watsonwyatt.com Actuarial Teachers and Researchers Conference 2008 Investment Risk Management in the Tails of the Distributions Chris Sutton 3 rd July 2008
Agenda Brief outline of current quantitative practices Weaknesses of these approaches Two key questions and two areas of research How to improve the risk measures Adding qualitative to quantitative
Defining Moments* Research Looking out over the future of the pension and investment industry using a complexity model We are seeing/ we will see a series of defining moments Absolute return 2003/ end of bear market Governance 2005/ Philips Pension fund Risk / Black swan 2007/ Goldman 25 sigma Regulation 2009 and after/ SOx for banking/ finance Defining Moments One small event but with massive impact The world is different afterwards but this only emerges with time 20 th Century Defining Moment - Wright Brothers first flight * Defining Moments, Watson Wyatt Thinking Ahead Group, June 2008
Moment Based Risk Measures Most asset managers and owners live in a Markowitz mean-variance world and use risk measures that assume stationary, normal distributions and stable covariance matrices: Tracking error: the standard deviation of the differences in return between a portfolio and its benchmark (or risk free rate) Value at Risk (VaR): the maximum potential loss that a portfolio could suffer over a pre-specified horizon with a concrete probability of occurrence Three main VaR calculation methodologies: Delta Normal Historic simulation Monte Carlo
Three elements in definition of VaR Variance VaR Frequency t n + 0 Value Horizon t o Source: Mag.Eugen PUSCHKARSKI, Riskmeasurement and decomposition.
Good things about VaR VaR has gained broad acceptance by regulators, investors and management teams in recent years because it can be expressed in money terms, and hence consistently calculates the risk arising from the short or long positions, and from different securities. For example, investors have traditionally denoted the risk of a stock by beta and risk of a bond by duration. An advantage of expressing VaR in money terms is that one can compare or combine risk across different securities. VaR can be used for a security, a portfolio, a trading desk, or a balance sheet.
Agenda Brief outline of current quantitative practices Weaknesses of these approaches Two key questions and two areas of research How to improve the risk measures Adding qualitative to quantitative
Not so good things about VaR Many applications required normal (or lognormal) underlying loss distributions Probability distributions for many assets exhibit fatter tails than the normal distribution Historic data is usually required, but is not always available especially at extremes so estimation error is high Models are often highly sensitive to variance/covariance assumptions When things go wrong, VaR has nothing to say about by how much this might be. VaR is not a coherent risk measure Which in turn means that VaR can be manipulated!
Coherent Measures of Risk VaR has been criticised on the grounds of its poor aggregation properties. It is not a coherent risk measure. In particular, it violates the property of subadditivity. A coherent risk measure is a risk measure ρ that satisfies properties of monotonicity, subadditivity, homogeneity, and translational invariance. Monotonicity: ρ(x) ρ(y) whenever X Y When comparing two risky assets with the first always have a larger payoff than the second, it is obvious that its potential losses and therefore its risk are smaller. Subadditivity: ρ(x1 + X2) ρ(x1)+ ρ(x2) The risk of a diversified portfolio must not exceed those of its components. Positive homogeneity: ρ(λx) = λρ(x) wheneverλ 0 There is no diversification effect when only the scale of investment is changed. Therefore, the risk should increase by the same scale as the investment Translation invariance: ρ(x + a) = ρ(x) a The first investment comprises one of the risky assets and an investment into he riskless asset, while the second investment only goes into the same risky asset. The desired risk measure should ensure that the risk of first investment is smaller because the potential losses, reduced by the certain profit from riskless asset, of first investment should always be less than the second one In contrast Expected Shortfall (ES), Weighted Value at Risk and Beta Value at Risk are coherent risk measures.
Correlations Mean-variance optimisation is widely used which relies heavily on correlations / covariance matrix VaR calculations use estimates of correlation across assets Explicitly in Delta Normal and Monte Carlo Implicitly in Historic simulations History shows though that correlations between asset classes are nonstationary, varying enormously over time (sometimes almost between +1 and -1) Point estimate correlations between assets are therefore almost useless in asset management Though correlations between e.g. an asset and economic conditions may show greater stability
Agenda Brief outline of current quantitative practices Weaknesses of these approaches Two key questions and two areas of research How to improve the risk measures Adding qualitative to quantitative
Diving in Leads to 2 important questions Q1: what do things look like when you are in the tails? Q2: how reliable are the models there anyway?
Where we are Emerging Leads to 2 areas for research A: improving risk measures B: adding qualitative to quantitative risk management
Agenda Brief outline of current quantitative practices Weaknesses of these approaches Two key questions and two areas of research How to improve the risk measures Adding qualitative to quantitative
What can be done to improve on VaR? Improve the input Employ fat-tailed distribution e.g. Student-t, Levy Use a GARCH family to better measure the volatility Extreme Value Theory (EVT) to better model the tail Extend the risk measures Expected Shortfall (ES) Weighted Value at Risk (WVaR) Beta Value at Risk (BVaR)
Improving the inputs: Fat Tails Probability distributions for asset returns often exhibit fatter tails than the standard normal distribution. Look at distributions with excess kurtosis such as Student-t
GARCH Models GARCH ( Generalized Autoregressive Conditional Heteroskedasticity) models are designed to capture certain characteristics that are commonly associated with financial time series: Fat tails Volatility clustering A GARCH model proposes that the volatility is a function for both past squared errors and past conditional variances, which can effectively fit the financial time series with volatilities tending to cluster.
Extreme Value Theory VaR is used to characterize losses up to a cut-off in the left tail of the loss distribution, but does not provide any additional information on the magnitude of loss beyond the threshold level. By offering a potential solution to this problem, EVT is gaining popularity. Because extreme events happen with small probability, we have to operate with very small data sets, and this means that our risk measures and the probabilities associated with them, are inevitably very imprecise. According to EVT, the normal distribution is the least conservative model of tail risk.
Extreme Value Theory The most traditional models are the block maxima models, which are models for the largest observations collected from successive periods of identically distributed observations. A more modern and powerful group of models are the peaks-overthreshold (POT) models, which are models for all large observations which exceed a high threshold. The POT models are generally considered to be the most useful for practical applications, due to their more efficient use of the data on extreme value.
Copula Interdependence of returns of two or more assets is usually calculated using the correlation coefficient. However, correlation only works well with normal distributions, while distributions in financial markets are mostly skewed. Furthermore, in time of crisis, correlations increase (sometimes substantially) and strategies that rely on normal correlations fall apart. The copula, therefore, has been applied to areas of finance susceptible to fat tails such as option pricing, portfolio VaR, and credit risk. Copula is a statistical measure that represents a multivariate uniform distribution, which examines the association or dependence between many variables.
Extended measures: Expected Shortfall Artzner et al(1997) proposes the use of Expected Shortfall (ES), also known as Conditional VaR (C-Var) or tail VaR or Tail-Conditional Expectation (TCE), to alleviate the problems inherent in VaR. ES is the conditional expectation of loss given that the loss is beyond the VaR level. ESα(L ) = E[L L VaRα(X )] ES measures how much one can lose on average in states beyond the VaR level. ES is a coherent risk measure.
Comparing VaR & Expected Shortfall VaR is designed to answer the question of what is the minimum loss incurred in the α% worst cases of the investment. It is indifferent of how serious the losses beyond that threshold of the possible α% losses Expected shortfall can answer the modified question what is the expected loss incurred in the α% worst cases of the investment. ES is always higher and more conservative than VaR. Source: Yasuhiro Yamai and Toshinao Yoshiba, On the Validity of Value-at-risk: Comparative Amalyses with Expected Shortfall
Merits and Limitations of VaR / ES Strength Weakness Value at Risk Related to the firm s own default probability. Easily applied to back testing. Established as the standard risk measure and equipped with sufficient infrastructure (including software and systems). Not in general a coherent risk measure. Pays no attention to magnitude of losses beyond VaR value (tail risk). Difficult to apply to portfolio optimizations when VaR is calculated by simulation. Expected Shortfall Able to consider loss beyond the VaR level. Less likely to give perverse incentives to investors. Sub-additive. Easily applied to portfolio optimizations. Not related to the firm s own default probability. Not easily applied to efficient back testing method Insufficient infrastructure (including software and systems) Effectiveness depends too much on estimation of the tail of distribution. Source: Yasuhiro Yamai and Toshinao Yoshiba, On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall
Weighted and Beta Value at Risk Loss Loss Source: Watson Wyatt qα qα The two distributions have the same α-tails (α is fixed), so that ES is the same for them. However, the distribution at the left is clearly better than the right one, because the expected loss of right distribution is higher. Sometimes we need more information about the distribution of outcomes than we can get from simply looking at the tail risk. Weighted and Beta Value at Risk consider the whole distribution.
Weighted Value at Risk Let u be a probability measure on (0,1]. Weighted Value at Risk (WVaR) with the weighting measure u can be defined in the following way: W VaR = E( L L (0,1] qα ( L)) u( dα ) where q α is the α quantile risk measure for loss L. q α ( L) = inf{ L F ( L) α} L Weighted Value at Risk is coherent risk measure.
Beta Value at Risk Beta VaR (BVaR) measures risk by the expectation of the average of the β biggest of α, independent copies of random loss, where α 1 and α β. Beta VaR takes the whole distribution into account. Higher α increases the risk measure because we consider the biggest of more copies of L. Thus, risk average is increased. Higher β reduces the risk measure as a bigger number of lower losses enter the averaging. Then Beta Value at Risk can be defined as B VaR ( L) = E ( 1 β = L 1 ( : α ) ) i i β Beta VaR can be viewed as a special case of weighted VaR.
Agenda Brief outline of current quantitative practices Weaknesses of these approaches Two key questions and two areas of research How to improve the risk measures Adding qualitative to quantitative
Credit crunch contagion Sub-prime/ credit crunch explained by: Credit super-cycle Markets over regulators 3-coms issue excess competition, complexity and compensation Over-competition for 10%+ returns Leverage the master not the servant Over-complexity in the chain Securitisation and opaque derivatives Over-compensation Too many not paid fairly for their work Systemic human-induced problems Frequent and prolonged crises Organisations self-destruct Black Swans The Black Swan By Taleb A defining moment risk redux
Governance primer Challenge to the current governance model is it fit for purpose? Most references to the governance of funds focus on basic issues of good practice where there s been progress doing things right These don t address the value creating goals of funds where the governance gap seems to be getting wider - doing the right things The WW/ Oxford research identified factors that could move a fund from good to great 600 pound gorilla in the room subject Clark/ Urwin Study* of top 10 global funds cherry-picked by results/ reputation Their governance had parallels and differences to investment firms They shared 12 best-practice factors to their performance success * Clark and Urwin: Best-practice investment. October 2007
Risk management process Real-time investment over calendar-time handled by CIO function Risk management depth and breadth Risk budget as the central discipline Dealing with Black Swans Agent sustainability factors Quant -Known Unknowns -Model based - Optimisation - Power laws Qual -Unknown Unknowns -Scenarios -Diversity - Black Swans With acknowledgements to Accenture Risk dashboard
Example Risk Dashboard KPI measures Overall assessment of AL risk Markets downside (beta) Markets unknown downside Macro factors risk Manager risk (TE) Manager risk (Soft risk) Funding status Covenant Unrewarded liability risks Operational/ counterparty Longevity/ Benefits/ Risk dashboard content Progress vs KPIs and key experienced variances from plan Belief structure based measures Risk budget model Include core risk return driver exposures Include CVaRs - 50 and 95 Belief structure hierarchy with macro view from CIO/managers CVaR - 50 and 95 Scenarios and stress tests Risk management preparation/ black swan section Impacts of secular big picture issues on downside/ upside Experienced TE and budgeted TE Organisational and sustainability issues Risk taking opportunity or constraint in the journey plan Risk taking opportunity or constraint in the journey plan Current position; interest rate/ inflation shock exposures Overview and report by exception Overview and report by exception
Which version of Risk do you want to use? Risk v1 Look back and extrapolate Calculate VaR from history Stationary distributions with constant means and volatilities Normal distribution of returns Risk comes from outside shocks All risks can be modelled Risk v2 Look back and forward Stress test. Jumps Model VaR forward-looking basis Estimated Shortfall / CVaR Non-stationary distributions GARCH Models Fat tails. Extreme Value Theory Risk includes players actions Some risks cannot be modelled Black Swans
Contacts and Limitations of Reliance Contact details Chris Sutton Thinking Ahead Group Watson Wyatt Investment Consulting +44 1737 241144 christopher.sutton@watsonwyatt.com Limitations In preparing this presentation at times we have relied upon data supplied to us by third parties. While reasonable care has been taken to gauge the reliability of this data, this presentation therefore carries no guarantee of accuracy or completeness and Watson Wyatt cannot be held accountable for the misrepresentation of data by third parties involved. This presentation is based on information available to Watson Wyatt at the date given and takes no account of subsequent developments after that date. It may not be modified or provided to any other party without Watson Wyatt s prior written permission. It may also not be disclosed to any other party without Watson Wyatt s prior written permission except as may be required by law. In the absence of our express written agreement to the contrary, Watson Wyatt accepts no responsibility for any consequences arising from any third party relying on this presentation or the opinions we have expressed. This presenation is not intended by Watson Wyatt to form a basis of any decision by a third party to do or omit to do anything.