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Dynamic Model for Stock Market Risk Evaluation
Kasimir Kaliva and Lasse Koskinen
Insurance Supervisory Authority
Finland
Goal: Stock Market Risk Modelling in Long Horizon
• Phenomenon: Stock market bubble
• Model should work from one quarter to several years– Prices should be mean reverting
• Fundament: Dividend / Price – ratio
• Explanatory factor: Inflation
• Usage: Risk Assessment and DFA
Background• Theory: Gordon growth model for dividend
Dynamics: Campbell et al, also near Wilkie– Dividend-price-ratio (P/D) time-varying,
stationary=>Mean reversion in stock pricesInflation expectation: Modigliani and Cohn (79)
• Statistical model: – Logistic Mixture Autoregression with
exogenous variable (Wong and Li (01))– Conditional (dynamic) on P/D -ratio
Data
• U.S. quarterly stock market (SP500) and inflation series; Log returns and dividens– Prices and dividends
• Period: 1959 –1994
• Structural breaks in dividend series and price/dividend –series in 1958 and 1995
• 1995- 2001 – share repurchases and growth strategies won popularity
Final Model Structure• Two state (S(t)) regime-switching model: If S(t) = 1: Δ p(t) = a1 + 1(t), (RW) If S(t) = 2: Δ p(t) = a2 – by(t-1)+ (t), causes mean reversion
- 1(t) N(0,σ∼ 1), 2(t) N(0,σ∼ 2), - y(t) = dividend/price -ratio
• State hidden: Prob{ S(t) = 1} = ( f(inflation) ); is
normal distribution, f is a function
Statistical Model for Dividend and Inflation
• Dividend: AR(2)-ARCH(4) –model
- Dividend is the driving factor.
• Inflation: AR(4) –model where dividend is explanatory variable
- Note! This is just statistical relation, not causal. See fig on cross-correlation!
Price Dynamics
Log-Likelihood method results in the
following significant relation:
S(t) = 1: Δ p(t) = 0.027 + 1(t)
S(t) = 2: Δ p(t) = 1.078 – 0.357y(t-1)+ (t),
- 1(t) N(0,0.052), ∼ 2(t) N(0,0.077)∼• Return distribution conditional:
State S(t) and y(t) = log(P(t) /D(t))
Model Testing
• The model is compared to 1) more general and 2) linear alternatives:
- More general LMARX (that include standard RW and more complicated models) is rejected at 5 % level
- Information criterion AIC and BIC select the nonlinear model instead of linear one
Model Diagnostic
• Quantile (QQ) –plot shows:
- Normal distribution assumption for the residuals of the linear model is wrong (fig)
• Quantile residual plot shows:
- Excellent fit for LMARX
See fig. (Can see that it is not from normal distribution?)
Intrerpretation• Model operates much more often in state 1
than in state 2; that is RW is a good description most of the time
• E(Δ d) = 0.014 < 0.027 = E(RW | S =1).
=> process generates bubbles
=> switch from S=1 to S=2 causes a market crash, since b < 0 (b is the coefficient of log(P(t) /D(t)) in state 2)
=> process is mean reverting
UK – data (not in the paper)
• Overfitting is a danger especially when nonlinear model is used
=> We tested also the UK -data
=>The model structure remains invariant (heteroscedastic residuals)
Risk Assessment
• The proposed model has shape-changing predictive distribution
• Shape depends on
- prediction horizon
- inflation
- P/D –ratio
=> Risk is time-varying