1. REGIME CHANGES AND FINANCIAL MARKETS Prepared for Topics in Quantitative Finance | Abhishek Rane - Andrew Ang and Allan Timmermann

2. PAPER STRUCTURE Introduction to Economic Regimes Model Specification Capturing Statistical Properties Asset Pricing with Regimes Applications in Finance Conclusion

3. INTRODUCTION Regime Switching means Volatilities, autocorrelations and cross covariance's vary across economic regimes. In equilibrium models, regimes in fundamental processes, like consumption or dividend growth, strongly affect the dynamic properties of equilibrium asset prices and can induce non- linear risk-return trade-offs. Regimes can cause large consequences to investors optimal Portfolio choice. For example, the mean, volatility and correlation patterns in stock returns changed dramatically at the start of, and persisted through the global financial crisis. Regime Switching Models have become popular because of their intuitive nature.

4. When applied to financial series, regimes identified by econometric methods often correspond to different periods in regulation, Policy and other secular changes. For example, interest rate behavior markedly change from 1979 through 1982, during which the FED changed its operating procedure to target monetary aggregates. In equities, different regimes correspond to periods of high and low volatility, and long bull and bear market periods. Thus, regime switching models can match narrative stories of changing fundamentals that sometimes can only be interpreted ex post, but in a way that can be used for ex-ante

5. MODEL SPECIFICATION Consider a variable y t, which depends on its own past history, y t−1, random shocks, ε t, and some regime process, s t. Regimes are generally modeled as discrete st ∈ {0, 1,..., k}, tracking the particular regime inhabited by the process at a given point in time. We are looking at at a two regime model. In this model Regimes are limited to affect the intercept, μst, autocorrelation, φ st, and volatility, σ st, of the process.

6. It is common to assume that st follows a homogenous first-order Markov chain, Π[i,j] = Pr(s t = j|s t−1 = i) = p ij One can also use models with multiple regimes ( by using a larger transition matrix) or where underlying regimes are not known. Estimation Techniques: Using Expectation Maximization algorithms or Max Likely hood methods. Maximum likely methods use Bayesian updating procedure which infers the probability of being in a regime given all available information up until that time, Pr(s|I t ), where I t is the information set at time t.

7. STATISTICAL PROPERTIES Skew ness and Tails :An attractive feature of regime switching models is that they capture central statistical features of asset returns. Assume a simple 2 regime model Model: y t = μ t +σ st * εt, εt ∼ iidN(0,1), Probability s t = 0 is π 0 and s t = 1 is 1 − π 0 Figure1 plots the probability density functions (pdfs) corresponding to this mixture of two normals for (μ 0 = 1,σ 0 = 1), (μ 1 = −2,σ 1 = 2), and π 0 = 0.8 While the two distributions separately have no fat tails, the mixture of the two normals produces pronounced negative skewness and excess kurtosis

9. The first 4 central Moments are given by: Differences in means across regimes, μ 0 − μ 1, enter the higher moments such as variance, skew, and kurtosis. Intuitively, the possibility of changing to a new regime with a different mean introduces an extra source of risk.

10. Time – Varying Correlations Correlations increase during market downturns as shown by Longin and Solnik (2001), Ang and Chen (2002), and others. Regime switching models are able to match these patterns well through persistence in the probabilities of staying in a regime with low means, high volatilities, and high correlations. Exceedence Correlations: Consider {(y1, y2)} drawn from a bivariate variable Y = (y1, y2) Exceedance level θ is positive (negative) means that the draws are greater (less) that their empirical means. {(y1 y2)|y1 ≥ (1 + θ)y1’ and y2 ≥ (1 + θ)y2’} for θ ≥ 0 and {(y1 y2)|y1 ≤ (1 + θ)y1 and y2 ≤ (1 + θ)y2} for θ ≤ 0, where yj’ is the mean of yj. The correlation of this subset of points is termed the exceedance correlation.

11. Figure 2 shows that the exceedance correlations of US-UK returns in the data exhibit a pronounced asymmetric pattern, with negative exceedance correlations higher than positive exceedance correlations.

12. ASSET PRICING We start with the conventional asset pricing model based on a representative agent with utility over consumption, Ct, following Lucas (1978): P t U′(C t ) = β * E t [U′ (C t+1 )(P t+1 + D t+1 )] (7) Β – Subjective Discount factor We assume power utility U (C ) = C (1+γ) /(1 + γ ).. γ ̸ = −1 Equity pays dividend D t and each period C t = D t Dividend process:

13. St ={0,1} and it is independent of past values of εt Investors are assumed to know st at time t (but not st+1) and so set prices conditional on which regime prevails today, st. π t0 = 1 if s t = 0, otherwise s t = 1, so that πt0 is an indicator tracking the current regime. Conjecture that the Solution takes the Form: P t = ρ(s t ) * D t, s t = {0, 1} (10) ρ – price dividend ratio which takes finite no of values (regimes) and is regime.

14. Recovering ρ: Gross Returns: Prepared for Topics in Quantitative Finance | Abhishek Rane

15. CONCLUSION Regimes are caused by a change in economic policy, e.g. a shift in monetary or exchange rate regime. In other cases, a major event, such as the bankruptcy of Lehman in September 2008, or the overthrow of the Shah in Iran and the associated spike in oil prices, may be the trigger. They are also caused by investor expectations Simple analysis of an equilibrium asset pricing model showed that regimes in consumption or dividend growth translate into regimes in asset returns. Prepared for Topics in Quantitative Finance | Abhishek Rane