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

Structural Break in Dividends in 1955

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!

Cross-Correlation Inflation vs Dividend Growth

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?)

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) = 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

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.