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 prices Inflation 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 – share repurchases and growth strategies won popularity

Structural Break in Dividends in 1955

Final Model Structure Two state (S(t)) regime-switching model: If S(t) = 1: Δ p(t) = a 1 +  1 (t), (RW) If S(t) = 2: Δ p(t) = a 2 – 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!

Cross-Correlation Inflation vs Dividend Growth

Price Dynamics Log-Likelihood method results in the following significant relation: S(t) = 1: Δ p(t) =  1 (t) S(t) = 2: Δ p(t) = – 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?)

QQ-plot for linear model (residuals)

QQ-plot for LMARX (quantile residuals)

Prob{S=2 } as a function of inflation - High inflation is tricker for state-switch

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) = < = 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

1-year Predictive Distribution. Log(P/D): a) low 3.2 b) high 3.8

1-year Predictive Distribution. Log(P/D) is 3.8, Inflation 4 %

5-year Predictive Distribution. Log(P/D) is 3.8, Inflation 4 %

10-year Predictive Distribution. Log(P/D) is 3.8, Inflation 4 %

Present Situation Good: Stock market risk is much lower than in 2000 Bad: P/D –ratio still high in the U.S.