A Real-time Trading Rule Dooruj Rambaccussing Department of - - PowerPoint PPT Presentation

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

50 100 150 200 250 01/1904 01/1916 01/1928 01/1940 01/1952 01/1964 01/1976 01/1988 01/2000 Time Plot Buy and Hold Rule

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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 > Pt 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 > Pt Go long on Equity

P∗

t < Pt 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 > Pt Go long on Equity

P∗

t < Pt Go long on Bonds

P∗

t =

1 Et[rt+1 − ∆dt+1]Et[Dt+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 > Pt Go long on Equity

P∗

t < Pt Go long on Bonds

P∗

t =

1 Et[rt+1 − ∆dt+1]Et[Dt+1] Issue : Unobservable values of Et[Dt+1], Et[rt+1] and Et[∆dt+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 > Pt Go long on Equity

P∗

t < Pt Go long on Bonds

P∗

t =

1 Et[rt+1 − ∆dt+1]Et[Dt+1] Issue : Unobservable values of Et[Dt+1], Et[rt+1] and Et[∆dt+1] Observable : ∆dt, pdt

(Department of Economics, Business School, University of Exeter) 08 June 2010 4 / 14

Basic Definitions

rt=ln(Pt+1+Dt+1 Pt )

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Basic Definitions

rt=ln(Pt+1+Dt+1 Pt ) pdt= Pt Dt

(Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14

Basic Definitions

rt=ln(Pt+1+Dt+1 Pt ) pdt= Pt Dt ∆dt+1= ln(Dt+1 Dt )

(Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14

Basic Definitions

rt=ln(Pt+1+Dt+1 Pt ) pdt= Pt Dt ∆dt+1= ln(Dt+1 Dt ) Campbell and Shiller Log-Linearized Form: rt+1 = κ + ρpdt+1 + ∆dt+1 − pdt pd = E[log(PDt)], κ = log(1+ exp(pd)) − ρpd andρ = exp(pd) 1 + exp(pd) pdt = κ 1 − ρ + ρ∞pd∞ +

∞

∑

i=1

ρi−1(∆dt+i − rt+i)

(Department of Economics, Business School, University of Exeter) 08 June 2010 5 / 14

Time Varying Asset Pricing Model

Environment (unobservables) µt+1 − δ0 = δ1(µt − δ0) + εµ

t+1

gt+1 − γ0 = γ1(gt − γ0) + εg

t+1

(Department of Economics, Business School, University of Exeter) 08 June 2010 6 / 14

Time Varying Asset Pricing Model

Environment (unobservables) µt+1 − δ0 = δ1(µt − δ0) + εµ

t+1

gt+1 − γ0 = γ1(gt − γ0) + εg

t+1

Measurement Equation ∆dt+1 = γ0 + gt + εd

t+1

pdt = A − B µt + B gt A = κ 1 − ρ + γ0 − δ0 1 − ρ , B1 = 1 1 − ρδ1 , B2 = 1 1 − ργ1 .

(Department of Economics, Business School, University of Exeter) 08 June 2010 6 / 14

General Form of Model

State Equation

• gt+1 = γ1

gt + εg

t+1

(Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14

General Form of Model

State Equation

• gt+1 = γ1

gt + εg

t+1

Measurement Equation ∆dt+1 = γ0 + gt + εd

t+1

pdt+1 = (1 − δ1)A − B2(γ1 − δ1) gt + δ1pdt − B1εµ

t+1 + B2εg t+1

(Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14

General Form of Model

State Equation

• gt+1 = γ1

gt + εg

t+1

Measurement Equation ∆dt+1 = γ0 + gt + εd

t+1

pdt+1 = (1 − δ1)A − B2(γ1 − δ1) gt + δ1pdt − B1εµ

t+1 + B2εg t+1

Θ = (γ0, δ0, γ1, δ1, σg, σµ, σD, ρg µ, ρgD, ρµD) Θ = I 2

0 x R3 +x

I 3

c x R2

I 2

0 ∈ (−1, 1)

I 3

c ∈ [−1, 1]

(Department of Economics, Business School, University of Exeter) 08 June 2010 7 / 14

Matrix Structure

Xt = FXt−1 + Rεt Yt = M0 + M1Yt−1 + M2Xt Kalman Equations ηt = Yt − M0 − M1Yt−1 − M2(FXt−1t−1) St = M2(FPt−1|t−1F + RΣR)M

2

Kt = (FPt−1|t−1F + RΣR)M

2S−1 t

Xt|t = FXt−1|t−1 + Ktηt Pt|t = (I − KtM2)(FPt−1|t−1F + RΣR)

(Department of Economics, Business School, University of Exeter) 08 June 2010 8 / 14

Matrix Structure

Xt = FXt−1 + Rεt Yt = M0 + M1Yt−1 + M2Xt Kalman Equations ηt = Yt − M0 − M1Yt−1 − M2(FXt−1t−1) St = M2(FPt−1|t−1F + RΣR)M

2

Kt = (FPt−1|t−1F + RΣR)M

2S−1 t

Xt|t = FXt−1|t−1 + Ktηt Pt|t = (I − KtM2)(FPt−1|t−1F + RΣR) Log Likelihood: L = −

T

∑

t=1

log(det(St)) −

T

∑

t=1

η

tS−1 t

ηt

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Optimization Results

Parameter Coefficient Std error γ0 0.0012 0.001 δ0 0.0405 0.0503 γ1 0.8056 0.0246 δ1 0.9688 0.0032 σg 0.0066 0.0005 σd 0.0023 0.0002 σµ 0.0071 0.0003 ρg µ 0.6006 0.0326 ρµD 0.1013 0.0321

Table:

Optimization Results. The above table reports the results of the

• ptimization problem solved for monthly data starting Jan 1900 to Dec

2008.

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Results

Period RBH RTR S.EBH S.ETR 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 RBH(k) − RTR(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.

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Test of Riskiness

Sweeney X Statistic X = Rtr − (1 − f )RBH σx = σ[f (1 − f )/N]

1 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 σx =5.4 x 10−5.

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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.

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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.

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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. 3) Random Sampling shows that Rule works well for short periods of time : 12 and 24 months

(Department of Economics, Business School, University of Exeter) 08 June 2010 14 / 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. 3) Random Sampling shows that Rule works well for short periods of time : 12 and 24 months 4) Probability Distribution of P ∗

t

Pt is Right Skewed : Market is

underpriced more often.

(Department of Economics, Business School, University of Exeter) 08 June 2010 14 / 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. 3) Random Sampling shows that Rule works well for short periods of time : 12 and 24 months 4) Probability Distribution of P ∗

t

Pt is Right Skewed : Market is

underpriced more often. 5) Periods of crises command enormous opportunities to make arbitrage Gains !

(Department of Economics, Business School, University of Exeter) 08 June 2010 14 / 14