### Life Insurance Products and Financing (Atkinson/Dallas) - Chapter 15 - STOCHASTIC MODELING

Introduction
• chapter focus on practical applications of mathematical and statistical techniques
• best estimate assumptions can usually be established w/ fair degree of confidence
• int rate is hard to predict w/ confidence
• strategies to reduce effect of int rate unpredictability
• conservative assumptions for i
• offer products that adjust benefits when int rates change
• match asset & liab cash flows
• stochastic modeling produces distribution of likely results
• allows answers to questions such as "what is prob of ROI < 7%" or "avg roi for worst 10% of scenarios"
Overview of Stochastic Modeling
• Uses of Stochastic modeling
• single product, portfolio of products, or entire company
• increasingly used for both pricing and ongoing mgmt of A/L
• can be applied to any variable/assumption - most commonly int rate/inv returns
• Steps Involved in Stochastic Modeling
• select a distribution function
• choose a random number generator
• stochastically generate sets of variables
• calc results for each set of rates
• liab or prodcut related results
• asset-related results
• Select a Distribution Function
• should generate values that best fit the range/freq/deviation of possible outcomes
• shoudl reproduce expected mean and variance
• mortality - typically a binomial dist function
• int - often normal or lognormal dist to reflect the change in int rate
• should test to see how well it fits w/ experience over a number of years
• parameters adjusted to improve fit
• Choose a Random Number Generator
• used in conjunction w/ dist function to generate random values
• Stochastically Generate Sets of Variables
• random number generator applied to dist function to create many sets of variable being modeled
• one set of rates is however many rates are needed to run the model once
• ex. mortality, sep rates for each cell in each period of the model
• ex. int rates - sep yield rates for each type of asset avail for purch in each future time period (in practice - only key assets & rest are determined via reference to key rates
• Calcualte Results for Eaach Set of Int Rates
• A/L Calcs perfomred as of deterministic model (w/ currently generated sets of rates)
• make sure affected variables adjusted
• ex. int rate assumptions can/will affect
• credit rate and acct values for dynamic products
• divs for par products
• lapse rates
• sales levels
• asset mkt values
• asset calls and prepayments
• some models will adjust for these automatically
Random Variables
• developed by combine dist function w/ random number generator
• general steps
• develop cdf F(x) that reflects distribution
• calc random number S ~ U(0,1)
• calc x for each S F(x) = S (if continuous) F(x-1) < S <= F(x) (if discrete)
• Binomial Distribution
• X(n) -> sum of n trials where P(x=1) = p
• mean (mu) = np variance (sigma squared) = npq std dev (sd or sigma) = (npq)^0.5
• Normal Distribution
• for sufficiently large values of n (book suggests minimum value of n = 30)
• Z(n) = (X(n) - 0.5n) / 0.5*(n)^0.5 ~ N(0,1)
• ex. generate 30 random values to calc X(30)
• so if you need 2000 values, need 2000 * 30 = 60000 random numbers
• then determine dist function f(x) (prob X(n) = x) ^ CDF F(x) (prob X >= x)
• otherwise can be used to generate values of Z
Stochastic Mortality
• mortality fluctuations can be quite significant
• stochastic mort models can help to understand likely variability of mort results and design products or programs (such as reins) to stabilize results
• models presume all lives independent
• not quite true, but good enough
• multiple policies on same insured
• disaster can simultaneously kill many insureds that work/travel together
• common accidents on family members
• lonely heart syndrome
• Seriatim Stochastic Modeling
• simplest approach is model 1 policy at a time
• perform for each policy each period
• qd and qw for that policy that period
• S ~ U(0,1)
• is S <= qd policy is marked as dead (set qd = 1)
• if not dead, S ~ U(0,1)
• if S <= qw policy is lapsed (set qw = 1)
• once policy terminated, s/b removed from inforce for future periods
• essentially, each policy is its own cell
• Alternate to Stochastic Modeling
• volatility of largest policies modeled using seriatim apporach
• volatility of remaing policies modeled as follows
• deterministic model to determine expected deaths each period
• calc avg expected mort rate (q(t) for each period
• total variance for each period approx = (#pols)*(q(t)*(1 - q(t))*(avgDB)^2
• assumes identical policies
• better estimate - calc variance for each cell and sum variance
• best estimate - calc variance for each cell and sum results
• law of large numbers - mean and variance above and normal dist allown prediction of distribution of DC
• Binomial Stochastic Modeling
• best fits a group of independend lives w/ same mort & DV
• often the case for a single cell
• useful wehn seriatim approach not feasible
• Applying the Binomial Distribution
• allows use of 1 random number to determine outcome for n policies at once
• how-to for 1 cell in 1 period:
• determine assumed qd and qw and # policies in force (no) (# pols will decrease from period to period)
• using Bin Dist, create CDF F(x) for # deaths (x) in peroid based on n pols in force
• S ~ U(0,1)
• if F(x-1) < s <= F(x) then x is # deaths and qd = x/n
• using Bin Dist, create CDF F(x) for # lapses (y) in period based on n - x pols in force
• S ~ U(0,1)
• if F(y-1) < s <= F(y) then y is # lapses and qw = y / (n-x)
• # pol inforce beg next period is n-x-y
• Calculating teh Binomial Dist Function
• apply bin dist to cell n w/ prob death = q
• f(x) = nCx*q^x*(1-q)^(n-x) where nCx = n! / ((n-x)!*x!)
• nCx can be gotten from Pascal's triangle
• fratio(x) = f(x) / f(x-1)
• fratio(x)
• f(x) can be calculated directly (see above) but more efficient to calc iteratively
• f(x) = f(x-1) * fratio(x) where f(0) = (1-q)^n
• fratio(x) = q/(1-q) * (n-x+1)/x
• Cumulative Dist Function F(x)
• F(s) = sum(f(x)) from 0 to s
Stochastic Interst Rates
• Overview
• more scenarios created, more credibel the results
• # scenarios limited by speed of software/hardware
• stochastic modeling of int rates best performed in aggregate
• complexities include
• must product yield rates for all possible future asset purchases, not just one rate for each future period
• int rates driven by world events which can have long term effects on rates
• can randomly generated rates adequately reflect this?
• int rates also driven by supply/demand
• to handle great variety of yields available on different investments, yields assumed to be some of two pieces
• gov't yield rate from same maturity of an asset
• spread over that yield rate (in BPS)
• int'l standard for comparing s/t rates - LIBOR
• In US, spread usually vs Treasuries
• Assuming spread is fixed and unchanging for each type of asset
• Yield Curves
• shows yield rates on one axis and time to maturity on other axis
• normally slope upwards w/ increasing time to maturity (normal yield curve)
• inverted yield curve - yield curves that slope downward w/ increasing time to maturity
• in practice, yield curves defined by one S/T and one L/T rate (90 day and 10 yr) and other rates determined from these
• interpolated rate = (1-Factor)*[90 day rate] + factor*[10 year rate]
• sample factors
• 90 180 1yr 2yr 3yr 5yr 7yr 10yr 20yr 30yr
• 0 .1694 .3600 .5671 .6706 .7647 .9059 1.0000 1.0784 1.1176
• Interest Rate Scenarios
• int rate scenario consists of one yield curve for each future period in model
• int rates from one period to the next are highly correlated so we can't jut generate two random int rates
• 90 day adn 10yr rate are correlated - often fall and rise in unison or partial unison
• three approaches to handling (among many)
• arbitrary method
• probablistic method
• successive ratios method
• Arbitrary Method
• not a stochastic model
• involves manually creating a set of int rate scenarios in an arbitrary fashion
• different scenarios may test effect of gradual or sudden incr/dec in int rate
• of limited value - rarely sufficient # to be credible
• arbitrary input -> arbitrary output
• ex. NY 7
• Probablistic Approach
• assume every curve defined by level (10 yr rate) and slope (90 day/10yr)
• using historic info, develop prob of each level changing to any other level during next period
• probabilities arranged into grid
• in example grid
• sum(prob) = 1.0 for each row
• min i = 2% and max i = 15%
• rates equally likely to move up or down near middle of grid
• near edges, more likely to move away from edge
• similar grid showing probability of each slope changing to any other slope
• randomly determine change in level and slope, still end up w/ 10yr and 90day rates that are related
• develop CDF F(x) for each level and slope
• can combine F(x) for current level and rand number to stochastically generate teh level for the next period's yield curve
• can combine F(x) for current slope and another rand numberto generate slope for next period's yield curve
• Jetton - single grid w/ yield curves and probabilities of moving from one curve to the next - only need 1 rand #
• Successive Ratios Approach
• assume ln of rato of successive int rates is normally distributed
• ln(i(t+1)/i(t)) ~ N
• don't want to apply to both 10yr adn 90 day since would end up w/ 2 ind rates
• use one RV w/ successive ratios approacht to generate next 10 yr rate
• combo of RV_10 adn RV_90 to generate 90 day
• RV_10*weight_10 + RV_90*weight_90 = 90 day rate
• weights reflect degree of correlation between 10 and 90 day
• alternate approach uses volatility factor (VolFactor)
• Z1,Z2 - 2 sep RV ~ N(0,1)
• Correlation - degree of correlation between changes in i90day(t) and i10yr(t)
• Z10yr - RV used to generate changes in i10yr(t) (reflects some correlation w/ i90day(t))
• i90day(t+1) = i90day(t)e^(Z1-VolFactor)
• Z10year = Z1*Correlation + Z2*(1-Correlation^2)^0.5
• i10yr(t+1) = i10yr(t)*e^(Z10yr-VolFactor)
• Advantages (compared to probablistic method)
• not limited to predetermined # of int rates or yield curves
• not necessary to research adn create large tables of probabilities
• only need volatility factor and correlation factor
• no min/max int rate
• no corridor...if you feel int rates will gravitate toward certain level
• tends to product more inverted yield curves than you'd normally expect
• Might need to bias formula towards normal curves
• differnece between 90day and 10yr can grow to unrealistic extremes
Effect on Liabilities
• overview
• Int Scenarios ->Int Rates->Product Cash Flows->Asset Cash Flows
• four rates determined beginning of each period
• avg rate earned on exiting assets - prior period inv income / prior period avg assets
• int rates avail on new investments - scenario yield curve
• int rates avail on competing products - aka Mkt Rates
• int rates creditd to company's products being modeled - credited rate
• Modeling Interest rates
• mkt rate - rate avail from financial alternatives
• if co is crediting rates in line w/ mkt rates
• surrenders, w/d s/b normal
• if co is crediting > mkt rates - improved persistency
• if co is crediting < mkt rates - worse persistency
• if big difference, could see large cash outflows
• partial w/d adn loans also affected by mkt rates (vs credited rates)
• term generally immune to int
• however prolonged inflation can erode value leading to lapses
• possibly reflect non-ins product int rates
• money mkt fund, 5yr bank CDs 10 yr govt bonds
• new money vs portfolio rate
• relevant mkt rate might be max(new money, portfolio)
• probably approximated fairly well as constant spread from gov't yeild rate
• Modeling Credit Int Rates
• function of 4 int rates
• portfolio rate (avg net int rate earned on products backing portfolio)
• new money rate (avg net int rate on new investments)
• guaranteed int rate (for product)
• mkt rate
• int guarantees & mkt rate act as constraints on what co can credit
• if segmentation method
• new deposits get new money rate
• existing deposits - net int earned on assets backing segment
• if portfolio method - earned rate => portfolio rate (w/ some adjustment for new funds @ new money rate)
• a product can have one or more guarantees
• long-term guaranteed rate
• short term current int rate guarantee
• bailout rate
• credited rate can't be less that LT or ST guarantees
• can be < bailout rate if willing to waive SC
• most co's have targeted spread they wish to earn
• credited rate = earned rate - targeted spread
• some cos have strategy of largest spread mkt will allow, subject to a max spread while maintaining a credited rate subject ot a min spread
• most cos - credited rate w/in certain range of mkt rates
• sample formula
• Max Possible Spread (MPS) = earned rate - 90% mkt rate
• credited rate = 90% mkt
• if MPS > 2%, credited rate = earned rate - 2%
• if credit rate > 110% of mkt rate, credited rate = 110% mkt rate
• if mps < 1%, credited rate = earned rate - 1%
• credited rate formula could reflect SC
• existence of SC allows co to credit slightly lower rate than if no SC
• Modeling the Effect on Lapses
• Life Ins products sole as investement vehicles generally have lapses quite sensitive to diff between mkt adn credit rates
• especailly sensitive if product has explicit credited rate
• study industry and company experience to develop formulas to help predict changes in lapse based on diff between mkt and credited rates
• change large and positive - lapse rates should increase
• change large and negative - lapse rates should improve
• sample lapse formulas
• qw(t) = qwBasic(t)(1+0.5*(100-difference)^2) - SC% min(0.01), max(0.50)
• qw(t) = qwbasic + 1.25*difference*3.25^|100*difference| - SC%, min(0.01), max(0.60)
• in general, a formula should cause lapses to fall below the base rate if credited rate exceeds mkt rate
• lapses should increase/decrease exponentially as the spread widens
• existence of SC should lower the lapse rate
• Modeling other Product Cash Flows
• credited rates affect dynamic CV and reserves
• div int rate affects amt divs paid and amt applied to div options
• partial w/d and prem persistency affected by spread for flex prem products
• pol loan utilization increases as mkt rates increase (esp if fixed LIR)
• expenses might inflate faster than expected
• could model inflation = mkt rate - constant
• anti-selection as unhealthy lives persist while healthy one lapse for more competitive products
Effects on Assets
• Steps applied to assets each period
• asset cash flows are determined, reflecting effect of current yield curve
• net CF determined = asset CF + product CF - dist earnings
• if net CF > 0, new assets purchased based on inv strategy
• if net CF < 0, model rules dictate sell assets or borrow cash
• book and mkt values determined for all assets @ EOP
• inv income, cap g/l determined for period
• Major Asset Classes
• questions for each asset adn how it relates to an ins company
• what are typical cash flows
• what are unusual cash flows and when can you expect them (what triggers them)
• does borrower have any rights to alter CF (by delaying or accelerating payments)
• does co have any rights to alter cash flow (puts)
• what expenses will co incur for mgmt/accting of each asset
• what % of investmetn will be lost to defaults/devaluation
• how liquid is the asset
• Gov't Securities
• if purchased @ discount, bond's yield > coupon rate
• if purchased @ premium, bond's yield < coupon rate
• amount of discont/premium amortized to 0 over life of bond in a fashion that results in a constant yield to maturity
• gov't bonds generally not callable
• assumed default rate often 0%
• usually lowest yielding asset
• expenses s/b consisten w/ corp bonds
• very liquid (most) - active mkt allows efficient trading
• Corp Bonds
• can have call adn put options
• call - borrow can repay early
• put - co can ask for early redemption
• usually issue non-callable when rates low
• penalty on borrow for early repayment
• diff between call price and mat value
• helps reimburse bondholder for lost int
• call price that decreases over time ex 104%, 103, 102, 101, 100
• private bonds have more substantial call premium
• typically PV of all remaing int & prin payments calced using spread over yield rate on govt securities
• model needs to make assumption as to when bond will be called
• futher away from original yield, greater prob. @ 1%-2% change, most bonds usually called
• puts less common
• allows borrow to repay @ less than full maturity value
• put option valuable for matching A/L
• ins co can liquidate assest @ favorable prices when rates high and reinvest @ higher rates or fund outflows
• most bonds in public mktplace
• u/w and sale managed by one or more investment banking groups
• privates negotiate and issued directly between borrower and lender
• Inc Co's like privates b/c investment banker fees saved
• privates shoudl have slightly higher yield b/c no inv banker fees
• privates often have sinking fund provision
• sinking fund provision clearly affects CFT timing
• cost to borrower for call option, therefore then to have higher rate
• put options tend to have lower rate
• public bonds more liquid that privates (day vs week w/ higher trading costs on private)
• diff in liquidity another reason why privates should have higher rate
• bond have different levels of seniority. Higher seniority, less default risk, lower rate
• BBB or higher - inv grade - majority of what ins cos purchase
• public less expensive to manage in portfolio b/c info more readily available
• High Yield Bonds (Junk)
• rated lower than BBB
• higher probability of default (5-10%) therefore higher yields to compensate for higher risk
• historically higher yield has more than compensated for higher defaults
• call option on high yield should have lower chance of being exercised
• like mortality anti-selection
• Commercial Mortgage
• large loans on commercial real estate (retail/office buildings)
• normally prin & int over 20 years, due @ 10 (10 year ballon)
• borrow usually pays origination fees
• usually contain prepayment provision
• prepayment penalties usually modest
• default risk usually highly correlated by geographic area
• not as risky as junk, but don't want too high a concentration
• fairly illiquid w/ no active mkt
• can usually sell a group of comm mort in about a month
• high asset monitoring costs
• Residential Mortgages
• loans on residential real estate - typically 50-80% of mkt value of real estate
• monthly payments of int and prin
• most 15 adn 30 year
• fixed of variable int rates
• mortgage may be purchased at either premium or discount
• amortized over life of mortgage, altering yield somewhat
• can be prepaid, usually w/o penalty
• if penalty exists, only first couple years
• level of prepayments based on multiple factors
• as int rates drop, more refinancing
• default risk
• usually related to unemployment, dis, death
• increases during recession (like junk and comm mortgages)
• fairly illiquid w/ no active market
• Collateralized Mortgage Obligations (CMOs)
• mortgage specialists assemble pools of thousands of indiv mortgages, then sell slices to investors
• securitization - creation of a new financial security that is backed by underlying cash flows
• by purchasing a slice of thousands of geographicly diverserve mortgages, buy can diversify risk
• tranche - slice of CMO
• payment and interest of each tranche varys depending on underlying cash flow
• each int paymetn dependent on underlying morgage prin outstanding
• each tranche receives its principle (subject to default) but amt of int rec'd depends on prepayment speed
• shorter tranches generally receive most of principle repayments first
• modeling is difficult b/c sensitivity of prepayments to change in i
• PAC (planned amortization class) - specialty tranche developed to address stability issues for investors
• PAC investure assured of getting fixed, pre-scheduled payments over speciifed period over a wide range of prepayment scenarios
• more volatile tranches created to absorb flucuations
• PAC has lower yield therefore other tranches can have higher yield for higher uncertainty
• many CMOs backed by gov't securities therefore no defaults and AAA rating
• CMOs actively traded and liquid (some volatile tranches may be difficult to sell)
• Asset-Backed Securities (ABS)
• consumer/corp debt - ex. credit card balances, auto loans, home equity loans, bank loans, commercial mortgages
• similar to CMOs
• some unique regulatory/acctg issues make these not as popular w/ ins cos
• CBO - collateralized bond obligation - corp bonds
• Real Estate
• most produce rental income - modest and uncertain compared to bonds and mortgages
• cash outflows req'd to maintain property
• largest cash flow is from sale of property
• sizable portion of return from apprectiate in property value
• some depreciate
• perhaps most illiquid asset
• poor match for most ins liab (modest/variable CF)
• makes sense when matched against very long-term liab or portion of co's capital
• Common Stock
• trend toward smaller divs (as % of stock price)
• rarely purchased for ongoing cash flows
• appreciation in price is main attraction
• prices highly variable
• very liquid - related to # shares outstanding (more shares, more liquid)
• historically 9-11% total return over 20-year periods
• poor match for most liab (low CF, volatility in price, acctg treatment)
• makes sense when matched against very long-term liab or portion of co's capital
• liquidity could be welcome addition
• Preferred Stock
• similar to bond w/ no maturity date
• junior to all bonds
• some have options to convert to common stock
• potentially valuable is stock price increases
• tax treatment of Pref Stock divs different from bond int
• fairly liquid, but ont as liquid as CS or bonds
• prices behave much like prices for 30+ year bonds
• if yield/quality acceptable, non-callable PS could make excellent match for longest LT liab, esp > longest bond mat avail
• Policy Loans
• if fixed int rate, utilization increases when PO can earn higher rate elsewhere
• some co's reduce credited rate on portion of value loaned
• par products make adjustments for loaned policies as well
• if variable rate, utilization will be more stable
• an increase in int rates can cause
• increase in surrenders (which repays policy loans)
• increase in policy loan activity
• Summary of Asset Cash Flows
• Positive Asset cash flows
• sales of any asset type
• bonds
• coupon payments
• calls (mat val + call prem)
• puts (mat val - put discount)
• sinkng fund payments
• maturity payments
• mort/cmo/abs
• regular payments of prin & int
• prepayments of prin and any prepayment penalties
• maturity payments
• real estate rental income
• CS & PS dividends
• policy loans
• int payments
• prin repayments
• Negative Asset cash flows
• investment expenses for all types, incl real estate maint
• improvements to real estate
• asset defaults
• Stochastic Modeling of Asset Cash Flows
• overview
• extrememly difficult undertaking
• basic groupings
• sales of assets
• prescheduled CF
• premature CF
• asset defaults
• prescheduled can be easily reflected once others addressed
• Sales of Assets
• many models assume assets held until they mature/prematurely repaind
• often @ odds w/ actual mgmt sime portfolios often actively managed
• best to reflect reality
• if activley managed portfolio, strategy s/b discussed & reflected in model
• Premature CF
• bond calls driven almost entirely by int rates
• to model bond calls, build grid of bond call rates that vary by change in i and bond quality rating
• bond puts
• controlled by insurer
• establish some parameters for exercising puts
• ex. if put price >10% above mkt value or neg CF & put price > mkt value
• extra sinking fund payments - modeled similarly to calls
• mortgage prepayments - similar to calls
• residential mortgages - add'l level of prepayments unrelated to delta i (relocations, bigger homes, etc)
• CMO/ABS prepayments - brute force approach - model all tranches to figure out what your trache will do
• recommended alt - table of prepayment rates that vary w/ delta i
• Default Assumptions
• CF interrupted on first default
• many times, assets in default are rehabbed and missed payments made up
• other times, asset sold @ reduced price
• reduction in price & missed payments is true cost of default
• simplest approach is treat defaults as perm loss of % of asset
• ex. if 0.4% annual default, carried in book @ 99.6%, 99.2%, etc
• CF reduced accordingly
• economic downturn will incr default rates for many asset types
• if possible, vary default rates w/ economic activity
• inventory of possible new assets part of input into model
• need to purchase from this inventory, if positive net cash flows
• ways to apply co's inv strategy
• input data may specify % of new assets to be invested in various classes of various quality and maturity
• input data may specify dist of new assets by asset class & quality and let model determine maturity to better match A/L CF
• might have different strategies for different LOB
• Covering Cash Shortfalls
• what to do when net CF is negative - borrow \$ or sell assets
• sell assets closest to mat date - MV least affected by delta i
• if investors don't care about unrealized g/l, sell assets w/ offsetting cap g/l
• sell assets that help co better match A/L
• ex. if assets longer than liab, sell LT assets
• modify strategy to min net cap g/l
• Calculation of Investment Results
• Book Values
• for many assets - price originally paid
• bonds, mort, cmo, abs - starts equal to price paid
• bonds: book = mat val + unamortized prem/discount
• mort/cmo/abs: book = loan priinciple + unamortized delta(price paid/principle)
• Capital G/L
• delta price sold and book value @ time of sale
• defaults recorded as capital losses
• Investment Income
• includes capital g/l
• InvIncome = Net Asset CF + increase in book value during period
• if new assets included in net asset CF as negative, will be included in delta book as positive
• no effect if purchased @ EOY
• 6 mo growth if mid-year purchase
Summarizing Stochastic Results
• # of scenarios depends on variablity of results
• depends on speed of modeling software and complexity of model being tested
• once all scenarios tested, results arranged sequentially, summarized by percentile adn sometimes gruoped to see range of possible results
Exercises