & 9,887 & 35 \\ 87 | 34 & 9,889 & 9,886 & 9,882 & 36 \\ 88 | \hline 89 | \end{array}$ 90 | 91 | Calculate: 10,000${}_{1|}q_{[30]}$. 92 | 93 | \(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 94 | 95 | 2.2. Suppose mortality follows the Standard Select Survival Model. 96 | 97 | Calculate: ${}_{1|2}q_{[70]+1}$. 98 | 99 | \(A) 0.025 (B) 0.027 (C) 0.029 (D) 0.031 (E) 0.033 100 | 101 | 2.3. Consider the following select-and-ultimate life table: 102 | 103 | $\begin{array}{r|r|r|r|r|r} 104 | \hline 105 | \mathbf{x} & \mathbf{q_{[x]}} & \mathbf{q_{[x] + 1}} & \mathbf{q_{[x] + 2}} & \mathbf{q_{x + 3}} & \mathbf{x + 3} \\ 106 | \hline 107 | 60 & 0.09 & 0.11 & 0.13 & 0.15 & 63 \\ 108 | 61 & 0.10 & 0.12 & 0.14 & 0.16 & 64 \\ 109 | 62 & 0.11 & 0.13 & 0.15 & 0.17 & 65 \\ 110 | 63 & 0.12 & 0.14 & 0.16 & 0.18 & 66 \\ 111 | 64 & 0.13 & 0.15 & 0.17 & 0.19 & 67 \\ 112 | \hline 113 | \end{array}$ 114 | 115 | Assume that deaths follow the uniform distribution of deaths assumption 116 | between integer ages. 117 | 118 | Calculate: ${}_{1.6}q_{[61] + 0.75}$. 119 | 120 | \(A) 0.1855 (B) 0.1856 (C) 0.1857 (D) 0.1858 (E) 0.1859 121 | 122 | 2.4. A select-and-ultimate life table with a select period of 2 years is 123 | based on probabilities that satisfy the following relationship: 124 | 125 | $q_{[x - i] + i}$ = $\frac{3}{5 - i} \times q_x$ for $i$ = 0, 1. 126 | 127 | You are given that $l_{68}$ = 10,000, $q_{66}$ = 0.026, and $q_{67}$ = 128 | 0.028. 129 | 130 | Calculate: $l_{[65] + 1}$. 131 | 132 | \(A) 10,414 (B) 10,451 (C) 10,479 (D) 10,493 (E) 11,069 133 | 134 | 2.5. Suppose mortality follows the Standard Select Survival Model. 135 | 136 | Calculate: $e_{[60] :\overline{5}|}$. 137 | 138 | \(A) 4.928 (B) 4.932 (C) 4.936 (D) 4.940 (E) 4.944 139 | 140 | 2.6. Consider a select-and-ultimate life table with a 2-year select 141 | period. 142 | 143 | You are given: 144 | 145 | \(i) $l_{[35]}$ = 1500 146 | 147 | \(ii) $l_{36}$ = 1472.31 148 | 149 | \(iii) $q_{[35]}$ = 0.0240 150 | 151 | \(iv) $q_{[35] + 1}$ = 0.0255 152 | 153 | Calculate: $l_{35}({}_{1|}q_{35})$. 154 | 155 | \(A) 42 (B) 44 (C) 46 (D) 48 (E) 50 156 | 157 | 2.7. Quinn is currently age 60. He was selected by the PlzDntDie Life 158 | Insurance Company one year ago. Quinn has mortality that follows a 159 | select-and-ultimate life table with a 2-year select period: 160 | 161 | \(i) The ultimate part of the model is such that: 162 | 163 | $\mu_x$ = 0.0002$(1.1)^x$ for $x$ $\ge$ 0. 164 | 165 | \(ii) The select part of the model is such that: 166 | 167 | $\mu_{[x] + s}$ = $(0.8)^{2 - s}\mu_{x + s}$ for $x$ $\ge$ 0, 0 $\le$ 168 | $s$ $\le$ 2. 169 | 170 | Calculate the probability that Quinn survives to age 61. 171 | 172 | \(A) 0.938 (B) 0.944 (C) 0.950 (D) 0.956 (E) 0.962 173 | 174 | 2.8. You are given: 175 | 176 | \(i) Mortality follows the Standard Select Survival Model. 177 | 178 | \(ii) Deaths are uniformly distributed over each year of age. 179 | 180 | Calculate: $\mathring{e}_{[75] + 1 :\overline{1.3}|}$. 181 | 182 | \(A) 1.277 (B) 1.280 (C) 1.283 (D) 1.287 (E) 1.290 183 | 184 | 2.9. For a select and ultimate life table with a 1-year select period: 185 | 186 | \(i) $\mu_{[55] + t}$ = 0.5$\mu_{55 + t}$ for 0 $\le$ $t$ $\le$ 1 187 | 188 | \(ii) $e_{55}$ = 18.02 189 | 190 | \(iii) $e_{[55]}$ = 18.33 191 | 192 | Calculate: $e_{56}$. 193 | 194 | \(A) 17.60 (B) 17.65 (C) 17.70 (D) 17.75 (E) 17.80 195 | 196 | 2.10. Consider a population of lives each age 55 and selected at that 197 | age, where 70% are non-smokers and 30% are smokers. 198 | 199 | The force of mortality is: 200 | 201 | $\mu_{[55] + t}$ = 0.009$t$(0.4 + 0.1$S$) for $t$ $\ge$ 0, 202 | 203 | where $S$ = 0 for a non-smoker and $S$ = 1 for a smoker. 204 | 205 | Calculate the probability that a randomly chosen life from the above 206 | population will die before age 75. 207 | 208 | \(A) 0.51 (B) 0.52 (C) 0.53 (D) 0.54 (E) 0.55 209 | 210 | 2.11. For a select-and-ultimate life table with a 2-year select period: 211 | 212 | \(i) Ultimate mortality follows the Illustrative Life Table. 213 | 214 | \(ii) $q_{[x]}$ = 0.5$q_x$, where $q_x$ is from the Illustrative Life 215 | Table. 216 | 217 | \(iii) $q_{[x]+1}$ = 0.25$q_{x + 1}$, where $q_{x + 1}$ is from the 218 | Illustrative Life Table. 219 | 220 | Calculate the probability that a select life aged 70 will die after age 221 | 75. 222 | 223 | \(A) 0.79 (B) 0.81 (C) 0.83 (D) 0.85 (E) 0.87 224 | 225 | 2.12. Consider the setup in Exercise 2.11. Calculate the 4-year 226 | temporary curtate expectation of life for a select life aged 70. 227 | 228 | \(A) 3.79 (B) 3.81 (C) 3.83 (D) 3.85 (E) 3.87 229 | 230 | ### Answers to Exercises 231 | 232 | 2.1. C 233 | 234 | 2.2. B 235 | 236 | 2.3. C 237 | 238 | 2.4. D 239 | 240 | 2.5. E 241 | 242 | 2.6. C 243 | 244 | 2.7. B 245 | 246 | 2.8. C 247 | 248 | 2.9. B 249 | 250 | 2.10. D 251 | 252 | 2.11. D 253 | 254 | 2.12. A 255 | 256 | ## Past Exam Questions 257 | 258 | - Exam MLC, Fall 2014: \#20 259 | 260 | - Exam MLC, Spring 2014: \#2 261 | 262 | - Exam MLC, Fall 2013: \#3 263 | 264 | - Exam MLC, Spring 2013: \#19 265 | 266 | - Exam MLC, Fall 2012: \#2 267 | 268 | - Exam MLC, Spring 2012: \#1, 13 269 | 270 | - Exam MLC, Sample Questions: \#66, 73, 136, 168 271 | 272 | - Exam MLC, Spring 2007: \#18 273 | -------------------------------------------------------------------------------- /LifeCon2/Chapters/15-MultipleLivesII.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Multiple Lives II" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | 12 | # Multiple Lives II 13 | 14 | ## Key Concepts 15 | 16 | ## Markov Representation 17 | 18 | \Large 19 | 20 | - The multiple life model described so far can be expressed as a 21 | 4-state Markov Chain. Consider two lives, (x) and (y). The following 22 | 4-state model will hereafter be referred to as the **Multiple Life 23 | Model** diagrammed in Examples of Multiple State Models: 24 | \vspace{2mm} 25 | 26 | - State 0: Both (x) and (y) are alive \vspace{2mm} 27 | 28 | - State 1: (x) is alive and (y) is dead \vspace{2mm} 29 | 30 | - State 2: (x) is dead and (y) is alive \vspace{2mm} 31 | 32 | - State 3: Both (x) and (y) are dead \vspace{10mm} 33 | 34 | - The forces of transition are: \vspace{2mm} 35 | 36 | - $\mu^{13}_{x + t}$ \vspace{2mm} 37 | 38 | - $\mu^{23}_{y + t}$ \vspace{2mm} 39 | 40 | - $\mu^{02}_{x + t:y + t}$, with independence (x) and (y), = 41 | $\mu^{13}_{x + t}$ = $\mu_x(t)$ \vspace{2mm} 42 | 43 | - $\mu^{01}_{x + t:y + t}$, with independence of (x) and (y), = 44 | $\mu^{23}_{y + t}$ = $\mu_y(t)$ 45 | 46 | \vspace{10mm} 47 | 48 | - In the Multiple Life Model, it is assumed that 49 | $\mu^{03}_{x + t:y + t}$ = 0. That is, it is impossible for (x) 50 | and (y) to die simultaneously. \vspace{2mm} 51 | 52 | \newpage 53 | ## Multiple Life Functions 54 | 55 | The multiple life functions described in **Multiple Lives I** can be 56 | expressed using the Multiple Life Model. \vspace{5mm} 57 | 58 | - **Probabilities:** \vspace{2mm} 59 | 60 | - ${}_{t}p^{00}_{xy}$ = ${}_{t}p_{xy}$ \vspace{4mm} 61 | 62 | - ${}_{t}p^{01}_{xy}$ + ${}_{t}p^{02}_{xy}$ + ${}_{t}p^{03}_{xy}$ 63 | = ${}_{t}q_{xy}$ \vspace{4mm} 64 | 65 | - ${}_{t}p^{00}_{xy}$ + ${}_{t}p^{01}_{xy}$ + ${}_{t}p^{02}_{xy}$ 66 | = ${}_{t}p_{\overline{xy}}$ \vspace{4mm} 67 | 68 | - ${}_{t}p^{03}_{xy}$ = ${}_{t}q_{\overline{xy}}$ \vspace{4mm} 69 | 70 | - $\mu^{01}_{x + t:y + t}$ + $\mu^{02}_{x + t:y + t}$ = 71 | $\mu_{xy}(t)$ \vspace{6mm} 72 | 73 | 74 | 75 | - **Insurance and Annuity Functions:** \vspace{2mm} 76 | 77 | - $\bar{A}_x$ = 78 | $\int^{\infty}_0 v^t({}_{t}p^{00}_{xy}\mu^{02}_{x + t:y + t} + 79 | {}_{t}p^{01}_{xy}\mu^{13}_{x + t}) dt$ \vspace{4mm} 80 | 81 | - $\bar{A}_y$ = 82 | $\int^{\infty}_0 v^t({}_{t}p^{00}_{xy}\mu^{01}_{x + t:y + t} + 83 | {}_{t}p^{02}_{xy}\mu^{23}_{y + t}) dt$ \vspace{4mm} 84 | 85 | - $\bar{A}_{xy}$ = 86 | $\int^{\infty}_0 v^t{}_{t}p^{00}_{xy}(\mu^{01}_{x + t:y + t} + \mu^{02}_{x + t:y + t}) dt$ 87 | \vspace{4mm} 88 | 89 | - $\bar{A}_{\overline{xy}}$ = 90 | $\int^{\infty}_0 v^t({}_{t}p^{01}_{xy}\mu^{13}_{x + t} + 91 | {}_{t}p^{02}_{xy}\mu^{23}_{y + t}) dt$ \vspace{4mm} 92 | 93 | - $\bar{a}_x$ = $\int^{\infty}_0 v^t({}_{t}p^{00}_{xy} + 94 | {}_{t}p^{01}_{xy}) dt$ \vspace{4mm} 95 | 96 | - $\bar{a}_y$ = $\int^{\infty}_0 v^t({}_{t}p^{00}_{xy} + 97 | {}_{t}p^{02}_{xy}) dt$ \vspace{4mm} 98 | 99 | - $\bar{a}_{xy}$ = $\int^{\infty}_0 v^t{}_{t}p^{00}_{xy} dt$ 100 | \vspace{4mm} 101 | 102 | - $\bar{a}_{\overline{xy}}$ = 103 | $\int^{\infty}_0 v^t({}_{t}p^{00}_{xy} + {}_{t}p^{01}_{xy} + 104 | {}_{t}p^{02}_{xy}) dt$ \vspace{4mm} 105 | 106 | - $\bar{a}_{x|y}$ = $\int^{\infty}_0 v^t{}_{t}p^{02}_{xy} dt$ 107 | \vspace{4mm} 108 | 109 | \newpage 110 | ## Contingent Functions 111 | 112 | \Large 113 | 114 | - A **contingent function** is such that the function value depends on 115 | the order in which the lives die. Assume the Multiple Life Model. 116 | \vspace{4mm} 117 | 118 | - ${}_{t}\lcfirst{q}{x}{y}$ = 119 | $\int^{t}_0 {}_{s}p^{00}_{xy}\mu^{02}_{x + s:y + s} 120 | ds$ \vspace{4mm} 121 | 122 | “The probability that (x) dies both before (y) and within $t$ 123 | years.” \vspace{4mm} 124 | 125 | If (x) and (y) are independent $\implies$ ${}_{t}\lcfirst{q}{x}{y}$ 126 | = $\int^{t}_0({}_{s}p_y){}_{s}p_x\mu_x(s)ds$ \vspace{4mm} 127 | 128 | - ${}_{t}\lcsecond{q}{x}{y}$ = 129 | $\int^{t}_0 {}_{s}p^{02}_{xy}\mu^{23}_{y + s} 130 | ds$ \vspace{4mm} 131 | 132 | “The probability that (y) dies both after (x) and within $t$ years.” 133 | This is different than ${}_{t}\lcfirst{q}{x}{y}$; for 134 | ${}_{t}\lcfirst{q}{x}{y}$ the event of interest is (y) dying after 135 | (x), but that does not necessarily have to occur within $t$ years. 136 | \vspace{4mm} 137 | 138 | If (x) and (y) are independent $\implies$ ${}_{t}\lcsecond{q}{x}{y}$ 139 | = $\int^{t}_0({}_{s}q_x){}_{s}p_y\mu_y(s) ds$ \vspace{4mm} 140 | 141 | - Furthermore: \vspace{4mm} 142 | 143 | - ${}_{t}q_x$ = ${}_{t}\lcfirst{q}{x}{y}$ + 144 | ${}_{t}\lcsecond{q}{y}{x}$ \vspace{2mm} 145 | 146 | - ${}_{t}q_{xy}$ = ${}_{t}\lcfirst{q}{x}{y}$ + 147 | ${}_{t}\lcfirst{q}{y}{x}$ \vspace{2mm} 148 | 149 | - ${}_{t}q_{\overline{xy}}$ = ${}_{t}\lcsecond{q}{y}{x}$ + 150 | ${}_{t}\lcsecond{q}{x}{y}$ \vspace{2mm} 151 | 152 | - ${}_{t}\lcfirst{q}{x}{y}$ = ${}_{t}\lcsecond{q}{x}{y}$ + 153 | ${}_{t}p^{02}_{xy}$ \vspace{2mm} 154 | 155 | - ${}_{\infty}\lcfirst{q}{x}{y}$ = ${}_{\infty}\lcsecond{q}{x}{y}$ 156 | \vspace{2mm} 157 | 158 | - For independent lives (x) and (y) each with de Moivre’s 159 | Law/uniform distribution with limiting age $\omega$: 160 | \vspace{2mm} 161 | 162 | - ${}_{t}\lcfirst{q}{x}{y}$ = $({}_{t}q_x)$$({}_{t/2}p_y)$ and 163 | ${}_{t}\lcsecond{q}{y}{x}$ = 164 | $\frac{1}{2}$$({}_{t}q_x)$$({}_{t}q_y)$ \vspace{2mm} 165 | 166 | - For independent lives (x) and (y) with constant forces of 167 | mortality $\mu_x$ and $\mu_y$, respectively: \vspace{2mm} 168 | 169 | - ${}_{t}\lcfirst{q}{x}{y}$ = $\frac{\mu_x}{\mu_x + 170 | \mu_y}$$({}_{t}q_{xy})$ and ${}_{\infty}\lcfirst{q}{x}{y}$ = 171 | $\frac{\mu_x}{\mu_x + \mu_y}$ 172 | 173 | \newpage 174 | - $\lcfirst{\bar{A}}{x}{y}$ = 175 | $\int^{\infty}_0v^s{}_{s}p^{00}_{xy}\mu^{02}_{x + s:y + s}ds$ 176 | \vspace{4mm} 177 | 178 | “This is the expected present value of a life insurance that pays 1 179 | at the moment of death of (x) if (x) is the first to die.” 180 | \vspace{4mm} 181 | 182 | If (x) and (y) are independent $\implies$ $\lcfirst{\bar{A}}{x}{y}$ 183 | = $\int^{\infty}_0 v^s ({}_{s}p_y){}_{s}p_x\mu_x(s)ds$ \vspace{10mm} 184 | 185 | - $\lcsecond{\bar{A}}{x}{y}$ = 186 | $\int^{\infty}_0v^s{}_{s}p^{02}_{xy}\mu^{23}_{y + s}ds$ \vspace{4mm} 187 | 188 | “This is the expected present value of a life insurance that pays 1 189 | at the moment of death of (y) if (y) is the second to die.” 190 | \vspace{4mm} 191 | 192 | If (x) and (y) are independent $\implies$ $\lcsecond{\bar{A}}{x}{y}$ 193 | = $\int^{\infty}_0 v^s ({}_{s}q_x){}_{s}p_y\mu_y(s) ds$ 194 | \vspace{10mm} 195 | 196 | - Furthermore: \vspace{4mm} 197 | 198 | - $\bar{A}_x$ = $\lcfirst{\bar{A}}{x}{y}$ + 199 | $\lcsecond{\bar{A}}{y}{x}$ \vspace{4mm} 200 | 201 | - $\bar{A}_{xy}$ = $\lcfirst{\bar{A}}{x}{y}$ + 202 | $\lcfirst{\bar{A}}{y}{x}$ \vspace{4mm} 203 | 204 | - $\bar{A}_{\overline{xy}}$ = $\lcsecond{\bar{A}}{x}{y}$ + 205 | $\lcsecond{\bar{A}}{y}{x}$ \vspace{6mm} 206 | 207 | - For independent lives (x) and (y) with constant forces of 208 | mortality $\mu_x$ and $\mu_y$, respectively: \vspace{2mm} 209 | 210 | - $\lcfirst{\bar{A}}{x}{y}$ = $\frac{\mu_x}{\mu_x + 211 | \mu_y + \delta}$ \vspace{2mm} 212 | 213 | - $\lcsecond{\bar{A}}{x}{y}$ = $\frac{\mu_y}{\mu_y + 214 | \delta}$$\frac{\mu_x}{\mu_x + \mu_y + \delta}$ 215 | 216 | \newpage 217 | ## Common Shock Model 218 | 219 | \Large 220 | 221 | - Under the **Common Shock Model**, is possible to make a direct 222 | transition from State 0 to State 3 in the Multiple Life Model, that 223 | is, $\mu^{03}_{x + t:y+t}$ $\neq$ 0. \vspace{10mm} 224 | 225 | - It is assumed that there is an event called the **common shock** 226 | that could kill both (x) and (y) at the same time, such as a car 227 | accident or a natural disaster. \vspace{10mm} 228 | 229 | - ${}_{t}p^{11}_{x + s}$ = $\exp[- \int^t_0 \mu^{13}_{x + s + u} du]$ 230 | \vspace{2mm} 231 | 232 | “This is the probability that (x + s) survives $t$ years ((y) has 233 | previously died).” \vspace{6mm} 234 | 235 | - ${}_{t}p^{22}_{y + s}$ = $\exp[- \int^t_0 \mu^{23}_{y + s + v} dv]$ 236 | \vspace{2mm} 237 | 238 | “This is the probability that (y + s) survives $t$ years ((x) has 239 | previously died).” \vspace{6mm} 240 | 241 | - ${}_{t}p^{00}_{xy}$ = 242 | $\exp[- \int^t_0 (\mu^{01}_{x + s:y + s} + \mu^{02}_{x + s:y + s} + \mu^{03}_{x + s:y + 243 | s}) ds]$ = ${}_{t}p_{xy}$ \vspace{2mm} 244 | 245 | “This is the probability that both (x) and (y) survive $t$ years.” 246 | \vspace{6mm} 247 | 248 | - ${}_{t}p^{01}_{xy}$ = 249 | $\int^t_0 {}_{s}p^{00}_{xy}(\mu^{01}_{x + s:y + s}){}_{t - s}p^{11}_{x + s} ds$ 250 | \vspace{2mm} 251 | 252 | “This is the probability that only (x) survives $t$ years.” 253 | \vspace{6mm} 254 | 255 | - ${}_{t}p^{02}_{xy}$ = 256 | $\int^t_0 {}_{s}p^{00}_{xy}(\mu^{02}_{x + s:y + s}){}_{t - s}p^{22}_{y + s} ds$ 257 | \vspace{2mm} 258 | 259 | “This is the probability that only (y) survives $t$ years.” 260 | \vspace{6mm} 261 | 262 | - ${}_{t}p^{03}_{xy}$ = 1 - ${}_{t}p^{00}_{xy}$ - 263 | ${}_{t}p^{01}_{xy}$ - ${}_{t}p^{02}_{xy}$ = 264 | ${}_{t}q_{\overline{xy}}$ \vspace{2mm} 265 | 266 | “This is the probability that (x) and (y) do not survive $t$ years.” 267 | \vspace{6mm} 268 | 269 | \newpage 270 | - Consider a typical **special case** of the common shock model: 271 | \vspace{2mm} 272 | 273 | \(i) $\mu^{03}_{x + s:y + s}$ = $\lambda$, a constant \vspace{2mm} 274 | 275 | \(ii) $\mu^{02}_{x + s:y + s}$ + $\mu^{03}_{x + s:y + s}$ = 276 | $\mu^{13}_{x + s}$ = $\mu_x$, a constant \vspace{2mm} 277 | 278 | \(iii) $\mu^{01}_{x + s:y + s}$ + $\mu^{03}_{x + s:y + s}$ = 279 | $\mu^{23}_{y + s}$ = $\mu_y$, a constant \vspace{2mm} 280 | 281 | Then: \vspace{2mm} 282 | 283 | - ${}_{t}p_x$ = $\exp[- \mu_x t]$ \vspace{2mm} 284 | 285 | - ${}_{t}p_y$ = $\exp[- \mu_y t]$ \vspace{2mm} 286 | 287 | - ${}_{t}p_{xy}$ = $\exp[- (\mu_x + \mu_y - \lambda) t]$ 288 | \vspace{2mm} 289 | 290 | - ${}_{t}p_{\overline{xy}}$ = ${}_{t}p_x$ + ${}_{t}p_y$ - 291 | ${}_{t}p_{xy}$ \vspace{10mm} 292 | 293 | - The probability that (x) and (y) are killed simultaneously by the 294 | common shock is: \vspace{2mm} 295 | 296 | $\int^{\infty}_0 {}_{s}p^{00}_{xy}\mu^{03}_{x + s:y + s} ds$. 297 | \vspace{2mm} 298 | 299 | “In order for both (x) and (y) to die at the same time, both have to 300 | survive for $s$ years and then immediately die. Integration 301 | considers all possible times $s$.” \vspace{4mm} 302 | 303 | - Consider the typical special case of the common shock model 304 | previously described; the probability that (x) and (y) die 305 | simultaneously is: \vspace{2mm} 306 | 307 | $\frac{\lambda}{\mu_x + \mu_y - \lambda}$. 308 | 309 | \newpage 310 | ## Exercises 311 | 312 | \Large 313 | 314 | 15.1. Consider the Multiple Life Model in Examples of Multiple State 315 | Models, with a husband aged x and a wife aged y. \vspace{2mm} 316 | 317 | You are given the following transition intensities for $t$ $>$ 0: 318 | \vspace{4mm} 319 | 320 | \(i) $\mu^{01}_{x + t:y + t}$ = 0.03 \vspace{2mm} 321 | 322 | \(ii) $\mu^{02}_{x + t:y + t}$ = 0.04 \vspace{2mm} 323 | 324 | \(iii) $\mu^{13}_{x + t}$ = 0.07 \vspace{2mm} 325 | 326 | \(iv) $\mu^{23}_{y + t}$ = 0.06 \vspace{4mm} 327 | 328 | Calculate the probability that the wife dies within 10 years. 329 | \vspace{4mm} 330 | 331 | \(A) 0.26 \hspace{0.2in} (B) 0.27 \hspace{0.2in} (C) 0.28 \hspace{0.2in} 332 | (D) 0.29 \hspace{0.2in} (E) 0.30 \vspace{10mm} 333 | 334 | 15.2. Consider the setup provided in Exercise 15.1. Calculate the 335 | probability that the husband dies both before the wife dies and within 336 | 10 years. \vspace{4mm} 337 | 338 | \(A) 0.26 \hspace{0.2in} (B) 0.27 \hspace{0.2in} (C) 0.28 \hspace{0.2in} 339 | (D) 0.29 \hspace{0.2in} (E) 0.30 \vspace{10mm} 340 | 341 | 15.3. Consider the setup provided in Exercise 15.1. Given $\delta$ = 342 | 0.03, calculate the expected present value of a life annuity that pays 343 | 1000 per year continuously each year while the husband is alive. 344 | \vspace{4mm} 345 | 346 | \(A) 13,000 \hspace{0.2in} (B) 13,500 \hspace{0.2in} (C) 14,000 347 | \hspace{0.2in} (D) 14,500 \hspace{0.2in} (E) 15,000 \vspace{10mm} 348 | 349 | 15.4. Consider independent lives (25) and (50). Each life has mortality 350 | such that: $l_x$ = 100(100 - $x$) for $x$ $\le$ 100. \vspace{2mm} 351 | 352 | Calculate the probability that (25) dies after (50) and within 8 years. 353 | \vspace{4mm} 354 | 355 | \(A) 0.007 \hspace{0.2in} (B) 0.009 \hspace{0.2in} (C) 0.011 356 | \hspace{0.2in} (D) 0.013 \hspace{0.2in} (E) 0.015 357 | 358 | \newpage 359 | 15.5. Consider independent lives (25) and (50), each with force of 360 | mortality $\mu_x$ = $\frac{1}{100 - x}$ for 0 $\le$ $x$ $<$ 100. 361 | \vspace{2mm} 362 | 363 | Calculate the probability that (50) dies before (25) and within 25 364 | years. \vspace{4mm} 365 | 366 | \(A) 0.08 \hspace{0.2in} (B) 0.17 \hspace{0.2in} (C) 0.25 \hspace{0.2in} 367 | (D) 0.33 \hspace{0.2in} (E) 0.42 \vspace{20mm} 368 | 369 | 15.6. Consider (40) and (50) with independent future lifetime random 370 | variables: \vspace{4mm} 371 | 372 | \(i) (40) has mortality that follows: $l_x$ = 95 - $x$ for 0 $\le$ $x$ 373 | $\le$ 95. \vspace{2mm} 374 | 375 | \(ii) (50) has a constant force of mortality equal to 0.05. \vspace{4mm} 376 | 377 | Calculate the probability that (40) dies after (50) and within 10 years. 378 | \vspace{4mm} 379 | 380 | \(A) 0.03 \hspace{0.2in} (B) 0.04 \hspace{0.2in} (C) 0.05 \hspace{0.2in} 381 | (D) 0.06 \hspace{0.2in} (E) 0.07 \vspace{20mm} 382 | 383 | 15.7. For independent lives Shane and Britney: 384 | 385 | \vspace{4mm} (i) Shane, aged 50, has mortality such that: ${}_{t}q_{50}$ 386 | = 0.030$t$ and ${}_{t}q_{51}$ = 0.031$t$ for 0 $\le$ $t$ $\le$ 1. 387 | 388 | \vspace{2mm} (ii) Britney, aged 40, has mortality that follows the 389 | Illustrative Life Table with the assumption of a uniform distribution of 390 | deaths within each year of age. 391 | 392 | \vspace{4mm} Calculate the probability that Britney dies both within 2 393 | years and before Shane. \vspace{4mm} 394 | 395 | \(A) 0.003 \hspace{0.2in} (B) 0.004 \hspace{0.2in} (C) 0.005 396 | \hspace{0.2in} (D) 0.006 \hspace{0.2in} (E) 0.007 \vspace{12mm} 397 | 398 | \newpage 399 | 15.8. Consider independent lives (25) and (50): \vspace{4mm} 400 | 401 | \(i) Each life has mortality such that: $l_x$ = 100(100 - $x$) for 0 402 | $\le$ $x$ $\le$ 100. \vspace{2mm} 403 | 404 | \(ii) $\delta$ = 0.05 \vspace{4mm} 405 | 406 | Calculate the actuarial present value of a life insurance that pays 1 at 407 | the moment of death of (25) if (25) is the second to die. \vspace{4mm} 408 | 409 | \(A) 0.07 \hspace{0.2in} (B) 0.08 \hspace{0.2in} (C) 0.09 \hspace{0.2in} 410 | (D) 0.10 \hspace{0.2in} (E) 0.11 \vspace{10mm} 411 | 412 | 15.9. Consider the setup provided in Exercise 15.8. Calculate the 413 | actuarial present value of a life insurance that pays 1 at the moment of 414 | death of (25) if (25) is the first to die. \vspace{4mm} 415 | 416 | \(A) 0.16 \hspace{0.2in} (B) 0.17 \hspace{0.2in} (C) 0.18 \hspace{0.2in} 417 | (D) 0.19 \hspace{0.2in} (E) 0.20 \vspace{10mm} 418 | 419 | 15.10. For independent lives (30) and (35): 420 | 421 | \vspace{4mm} (i) $\mu_{30}(t)$ = 0.05 for $t$ $>$ 0 422 | 423 | \vspace{2mm} (ii) $\mu_{35}(t)$ = $\frac{1}{55 - t}$ for 0 $<$ $t$ $<$ 424 | 55 425 | 426 | \vspace{4mm} Calculate the probability that (35) dies before (30). 427 | \vspace{4mm} 428 | 429 | \(A) 0.34 \hspace{0.2in} (B) 0.40 \hspace{0.2in} (C) 0.46 \hspace{0.2in} 430 | (D) 0.54 \hspace{0.2in} (E) 0.60 \vspace{10mm} 431 | 432 | 15.11. Two lives (x) and (y) have mortality such that: \vspace{4mm} 433 | 434 | \(i) The Common Shock Model applies. \vspace{2mm} 435 | 436 | \(ii) $\mu^{03}_{x+t:y+t}$ = 0.005 for $t$ $>$ 0 \vspace{2mm} 437 | 438 | \(iii) $\mu^{02}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{13}_{x+t}$ = 439 | 0.015 for $t$ $>$ 0 \vspace{2mm} 440 | 441 | \(iv) $\mu^{01}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{23}_{y+t}$ = 442 | 0.015 for $t$ $>$ 0 \vspace{4mm} 443 | 444 | Calculate the probability that both (x) and (y) are dead within 10 445 | years. \vspace{4mm} 446 | 447 | \(A) 0.05 \hspace{0.2in} (B) 0.06 \hspace{0.2in} (C) 0.07 \hspace{0.2in} 448 | (D) 0.08 \hspace{0.2in} (E) 0.09 \vspace{10mm} 449 | 450 | \newpage 451 | 15.12. Two lives (x) and (y) have mortality such that: \vspace{4mm} 452 | 453 | \(i) The Common Shock Model applies. \vspace{2mm} 454 | 455 | \(ii) $\mu^{01}_{x+t:y+t}$ = 0.03 for $t$ $>$ 0 \vspace{2mm} 456 | 457 | \(iii) $\mu^{02}_{x+t:y+t}$ = 0.06 for $t$ $>$ 0 \vspace{2mm} 458 | 459 | \(iv) $\mu^{03}_{x+t:y+t}$ is a constant for $t$ $>$ 0 \vspace{2mm} 460 | 461 | \(v) $\mu^{02}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{13}_{x+t}$ for 462 | $t$ $>$ 0 \vspace{2mm} 463 | 464 | \(vi) $\mu^{01}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{23}_{y+t}$ for 465 | $t$ $>$ 0 \vspace{2mm} 466 | 467 | \(vii) The probability that (x) survives 5 years is 0.67. \vspace{2mm} 468 | 469 | Calculate the probability that (x) and (y) die simultaneously. 470 | \vspace{4mm} 471 | 472 | \(A) 0.12 \hspace{0.2in} (B) 0.14 \hspace{0.2in} (C) 0.16 \hspace{0.2in} 473 | (D) 0.18 \hspace{0.2in} (E) 0.20 \vspace{8mm} 474 | 475 | 15.13. Consider the setup in Exercise 15.12. Estimate 476 | ${}_{0.4}p^{03}_{xy}$ using Euler’s forward method with step 0.2 to 477 | numerically solve Kolmogorov’s forward equations. \vspace{4mm} 478 | 479 | \(A) 0.004 \hspace{0.2in} (B) 0.006 \hspace{0.2in} (C) 0.008 480 | \hspace{0.2in} (D) 0.010 \hspace{0.2in} (E) 0.012 \vspace{8mm} 481 | 482 | 15.14. For a special last-survivor insurance of 100,000 on (x) and (y): 483 | \vspace{4mm} 484 | 485 | \(i) The death benefit is payable at the moment of the second death. 486 | \vspace{2mm} 487 | 488 | \(ii) Level annual benefit premiums of $\pi$ are payable continuously 489 | each year only while exactly one of (x) and (y) is alive. \vspace{2mm} 490 | 491 | \(iii) The Common Shock Model applies. \vspace{2mm} \vspace{2mm} 492 | 493 | \(iv) For $t$ $>$ 0: $\mu^{03}_{x+t:y+t}$ = 0.02, 494 | 495 | $\mu^{02}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{13}_{x+t}$ = 0.05, 496 | 497 | $\mu^{01}_{x+t:y+t}$ + $\mu^{03}_{x+t:y+t}$ = $\mu^{23}_{y+t}$ = 0.07 498 | \vspace{4mm} 499 | 500 | \(v) $\delta$ = 0.06 \vspace{4mm} 501 | 502 | Calculate: $\pi$. \vspace{4mm} 503 | 504 | \(A) 8400 \hspace{0.2in} (B) 8500 \hspace{0.2in} (C) 8600 \hspace{0.2in} 505 | (D) 8700 \hspace{0.2in} (E) 8800 506 | 507 | \newpage 508 | 15.15. Consider a special last survivor insurance of 1 on (x) and (y): 509 | \vspace{4mm} 510 | 511 | \(i) The Multiple Life Model applies. \vspace{2mm} \vspace{2mm} 512 | 513 | \(ii) $\delta$ = 0.08 \vspace{2mm} 514 | 515 | \(iii) $\mu^{01}_{x+t:y+t}$ = $\mu^{02}_{x+t:y+t}$ = 0.04 for $t$ $>$ 0 516 | \vspace{2mm} 517 | 518 | \(iv) $\mu^{13}_{x+t}$ = $\mu^{23}_{y+t}$ = 0.06 for $t$ $>$ 0 519 | \vspace{2mm} 520 | 521 | \(v) $\mu^{03}_{x+t:y+t}$ = 0 \vspace{4mm} 522 | 523 | Calculate the expected present value of the last survivor insurance. 524 | \vspace{4mm} 525 | 526 | \(A) 0.21 \hspace{0.2in} (B) 0.23 \hspace{0.2in} (C) 0.25 \hspace{0.2in} 527 | (D) 0.27 \hspace{0.2in} (E) 0.29 528 | 529 | \newpage 530 | ### Answers to Exercises 531 | 532 | \Large 533 | 534 | 15.1. D \vspace{2mm} 535 | 536 | 15.2. D \vspace{2mm} 537 | 538 | 15.3. A \vspace{2mm} 539 | 540 | 15.4. B \vspace{2mm} 541 | 542 | 15.5. E \vspace{2mm} 543 | 544 | 15.6. B \vspace{2mm} 545 | 546 | 15.7. D \vspace{2mm} 547 | 548 | 15.8. C \vspace{2mm} 549 | 550 | 15.9. B \vspace{2mm} 551 | 552 | 15.10. A \vspace{2mm} 553 | 554 | 15.11. B \vspace{2mm} 555 | 556 | 15.12. D \vspace{2mm} 557 | 558 | 15.13. C \vspace{2mm} 559 | 560 | 15.14. C \vspace{2mm} 561 | 562 | 15.15. A 563 | 564 | \newpage 565 | ## Past Exam Questions 566 | 567 | - Exam MLC, Spring 2015: \#3 \vspace{2mm} 568 | 569 | - Exam MLC, Spring 2014: \#7 \vspace{2mm} 570 | 571 | - Exam MLC, Fall 2013: \#2 \vspace{2mm} 572 | 573 | - Exam MLC, Spring 2013: \#5 \vspace{2mm} 574 | 575 | - Exam MLC, Fall 2012: \#21 \vspace{2mm} 576 | 577 | - Exam MLC, Sample Questions: \#53, 122, 194, 225, 249, 261, 262, 263, 578 | 265, 266, 268, 271, 279 \vspace{2mm} 579 | 580 | - Exam MLC, Spring 2007: \#14 \vspace{2mm} 581 | -------------------------------------------------------------------------------- /LifeCon2/Chapters/18-MAFAppendix.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Appendix" 3 | output: 4 | html_document: 5 | toc: yes 6 | toc_depth: 3 7 | # pdf_document: 8 | # toc: yes 9 | # toc_depth: '3' 10 | --- 11 | 12 | # Appendix 13 | 14 | ## Appendix A: Exam Syllabi, Exam Tables, and Past Exam Questions 15 | 16 | All relevant materials regarding SOA Exam MLC can be found at: 17 | 18 | [http://www.soa.org/education/exam-req/edu-exam-m-detail.aspx](http://www.soa.org/education/exam-req/edu-exam-m-detail.aspx). 19 | \vspace{5mm} 20 | 21 | All relevant materials regarding the former CAS Exam LC can be found at: 22 | 23 | [http://www.casact.org/admissions/syllabus/index.cfm?fa=LCsyllabi&parentID=332](http://www.casact.org/admissions/syllabus/index.cfm?fa=LCsyllabi&parentID=332). 24 | \vspace{10mm} 25 | 26 | ## Appendix B: Multi-State Model Examples 27 | 28 | ![](Figures/Appendix_MSM.pdf) 29 | 30 | -------------------------------------------------------------------------------- /LifeCon2/Figures/Appendix_MSM.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ewfrees/LifeCon/66e09a97ef093c46778c3d5724f196abdea845f0/LifeCon2/Figures/Appendix_MSM.pdf -------------------------------------------------------------------------------- /LifeCon2/LICENSE: -------------------------------------------------------------------------------- 1 | CC0 1.0 Universal 2 | 3 | Statement of Purpose 4 | 5 | The laws of most jurisdictions throughout the world automatically confer 6 | exclusive Copyright and Related Rights (defined below) upon the creator and 7 | subsequent owner(s) (each and all, an "owner") of an original work of 8 | authorship and/or a database (each, a "Work"). 9 | 10 | Certain owners wish to permanently relinquish those rights to a Work for the 11 | purpose of contributing to a commons of creative, cultural and scientific 12 | works ("Commons") that the public can reliably and without fear of later 13 | claims of infringement build upon, modify, incorporate in other works, reuse 14 | and redistribute as freely as possible in any form whatsoever and for any 15 | purposes, including without limitation commercial purposes. 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  • Loss Data Analytics
    • 18 | # after: | 19 | #
  • Published with bookdown
    • 20 | # edit: https://github.com/rstudio/bookdown-demo/edit/master/%s 21 | -------------------------------------------------------------------------------- /LifeCon2/disqus.html: -------------------------------------------------------------------------------- 1 |
    2 | 20 | -------------------------------------------------------------------------------- /LifeCon2/index.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Actuarial Mathematics Analytics" 3 | author: "An open text authored by the Actuarial Community" 4 | date: "`r Sys.Date()`" 5 | site: bookdown::bookdown_site 6 | #output: 7 | # bookdown::gitbook: 8 | # includes: 9 | # after_body: disqus.html 10 | documentclass: book 11 | bibliography: [Bibliography/LDAReference.bib, Bibliography/packages.bib] 12 | biblio-style: apalike 13 | link-citations: yes 14 | #github-repo: rstudio/bookdown-demo 15 | description: "Actuarial Mathematics Analytics is an interactive, online, freely available text. 16 | - The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. 17 | - A subset of the book will be available in pdf format for low-cost printing. 18 | - The online text will be available in multiple languages to promote access to a worldwide audience." 19 | --- 20 | 21 | # Preface {-} 22 | 23 | ####Book Description {-} 24 | 25 | **Actuarial Mathematics Analytics** is an interactive, online, freely available text. 26 | 27 | - The online version contains many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote *deeper learning*. 28 | 29 | - A subset of the book is available for *offline reading* in pdf and EPUB formats. 30 | 31 | - The online text will be available in multiple languages to promote access to a *worldwide audience*. 32 | 33 | 34 | 35 | ####What will success look like? {-} 36 | 37 | The online text will be freely available to a worldwide audience. The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. Moreover, a subset of the book will be available in pdf format for low-cost printing. The online text will be available in multiple languages to promote access to a worldwide audience. 38 | 39 | ####How will the text be used? {-} 40 | 41 | This book will be useful in actuarial curricula worldwide. It will cover the loss data learning objectives of the major actuarial organizations. Thus, it will be suitable for classroom use at universities as well as for use by independent learners seeking to pass professional actuarial examinations. Moreover, the text will also be useful for the continuing professional development of actuaries and other professionals in insurance and related financial risk management industries. 42 | 43 | ####Why is this good for the profession? {-} 44 | 45 | An online text is a type of open educational resource (OER). One important benefit of an OER is that it equalizes access to knowledge, thus permitting a broader community to learn about the actuarial profession. Moreover, it has the capacity to engage viewers through active learning that deepens the learning process, producing analysts more capable of solid actuarial work. 46 | Why is this good for students and teachers and others involved in the learning process? 47 | 48 | Cost is often cited as an important factor for students and teachers in textbook selection (see a recent post on the [$400 textbook](https://www.aei.org/publication/the-new-era-of-the-400-college-textbook-which-is-part-of-the-unsustainable-higher-education-bubble/)). Students will also appreciate the ability to “carry the book around” on their mobile devices. 49 | 50 | 51 | ####Why loss data analytics? {-} 52 | 53 | Although the intent is that this type of resource will eventually permeate throughout the actuarial curriculum, one has to start somewhere. Given the dramatic changes in the way that actuaries treat data, loss data seems like a natural place to start. The idea behind the name *loss data analytics* is to integrate classical loss data models from applied probability with modern analytic tools. In particular, we seek to recognize that big data (including social media and usage based insurance) are here and high speed computation s readily available. 54 | 55 | 56 | ####Project Goal {-} 57 | 58 | The project goal is to have the actuarial community author our textbooks in a collaborative fashion. 59 | 60 | To get involved, please visit our 61 | [Online Actuarial Textbooks Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/). 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | -------------------------------------------------------------------------------- /LifeCon2/lifecon.sty: -------------------------------------------------------------------------------- 1 | \NeedsTeXFormat{LaTeX2e} 2 | 3 | \ProvidesPackage{lifecon}[2004/07/27 v2.1 (Eddy Trivedi)] 4 | \PackageInfo{lifecon}{Provides Commands to typeset Life 5 | Contingencies' Symbols} 6 | 7 | \RequirePackage{amsmath} 8 | 9 | \newlength{\LC@temph} 10 | \newlength{\LC@tempw} 11 | 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| - chmod +x ./_deploy.sh 11 | 12 | script: 13 | - ./_build.sh 14 | - ./_deploy.sh 15 | -------------------------------------------------------------------------------- /LifeCon3/Bibliography/Bibliography.Rmd: -------------------------------------------------------------------------------- 1 | `r if (knitr:::is_html_output()) '# Bibliography {-}'` 2 | -------------------------------------------------------------------------------- /LifeCon3/Bibliography/LDAReference.bib: -------------------------------------------------------------------------------- 1 | @Book{xie2015, 2 | title = {Dynamic Documents with {R} and knitr}, 3 | author = {Yihui Xie}, 4 | publisher = {Chapman and Hall/CRC}, 5 | address = {Boca Raton, Florida}, 6 | year = {2015}, 7 | edition = {2nd}, 8 | note = {ISBN 978-1498716963}, 9 | url = {http://yihui.name/knitr/}, 10 | } 11 | 12 | @InProceedings{III2015, 13 | Title = {International Insurance Fact Book}, 14 | Author = {{Insurance~Information~Institute}}, 15 | Year = {2015}, 16 | Note = {[Retrieved on May 10, 2016]}, 17 | Publisher = {Insurance Information Institute}, 18 | Url = {http://www.iii.org/sites/default/files/docs/pdf/international_insurance_factbook_2015.pdf} 19 | } 20 | 21 | @Book{bowers1986actuarial, 22 | Title = {Actuarial Mathematics}, 23 | Author = {Bowers, Newton L. and Gerber, Hans U. and Hickman, James C. and Jones, Donald A. and Nesbitt, Cecil J.}, 24 | Publisher = {Society of Actuaries Itasca, Ill.}, 25 | Year = {1986} 26 | } 27 | 28 | @Book{dickson2013actuarial, 29 | Title = {Actuarial Mathematics for Life Contingent Risks}, 30 | Author = {Dickson, David C. M. and Hardy, Mary and Waters, Howard R.}, 31 | Publisher = {Cambridge University Press}, 32 | Year = {2013} 33 | } 34 | 35 | @InProceedings{survey2013, 36 | Title = {2013 Insurance Predictive Modeling Survey}, 37 | Author = {Earnix}, 38 | Year = {2013}, 39 | Note = {[Retrieved on July 7, 2014]}, 40 | Publisher = {Earnix and Insurance Services Office, Inc.}, 41 | Url = {http://earnix.com/2013-insurance-predictive-modeling-survey/3594/} 42 | } 43 | 44 | @Article{bailey1960, 45 | Title = {Two studies in automobile ratemaking}, 46 | Author = {Bailey, Robert A. and LeRoy, J. Simon}, 47 | Journal = {Proceedings of the Casualty Actuarial Society Casualty Actuarial Society}, 48 | Year = {1960}, 49 | Number = {I}, 50 | Volume = {XLVII} 51 | } 52 | 53 | @InProceedings{SASsurvey, 54 | Title = {Building believers: How to expand the use of predictive analytics in claims}, 55 | Author = {Gorman, Mark and Swenson, Stephen}, 56 | Year = {2013}, 57 | Note = {[Retrieved on August 17, 2014]}, 58 | Publisher = {SAS}, 59 | Url = {http://www.sas.com/resources/whitepaper/wp_59831.pdf} 60 | } 61 | 62 | @inproceedings{venter1983transformed, 63 | title={Transformed beta and gamma distributions and aggregate losses}, 64 | author={Venter, Gary}, 65 | booktitle={Proceedings of the Casualty Actuarial Society}, 66 | volume={70}, 67 | number={133 \& 134}, 68 | pages={289--308}, 69 | year={1983} 70 | } 71 | 72 | @article{mcdonald1984some, 73 | title={Some generalized functions for the size distribution of income}, 74 | author={McDonald, James B}, 75 | journal={Econometrica: journal of the Econometric Society}, 76 | pages={647--663}, 77 | year={1984}, 78 | publisher={JSTOR} 79 | } 80 | -------------------------------------------------------------------------------- /LifeCon3/Bibliography/LossDataAnalyticsReference.bib: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/ewfrees/LifeCon/66e09a97ef093c46778c3d5724f196abdea845f0/LifeCon3/Bibliography/LossDataAnalyticsReference.bib -------------------------------------------------------------------------------- /LifeCon3/Bibliography/packages.bib: -------------------------------------------------------------------------------- 1 | @Manual{R-base, 2 | title = {R: A Language and Environment for Statistical Computing}, 3 | author = {{R Core Team}}, 4 | organization = {R Foundation for Statistical Computing}, 5 | address = {Vienna, Austria}, 6 | year = {2016}, 7 | url = {https://www.R-project.org/}, 8 | } 9 | @Manual{R-bookdown, 10 | title = {bookdown: Authoring Books and Technical Documents with R Markdown}, 11 | author = {Yihui Xie}, 12 | year = {2016}, 13 | note = {R package version 0.3}, 14 | url = {https://CRAN.R-project.org/package=bookdown}, 15 | } 16 | @Manual{R-knitr, 17 | title = {knitr: A General-Purpose Package for Dynamic Report Generation in R}, 18 | author = {Yihui Xie}, 19 | year = {2016}, 20 | note = {R package version 1.15.1}, 21 | url = {https://CRAN.R-project.org/package=knitr}, 22 | } 23 | @Manual{R-rmarkdown, 24 | title = {rmarkdown: Dynamic Documents for R}, 25 | author = {JJ Allaire and Joe Cheng and Yihui Xie and Jonathan McPherson and Winston Chang and Jeff Allen and Hadley Wickham and Aron Atkins and Rob Hyndman}, 26 | year = {2016}, 27 | note = {R package version 1.2}, 28 | url = {https://CRAN.R-project.org/package=rmarkdown}, 29 | } 30 | -------------------------------------------------------------------------------- /LifeCon3/Figures/Ch1_01_ex1.1.16.JPG: 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| 8 | git config --global user.email "xie@yihui.name" 9 | git config --global user.name "Yihui Xie" 10 | 11 | git clone -b gh-pages https://${GITHUB_PAT}@github.com/${TRAVIS_REPO_SLUG}.git book-output 12 | cd book-output 13 | cp -r ../_book/* ./ 14 | git add --all * 15 | git commit -m"Update the book" || true 16 | git push origin gh-pages 17 | -------------------------------------------------------------------------------- /LifeCon3/_output.yml: -------------------------------------------------------------------------------- 1 | bookdown::pdf_book: 2 | includes: 3 | in_header: preamble.tex 4 | toc: TRUE 5 | toc_depth: '3' 6 | latex_engine: xelatex 7 | citation_package: natbib 8 | keep_tex: yes 9 | bookdown::epub_book: default 10 | bookdown::gitbook: 11 | css: style.css 12 | includes: 13 | in_header: ShowHide.js 14 | config: 15 | toc: 16 | before: | 17 |
  • Loss Data Analytics
    • 18 | after: | 19 |
  • Published with bookdown
    • 20 | edit: https://github.com/rstudio/bookdown-demo/edit/master/%s 21 | -------------------------------------------------------------------------------- /LifeCon3/disqus.html: -------------------------------------------------------------------------------- 1 |
    2 | 20 | -------------------------------------------------------------------------------- /LifeCon3/index.Rmd: -------------------------------------------------------------------------------- 1 | --- 2 | title: "Actuarial Mathematics Analytics" 3 | author: "An open text authored by the Actuarial Community" 4 | date: "`r Sys.Date()`" 5 | site: bookdown::bookdown_site 6 | output: 7 | bookdown::gitbook: 8 | includes: 9 | after_body: disqus.html 10 | documentclass: book 11 | bibliography: [Bibliography/LDAReference.bib, Bibliography/packages.bib] 12 | biblio-style: apalike 13 | link-citations: yes 14 | #github-repo: rstudio/bookdown-demo 15 | description: "Actuarial Mathematics Analytics is an interactive, online, freely available text. 16 | - The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. 17 | - A subset of the book will be available in pdf format for low-cost printing. 18 | - The online text will be available in multiple languages to promote access to a worldwide audience." 19 | --- 20 | 21 | # Preface {-} 22 | 23 | ####Book Description {-} 24 | 25 | **Actuarial Mathematics Analytics** is an interactive, online, freely available text. 26 | 27 | - The online version contains many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote *deeper learning*. 28 | 29 | - A subset of the book is available for *offline reading* in pdf and EPUB formats. 30 | 31 | - The online text will be available in multiple languages to promote access to a *worldwide audience*. 32 | 33 | 34 | 35 | ####What will success look like? {-} 36 | 37 | The online text will be freely available to a worldwide audience. The online version will contain many interactive objects (quizzes, computer demonstrations, interactive graphs, video, and the like) to promote deeper learning. Moreover, a subset of the book will be available in pdf format for low-cost printing. The online text will be available in multiple languages to promote access to a worldwide audience. 38 | 39 | ####How will the text be used? {-} 40 | 41 | This book will be useful in actuarial curricula worldwide. It will cover the loss data learning objectives of the major actuarial organizations. Thus, it will be suitable for classroom use at universities as well as for use by independent learners seeking to pass professional actuarial examinations. Moreover, the text will also be useful for the continuing professional development of actuaries and other professionals in insurance and related financial risk management industries. 42 | 43 | ####Why is this good for the profession? {-} 44 | 45 | An online text is a type of open educational resource (OER). One important benefit of an OER is that it equalizes access to knowledge, thus permitting a broader community to learn about the actuarial profession. Moreover, it has the capacity to engage viewers through active learning that deepens the learning process, producing analysts more capable of solid actuarial work. 46 | Why is this good for students and teachers and others involved in the learning process? 47 | 48 | Cost is often cited as an important factor for students and teachers in textbook selection (see a recent post on the [$400 textbook](https://www.aei.org/publication/the-new-era-of-the-400-college-textbook-which-is-part-of-the-unsustainable-higher-education-bubble/)). Students will also appreciate the ability to “carry the book around” on their mobile devices. 49 | 50 | 51 | ####Why loss data analytics? {-} 52 | 53 | Although the intent is that this type of resource will eventually permeate throughout the actuarial curriculum, one has to start somewhere. Given the dramatic changes in the way that actuaries treat data, loss data seems like a natural place to start. The idea behind the name *loss data analytics* is to integrate classical loss data models from applied probability with modern analytic tools. In particular, we seek to recognize that big data (including social media and usage based insurance) are here and high speed computation s readily available. 54 | 55 | 56 | ####Project Goal {-} 57 | 58 | The project goal is to have the actuarial community author our textbooks in a collaborative fashion. 59 | 60 | To get involved, please visit our 61 | [Loss Data Analytics Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/). 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | -------------------------------------------------------------------------------- /LifeCon3/preamble.tex: -------------------------------------------------------------------------------- 1 | \usepackage{booktabs} 2 | \setcounter{secnumdepth}{2} 3 | -------------------------------------------------------------------------------- /LifeCon3/style.css: -------------------------------------------------------------------------------- 1 | p.caption { 2 | color: #777; 3 | margin-top: 10px; 4 | } 5 | p code { 6 | white-space: inherit; 7 | } 8 | pre { 9 | word-break: normal; 10 | word-wrap: normal; 11 | } 12 | pre code { 13 | white-space: inherit; 14 | } 15 | -------------------------------------------------------------------------------- /LifeCon3/toc.css: -------------------------------------------------------------------------------- 1 | #TOC ul, 2 | #TOC li, 3 | #TOC span, 4 | #TOC a { 5 | margin: 0; 6 | padding: 0; 7 | position: relative; 8 | } 9 | #TOC { 10 | line-height: 1; 11 | border-radius: 5px 5px 0 0; 12 | background: #141414; 13 | background: linear-gradient(to bottom, #333333 0%, #141414 100%); 14 | border-bottom: 2px solid #0fa1e0; 15 | width: auto; 16 | } 17 | #TOC:after, 18 | #TOC ul:after { 19 | content: ''; 20 | display: block; 21 | clear: both; 22 | } 23 | #TOC a { 24 | background: #141414; 25 | background: linear-gradient(to bottom, #333333 0%, #141414 100%); 26 | color: #ffffff; 27 | display: block; 28 | padding: 19px 20px; 29 | text-decoration: none; 30 | text-shadow: none; 31 | } 32 | #TOC ul { 33 | list-style: none; 34 | } 35 | #TOC > ul > li { 36 | display: inline-block; 37 | float: left; 38 | margin: 0; 39 | } 40 | #TOC > ul > li > a { 41 | color: #ffffff; 42 | } 43 | #TOC > ul > li:hover:after { 44 | content: ''; 45 | display: block; 46 | width: 0; 47 | height: 0; 48 | position: absolute; 49 | left: 50%; 50 | bottom: 0; 51 | border-left: 10px solid transparent; 52 | border-right: 10px solid transparent; 53 | border-bottom: 10px solid #0fa1e0; 54 | margin-left: -10px; 55 | } 56 | #TOC > ul > li:first-child > a { 57 | border-radius: 5px 0 0 0; 58 | } 59 | #TOC.align-right > ul > li:first-child > a, 60 | #TOC.align-center > ul > li:first-child > a { 61 | border-radius: 0; 62 | } 63 | #TOC.align-right > ul > li:last-child > a { 64 | border-radius: 0 5px 0 0; 65 | } 66 | #TOC > ul > li.active > a, 67 | #TOC > ul > li:hover > a { 68 | color: #ffffff; 69 | box-shadow: inset 0 0 3px #000000; 70 | background: #070707; 71 | background: linear-gradient(to bottom, #262626 0%, #070707 100%); 72 | } 73 | #TOC .has-sub { 74 | z-index: 1; 75 | } 76 | #TOC .has-sub:hover > ul { 77 | display: block; 78 | } 79 | #TOC .has-sub ul { 80 | display: none; 81 | position: absolute; 82 | width: 200px; 83 | top: 100%; 84 | left: 0; 85 | } 86 | #TOC .has-sub ul li a { 87 | background: #0fa1e0; 88 | border-bottom: 1px dotted #31b7f1; 89 | filter: none; 90 | display: block; 91 | line-height: 120%; 92 | padding: 10px; 93 | color: #ffffff; 94 | } 95 | #TOC .has-sub ul li:hover a { 96 | background: #0c7fb0; 97 | } 98 | #TOC ul ul li:hover > a { 99 | color: #ffffff; 100 | } 101 | #TOC .has-sub .has-sub:hover > ul { 102 | display: block; 103 | } 104 | #TOC .has-sub .has-sub ul { 105 | display: none; 106 | position: absolute; 107 | left: 100%; 108 | top: 0; 109 | } 110 | #TOC .has-sub .has-sub ul li a { 111 | background: #0c7fb0; 112 | border-bottom: 1px dotted #31b7f1; 113 | } 114 | #TOC .has-sub .has-sub ul li a:hover { 115 | background: #0a6d98; 116 | } 117 | #TOC ul ul li.last > a, 118 | #TOC ul ul li:last-child > a, 119 | #TOC ul ul ul li.last > a, 120 | #TOC ul ul ul li:last-child > a, 121 | #TOC .has-sub ul li:last-child > a, 122 | #TOC .has-sub ul li.last > a { 123 | border-bottom: 0; 124 | } 125 | #TOC ul { 126 | font-size: 1.2rem; 127 | } 128 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | 2 | ## Open Actuarial Textbooks Project Goal 3 | The goal is to have the actuarial community author our textbooks in a collaborative fashion. *GitHub* provides a natural development environment to achieve this goal. See the [Open Actuarial Textbooks Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/) for more information about this project. 4 | 5 | ### Loss Data Analytics 6 | Most of the work has been done on modeling **Loss Data**. See our [GitHub Site](https://github.com/ewfrees/Loss-Data-Analytics ) for more information about these efforts. 7 | 8 | ### Life Contingencies 9 | 10 | We are also starting to organize several sets of notes for *Life Contingencies*, also known as *Actuarial Mathematics* and *Long Term Actuarial Mathematics*. 11 | 12 | The folder, *LifeCon1*, contains lecture notes from Edward (Jed) Frees, University of Wisconsin-Madison. The folder, *LifeCon2*, contains lecture notes from Paul Johnson, also University of Wisconsin-Madison. You can look over these lecture notes in .pdf format by clicking [lifeCon1](https://docs.google.com/a/wisc.edu/viewer?a=v&pid=sites&srcid=d2lzYy5lZHV8bG9zcy1kYXRhLWFuYWx5dGljc3xneDo3MGQ5YWY2NmYwMDM5ZjQw) and [lifeCon2](https://docs.google.com/a/wisc.edu/viewer?a=v&pid=sites&srcid=d2lzYy5lZHV8bG9zcy1kYXRhLWFuYWx5dGljc3xneDpiNjBkNWM1N2JjZjBmNGY), respectdively. Epub versions are available at our [Life Contingences Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/chapter-development/life-contingencies) (hit the down-arrown next to the file that you wish to download). We are currently working on converting lecture notes from Jay Vadiveloo, University of Connecticut. His files (in Microsoft Word) are also available at our [Life Contingences Project Site](https://sites.google.com/a/wisc.edu/loss-data-analytics/chapter-development/life-contingencies). 13 | 14 | ### Examples from Loss Data Analytics 15 | 16 | With the source code on this *GitHub* site, you can modify, combine and improve these source codes. The folders *LifeCon1* and *LifeCon2* allow users to compile output in .pdf, .epub, and .html formats. See our [Getting Started](https://github.com/ewfrees/Loss-Data-Analytics/tree/master/GettingStarted) folder in the **Loss Data Analytics** repository. We are using an *R* package called *bookdown* to produce our books. A major advantage of this approach is that it allows us to automatically produce .html output for interactive use, a .pdf version for offline viewing, and an .epub version for mobile devices. To illustrate the .html output, you can check out our draft of the *Loss Data Analytics* book at [Online Version from R Bookdown](http://instruction.bus.wisc.edu/jfrees/UWCAELearn/LossDataAnalytics/index.html) . 17 | 18 | ## Your Help 19 | As described in our [License](https://github.com/ewfrees/Loss-Data-Analytics/tree/master/GettingStarted/LICENSE.md), anyone can use these files. Especially at this early stage of the project, if you want to help and would like more info, please write Jed Frees . 20 | 21 | 22 | --------------------------------------------------------------------------------