1

Risk Indicators in the equity marketRisk Indicators in the equity market

giorgio.consigli@unibg.it

Joint work with L. MacLean, Y. Zhao and W.T. Ziemba

X workshop on Quantitative Finance

Politecnico di Milano, 29-30.1.2009

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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 90’s 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)

S&P500 valuation modelmarket price -- blue diamond

theoretical price -- green diamondsmisvaluation -- red line

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17/1

1/19

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06/1

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23/1

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05/0

1/19

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1/19

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04/0

8/19

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2/19

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21/0

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27/0

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ove

r/u

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ion

map into a risk premium ?

)()(/)()(~

/)(

)(/)()(

)(/)()(~

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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 S&P500.

• 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 S&P500 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:

),,( P

ttX

ttX

ttB

ttP

ttS

at index VIX :)(

at yieldmarket stock -bond :)(

at timereturn cash :)(

at time price Note)Treasury year (10 bond :)(

at time price index) P&(Sstock :)(

2

1

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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:

)ln(),ln( 21 tttt SYPY

)(),,0(),(

)),,((),,()),,((),,()(

)()()(

)()(

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1111

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tdRdWtdttdY

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tt

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,,,,,

),1,0( ,

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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

0)(

0)(

)),,((),,()),,((),,()(

2

1

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j

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wwtXt j

)ln())(ln()(

1,0,)()(1

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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 i=1,2 (up and down shocks respectively)

i

2,1 )()()(

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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

21212121 ,,,,,,,

RX

212121 ,,,,,

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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:

tsyyyyye ssssss ,...,1),,(:, 21'

1

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t

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1

2

1

1

12

1

1

),(),(

),(),(),(

),(

),(),(

))()(())((2

1exp)()2()|,(

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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:

iA

i

i

iA

i

iAiA

i

i

iA

n

jijiA

n

jiji

n

jij

n

jijAii

inii

xnxxxnxG

ixxx

1

12

111

1

)ln(|

2,1,,...,

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parameters estimation

• The method proceeds as follows:1. Set stress weights and calculate stress values2. Calculate the empirical distributions for positive and negative

shocks over the sample period3. Specify a grid size, an initial tail area and a step, identify

positive and negative shock times4. Calculate for the given shock sequence the cond ML coefficients5. 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

),( Il

),( Il

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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

2,1 ,))ˆ(()1)((

))ˆ((ˆ))ˆ((ˆˆ~

1,

1,2221,11122

iNEtNP

NENEY

tiii

ttt

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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.

S&P500 (blue)

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computational evidence• In most cases the fits are good. The weights which gave the

best fit, w=0 (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 w=0.75

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Shocks probability

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Results – Tests on market hypotheses over the entrie sample

Risk premium• We estimate the S&P 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 > 6.6349,

X^2(1) with 99% confidence

012 v

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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 26:7, 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, 63—80

• 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,