Life Insurance Products and Financing (Atkinson/Dallas) - Chapter 14

Life Insurance Products and Financing (Atkinson/Dallas) - Chapter 14 - FINANCIAL MODELING

Modeling of Liabilities

Overview
- create financial model by combing results of numerous ages and risk classes
- multiply per-unit-issued results by wieghts for dist of business by age adn risk class and summarize the weighted results
- General Steps
  - calc results on per-unit issued basis for a number of cells
  - multiply each cell's per-unit-issued results by appropriate # units -> gross results/cell
  - sum up gross results to calc total model results
Uses of Liability Models
- heavily for product decisions - including feature design and price structure
- combined from several products -> entire product line
- decide to introduce new product/discontinue existing one
- assess value of block of business to be acquired/sold
- essential tools to decide to continue/sell/close down LOB or Co
Types of Liab Models - purpose of model will determine scope/size/complexity
- Pricing Models
  - prelim models typically few representative IA/sex/risk/pol size cells w/ highest expected sales
  - final models include complete range of representative ages
  - often use refined models for bulk of pricing and simplified methods to handle rest
  - usually necessary to develop rates for all rep IA for both genders for at least some subset of cells
  - allows you to cross-subsidize results between pricing cells
    - higher profits from most cells may support lower profits at one or two key cells
- New Business Models
  - for planning and budgeting purposes
  - project amt and fin impact of NB
  - more useful - model that reflects effect of exp NB from all major products
  - usually sufficient to model results of co's best sellers and gross up
- InForce Models
  - uses
    - combined with NB models for planning/budgeting
    - calc value of block to be sold/acquired
    - determine adequacy of reserves
    - test reasonableness/equity of revised div scale/COI rates/int crediting strategy
    - project future liab CF to maatch asset CF
  - similar products usually grouped together
  - law of diminishing returns - additional accuracy gained not worth time it takes to model the next product
  - maybe vary issue ages modeled - fewer for smaller products
Building Data for Liab Model
- type type of data input: aggregate and cell data
- aggregate data
  - few assumptions, assumption multipliers and other parameters
  - easily changable
- cell data
  - mainly product parameters and assumptions
  - assumptions in cell data include mort rates/lapse rates/expense rates/avg size/# units
- many assumptions from pricing assumptions or recent experience studies
- avg size/# units usually from inforce data
- many cos use purchased modeling software
  - extract used to input to liab model - normally cnotains plancode/issuedate/IA (or DOB)/sex/risk class/units/SAPVx/TaxVx/AcctVal(ul)/Amt Reins
- Liability Model Calculations
  - want model to reasonably approximate fin statement results, therefore cal yr results
  - may need quarterly projects for plan/budget
  - book assumes 1/1 issue date to ease calendar year reporting
  - can group data so 1/1 is avg issue date
- Model Variables
  - calyr = issyr + t - 1 where t is policy year
  - issyr = calyr - t + 1
  - t = calyr - issyr + 1
- Number of Units
  - reflects distribution of business among cells
  - NumUnits(c) - # units for cell c (# issued)
  - NumUnitsIF(c) - # units inforce @ start of model for cell c
  - NumUnits(c) = NumUnitsIF(c) / SurvFactor(c,begyr-1)
  - if SurvFactor(c,begyr-1) = 1, above adjustment not necessary
- Total Results
  - variable_tot(calyr) = sum(variable(c,calyr)*NumUnits(c))
- Cell and Total Calculations
  - cash flows, reserves, reins calced @ cell level
  - ii, taxes, req'd cap, profit measure could be calced @ cell or total level
    - calcing @ cell level
      - probably slower
      - makes summarization simpler adn more flexible
    - calcing @ total level
      - more contol over final results
      - Quickly adjust total only parameters adn recalc results
Validating a Liability Model
- essential to validate prior to use
- NB or Pricing - compare ratios/patterns over time
- Inforce Model - reproduce starting inforce #s
  - static validation - reproducing actual values @ given point in time
- ways to improve model
  - changing representative issue ages
  - adjusting avg issue date
  - splitting into more issue periods
- after initial inforce validated - reasonability of model going forward
  - test model using inforce from year-1 and compare to actual results
- validating over a period of time - dynamic validation
- instead of tuning assumptions, sometimes just make adjustments @ total level
Liability Model Output
- commonly rows for each result and coulms for time periods
- s/b organized into familiar and useful formats, such as I/S or B/s
- other common outputs include
  - product cash flows - useful for planning inv strategies
  - inventory reports - ins inforce, # pols inforce
  - AmtInfIF(c,calyr) = DB_pu(c,calyr)*NumUnits(c)*SurvFactor(c,calyr) (_pu is per unit)
  - NumPol(c) = NumUnits(c) / AvgSize(c) where AvgSize(c) = avg # units / policy issued
  - NumPolIF(c) = NumPol(c)*SurvFactor(c,calyr)
  - AmtInsLapsed(c,calyr) = DB_pu(c,calyr)*Lapses(c,calyr)
  - Annualqw(calyr) = AmtInsLapsed_tot(calyr) / AmtInsIF_tot(calyr - 1) -> annualized lapse rate
Total Profit Measures
- w/inforce data included, can onlyl calc meaningful results for ROI, ROE, adn EV
- EV can be calced sep for inforce and each future issue year
Aggregate Models
- over short-term - relatively simple aggregate models commonly outperform elaborate cell-based models
- focus on growth rates adn trends of ratios
- ex. Income Stmt items as % of prem - probably see stable relationships and clear trends
- some kinds of business better predicted as % of assests or some other base
- no std approaches to aggregate models
- aaggregate models are poor predictor if co makes significant changes
- best models combine s/t fit of simple aggregate models w/ long term predictive capability of cell-based models
- once it is understood how to adjust s/t results to better match aggregate results, same techniques can be used to adjust l/t results

Asset/Liab Modeling

Purpose of A/L Modeling
- design investmetn strategy that fits product/liab portfolio
- more accurately predict inv income
- determine potential effect of diff int rate scenarios
- test strategies used to set credited rates
- asset modeling is driven by inv strategy, but informed inv strategy can only be develped once cash flows have been estimated
Designing an Investment Strategy
- most non term products depend heavily on investment returns
- Predicting Investment Income more Accurately
  - calc investment income from an asset model tied to liab cash flows
  - already know a lot about own portfolio - use this knowledge to predict future ii
  - ii from asset model has two parts
    - income from existing assets
    - income from future assets
- Testing the Effect of Interest Rate Scenarios
  - can predict how cash flows change to various int rate scenarios
    - modeling assets allows testing of investmetn strategies
  - rise in int rates can cause losses for a company
    - if co subsidized the credited rate to stay competitive, reduces profit but keeps business
    - if rate not competitive, PO surrender, minimal SC, large outflow when L/T assets have depress mkt value
  - testing of int rate scenarios can have several positive effects
    - makes co more aware of significant risks it is taking
    - may change inv strategies to reduce exposure to certain risks
    - may change the products offered to reduce exposre ot certain risks
    - may limit total amt of certain kinds of business it will accept to limit aggregate risk
    - may increase certain kinds of business to better balance and diversify its risks
    - some type and levels of risk acceptable
- Developing Int Crediting Strategies
  - mostly some capability to test int rate scenarios needed to develop and test int crediting strategies
  - need to estimate how competitor's credited rates, our credited rates and prodcut CF will vary w/ diff int rate scenarios
Introduction to Asset Modeling
- asset model used to project CF, mkt values, book values and otherh items for portfolio of assets
- assuming that asset portfolio being modeled is tied to specific liab portfolio
- assumptions
  - bonds can be purchased at time in reality
  - all bonds purchased at end of quarter, immediately after coupon payment made
  - all coupon and maturity payments at end of quarter
- BookValue(b,cyq) = Price(b) - when bond purchased @ cyq (calyr qtr)
- BookValue(b,cyq) = BookValue(b,cyq - 1)*(1+Qtryield(b)) - GrossCashFlow(b,cyq)
- Gross Cash Flow includes coupons and par @ maturity
- Book Values can also be calced as PV(future gross cash flows) @ yield rate
- from book value, can calc net cash flows (CumCashFlow)
- CumCashFlow(b,cyq) = GrossCashFlow(b,cyq)-[DefaultRate(b) + InvExpRate(b)]*BookValue(b,cyq-1)/4 (/4 since quarterly cf)
- InvIncome = NetCF + delta BookValue
- InvIncome(b,cyq) = CumCashFlow(b,cyq) + BookValue(b,cyq) - BookValue(b,cyq-1)
Assembling Data for an Asset Model
- Items needed (in addition to liability items)
  - prelim asset strategy
  - inventory of assets available for purchase
  - inventory of assets currently backing liab (for inforce block)
  - assumptions that describe future int rate patterns
  - strategy for dealing w/ negative CF
- Preliminary Inv Strategy
  - need to narrow the universe of assets to consider for model
  - attributes of assets to consider (acceptable classes, quality, dur, maturity)
  - used to guide model to types and mix of assets to purchase from future positive CF
  - s/b joint effort between co's inv mgr and those responsible for liab
  - characteristics will depend on
    - co's general inv philosophy
    - A/L already on books
    - regulatory restrictions
- Assets Available for Purchase
  - inventory of assets avail for purchase is needed
  - s/b consistent w/ prelim inv strategy
  - select a relatively small # of representative assets
  - "model assets" that reflect mix of quality ratings and assoc yields
  - do NOT mix assets w/ diff maturities - mat date greatly affects CF pattern
- Assets Currently backing Liabs
  - Inventory of existing assests hould include all infor needed to project future asset CF, book values and mkt values
- Future Int Rate Patterns
  - Assuming future int rates modeled one set at a time
- Handling Netative Cash Flows
  - Two common strategies
    - borrow money
      - typically from other porduct lines w/in same co
      - need assumption as to int rate charged when borrowing is necessary
      - rate for external borrowing should reflect co's credit quality
      - borrowed funds to be repaoid at earliest opportunity from pos CF
      - S/T rates appropriate
    - selling assets
      - model must calc MV adn needs rules for which to sell first, such as
        
        assets w/ largest cap gains first
        assets w/ shortest time to mat first
        assets held for at least 1 year first
Asset Adequacy or Free Cash Flows
- Asset Adequacy
  - performed on block of inforce to test adequacy of assets allocated to block
  - projection of both a/l under various int rate scenarios
  - total assets > total liab, assets adequate for scenario
  - if insufficient under many scenarios, actuary can require more assets allocated to back block
  - can be performed w/ beginning assets < or > beg reserves
- Free Cash Flow
  - assets rebalances @ end of each period to match solvency reserves + req'd capital
  - free cash flow - fee to be uses as co chooses OR
    - req'd to be contributied to support business
- Asset Modeling Process under Free Cash Flow Methodology
  - focus on meling assets of policy year basis
  - for model, assum CF only at BOY, middle, EOY and rebalancing only @EOY
  - at BOY and Mid-year
    - cum CF is determined
    - if cum CF > (<) 0, int rec (paid) next CF date = 1/2 years int on this cum CF
  - Beginning of Year
    - AssetsReq(t-1) = SolvRes(t-1) + ReqCap(t-1)
    - CumCashFlowBeg(t) = AssetCashFlowBeg(t) + LiabCashFlowBeg(t)
    - if CumCashFlowBeg(t) < 0
      - IntReceivedMid(t) = 0
      - IntPaidMid(t) = CumCashFlowBeg(t)*IntPaidRate(t)
    - if CumCashFlowBeg(t) > 0
      - IntReceivedMid(t) = CumCashFlowBeg(t)*IntReceivedRate(t)
      - IntPaidMid(t) = 0
  - Middle of Year
    - CumCashFlowMid(t) = CumCashFlowBeg(t) + AssetCashFlowMid(t) + LiabCashFlowMid(t) + IntReceivedMid(t) - IntPaidMid(t)
    - IntPaid[Rec'd]End(t) = CumCashFlowMid(t)*IntPaid[Rec'd]Rate(t) (and other is 0)
  - End of Year
    - CumCashFlowEnd(t) = CumCashFlowMid(t) + AssetCashFlowEnd(t) + LiabCashFlowEnd(t) + IntReceivedEnd(t) - IntPaidEnd(t)
    - AssetsEnd(t) = AssetsReq(t-1) + InvIncome(t) - AssetCashFlow(t)
  - Free Cash Flow
    - AssetsReq(t) = SolvRes(t) + ReqCap(t)
    - AssetsEnd(t) + CumCashFlowEnd(t) - FreeCashFlow(t) = AssetsReq(t)
    - FreeCashFlow(t) = SolvRest(t-1) + ReqCap(t-1) - [SolvRes(t) + ReqCap(t)] + CumCashFlowEnd(t) + InvIncome(t) - AssetCashFlow(t)
      - = CumCashFlow(t) + InvIncome(t) - AssetCashFlow(t) - SolvResIncr(t) - ReqCapIncr(t)
  - Free Cash Flow and Dist Earnings
    - FreeCashFlow(t) = LiabCashFlow(t) + InvIncome(t) + IntReceived(t) - IntPaid(t) - SolvResIncr(t) - ReqCapIncr(t)
    - new formula for dist earnings
- Validation of Asset Model
  - compare avg int rates for each period w/ assumed yield on ne and existing assets
  - compare asset purcahses w/ resulting asset net CF taht follow
    - amts not expected to match exactly, s/b saem range
  - par value, book value and avg yield @ beg of model should match inforce portfolio
- Asset Model Output
  - liab cash flows
  - assets purchased
  - asset cash flows
  - loans to fund cash shortfalls & repayment of prin and int
  - inv income
  - book value of assets
  - mkt value of assets
  - avg yeild on new assets purchased
  - avg yield on entire asset portfolio
  - avg dur of new assets purchased
  - avg dur of entire asset portfolio

Asset/Liability Matching

Overview
- immunization - matching of assets and liabs
- reduced financial effect of changes inint rates
- assuming liab CF not affected by int rate changes
Exact Matching
- begin by matching final liab CF and working backwards to curretn time
- cant to exact matching for very L/T (> 30 yr) liabs
- Practical Problems w/ exact matching
  - future pos cash flows need ot purchase assets. Cant purchase future assets until then
    - need to factor future asset purchases into matching plans
    - if significant disintermediation risk, migh need to match shortest dur first
  - asset defaults or calls, matching is out of balance. more assets need to be purchased
  - if liab CF deviates significant from expectd, portfolio needs rebalancing
- Exact matching case study - pp 756-759
Duration Matching
- exact matching is not usually practical
- duration matching mor common
- secondary use: excellent predictor of effect of small int rate changes
- if both A/L durations matche, small change in int rates should have equal effect on A & L
- company will have to rebalcnae occassionally to maintain matching of A/L
- Macauly Duration
  - duration - a measure of average time of a series of CF
  - Macauly dur - weighted avg w/ PV(CF) used as weights
  - MacDuration(i) = sum(t*v^t*CashFlow(t)) / sum(v^t*CashFlow(t))
  - MacDur of a single cash flow is that cash flow's time
  - MacCur of multiple CF is wtg avg time of the CF
  - when matching A/L dur, both need to use same int rate stream - usually current rate
- Modified Duration
  - used to estimate the effect of a small change in int rates on PV of CF
  - useful for predicting changes in PV due to int rate changes
  - mod dur is what most people mean when they say "duration"
  - PVCashFlow(i) = sum(v^t*CashFlow(t))
  - ModDuration(i) = d(PVCashFlow(i))/di / PVCashFLow(i)
    - = sum(t*v^(t+1)*CashFlow(t)) / sum(v^t*cashFlow(t))
    - = v*MacDuration(i)
    - %change in PVCashFlow(i) = -ModDuration(i)*delta(i)
- w/ duration matching - not exact matching of CFs
- dur matching implicitly assumes mismatches can be offest by investing/borrowing @ int rate used for dur calc
- possible to match dur and have terrible mismatch of CF
- Convexity
  - 2nd order deriviative of PVCashFLow(i) (modDur is 1st order)
  - Convexity(i) = sum(t*(t+1)*v^(t+2)*CashFlow(t)) / sum(v^t*CashFlow(t))
  - Mod Dur and convexity combined - more accurately calc effect of change in i on PVCashFLow(i)
  - % change in PVCashFlow(i) = -ModDuration(i)*delta(i) + 1/2*Convexity(i)*delta(i)^2
    - cannot expect 2 term formula to reproduce complexity of n cash flows
  - when matching using convexity
    - calc dur and convexity for liab CF
    - create 2 asset portfolios w/ same dur
    - blend to get convexity to match liab convexity
Horizon Matching
- exact matching used first few years (5-10) adn remaing CF matched using dur matching
- as time progressed, rebalcned w/ migration of some CF to exact matching group
Summary of A/L Matching
- requires collaboration between actuaries (liab experts) adn investmetn dept (asset experts)
- year-by-year CF can be significantly mismatched
  - if disintermediation is most significant risk - asset dur s/b < laib dur
  - if reinvestment risk is most significant risk - asset dur s/b > liab dur
- discussions ingored fact that most liab CF ARE affected by int rate changes
  - policyowner optionality - PO elecing options that affect CF
    - partial w/d, pol loans, surrenders

Exercises - beginning p 772

Copyright © 2004 Steve Welander.
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