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
- premium 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
- time for additional scenarios vs value of additional information
- 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
- 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
- Disadvantages (can be corrected w/ adjustments)
- 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
- formula should produce no additional lapses when spread is small
- 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
- diff between purchase price adn par is discount/premium
- 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
- call premium
- 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)
- considered investment grade
- 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
- 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
- nwe/add'l policy loans
- 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
- economic downturn will incr default rates for many asset types
- if possible, vary default rates w/ economic activity
- Purchasing New Assets
- 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
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'.