- 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

- uses

- Pricing Models
- 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

- calcing @ cell level

- 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

- 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

- can predict how cash flows change to various int rate scenarios
- 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

- model must calc MV adn needs rules for which to sell first, such as

- borrow money

- Two common strategies

- Items needed (in addition to liability items)
- 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 Adequacy

- 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

- future pos cash flows need ot purchase assets. Cant purchase future assets until then
- 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

- policyowner optionality - PO elecing options that affect CF

Copyright © 2004 Steve Welander.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled 'GNU Free Documentation License'.