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Business School

ACTL4303 AND ACTL5303 ASSET LIABILITY MANAGEMENT

Week 10 Asset Liability Models

Dynamic Financial Analysis (DFA) applies financial models to analyse pricing, reserving, and capital adequacy in insurance (and similar issues in superannuation)

Three basic components

1.  Economic Scenario Generator (ESG)

2.  Liability Projection Model (usually dependent on bond yields and inflation from ESG)

3.  Decision Making Model (analysis of key variables)

Stochastic Asset and Liability Models

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Simulate the economic and financial environment globally including for example: •  Long term interest rates •  Short term interest rates •  Stock market returns •  Real estate returns •  Inflation (consumer, medical, wages) •  Exchange rates

The objective of an ESG is not prediction or asset pricing

It is to enable probabilistic conclusions by simulating a realistic distribution of outcomes

Economic Scenario Generators (1)

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Common design variants are: •  Cascade models. Variables depend only on lagged values of

themselves and variables of higher order, usually incorporating some form of longer term mean reversion to equilibrium.

•  Efficient market models. Often derived from stochastic differential equations involving Weiner processes (time independent).

•  Vector Auto-Regressive Models. These are non-prescriptive about the structure of relationships and contemplate a complex web of autoregressive relationships, prone to unstable estimation

Economic Scenario Generators (2)

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In evaluating an ESG, consider:

•  What is the purpose? (eg Long-term strategy assessment or short-term solvency tests)

•  Critically, evaluate the theory and thinking behind the model structure

•  Is the model calibrated on a relevant dataset (eg inflation since inflation targeting monetary policy regime)?

•  How does the model contemplate extreme equity behavior?

•  Are the linkages between asset classes and inflation reasonable and practical?

•  Is the model hostage to theory and in contradiction of data or vice versa?

Economic Scenario Generators (3)

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The Wilkie Model ( a cascade example)

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•  This is the original actuarial ESG

•  While widely criticised, Wilkie made the model development public, provoking much constructive discussion and examination

•  The focus of this model was originally on the maturity guarantees offered by life insurance offices (very long term)

•  It was not intended, and arguably inappropriate, for short-term applications involving tail probabilities

The Wilkie Model (1986, 1995)

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•  Assumes a single inflation regime

•  Dividend yield depends on prior values and inflation

•  Dividend growth depends on current and past inflation, past dividend yield shocks and a moving average of past values

•  Share prices derive from dividend growth and dividend yield

•  The equity structure contradicts a random walk of stock prices

•  Long term interest rates have an inflation component (exponentially weighted moving average) and an inflation-adjusted yield

•  Short-term interest rates modeled as a spread from long-term rates

The Wilkie Model (1986, 1995)

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Model Evolution post Wilkie

•  Frank Russell Stochastic Programming Models (1991) •  CAP:Link was an early Towers Perrin ESG (1996,2006) •  Aon Consulting Model 1999 •  Casualty Actuarial Society (CAS) and the Society of

Actuaries (SOA) made a model publicly available in 2005 •  American Academy of Actuaries (AAA) also supplies

scenarios for testing capital adequacy, particularly where variables annuities are involved

•  Following the merger between Towers Perrin and Watson Wyatt CAP:Link is now defunct

•  Towers Watson’s current model is Star ESG (2008)

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CAP:Link Economic Scenario Generator - Cascade structure (1996)

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CAP:Link subsequent evolution (2006)

Short and Long Interest Rates and Full Government Yield Curve

Price Inflation

Earnings Yield Earnings Growth

Real GDP

Currency Strength

Credit Spreads

Note emphasis on yield curve, incorporation of GDP, inflation shift to lower in the cascade and change from dividends to earnings to model equity returns

Towers Watson Star ESG (2008) •  Nominal interest rates The Norman (2009) Real World Extension of the Libor

Market Model. This model is also used, with minor adjustments, to model real yields and LIBOR spreads

•  Inflation Linear function of current and past short-term interest rates

and inflation Error term is AR(1) with conditional heteroskedasticity to

give higher volatility in higher inflation environments •  Equity returns Geometric Brownian motion with jumps •  Linkage via Student’s T Copula to address tail correlations

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Towers Watson Star ESG – CAP:Link successor

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Towers Watson Star ESG Inflation influenced by monetary policy

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Towers Watson Star ESG Equity Returns – no longer dividends or earnings

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Inflation modelling (1) The CAS-SOA (2005) model is similar to Wilkie (1986)

where is inflation in period t is the mean reversion parameter is the mean reversion level of inflation is a normal random variate N(0,1) is the volatility of inflation

Is the real-world inflation process stationary?

t+1q =tq + qk qµ −

tq( )Δt + tε qσ Δt

tqqkqµtεqσ

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Inflation modelling (2) The Star ESG (2008) recognises that the volatility of inflation is autoregressive and dependent on the level of inflation. Inflation is related to its prior value, the level of short-term interest rates and the recent change in interest rates:

where is the short-term interest rate is the error term is a normal random variate

Monetary policy drives inflation in this model, consistent with the Taylor rule

tq =αt−1q +β tr +γ tr − t−1r( )+δ + te

trte

te = p t−1e +σt−1

2q +2

ξ tε

tε

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Equity return modelling (1) •  Early long-term ESG models (eg Wilkie, CAP-Link)

separately projected equity income (earnings or dividends), and the multiple applied to that income (dividend yield or earnings yield)

•  As the focus of ESGs shifted to the short term, there was more attention to modelling extreme market behaviour

•  Regime switching models are a simple and effective way of mixing distributions for different market environments

•  They accord with the traditional perspective that bear markets have different return and risk characteristics

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•  Rather than one pair of parameters (µ, σ) to describe the return distribution, we think of a different distribution for each regime

•  Example Regime 1 : Positive expected return, moderate volatility Regime 2 : Negative expected return, high volatility

•  Markov switching between regimes – probability of changing regime depends only on the current regime, not history

•  Use maximum likelihood to estimate (µ, σ) for each regime and the two probabilities of remaining in each

Regime-switching models

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Equity return modelling (2) •  The CAS-SOA model uses a two regime log-normal model

structure with different parameters for large and small stocks

•  Regime switches for large and small stocks are correlated (natural extension for global equities)

•  Correlation of regime states (markets being in bear markets at the same time) will mimic the tendency for correlations to increase when markets are weak

•  Another way of allowing for extreme volatility is a jump model (Star-ESG)

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•  Assumed relationships between inflation and asset returns need to be carefully scrutinised

•  An invalid relationship may still find empirical support because of peculiarities of the period over which the model is calibrated

•  In a small open economy currency fluctuations, and policy response, add to the complexity of relationships

•  If an asset liability model incorporates a positive relationship between inflation and risky assets this will justify higher weightings to risky assets

•  The reliability of any such assumption needs to be well understood

Inflation and asset returns

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•  Bond yields price inflation expectations over the term of the bond

•  If inflation targeting is credible, variations in short-term inflation will affect the slope, moreso than level, of the yield curve (Taylor Rule, Nelson-Siegel)

•  Estimation of bond yield relationships is conditional on the monetary policy regime, and its perceived effectiveness

•  A supply-side shock (eg energy or food prices) might shake confidence in inflation targeting

Inflation and bond yields

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•  In a low inflation regime the equity market will be relatively insensitive to modest fluctuations in the inflation rate

•  In the transition to a high inflation regime bond yields will move higher and this may devalue equities, though not necessarily

•  If a high inflation regime is due to cost-push inflation, profit margins will compress and equity returns may suffer

•  If a high inflation regime is driven by a positive output gap (demand pull), profit margins may expand and equity returns may benefit

•  There is no clear connection between the inflation environment and equity returns in the short term

•  If inflation is due to currency weakness the relationship is more nuanced

Inflation and equity returns

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•  A weaker currency will increase the price of imported goods but the impact on consumer prices may be subdued and gradual

•  The currency may be weak due to inferior terms of trade and consequent weak demand

•  A weaker currency assists exporters and import competing companies •  Higher inflation arising from more expensive tradables may therefore

coincide with improved equity returns as domestic financial conditions ease

•  A weaker currency also improves the returns from unhedged overseas equity investment

Currency and equity returns

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•  Model risk arises from the model being an approximation to a complex economic system

•  Parameter risk arises from treating estimated parameters as the actual parameter values

•  Given model risk and parameter risk the uncertainty of outcomes is potentially wider than suggested by a model

•  ‘ I know my model is probably incomplete, and even if it is appropriate, the parameter estimation is inaccurate. The future may surprise relative to simulated distribution.’

Model risk and parameter risk

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•  We can imagine a labyrinth of relationships moving beyond inflation and asset returns and encompassing a wide range of multi-country dynamics including policy response scenarios and economic shocks.

•  Very quickly the complexity of an imaginative model will overwhelm the statistical significance of estimation

•  There will also be questions about how the future may be structurally different to the data span of estimation

•  Understanding the inevitable shortcomings and inherent biases of an asset liability model is as important as the design of the model

Asset liability model simplification

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Market Consistency •  Generally economic scenario generators are not too

concerned with current market consistency (eg volatility of equity returns aligning with implied volatility of index options) as horizon lengthens

•  Instead calibration focuses on consistency with how markets behave over time

•  For example current bond yields are low compared to history and equilibrium real yields so the ESG may contemplate them drifting upward over time

•  An ESG is attempting to be realistic over a medium term horizon, not just the next year

•  Focus is on simulating the future, not mimicking the past

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Calibration •  Once a model structure is resolved the parameters are

adjusted so that equilibrium levels of inflation, real cash rates, yield curve slope, corporate bond spreads, and equity risk premiums etc are reasonable

•  Because of restricted data availability and changing economic structures, estimation for some elements may be based on a relatively short period (eg 20 years)

•  Is the frequency and severity of equity bear markets reasonable compared to long run history? What is the frequency of yield curve inversions? Do the sequences of market returns seem plausible? Does inflation turn negative

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Summary •  Understand the features of any model structure and

whether it is appropriate for the task at hand, especially the horizon

•  A model with fewer assumed linkages can be carefully considered and is likely to be more robust than an elaborate model

•  The calibration of the model is as important as structural features and should reflect a broad perspective on market behaviour

•  Does recent history of the current market environment show enough variation (eg inflation)?

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