Modelling Incomplete Markets for Long Term Asset Liability Management

Transcription

1 INI DQF Workshop on Modelling Philosophy 22 April 2005 Modelling Incomplete Markets for Long Term Asset Liability Management M A H Dempster Centre for Financial Research Judge Institute of Management University of Cambridge & Cambridge Systems Associates Limited mahd2@cam.ac.uk

2 Outline 1. Introduction 2. Alternative models for incomplete markets 3. Dynamic stochastic programming 4. Structural economic/capital market modelling 5. Case studies 6. Conclusions

3 1. Introduction Experience with a variety of actual applications including:- Long term asset allocation Asset liability management Derivative portfolio pricing and hedging strategies Risk management Capital allocation Real options evaluation Financially hedged logistics operations

4 Financial modelling for fund management What is needed for fund design in these applications? Capacity to perform long-term asset allocation Ability to guarantee returns over long time horizons, in the face of uncertainty about: changing economic and market conditions Need for active risk management changing demographic and actuarial conditions Need for unified asset liability management

5 Gather Data Statistical Analysis of Data Strategic Financial Planning Economic data Market data Econometric Modelling Liabilities model Model returns on investment classes Monte Carlo Simulation Liability forecasts Investment class return forecasts Visualization + Analysis Optimization Model and Fund Objectives and Constraints Dynamic optimization model for assets-liabilities Software generation of model + optimization Investment Decisions Software engineering Risk preferences Investment horizon Visualization + Analysis

6 Strategic asset liability management problem definition Given a set of uncertain asset returns and liabilities, a fixed planning horizon and set of rebalance dates, find the trading strategy that maximizes expected risk adjusted net return subject to the constraints

7 Dynamic stochastic programming (DSP) solution Discrete set of trading times Uncertainty represented by a finite set of scenarios Asset returns may be denominated in different currencies Fund operates from the point of view of one currency called the home currency Fund begins with an initial wealth in the home currency and may be subject to cash injections and withdrawals Fund faces market frictions such as proportional transaction costs and portfolio restrictions such as position limits

8 The real world can surprise us -- sometimes unfortunately with perfect storms

9 2 Alternative Models for Incomplete Markets Consider modelling financial instrument price processes under the real world measure for real i.e. incomplete -- markets where there are always unpriced uncertainties Three approaches have been taken: Use a risk neutral measure model augmented by appropriate unobservable market prices of risk which must be estimated by filtering or Markov chain Monte Carlo and then deleted for derivative pricing Babbs & Nowman (1999) Duffee (2002) Factor based discount factor or pricing kernel estimation for use with cash flow models for corporate modelling and capital allocation Rogers (1997) Smith (1998) Cochrane (2001) Structural models of price returns for asset allocation and asset liability management Wilkie (1986, 1995) Mulvey (1998) Each of these approaches has its advantages, drawbacks and natural time scales Effectiveness of any approach is ultimately an empirical matter

10 3 Dynamic Stochastic Programming Consider a financial planning problem formulated as a canonical linearly constrained dynamic stochastic programming (DSP) problem in recourse form m ax{ f ( x ) + E [m ax ( f ( x ) E [m ax f ( x )])] } 1 1 ω 2 2 ω ω Τ x ω 2 T ω x T x s.t. A x = B x + A x = b a.s B x + A x = b a.s l x u : B x = b a.s. T + 1 l x u a. s. t = 2,..., T + 1 t t t T T b T T + 1 (1) ω E : = { b, f, A, B } fo r t = 2,..., T + 1 t t t t t ω T ω Τ 1 denotes conditional expectation w rt the data process ω h istory ω τ 1 of the

11 Scenario trees Simulated data process paths termed scenarios must be represented in the form of a scenario tree Example 2 2 tree Each data process path through the tree corresponds to a scenario and each node in the tree represents a decision along one or more scenarios

12 Dynamically sampled scenario tree

13 Deterministic equivalent A large scale LP in the linear case with ω t ε Ω a realization of the random vector ω t corresponding to the sample path to a node of the scenario (data path) tree Dantzig (1955) { π : = min c x + p ( ω ) c ( ω ) x ( ω ) + p ( ω ) c ( ω ) x ( ω )] x ω Ω ω Ω T T T... + p ( ω ) c ( ω ) x ( ω ) s.t. A x = ( ω ) ( ω ) 2 ( ω b2 ω ω ω 2 ω + 33 ω 3 ω = 3 ω A x A x Large sparse matrix size increases exponentially with the number of decision times and linearly with the number of scenarios T T ω Ω T T 2 T T T T1 1 T 2 2 TT T T 1 T T T ) = ( ) a.s. A ( ) x A ( ) x ( ) A ( ) x ( ) b ( ) a.s. : :... : A ( ω ) x + A ( ω ) x ( ω ) A ( ω ) x ( ω ) = b ( ω ) a.s. l x u 1 1 t l x ( ω ) u t t t t t ω Ω t = 1,..., T } b

14 Scenario tree generation!! ""! # $%% "

15 Problem generation and solution methods Tractable dynamic stochastic programmes (DSPs) yield convex possibly nonlinear deterministic equivalent problems Problems are generated in a standard (MPS) or stochastic (SMPS) programming format using a modelling system Solution method depends on objective functions Downside-quadratic: nested Benders or interior point Exponential, Power/Log: nested Benders Linear: nested Benders, interior point or simplex

16 Implementation In practice a separatedsp is solved for each trading time and only the first stage solution is implemented since Realized values of variables are unlikely to coincide with simulated values There is an opportunity to update underlying dynamic statistical model at each decision time

17 Fundamentals of the Stochastics system Asset returns Liabilities Scenario Tree Estimation/Calibration Simulation Generation Optimization Sequential Sampling Visualization

18 Visualisation of data, problem & results Stochastics suite StochView Dynamic stochastic programme generator StochOpt StochGen StochSim Simulation of tree of future scenarios Dynamic stochastic programme optimizer (solver) StochLib Library of Stochastics components

19 3 Structural Economic/Capital Market Modelling Global asset return model US Japan Emerging Markets Euro Zone UK

20 Asset return model currency area structure PSBR Nominal GDP Economic Model Zero Yield Curve Price Inflation Real GDP Wages and Salaries Capital Markets Model Cash Fixed Income Equities Currency Exchange Rates Other Areas Pension Liability Models Benefits for DB Contributions for DC and DB

21 Nonlinear capital market model Home currency is USD 2nd order model for US stocks, long rate, short rate, GDP, inflation, wages and public sector borrowing UK stocks, long rate, short rate and UK/US exchange rate EU stocks, long rate, short rate and EU/US exchange rate Japanese stocks, long rate, short rate and JP/US exchange rate Emerging markets stocks and bonds Collection of country models linked by correlated innovation terms and exchange rate equations

22 Basic capital market model Stock index (S) Short term rate (r) Long term rate (l) Exchange rate (x) ds = dt + dz S dr = dt + dz S S S r r r dl = ldt + ldzl dx = dt + dz x x x x The drift µ i and volatility σ i parameters are defined as functions of the state variables and time t as µ ( S, r, l, x, t ) d t σ ( S, r, l, x, t ) d Z and i i dz i (i = S, r, l, x) are increments of correlated Wiener processes. All dependent variables are measured in rates -- proportional returns for S and x and changes (for r and l) t

23 Emerging capital markets model AR(1) model with GARCH(1,1) error structure y = α + α y β u + u t 0 1 t -1 1 t -1 t H = γ + ph qu 2 t t-1 t 1 ut : = H tt y t denotes the return on the stock or bond index and ε t are standard normal random variables correlated with other innovations

24 Asset return statistical model ( ) ( ) µ ϕ ( ) dx = diag x [ x + ΣdZ ] ϕ Σ!! "! #!!! $%&'!

25 Covariance/correlation matrices Covariance/correlation matrix of residuals 93:12 to 01:02 Covariance/correlation matrix of raw returns 93:12 to 01:02

26 Model estimation and simulation Estimated using monthly data (July 1974 February 2004) Estimated using regression/maximum likelihood Simulated using Monte Carlo methods with Gaussian and t innovations

27 Asset returns, exchange rates and economic variable growth rates Scenarios represent uncertain future asset returns exchange rates and economic variable growth rates Scenarios are generated by simulating from the underlying dynamic model of the variable returns 3 types of dynamic models considered Nonlinear model (BMSIM) Vector autoregressive (VAR) models (VARSIM and USMACRO) Historical bootstrap model (HSIM)

28 Comparative statistical testing Standard statistical performance tests compared the in-sample and out-of-sample performance of the alternatives with regard to: Goodness of fit Impulse response analysis Prediction error Superiority to naïve forecasts In all tests the nonlinear models outperformed the VAR and bootstrap models Such tests cannot establish any degree of predictability -- even of the variability of returns without out-of-sample testing of the full system

29 Visualization of scenarios

30 Ten year out-of-sample scenario forecasts

31 Asset return & liability simulation Cascade Model Risk Factor Estimation Macro Forecasts Historical Data Cyclical Patterns & Monetary Policy Sub-Model for Random Shocks Asset Returns & Liabilities Simulation

32 Cascade model for long-term forecasting Macro Forecasts Forecasting GDP and Inflation Analysing trend and cycle components and structural breaks Extrapolating long-term steady state equilibrium growth Cyclical Patterns & Monetary Policy Generation of scenarios for business-cycle sensitive variables e.g. slope of yield curve, investment cycle, inventories. Additional variables may be added e.g. oil price forecasts, commodities etc. Asset Returns& Liabilities Simulation Generation of trajectories for Equity, Credit spreads, FX and all liabilities

33 5 Case Studies I Optimal Dynamic Asset Allocation Fixed planning horizon: 3, 5, 10,, 40 years Portfolio rebalance dates: quarterly, semi-annually, annually, Dynamic investment strategy maximizes risk adjusted fund wealth subject to constraints such as position and loss limits maximize E[ u( w( x))] subject to Ax b Here u is a utility function representing fund risk attitude and x is a decision process representing the portfolio composition at each rebalance date in each scenario subject to the data (A,b) representing the constraints

34 Utility functions Utility functions are normally increasing to capture the investor s preference for a higher terminal wealth and concave to capture the investor s risk averse attitude the greater the curvature the greater the aversion to risk Exponential Power Log Downside quadratic Variance u( w) = e aw u( w) 1 = w a u( w) = log( w) a 2 u( w) = (1 a) w a( w tw) u( w) = (1 a) w a( w tw) 2 Linear u( w) = w

35 Cash balance Inventory balance Wealth definition Accounting constraints i I + p ( ω)( gx ( ω) fx ( ω)) = 0 it it it x ( ω) = x ( ω)(1 + v ( ω)) + x ( ω) x ( ω) + it it 1 it it it = w ( ω) p ( ω) x ( ω) t it it i I Non-negativity x ( ω), x ( ω) 0 + it it

36 Portfolio constraints Shorting/borrowing limits Position limits Turnover constraints Solvency constraints p ( ω) x ( ω) x it it i p ( ω ) x ( ω ) φ w ( ω ) it it i t p ( ω ) x ( ω ) p ( ω ) x ( ω ) α w ( ω ) it it it 1 it 1 i t w ( ω) 0 t

37 System asset allocation backtests Viewpoint of US dollar investor All portfolio rebalances subject to 2% transaction value tax (Eire) Monthly data available Three historical out-of-sample periods Period Length S&P500 Return Asset Return Model Rebalance Frequency years 7.41% p.a. 3 areas (ex Japan) Annual years 14.28% p.a. 4 areas Annual years 0% p.a. 4 areas 4 areas + emerging markets Semi-annual 2.5 years -2.30% p.a. 4 areas 4 areas + emerging markets Semi-annual Various fund objectives and attitudes to downside risk In all historical backtests system outperformed S&P500 by up to 10% p.a. All system returns were positive even through the recent high tech crash!

38 Initial Estimation Period Out-ofsample Period Length Asset Return Model Simulator Number of Scenarios k Rebalance Frequency Risk Management Criterion Horizon Constraint Annualised Return % (see Section 5.4) S&P 500 Benchmark Annualised Return % T1 T2 T years 3 areas (ex Japan) BMSIM 4 annual terminal telescoping years 4 areas BMSIM 4 annual terminal telescoping years 4 areas VARSIM 4 annual terminal telescoping years 4 areas BMSIM 8.2 semi-annual terminal telescoping years above + emerging markets BMSIM 8.2 semi-annual terminal telescoping years above + US economy BMSIM 8.2 semi-annual terminal telescoping years 4 areas VARSIM 8.2 semi-annual terminal telescoping years 4 areas BMSIM 8.2 annual all periods telescoping years 4 areas VARSIM 8.2 annual all periods telescoping years 4 areas HSIM 8.2 annual all periods telescoping years 4 areas VARSIM 8.2 annual all periods 5-year rolling # $ % $ & $

39 The pension gap problem There are different ways to solve this problem: A. Structured funds with a guaranteed return on investment B. Individual asset liability management C. New products for retirement Modelling individual liabilities and net cash flows is different from modelling at the pension fund or insurance company level higher degrees of uncertainty than at pension/insurance level are involved!

40 New products for retirement More research is needed to understand the stochastic liabilities Guaranteed fund products a first step towards the construction of a range of retirement products Dempster et al (2003) Alternatives requiring similar analysis are guaranteed annuity options Wilkie et al (2003, 2004) Boyle & Hardy (2003)

41 II Structured Funds with a Guaranteed Return on Investment This approach involves the design of a fund (fund of funds) with guarantees which are carefully matched to the risk profiles of the specified class of individual investors the guarantees are sufficient to cover the aggregated investors liabilities and their consumption

42 Wealth distribution of a Sharpe ratio maximising portfolio Goetzmann et al. (2002) Source: H Till & J Eagleeye, Quantitative Finance 3 (2003) C42-48

43 Risk management and capital guarantees max E[ β return (1 β ) risk] Terminal wealth distribution (from scenario tree) Portfolios are penalized for each scenario in which they under-perform relative to the target Target VaR Terminal wealth (as proportion of initial wealth)

44 Alternative to probability constraints H ( ) total ( ) max 0, ( ) ( ) h ω = L ω W ω ω Ω t T t t t ( ) m a x h ( ) ω = ω ω Ω t T to ta l t ( m a x ( ) 0 ) to ta l t P h > α t T

45 Alternative to probability constraints max β p ω W ω β p ( ω ) H ( ω ) ( 1 ) ( ) ( ) t + + x, ( ),, ( ),, ( ): d t a ω xt a ω xt a ω ω Ω t T { T} ω Ω d a A, ω Ω, t T { T} β

46 Graphical representation of scenarios t=0 t=1/4 t=1/2 t=3/4 t=1 t=5/4 t=3/2 t=7/4 t=2 s=1 s=2 s=3

47 Portfolio Allocations 1y 2y 3y 4y 5y 10y 30y Stock Jan Jan Jan Jan Jan

48 160 Backtest 99-04: = 8192 scenarios Expected Maximum Shortfall Jan-99 1-Jan Dec Dec Dec Dec-03 barrier EMS EMS MC eurostoxx 50

49 Monte Carlo parameters Optimization parameters Monte Carlo expected returns Results

50 III Individual Asset Liability Management Classic life cycle models cannot explain the individual behaviour of all investors

51 Individual asset liability management Modelling individual liabilities and net cash flows involves data gathering on individual households of different age groups and wealth with given income, liabilities and certain goals over life Classical consumption investment models Merton(1974) Dybvig(2005) far from capturing real life complexity Terminal Value Comparison % 10.7% % 5.4% % Wealth Age

52 Dynamic asset allocation Asset allocation should switch into less risky assets before a major liability to ensure that the investor has enough to pay the liability in adverse market conditions In this model investment assets cannot be sold down between rebalance points to meet liabilities so that cash must be kept to cover any liabilities occurring before the next rebalance point 2000 Liability 2026 Liability

53 Scenario clipping events Average for each tree of the percentage failure of K-S test at 5% level for 100 seeds percentage

54 Rare events Test for the difference of mean number of occurrences of events from zero number of std devs

55 The Markowitz investor E ffic ient Frontier: porfolio/return s pac e Efficient frontier: return/risk space 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% M LG ovt M LHY E A FE r2k r1k c as h Er/% Borrow Dynamic planner can take achieve a higher return without borrowing 0% Eσ We calculate a market portfolio based on risky assets and cash This portfolio is implemented at each rebalance point Between rebalances we allow deviation from this portfolio within a narrow band

56 7. Conclusions Strategic ALM and tactical risk systems for funds and individuals are a reality today Multiperiod model yields multiple advantages Robust portfolios in the face of dynamic uncertainty Significantly outperforms single period buy-and-hold models Best, worst and VaR limited what-if portfolios views available Forewarned is forearmed! All business structures and regulatory constraints can be accurately modelled Models involving millions of equations and variables can be solved in minutes on PCs Flexibility and visualization are the keys to effective decision support for strategic fund management