A Real-time Trading Rule Dooruj Rambaccussing Department of Economics Business School University of Exeter 08 June 2010 (Department of Economics, Business School, University of Exeter) 08 June 2010 1 / 14
Objective Can we beat the Buy and Hold Strategy by diversifying in the risk-free asset when equity markets are overpriced ? (Department of Economics, Business School, University of Exeter) 08 June 2010 2 / 14
Objective Can we beat the Buy and Hold Strategy by diversifying in the risk-free asset when equity markets are overpriced ? Trading Rule built up around Real time concept with EMH asumptions (Department of Economics, Business School, University of Exeter) 08 June 2010 2 / 14
Results Time Plot 250 Buy and Hold Rule 200 150 100 50 0 01/1904 01/1916 01/1928 01/1940 01/1952 01/1964 01/1976 01/1988 01/2000 (Department of Economics, Business School, University of Exeter) 08 June 2010 3 / 14
Model Rule :Compare theoretical price (NPV) with actual Price (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Model Rule :Compare theoretical price (NPV) with actual Price P ∗ t > P t Go long on Equity (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Model Rule :Compare theoretical price (NPV) with actual Price P ∗ t > P t Go long on Equity P ∗ t < P t Go long on Bonds (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Model Rule :Compare theoretical price (NPV) with actual Price P ∗ t > P t Go long on Equity P ∗ t < P t Go long on Bonds 1 P ∗ t = E t [ r t + 1 − ∆ d t + 1 ] E t [ D t + 1 ] (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Model Rule :Compare theoretical price (NPV) with actual Price P ∗ t > P t Go long on Equity P ∗ t < P t Go long on Bonds 1 P ∗ t = E t [ r t + 1 − ∆ d t + 1 ] E t [ D t + 1 ] Issue : Unobservable values of E t [ D t + 1 ] , E t [ r t + 1 ] and E t [ ∆ d t + 1 ] (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Model Rule :Compare theoretical price (NPV) with actual Price P ∗ t > P t Go long on Equity P ∗ t < P t Go long on Bonds 1 P ∗ t = E t [ r t + 1 − ∆ d t + 1 ] E t [ D t + 1 ] Issue : Unobservable values of E t [ D t + 1 ] , E t [ r t + 1 ] and E t [ ∆ d t + 1 ] Observable : ∆ d t , pd t (Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14
Basic Definitions r t = ln ( P t + 1 + D t + 1 ) P t (Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14
Basic Definitions r t = ln ( P t + 1 + D t + 1 ) P t pd t = P t D t (Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14
Basic Definitions r t = ln ( P t + 1 + D t + 1 ) P t pd t = P t D t ∆ d t + 1 = ln ( D t + 1 ) D t (Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14
Basic Definitions r t = ln ( P t + 1 + D t + 1 ) P t pd t = P t D t ∆ d t + 1 = ln ( D t + 1 ) D t Campbell and Shiller Log-Linearized Form : r t + 1 = κ + ρ pd t + 1 + ∆ d t + 1 − pd t exp ( pd ) pd = E [ log ( PD t )] , κ = log ( 1 + exp ( pd )) − ρ pd and ρ = 1 + exp ( pd ) ∞ κ 1 − ρ + ρ ∞ pd ∞ + ρ i − 1 ( ∆ d t + i − r t + i ) ∑ pd t = i = 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14
Time Varying Asset Pricing Model Environment (unobservables) µ t + 1 − δ 0 = δ 1 ( µ t − δ 0 ) + ε µ t + 1 g t + 1 − γ 0 = γ 1 ( g t − γ 0 ) + ε g t + 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 6 / 14
Time Varying Asset Pricing Model Environment (unobservables) µ t + 1 − δ 0 = δ 1 ( µ t − δ 0 ) + ε µ t + 1 g t + 1 − γ 0 = γ 1 ( g t − γ 0 ) + ε g t + 1 Measurement Equation g t + ε d ∆ d t + 1 = γ 0 + � t + 1 pd t = A − B � µ t + B � g t 1 − ρ + γ 0 − δ 0 κ 1 1 A = 1 − ρ , B 1 = , B 2 = . 1 − ρδ 1 1 − ργ 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 6 / 14
General Form of Model State Equation g t + ε g � g t + 1 = γ 1 � t + 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14
General Form of Model State Equation g t + ε g � g t + 1 = γ 1 � t + 1 Measurement Equation g t + ε d ∆ d t + 1 = γ 0 + � t + 1 g t + δ 1 pd t − B 1 ε µ t + 1 + B 2 ε g pd t + 1 = ( 1 − δ 1 ) A − B 2 ( γ 1 − δ 1 ) � t + 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14
General Form of Model State Equation g t + ε g � g t + 1 = γ 1 � t + 1 Measurement Equation g t + ε d ∆ d t + 1 = γ 0 + � t + 1 g t + δ 1 pd t − B 1 ε µ t + 1 + B 2 ε g pd t + 1 = ( 1 − δ 1 ) A − B 2 ( γ 1 − δ 1 ) � t + 1 Θ = ( γ 0 , δ 0 , γ 1 , δ 1 , σ g , σ µ , σ D , ρ g µ , ρ gD , ρ µ D ) Θ = I 2 0 x R 3 I 3 c x R 2 + x I 2 I 3 0 ∈ ( − 1 , 1 ) c ∈ [ − 1 , 1 ] (Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14
Matrix Structure X t = FX t − 1 + R ε t Y t = M 0 + M 1 Y t − 1 + M 2 X t Kalman Equations η t = Y t − M 0 − M 1 Y t − 1 − M 2 ( FX t − 1 t − 1 ) M 2 ( FP t − 1 | t − 1 F � + R Σ R � ) M � S t = 2 ( FP t − 1 | t − 1 F � + R Σ R � ) M � 2 S − 1 K t = t X t | t = FX t − 1 | t − 1 + K t η t ( I − K t M 2 )( FP t − 1 | t − 1 F � + R Σ R � ) = P t | t (Department of Economics, Business School, University of Exeter) 08 June 2010 8 / 14
Matrix Structure X t = FX t − 1 + R ε t Y t = M 0 + M 1 Y t − 1 + M 2 X t Kalman Equations η t = Y t − M 0 − M 1 Y t − 1 − M 2 ( FX t − 1 t − 1 ) M 2 ( FP t − 1 | t − 1 F � + R Σ R � ) M � S t = 2 ( FP t − 1 | t − 1 F � + R Σ R � ) M � 2 S − 1 K t = t X t | t = FX t − 1 | t − 1 + K t η t ( I − K t M 2 )( FP t − 1 | t − 1 F � + R Σ R � ) = P t | t Log Likelihood: T T η � t S − 1 ∑ ∑ L = − log ( det ( S t )) − η t t t = 1 t = 1 (Department of Economics, Business School, University of Exeter) 08 June 2010 8 / 14
Optimization Results Parameter Coefficient Std error γ 0 0.0012 0.001 δ 0 0.0405 0.0503 γ 1 0.8056 0.0246 δ 1 0.9688 0.0032 σ g 0.0066 0.0005 σ d 0.0023 0.0002 σ µ 0.0071 0.0003 ρ g µ 0.6006 0.0326 ρ µ D 0.1013 0.0321 Table: Optimization Results. The above table reports the results of the optimization problem solved for monthly data starting Jan 1900 to Dec 2008. (Department of Economics, Business School, University of Exeter) 08 June 2010 9 / 14
Results R BH R TR S.E BH S.E TR Period 1 year 0.044 0.052 0.0034 0.0027 2 year 0.092 0.107 0.0035 0.0028 3 year 0.142 0.166 0.0036 0.0028 4 year 0.194 0.230 0.0036 0.0029 5 year 0.249 0.296 0.0036 0.0029 Table: Cummulated Returns over Horizons. The table provides the cummulated returns over horizons of 12, 24, 36, 48 months for both the model and the Buy and Hold strategy. The standard errors are also reported to illustrate the riskiness of the returns. (Department of Economics, Business School, University of Exeter) 08 June 2010 10 / 14
Results Period Paired Correlation Mean Differences Std error t-statistic 1 year 0.368 -0.010 0.0007 -13.99 2 year 0.329 -0.021 0.0014 -14.88 3 year 0.297 -0.035 0.0022 -15.40 4 year 0.268 -0.049 0.0031 -15.69 5 year 0.247 -0.066 0.0041 -15.88 Table: Test of Correlated Means. The right hand side column illustrates the holding period (k). The correlation between the two return series are also reported. the R BH ( k ) − R TR ( k ) refers to the mean difference. The denominator in the test is given by the std error. The degrees of freedom for the 1,2,3, 4, 5 years of horizons were 1295, 1283,1271,1259 and 1247 respectively. (Department of Economics, Business School, University of Exeter) 08 June 2010 11 / 14
Test of Riskiness Sweeney X Statistic X = R tr − ( 1 − f ) R BH 1 σ x = σ [ f ( 1 − f ) / N ] 2 f: proportion of months the asset is held in the risk free asset. N is the number months the rule is put to the test. σ is the standard error of the monthly returns under the Buy and Hold strategy. X = 0.0022 σ x = 5.4 x 10 − 5 . (Department of Economics, Business School, University of Exeter) 08 June 2010 12 / 14
Sampling with Replacement Period 20 40 80 120 1 year 55 60 61.25 60 2 year 40 52.5 53.75 53.12 3 year 40 47.5 41.25 41.87 4 year 45 50 40 41.25 5 year 45 52.5 42.5 43.12 Table: Performance of Model versus Buy and Hold . This table illustrates the number of times that the rule beats or equals the Buy and Hold Strategy in Random Samples. (Department of Economics, Business School, University of Exeter) 08 June 2010 13 / 14
Conclusion The Rule Does not Work well !!! Mean Reversion takes time (Frequency of analysis) Cummulated returns 2.05 times more under the Rule at the end of the horizon. (Department of Economics, Business School, University of Exeter) 08 June 2010 14 / 14
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