Individual Asset Liability Management: Dynamic Stochastic Programming Solution

Transcription

1 EU HPCF Conference New Thinking in Finance Pensions & Insurance 1 Individual Asset Liability Management: Dynamic Stochastic Programming Solution Elena Medova joint work with Michael Dempster, Wajdi Tekaya & Philipp Ustinov Cambridge Systems Associates Limited Supported by the EU HPC Finance Project

2 2 Outline Unintended consequences of pension reforms and new rules for regulation of financial advisory companies New theory and technology for individual Asset Liability Management Stochastic programming solution of ialm Formulation of the optimal consumption/saving problem for households Examples of selected plans Technological challenges and progress so far Summary

3 3 The financial and economic crisis has seriously aggravated the underlying ageing challenge. Higher unemployment, lower growth, higher national debt levels and financial market volatility have made it harder for all systems to deliver on pension promises. EU Green paper on Pensions, July 2010 Many previous attempts to change rules of existing pensions led to massive demonstrations and protests from the public

4 Projections of Retirement Income US Preretirement Income Replacement Rate [PEW Mobility Project, May 2013] 4

5 Type of pensions Risk State pensions Government Reduced state social security guarantees due to Defined Benefit Defined Contribution Pensions and Risks Corporate Corporate and Individual high national debts Loss in value of institutional pension funds due to current crash in asset prices and low interest rates Low asset returns predicted for the next decade with the possibility of high inflation Loss in value of savings due to low saving rates 5 SIPP, 401K, individual savings, etc Individual Institutional ALM techniques for financial planning The asset management industry has yet to respond. This is the most challenging task theoretically and computationally - too complex for an individual

6 Financial Advice 6 Regulatory calls (across developed economies) for transparency and for a responsible investment approach suitable for a fund s specific member profile, liquidity needs and liabilities In the UK Financial Advice has changed from Only two types of financial adviser - Independent or Restricted All financial advisory companies are required to clearly set out the charges the client will pay prior to providing any advice - to ensure clients are fully aware of the costs involved: fees only (not commissions) for independent advisers Adviser s duty to clarify on what they are able, or are not able, to advise Unintended consequences Many big names on the high street dramatically cut back on face-to-face financial advisory services, or withdrew them altogether. This is particularly the case for clients with less than 100,000 invested Rapid growth Fund Platforms: fund wraps and fund supermarkets

7 Fund Platforms Technology risk? Model risk? 7 Fund Supermarkets DIY route to investment in ISA and SIPP Hargreaves Lansdown, Fidelity FundsNetwork, Alliance Trust Savings invest (Own), Barclays Stockbrokers MarketMaster (Own), J.P. Morgan Asset Management WealthManager+ (Own), Bestinvest Select (SEI),... strategic allocations matched to client s needs (Fidelity) How do Fund Platform tools assist users decisions regarding life long investments and savings? How is the new promise of goals-based investment implemented? It seems that all portfolio asset allocation models are based on some sort of meanvariance optimization (MVO) model

8 Optimal Consumption/ Saving Over Lifetime Is the static view of risks and their dependencies applicable for longterm investment decisions? The classic mean-variance analysis of Markowitz (1952) is a static analysis which assumes that investors care only about risk and return one period ahead The special case of myopic investors for long-term problems Utility theory dynamic models - Life cycle consumption investment models (Samuelson 1938, 1963, 1969; Merton 1969, 1971, 1973; Campbell 2002,...) Households dilemma in defining wealth over lifetime What is the risk in matching cash inflows from assets and outflows from liabilities? Is wealth the inflation-indexed real income that our assets can sustain over time? Is wealth the long-term spending that our portfolio can sustain? 8

9 The myth of risk attitudes Daniel Kahneman JPM Fall To understand an individual s complex attitudes towards risk we must know both the size of the loss that may destabilize them, as well as the amount they are willing to put in play for a chance to achieve large gains. Temporary perspectives may be too narrow for the purpose of wealth management. The theories - utility theory and its behavioural alternatives - assume that individuals correctly anticipate their reaction to possible outcomes and incorporate valid emotional prediction into their investment decisions. In fact, people are poor forecasters of their future emotions and future tastes they need help in this task and I believe that one of responsibilities of financial advisors should be to provide that help. DSP new theory and innovative technology - HPC Finance Project

10 New Meta-Model Model for Individual Financial Planning - ialm 10 The ialm system is a decision support tool based on the theory of dynamic stochastic optimization Principal ideas are brought together from behavioural and classical finance and from decision theory Due to the significant uncertainties involved in identification of future feasible consumption the user may interactively re-solve the problem with inputs varied in the individual s preferences and goal priorities. This allows comparing the consequences of these changes on the optimal long-term financial plan the gaming aspect of the solution process

11 Dynamic Stochastic Programming 11 General idea of dynamic stochastic programming Incorporate many alternative futures in the form of a tree of simulated scenarios for the discrete-time stochastic data process ω : = { ω t : t = t1,0,..., t T + 1,0} = { ω,..., ω, ω,..., ω,..., ω,..., ω }. t t t t t t 1,0 1, u 2,0 2, u T,0 T, u stage 1 stage 2 stage T Model future decisions and implement current ones to obtain a forward plan to the problem s horizon When the current recommended solution is implemented, then all future solutions across future scenarios which will follow with a certain probability are taken into account

12 12 Scenario Tree Schema 1st stage 2d stage 3d stage leaf node root node Scenario 8 t = A multi-period scenario tree

13 Multi-stage Dynamic Stochastic Programme 13 t t t t t min f ( x ) + E min f ( ω, x ) E min f ( ω, x ) u 2,0 2, u T,0 T,0 1 2,0 2 tt,0 tt 1,0,...,,..., T t x ω,..., 1,0 t x 1, t x ω ω u 2,0 t x 2, u t x T,0 tt, u x s. t. M A x 11 t 1 1,0 t t t t A x A x b a s 1,1 1,1 1,1 1,1 ( ω ) t + ( ω ) ( ) ( ).. 0,0 t ω = ω 0,1 t t t t Tu+ 1,1 t Tu+ 1, Tu t Tu+ 1 A x A x b a s T 1,0 T 1,0 T 1,0 T 1,0 ( ω + ) ( ω + ) ( ω + ) = ( ω + ).. 1,0 T, u 1, u min f1( x ) + pt ( ω 2,0 t ) f 2,0 t ( ω 2,0 t, x 2,0 t ( ω ),..., ( ))... ( ) (, ( ),..., ( )) 2,0 t x ω + + ω ω ω ω 2,0 t2, u t p 2,0 tt,0 t f T,0 tt,0 t x T,0 tt,0 t x T,0 tt, u tt,0 s. t. A x 1,0 t t2,0 t T,0 11 t 1 21 t t 22 t t t Ω A ( ω ) x + A ( ω ) x ( ω ) = b ( ω ) ω Ω M 1,1 1,0 1,1 1,1 1,1 Deterministic Equivalent Ω = b = b 1,1 1,1 1,1 A ( ω ) x A ( ω ) x ( ω ) = b ( ω ) ω Ω Tu+ 1,1 t t Tu+ 1, Tu t t t Tu+ 1 t t t T+ 1,0 1,0 T+ 1,0 T, u T+ 1,0 T+ 1,0 T+ 1,0 T+ 1,0 2 t t

14 Solution Methods 14 Scenario history ω t εωis a possible realization of the random vector ω t and corresponds to a node of the scenario tree Deterministic equivalent of the dynamic stochastic program (DSP) is a very large sparse linear programming (LP) problem coefficients and right hand sides in the constraints are realizations of the stochastic data process Solution method for linear objectives nested Benders or interior point The vector process for the optimal stochastic decision process is given by x : { :,..., } {,...,,,...,,...,,..., }. = xt t = t1,0 tt, u = xt x ,0 t x 1, u t x ,0 t x 2, u t x T,0 tt, u stage 1 stage 2 stage T Implementable decisions correspond to the root node of the scenario tree

15 Stochastic Programming Techniques for ialm 15 root node t = leaf node I. Simulation Generation of stochastic data with a discrete number of annual observations of a continuous time vector data process branching at specified times (decision times) in the future Scenario tree is a schema for forward simulation along each branch a multiple number of stochastic processes are simulated. Some are independent, other may be correlated. Simulation discrete time steps correspond to the data sampling frequency of the process of interest ialm involves simulation of asset returns and liabilities punctuated by life events II. Optimization Discrete time and state optimization giving a different optimization problem (given by its objective and constraints) at each node of the scenario tree dependent on both its predecessors and successors Major decision time points are stages of the tree Implementable decisions are at the root node which are the most constrained decisions robust against all alternative scenarios generated while the remainder allow what-if prospective analysis ialm solves a large scale linear optimization problem Consumption (goal) maximization at each decision time subject to constraints such as risk, budget, cash flow balance and so on annually Sustainable wealth maximization across all years and all generated scenarios simultaneously

16 16 Modelling Changing Financial Markets Investment securities Domestic and International Equities Government Bonds Corporate Bonds Alternatives T-bills and all bond coupons Treasury Inflation Protected Securities (TIPS) Cash CPI Other fixed assets Fundamental financial models Multi-dimensional GBM process d ln X i, t = µ i d t + σ i d W i, t Geometric Ornstein Uhlenbeck (OU) process d ln r = ( α β ln r ) d t + σ d W t t t OU process dr = ( α β r ) dt + σ dw t t t

17 17 Annual Returns of the S&P 500 Index

18 Gather Individual and Market Data Overview of individual ALM Personal data Market data 18 Econometric and Actuarial Modelling Events model Liabilities model Model returns on investment classes Scenario Tree Simulation Events Cash-out flows forecast Cash-in flows forecasts Optimization Model: Tailored Portfolios, Goal Spending, Cash flows Balances, etc Dynamic optimization model for assets & liabilities Objective: maximize spending on risk managed goals Visualization of decisions Various Constraints

19 Meta-Model Model Generation with STOCHASTICS TM 19

20 Key Features of the ialm Tool 20 Goal-oriented objective More uncertainty accounted for not just uncertainties of market returns but uncertainties of personal life events DSP methodologies implemented using Stochastics TM Optimal portfolio decisions correspond to the best feasible desirable consumption subject to existing and future liabilities Portfolio risks are managed by constraining portfolio drawdown in each scenario imposing limits on portfolio asset holdings in each scenario Multiple portfolio accounts simultaneously managed with optimal treatment of taxes (rule-based assumptions applicable to various jurisdictions)

21 ialm Behavioural Finance 21 Broad Framing: overall objective is to provide sustainable spending over a household s lifetime in terms of desired consumption on multiple life goals specified by preferences and their priorities Narrow Framing: maximization of goal consumption at given times (annually) Each single goal utility function is defined with respect to reference points chosen by the household in terms of spending on the goal

22 Value Function of the Prospect Theory 22 Recall the Value Function of Prospect Theory reference point

23 Individual Goal Utility Narrow Framing 23 Utility function for an individual goal is given by three reference points For each single goal the level of spending y is in the range between acceptable (s) and desirable (g) and minimum (h) spending subject to existing and foreseen liabilities. Together with goal priorities these values specify the piecewise linear shape of the utility function for each goal The objective is to maximize goal spending each year with a piecewise linear utility function u (utility) g 1 α g 1 α s h α h s g y (spending) 1

24 Overall Objective Broad Framing 24 To provide sustainable spending Optimization problem objective is to maximize the expected present value (over all scenarios) of lifetime consumption, i.e. spending on goals T E 1{any alive, t} u t ( C ) t= 1 1 where u ( C ) = u z + I xs xs τ i τ ( π π ) t g, t t t g G φ t xs τ Here z is excess borrowing, I is total tax payment and φ is the inflation index at t t t t Consumption refers to all elective spending on chosen goals alive d m alive C =, s, (, +, ) + ˆ t α g t φ Fg t Fg t α g t g, t φ g, ty g, t g G m g g G \ G m

25 Utility of Initial Wealth as Lifetime Consumption 25

26 Dynamic Stochastic ialm Formulation 26 Decision theory type overall objective - optimum resource allocation over a network of income and out going cash flows Optimal management of various portfolios subject to varieties of constraints Management and transaction fees constraints Initial wealth constraints Cashflow rebalancing constraints Random time horizon ALM problem Stochastic dynamic analysis taking into account many complex dependencies Number of goals Timing of liabilities and goals with significant values Dependency of the dynamic asset allocation on life expectancy: longevity hedging Health statistics...

27 27 One Goal Savings for Retirement Net wealth Transaction costs Returns Coupons and dividends Interest on bank deposits Regular income Employer pension contributions Qualified coupons and dividends CD +II zr 11 cash tt + po + II ρ+p qekqk qcqd +II Portfolio ()a x P qualified contributions q+p contributions qualified +P 401 qk +P Qualified account ()0 asset sales qa x q z qnr +P m Cash holding () +z q P asset purchases qa+ x Margin borrowing () m +m xs tz C xsxs 1(1) t +zr Excess borrowing xs z,it z () cashm r+mr Goal Equity (see below) z ++ IttI,11 (1) cashs rr goal spending Income borrowing I z Loans secured on assets +z IttI,11 () cashs rr ττ+if av qp qrτp xsxs 1t zr L Retirement Goal Liabilities Taxation Qualified returns Qualified portfolio ()qa x tx+tx qq + rr + xx aaaa Transaction costs (qualified portfolio)

28 Cash Flow Network 28 Net wealth Transaction costs Returns Coupons and dividends Interest on bank deposits Regular income Employer pension contributions Qualified coupons and dividends CD +II zr 11 cash tt + po + II qekqk ρ+p qcqd +II qualified contributions Portfolio ()a x P qualified contributions q+p +P 401 qk +P Qualified account ()0 asset sales qa x q z qnr +P m Cash holding z() +z q P asset purchases qa+ x asset purchases Margin borrowing () m +m + xs tz C xsxs 1(1) t +zr,it z Excess borrowing xs z () cashm r+mr Goal Equity (see below) z ++ IttI,11 (1) cashs rr Income borrowing I z Loans secured on assets +z IttI,11 () cashs rr ττ+if av qp qrτp xsxs 1t zr L Interest charges on margin loans Interest on goal loans Goal consumption (non capital) Liabilities Taxation Unauthorized qualified withdrawal penalty Interest charges on secured borrowing Interest charges on income loans Interest charges on excess borrowing Qualified returns Qualified portfolio ()qa x tx+tx qq + + xx aaaa rr Transaction costs (qualified portfolio)

29 Visual Summary of Profile Goals 29 Getting an Overview Cash Flows Portfolio Wealth

30 Example I 30 A 40 year old couple with 2 dependants Starting assets: Non-Qualified Asset Account: 71,000 SIPP Account: 15,000 ISA Account: 35,000 Tangible Assets(Family home): 400,000 (with a 20 year 150, 000 mortgage taken out in 2000) Income: Pre-Retirement 133,000 per annum Objective to maintain desirable level of consumption pre- and post retirement

31 Example II Basic household data as in Example I Additional objective to provide for the private education of children 31

32 Example I - Retirement Planning Solution 32

33 Example II Multiple Goals Solution 33

34 Sensitivity of Investment Decisions to Goals Number of Life Goals: More Goals Riskier Investment 34 Size Return Risk 129, % 9.6% Size 96,692 Return 6.0% Risk 10.6% Size Return Risk 96, % 10.6%

35 Sensitivity of Investment Decisions to Goals Goal Timing: Earlier Goals More Diversified Investment 35 Now Goal Retirement Size 106,433 Return 4.1% Risk 5.9% Now Goal Retirement Size 106,246 Return 6.1% Risk 10.9% Now Retirement Size 97,647 Return 6.0% Risk 10.5%

36 Sensitivity of Investment Decisions to Uncertainty of Life Events Life Expectancy: Greater Longevity Riskier Investment 36 SA Actuarial Life Table (Return: 5.45%, Vol: 8.58%) UK Actuarial Life Table (Return: 5.91%, Vol: 10.24%)

37 Sensitivity of Investment Decisions to Uncertainty of Life Events Greater Life Expectancy Greater Wealth 37 SA actuarial life table UK actuarial life table

38 38 Technology & HPC Techniques Impact

39 39 Technology & HPC Techniques Impact The improvement in run time from 2005 to 2012 is solely due to hardware improvement The run time in 2013 (approximately half of 2012) is the result of the use of optimization algorithms that benefit from an HPC parallel computing environment (Open MP) Benders decomposition type algorithms such as Stochastics GNBS typically benefit greatly from HPC techniques The Markovian property of the asset return models can allow more efficient implementation and numerical solution

40 ialm Summary 40 ialm provides optimum values for many decision variables spending, borrowing, saving, etc -- across time simultaneously for multiple scenarios of random processes representing uncertain markets and life circumstances Currently the ialm model includes 20 random processes that vary over the client s lifetime and around 200 mathematically formulated conditions (constraints) per node of the scenario tree Average solution time is around 1 minute involving a typical LP optimization problem with over 3 million non-zero entries Further significant improvements are being tested which combine algorithmic techniques and new parallel HPC technology