Risk Indicators in the equity market

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X workshop on Quantitative Finance Politecnico
di Milano, 29-30.1.2009
Risk Indicators in the equity market
giorgio.consigli_at_unibg.it Joint work with L.
MacLean, Y. Zhao and W.T. Ziemba
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• Financial instability
• The US equity market
• The asset pricing model
• Parameter estimation
• Market evidence
• Conclusions and future research

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1. Financial instability

• In 1996 FED Chairman Alan Greenspan postulated
  the long term tendency of stock market yields to
  fluctutate around the 10 year Treasury rate.
• Persistent divergence from this underlying
  process was then to be intepreted as a signal of
  either over or undervaluation of the equity
  market.
• This idea translates into an extremely practical
  and easy relative value principle for investment
  decisions at strategic level
• The presence of a bubble can in this setting be
  detected by a departure of the market index from
  its theoretical value determined, given current
  earning expectation, by the 10 year rate.

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Irrational exuberance

• Greenspan speech came after the 1987 WS crisis
  and shortly before the 1997 crisis and the 2000
  dot.com crisis.
• At the end of 1998, a widespread instability
  affected the Hedge fund industry, due to
  speculative international strategies across
  equity and bond markets.
• A speculative bubble also drove the surge and
  fall of far-east markets resulting in the early
  90s series of crises and the 1995 crisis in
  Japan.
• The list might continue and motivates this work,
  in which we propose a stochastic model for equity
  and bond returns, that under certain conditions
  is able to capture a growing, yet unexpressed
  source of instability

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Irrational exuberance
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Irrational exuberance
U(t)-r(t)
U(t)-r(t)
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Irrational exuberance
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The implied volatility index (VIX, CBOT)

• Since Jan 1990, the CBOT quotes daily estimates
  of an aggregate measure of implied volatility om
  ATM 30 day options on SP500.
• The index reflects agents expectations on forward
  (forthcoming) market movements and provides a gap
  measure between historical and forward equity
  returns
• According to the structural approach to credit
  risk, implied volatility is also a key variable
  to assess the credit cycle and provides a direct
  signal of market uncertainty over future
  corporate earnings
• We propose in the model this additional risk
  factor as driver, warning signal, of forthcoming
  market instability

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The implied volatility index (VIX, CBOT)
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Instability source

• We build on these ideas and propose an approach
  to risk control relying on a market model with
  endogenous instability factors.
• We focus primarily on the equity market. Bonds
  and cash complete the market model
• Price movements are defined by GBM for bonds and
  GBM plus a marked point process for stocks
• We wish to link the behaviour of the point
  process to the introduced instability factors

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2. The asset pricing model

• We consider a market including a cash account,
  the SP500 index and the 10 year Treasury note.
• The stock and bond prices are random processes
  defined in an appropriate probability space
  representing the uncertain market
  dynamics.
• The bond-stock yield differential and the VIX
  process may determine a departure of market
  values from a theoretical value.
• We use the following notation

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Asset pricing

• The dynamics of price movements are defined by a
  Wiener process for bonds and a Cox process for
  stocks. Let
• We capture the equity and bond correlation and
  the dependence of the equity process on the risk
  factors directly introducing a model with random
  drifts
• We assume that volatilies remain constant over
  time while

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Asset pricing

• The risk process dynamics for market instability
  are captured by dR
• We separate positive from negative shocks and
  employ a threshold regression method to estimate
  the significance of each risk factor
• Market stress is defined through a discordance
  measure

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Asset pricing (ctd)

• Shock intensities are assumed to depend
  monotonically on the stress generated by the risk
  factors
• An increasing intensity implies a Weibull process
  so that follows a Weibull distribution with
  density for i1,2 (up and down shocks
  respectively)

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Asset pricing (ctd)

• The distinguishing feature of the asset pricing
  model is the risk process
• The parameters of the market processes are
• It is assumed that the risk factors characterize
  market stress, which in turn affects shocks to
  equity prices through the model parameters

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3. Parameters estimation

• The estimation methodology employs a threshold
  regression methodology.
• A shock sequence is initially assumed relying on
  excesses beyond a pair of positive and negative
  threshold
• Given the shock sequence, conditional ML is
  performed
• Then the shock sequence is updated
• For every shock sequence the dependence on the
  stress factors is directly evaluated
• The procedure stops when the loglikelihood
  function is maximised for the given shock sequence

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distributions (ctd)

• Consider
• The conditional distribution of the increments
  given the jump sequence is bivariate normal with
  density

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parameter estimation (ctd)

• Again given the jump sequence it is
  straightforward to estimate
• The likelihood for given the jump sequence
  and actual shock times

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parameters estimation

• The method proceeds as follows
• 1. Set stress weights and calculate stress values
• 2. Calculate the empirical distributions for
  positive and negative shocks over the sample
  period
• 3. Specify a grid size, an initial tail area and
  a step, identify positive and negative shock
  times
• 4. Calculate for the given shock sequence the
  cond ML coefficients
• Back to 3 until the best shock sequence is
  identified
• The diffusion and jum size parameters are
  estimated by maximizing the loglikelihood
• The Weibull parameters are estimated from

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4. Computational evidence

• We present now estimation results and test two
  market hypotheses underlying the model
• Parameter estimation is based on the described ML
  estimation conditioning on ar given jump
  sequence.
• Starting from an initial 1 excess with steps
  0.05 we span the tail area

Diffusion parameters
Risk process parameters
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Results

• Given these estimates we can perform a forecast
  experiment over the entire sample 1990-2007
• Starting from January 1, 1990, the
  predicted/expected increments were calculated
  over the subsequent time peirod as

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Results

• We have simulated out of sample 2000 daily
  trajectories for the estimated Cox process with
  shock magnitude and frequency explicitly
  dependent on the BSYR over the 20 years.

SP500 (blue)
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computational evidence

• In most cases the fits are good. The weights
  which gave the best fit, w0 (full VIX) are the
  same for positive and negative shocks
• In many time points the BSYD is closer to actual
  price dynamics
• The best convex combination is given by w0.75

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Shocks probability
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Results Tests on market hypotheses over the
entrie sample

• Risk premium
• We estimate the SP implicit risk premium by
  testing the FED equilibrium condition the null
  hypothesis
• is rejected at the 95 with a difference of
  6,78 interpreted as a constant risk premium in
  the market
• Bubble
• The shock driver can very well be associated with
  risk sources other than the two here considered
• The logikelihood ratio test for the cox process
  is significant on the 99 confidence interval
• 2(A.loglikelihood B.loglikelihood) 124.19 gt
  6.6349, X2(1) with 99 confidence

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Conclusions and further work

• The presented market model integrates common
  practitioners beliefs within a satisfactory
  analytical framework
• The Cox process instantiates an endogenous source
  of instability and a novel estimation procedure
  has been implemented with the reported results
• We will then extend the analysis to other markets
  (Nasdaq, Eurostoxx, etc.)
• The solution of the associated stochastic control
  problem with alternative risk-return payoffs will
  follow

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References

• Consigli, G., 2002. Tail estimation and
  mean-variance portfolio selection in markets
  subject to financial instability. Journal of
  Banking and Finance 267, 1355-1382
• Koivu, M., Pennanen, T., Ziemba, W.T., 2005,
  Cointegration of the Fed model, Finance Research
  Letters 2, 248-259.
• K.Berge, G.Consigli and W.T.Ziemba (2008). The
  Predictive Ability of the bond-stock earnings
  yield differential in relation to the Equity risk
  premium, The Journal of Portfolio Management
  34.3, 6380
• G.Consigli, L.M.MacLean, Y. Zhao and W.T.Ziemba
  (2009). The Bond-Stock Yield Differential as a
  Risk Indicator in Financial markets. To appear in
  The Journal of Risk 11(3)
• L.M.MacLean, Y.Zhao, G.Consigli, W.T.Ziemba
  (2008). Estimating parameters in a pricing model
  with state dependent shocks. Handbook of
  Financial Engineering, P.M. Pardalos, M.Doumpos
  and C. Zopounidis (Eds), Springer-Verlag,