Bayesian Dynamic Factor Models with Shrinkage in Asset Allocation. Duke University

Transcription

1 Bayesian Dynamic Factor Models with Shrinkage in Asset Allocation Aguilar Omar Lynch Quantitative Research. Merrill Quintana Jose Investment Management Corporation. CDC West Mike of Statistics & Decision Sciences Institute Duke University

2 Dynamic Bayesian Partial Shrinkage Models for the Expected ffl Returns. Outline ffl International Exchange Rates. ffl Bayesian Dynamic Factor Models. ffl Multivariate Stochastic Volatility Components. ffl Dynamic Asset Allocation. ffl Extensions and Future Directions.

3 y jt = log(s jt =s j;t 1) [log(1 + i USA;t 1 ) log(1 + i j;t 1)]: Conditional independent and Normally distributed excess returns, ffl y ο N( t ; ± t ). t International Exchange Rates DATA: ffl Monthly spot exchange rates (s it ), with respect to the US Dollar. Country Currency Code Country Currency Code Germany Mark DEM France Franc FRF Great Britain Pound Netherlands Guilder NLG Australia Dollar Italy Lira ITL Japan Yen JPY Spain Peseta ESP Canada Dollar CAD New Zealand Dollar NZD Switzerland Franc CHF United States Dollar USD ffl One-month-ahead excess returns, from 2/76 to 12/99.

4 International Exchange Rates Returns DEM Returns % Returns % Returns % JPY CAD CHF Returns % Returns % Returns % FRF NLG ITL Returns % Returns % Returns % ESP NZD Returns % Returns %

5 Dynamic regressions with economic predictors to estimate and predict ffl expected returns t. the Explore patterns of variability and residual structure over time via ffl ± t. modeling Dynamic asset allocation in portfolio construction via sequential ffl updating. International Exchange Rates GOALS: ffl Find latent processes driving the changes in variances and correlations. ffl Improve short-term forecasts of means t and variances ± t.

6 Short Interest Rates DEM JPY CAD CHF FRF NLG ITL ESP NZD

7 Yield Curve DEM JPY CAD CHF FRF NLG ITL ESP NZD

8 Interest Rate Acceleration DEM JPY CAD CHF FRF NLG ITL ESP NZD

9 Bayesian Partial Shrinkage Model for t y t = t + ν t ; ν t ο N(; ± t ); ffl The linkage matrix L t References include Quintana, Chopra and Putnam (1995) and Zellner, ffl and Min (1991). Hong Dynamic SUR Model with Shrinkage: t = Z t fi t ; fi t = L t ff t + d t ; d t ο N(; U t ); ff t = G t ff t 1 +! t ;! t ο N(; W t ); ff t 1 ο N(m t 1; C t 1): determines the relationships between the beta coefficients.

10 Bayesian Partial Shrinkage Model for t 8 < Assume there is only one predictor z jt, for j = 1;:::q. y jt = jt + ν jt ; ν jt ο N(;ff 2 jt); jt = z jt fi jt ; fi jt = μ t + ffl jt ; μ t = μ t 1 + ο t ; ο t ο N(;u); ffl jt = ffi j ffl j;t 1 + jt ; t ο N(;fi j ): ffl These relationships imply that fi jt = ffi j fi j;t 1 + (1 ffi j )μ t 1 + jt : full shrinkage; ffl Shrinkage parameter» ffi j» 1 where ffi j = : 1 no shrinkage:

11 ffl f t ffl H t ffl ffl t ffl Ψ t Dynamic Factor Models for ± t Aguilar and West (2) dynamic factor model for y t ο N( t ; ± t ), y = t + X t f t + ffl t ; t t ο N(f t j; H t ); f ± t = X t H t X t + Ψ t : ffl t ο N(ffl t j; Ψ t ): ffl X t is the q k factor loadings matrix with q fl k, is a k 1 vector of conditionally independent latent common factors, = diag(h1t;:::;h kt ) instantaneous factor variances, is a q 1 vector of series-specific quantities, = diag(ψ1t;:::;ψ qt ) instantaneous, idiosyncratic" variances and ffl ffl t and f s are mutually independent for all t; s.

12 For z it = log(f 2 it) = it + ν it and it = log(h it ) we assume latent - models, VAR(1) z t = t + ν t ; where μ t = μ t 1 + t ; with correlated innovations across factors! t - These relationships imply ο N(! t j; U): Stochastic Volatility Components ffl Multivariate SV models for the latent factor processes f t ο N(; H t ). t = μ t + fl t ; fl t = Φfl t 1 +! t : t = μ t + Φ( t 1 μ t 1) +! t :

13 Global Risk Parameter: Common level of volatility across factors ffl μ jt = μ t ; 8j: when ffl Time-varying factor effects: Series μ t Independent univariate SV models for each one of the q idiosyncratic ffl ψ it. variances Stochastic Volatility Components COMMENTS: Mean reversion: marginal/overall volatility levels μ jt. ffl ffl Volatility Persistence: Φ high. determine the relative importance of each one of the factors over time.

14 X = B 1 1 x2;1 x k+1;1 x k+1;2 x k+1;3 x k+1;k x q;1 x q;2 x q;3 x q;k C Specifications IDENTIFICATION: ffl Constant or "slowly varying" loadings matrix X t = X, 1 B C C B... B C B C x k;1 x k;2 x k;3 1 B C : C B C B B C C B B C ffl Series order defines interpretation of factors. ffl Informative priors on SVM innovations variances.

15 Inference based on a fixed sample over t = 1;:::;T f f t ; t ;ψ it Sequential Analysis over t = T + 1;T + 2;;::: Model Fitting and Bayesian Analysis ffl Bayesian analysis via posterior simulations (GIBBS-Metropolis). ffl MCMC samples from the joint posterior for, shrinkage model parameters, dynamic factor model parameters and - latent processes: factors and volatilities - : t = 1;:::;T g ffl Sequential Particle Filtering to update posterior samples! p( jd t 1)! p( jd t )! ffl and compute/revise predictive distributions! p(y t jd t 1)! p(y t+1jd t )!

16 References Stochastic Volatility Models: ffl N Shephard (1994, 96), Harvey, Ruiz and Shephard (1994). ffl E Jacquier, NG Polson and PE Rossi (1994, 95). ffl S Kim, N Shephard and S Chib (1998). Dynamic Factor Models: ffl N Shephard and M Pitt (1999), O Aguilar and M West (2). Particle Filtering: ffl M Pitt and N Shephard (1999), O Aguilar and M West (2). ffl J Liu and M West (1999). ffl M West (1993), NJ Gordon, DJ Salmond and AFM Smith (1993).

17 Volatilities of Currency Returns DEM stdev stdev stdev JPY CAD CHF stdev stdev stdev FRF NLG ITL stdev stdev stdev ESP NZD stdev stdev

18 Latent Factors and Their Volatilities Factor DEM Factor Factor

19 .8 (.1) 1. (.). (.).15 (.2).16 (.5) 1. (.).2 (.).9 (.1).22 (.2) CAD 1. (.3).8 (.1).1 (.) CHF.95 (.1).3 (.2) -.1 (.1) FRF.99 (.1).1 (.2). (.1) NLG.93 (.1).5 (.2) -.1 (.1) ESP.17 (.1).34 (.1).84 (.1) NZD Factor Loadings Matrix X Factor 1 Factor 2 Factor 3 DEM 1. (.). (.). (.) JPY.52 (.3).23 (.3).4 (.4) ITL.92 (.1).2 (.2) -.2 (.1)

20 Idiosyncratic Idiosyncratic Idiosyncratic Idiosyncratic Idiosyncratic Volatilities DEM Idiosyncratic Idiosyncratic JPY Idiosyncratic Idiosyncratic CAD CHF FRF Idiosyncratic Idiosyncratic NLG ITL ESP Idiosyncratic NZD

21 % Idiosyncratic % Idiosyncratic % Idiosyncratic % Idiosyncratic Idiosyncratic Volatilities as % of Total DEM % Idiosyncratic % Idiosyncratic JPY % Idiosyncratic % Idiosyncratic CAD CHF FRF % Idiosyncratic % Idiosyncratic NLG ITL ESP % Idiosyncratic NZD

22 Global Risk Parameter Risk Time

23 Regression Parameter for Short Interest Rates.56 DEM JPY.56 CAD.56 CHF FRF.56 NLG.56 ITL ESP NZD -.45 Full Shrinkage 9% 8% 7% 6% 5% 4% 3% 2% 1% No Shrinkage

24 Regression Parameter for Short Interest Rates DEM JPY CAD CHF FRF NLG ITL ESP NZD Full Shrinkage 95% Shrunk 9% Shrunk

25 Regression Parameter for Yield Curve.58 DEM JPY.58 CAD.58 CHF FRF.58 NLG.58 ITL ESP NZD -.47 Full Shrinkage 9% 8% 7% 6% 5% 4% 3% 2% 1% No Shrinkage

26 Regression Parameter for Yield Curve DEM JPY CAD CHF FRF NLG ITL ESP NZD Full Shrinkage 95% Shrunk 9% Shrunk

27 Regression Parameter for Interest Rate Acceleration.32 DEM JPY.32 CAD.32 CHF FRF.32 NLG.32 ITL ESP NZD -.55 Full Shrinkage 9% 8% 7% 6% 5% 4% 3% 2% 1% No Shrinkage

28 Regression Parameter for Interest Rate Acceleration DEM JPY CAD CHF FRF NLG ITL ESP NZD Full Shrinkage 95% Shrunk 9% Shrunk

29 Expected Return Expected Return Expected Return Expected Return Expected Return Estimates DEM Expected Return Expected Return JPY Expected Return CAD Expected Return CHF FRF Expected Return NLG Expected Return ITL ESP Expected Return NZD

30 t = (a1t;a2t;:::;a qt ) be the one-step ahead portfolio where a it is Let a of wealth invested in the i-th currency. proportion the Forecast: distributions p(y ffl jd t 1) with g and predictive G t denoting t t predicitve mean and covariance matrix. corresponding the - unconstrained, transaction costs efficient, liquidity and turnover. Dynamic Asset Allocation ffl Portfolio return at time t, r t = a ty t. Optmization: - minimise risk: a tg t a, - for a specified "target" return: a tg t = m,

31 DEM a t Unconstrained Portafolio with m = : JPY CAD CHF FRF NLG ITL ESP NZD Full Shrinkage 95% Shrunk 9% Shrunk

32 DEM a t Unconstrained Portafolio with m = : JPY CAD CHF FRF NLG ITL ESP NZD Full Shrinkage 9% 8% 7% 6% 5% 4% 3% 2% 1% No Shrinkage

33 Cumulative Returns and Performance Cumulative Return 3 Years Returns Cumulative Return % Full Shrinkage 95% Shrunk 9% Shrunk Cumulative Returns 3 years Risk-Adjusted Returns Ratio Risk Sharpe Ratios 3 years Risk

34 Cumulative Returns and Performance Cumulative Return 3 Years Returns Cumulative Return % Full Shrinkage 9% No Shrinkage Cumulative Returns 3 years Risk-Adjusted Returns Ratio Risk Sharpe Ratios 3 years Risk

35 Summary and Future Research ffl Peaks in the volatilities are consistent with positive correlations in volatility across the factors (EU (DEM), +NZD and JPY+). ffl Non-negligible over-all idiosyncratic variations. ffl Improvements in short-term forecasting and decision making through partial shrinkage models. ffl Novel and customized MCMC algorithms for fitting and computation of posterior and predictive analyses. ffl Improved sequential particle filtering. ffl Change points: - Intervention analysis. - Heavy tailed components. ffl Model uncertainty: number of factors, order of factors.