Long and Short Run Correlation Risk in Stock Returns
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1 Long and Short Run Correlation Risk in Stock Returns Discussion by Ric Colacito Econometric Society Winter Meetings, Denver, 1/ / 10
2 Contribution 1 Background: market variance risk premium predicts the equity premium. 2 / 10
3 Contribution 1 Background: market variance risk premium predicts the equity premium. 2 Question: is this because of individual securities risk or because of correlation risk? 2 / 10
4 Contribution 1 Background: market variance risk premium predicts the equity premium. 2 Question: is this because of individual securities risk or because of correlation risk? 3 This paper: explanatory power is driven by correlations (especially long-run correlations!) 2 / 10
5 Empirical contribution in a nutshell 1 Market variance risk premium predicts the equity premium Monthly Return Regressions Quarterly Return Regressions Panel A: Predictability Regressions Excluding Traditional Predictors VRP Mt (2.33) (4.66) VRP t ( 0.53) ( 0.28) ( 1.18) ( 1.03) CRP t (2.81) (2.73) (3.88) (3.86) 2 3 / 10
6 Empirical contribution in a nutshell 1 Market variance risk premium predicts the equity premium 2 Explanatory power is driven by correlations and not by idiosyncratic risk Monthly Return Regressions Quarterly Return Regressions Panel A: Predictability Regressions Excluding Traditional Predictors VRP Mt (2.33) (4.66) VRP t ( 0.53) ( 0.28) ( 1.18) ( 1.03) CRP t (2.81) (2.73) (3.88) (3.86) 2 3 / 10
7 Including the usual suspects... Panel B: Predictability Regressions Including Traditional Predictors V RP Mt (2.93) (4.16) V RP t ( 0.16) ( 0.10) ( 1.02) ( 1.26) CRP t (2.01) (2.06) (2.84) (3.07) RV Mt (1.98) (2.21) RV t ( 0.34) ( 0.13) ( 1.24) ( 0.78) RC t (1.98) (2.02) (2.44) (2.26) CAY t (1.04) (2.53) (0.43) (0.37) (0.33) (2.57) ( 0.55) ( 0.04) DEF t ( 0.16) (0.21) ( 0.13) ( 0.13) ( 0.96) ( 0.39) ( 0.78) ( 0.71) log(p/e) t ( 2.19) ( 0.58) ( 0.91) ( 0.35) ( 2.33) (0.02) ( 0.55) (0.15) T ERM t (0.33) ( 0.48) ( 1.59) ( 1.42) (0.71) ( 0.45) ( 1.50) ( 1.72) Adj. R 2 (%) Usual suspects are not significant Does this hold also for other forecasting horizons? 4 / 10
8 Including the usual suspects... Panel B: Predictability Regressions Including Traditional Predictors V RP Mt (2.93) (4.16) V RP t ( 0.16) ( 0.10) ( 1.02) ( 1.26) CRP t (2.01) (2.06) (2.84) (3.07) RV Mt (1.98) (2.21) RV t ( 0.34) ( 0.13) ( 1.24) ( 0.78) RC t (1.98) (2.02) (2.44) (2.26) CAY t (1.04) (2.53) (0.43) (0.37) (0.33) (2.57) ( 0.55) ( 0.04) DEF t ( 0.16) (0.21) ( 0.13) ( 0.13) ( 0.96) ( 0.39) ( 0.78) ( 0.71) log(p/e) t ( 2.19) ( 0.58) ( 0.91) ( 0.35) ( 2.33) (0.02) ( 0.55) (0.15) T ERM t (0.33) ( 0.48) ( 1.59) ( 1.42) (0.71) ( 0.45) ( 1.50) ( 1.72) Adj. R 2 (%) Economic significance How do returns respond to a one standard deviation shock? 4 / 10
9 Economic significance β σ skewness kurtosis effect on returns CAY b.p. VRP M b.p. CRP b.p. 5 / 10
10 Economic significance β σ skewness kurtosis effect on returns CAY b.p. VRP M b.p. CRP b.p. 1 Returns even more sensitive to correlation shocks: why? 5 / 10
11 Economic significance β σ skewness kurtosis effect on returns CAY b.p. VRP M b.p. CRP b.p. 1 Returns even more sensitive to correlation shocks: why? 2 Adjusted R 2 are 3 to 4 times as large with correlations. 5 / 10
12 Economic significance β σ skewness kurtosis effect on returns CAY b.p. VRP M b.p. CRP b.p. 1 Returns even more sensitive to correlation shocks: why? 2 Adjusted R 2 are 3 to 4 times as large with correlations. Are we capturing something more? 5 / 10
13 Comments on time series analysis 1 Comment on statistical insignificance of usual suspects 2 Comment on economic significance of one standard deviation shock to correlation (check the units) 6 / 10
14 Comments on time series analysis 1 Comment on statistical insignificance of usual suspects 2 Comment on economic significance of one standard deviation shock to correlation (check the units) 3 Does the result hold at longer forecasting horizons? 6 / 10
15 Comments on time series analysis 1 Comment on statistical insignificance of usual suspects 2 Comment on economic significance of one standard deviation shock to correlation (check the units) 3 Does the result hold at longer forecasting horizons? 4 Can we decompose correlation into short- and long-run component also in the time series analysis? 6 / 10
16 Comments on time series analysis 1 Comment on statistical insignificance of usual suspects 2 Comment on economic significance of one standard deviation shock to correlation (check the units) 3 Does the result hold at longer forecasting horizons? 4 Can we decompose correlation into short- and long-run component also in the time series analysis? E.g. extract correlations from options with various maturities... 6 / 10
17 Comments on time series analysis 1 Comment on statistical insignificance of usual suspects 2 Comment on economic significance of one standard deviation shock to correlation (check the units) 3 Does the result hold at longer forecasting horizons? 4 Can we decompose correlation into short- and long-run component also in the time series analysis? E.g. extract correlations from options with various maturities... 5 Comment on the methodology for estimating equi-correlations... 6 / 10
18 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 7 / 10
19 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 7 / 10
20 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 3 Note that portfolio variance is V[w r] = n w 2 j=1 j V[r j ] + 2ρ n n j=1 i=1,i j w j w i V[r i ]V[r j ] 7 / 10
21 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 3 Note that portfolio variance is V[w r] = n w 2 j=1 j V[r j ] + 2ρ n n j=1 i=1,i j w j w i V[r i ]V[r j ] 4 Estimate separately individual variances (V[r 1 ],V[r 2 ],...) and portfolio variance (V[w r]). 7 / 10
22 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 3 Note that portfolio variance is V[w r] = n w 2 j=1 j V[r j ] + 2ρ n n j=1 i=1,i j w j w i V[r i ]V[r j ] 4 Estimate separately individual variances (V[r 1 ],V[r 2 ],...) and portfolio variance (V[w r]). 5 Use known portfolio weights to back out correlation (ρ) 7 / 10
23 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 3 Note that portfolio variance is V[w r] = n w 2 j=1 j V[r j ] + 2ρ n n j=1 i=1,i j w j w i V[r i ]V[r j ] 4 Estimate separately individual variances (V[r 1 ],V[r 2 ],...) and portfolio variance (V[w r]). 5 Use known portfolio weights to back out correlation (ρ) Comparison to DECO model by Engle and Kelly (2009)? 7 / 10
24 Methodology for estimating correlations 1 Assume that all correlations are identical (ρ) 2 Consider a vector of portfolio weights (w = [w 1,w 2,...,w n ] ) 3 Note that portfolio variance is V[w r] = n w 2 j=1 j V[r j ] + 2ρ n n j=1 i=1,i j w j w i V[r i ]V[r j ] 4 Estimate separately individual variances (V[r 1 ],V[r 2 ],...) and portfolio variance (V[w r]). 5 Use known portfolio weights to back out correlation (ρ) Comparison to DECO model by Engle and Kelly (2009)? Is this helpful for asset allocation in large systems? 7 / 10
25 Cross-sectional results RM (3.25) (2.08) (3.20) (2.15) (2.65) (1.74) (3.90) (2.20) (2.78) (1.88) HF MV OL ( 3.98) ( 3.92) LF MV OL ( 1.99) ( 1.87) HF IV OL (0.92) (0.93) (1.14) (1.14) LF IV OL ( 1.82) ( 1.72) ( 2.87) ( 2.25) HF COR ( 1.73) ( 1.70) ( 1.71) ( 1.72) LF COR ( 2.89) ( 2.86) ( 3.54) ( 3.32) SMB (0.62) (0.84) (0.92) (1.00) (0.73) (0.88) (0.30) (0.54) (0.44) (0.55) HML (2.15) (0.67) (1.32) (0.50) (1.64) (0.61) (2.82) (1.28) (1.79) (1.04) UMD ( 2.00) ( 1.33) ( 1.71) ( 1.99) ( 1.09) ILLIQ ( 0.86) ( 1.06) ( 0.96) ( 0.67) ( 0.63) Wald p-value (0.94) (0.91) (0.89) (0.87) (0.94) (0.92) (0.57) (0.51) (0.45) (0.38) Long-run correlation is significant, short-run correlation is not 8 / 10
26 Cross-sectional results RM (3.25) (2.08) (3.20) (2.15) (2.65) (1.74) (3.90) (2.20) (2.78) (1.88) HF MV OL ( 3.98) ( 3.92) LF MV OL ( 1.99) ( 1.87) HF IV OL (0.92) (0.93) (1.14) (1.14) LF IV OL ( 1.82) ( 1.72) ( 2.87) ( 2.25) HF COR ( 1.73) ( 1.70) ( 1.71) ( 1.72) LF COR ( 2.89) ( 2.86) ( 3.54) ( 3.32) SMB (0.62) (0.84) (0.92) (1.00) (0.73) (0.88) (0.30) (0.54) (0.44) (0.55) HML (2.15) (0.67) (1.32) (0.50) (1.64) (0.61) (2.82) (1.28) (1.79) (1.04) UMD ( 2.00) ( 1.33) ( 1.71) ( 1.99) ( 1.09) ILLIQ ( 0.86) ( 1.06) ( 0.96) ( 0.67) ( 0.63) Wald p-value (0.94) (0.91) (0.89) (0.87) (0.94) (0.92) (0.57) (0.51) (0.45) (0.38) Long-run correlation is significant, short-run correlation is not Economic significance? 8 / 10
27 Cross-sectional results RM (3.25) (2.08) (3.20) (2.15) (2.65) (1.74) (3.90) (2.20) (2.78) (1.88) HF MV OL ( 3.98) ( 3.92) LF MV OL ( 1.99) ( 1.87) HF IV OL (0.92) (0.93) (1.14) (1.14) LF IV OL ( 1.82) ( 1.72) ( 2.87) ( 2.25) HF COR ( 1.73) ( 1.70) ( 1.71) ( 1.72) LF COR ( 2.89) ( 2.86) ( 3.54) ( 3.32) SMB (0.62) (0.84) (0.92) (1.00) (0.73) (0.88) (0.30) (0.54) (0.44) (0.55) HML (2.15) (0.67) (1.32) (0.50) (1.64) (0.61) (2.82) (1.28) (1.79) (1.04) UMD ( 2.00) ( 1.33) ( 1.71) ( 1.99) ( 1.09) ILLIQ ( 0.86) ( 1.06) ( 0.96) ( 0.67) ( 0.63) Wald p-value (0.94) (0.91) (0.89) (0.87) (0.94) (0.92) (0.57) (0.51) (0.45) (0.38) Long-run correlation is significant, short-run correlation is not Economic significance? Robustness of results to alternative estimators of short- and long-run covariance matrices? 8 / 10
28 One more intriguing results HF LF HF LF HF LF Mean Std. Dev. RM MVOL MVOL IVOL IVOL COR COR SMB HML UMD ILLIQ RM HF MVOL LF MVOL HF IVOL LF IVOL HF COR LF COR SMB HML UMD ILLIQ Short-run: market volatility and correlation co-move a lot 9 / 10
29 One more intriguing results HF LF HF LF HF LF Mean Std. Dev. RM MVOL MVOL IVOL IVOL COR COR SMB HML UMD ILLIQ RM HF MVOL LF MVOL HF IVOL LF IVOL HF COR LF COR SMB HML UMD ILLIQ Short-run: market volatility and correlation co-move a lot Long-run: market volatility and correlation co-move way less 9 / 10
30 One more intriguing results HF LF HF LF HF LF Mean Std. Dev. RM MVOL MVOL IVOL IVOL COR COR SMB HML UMD ILLIQ RM HF MVOL LF MVOL HF IVOL LF IVOL HF COR LF COR SMB HML UMD ILLIQ Short-run: market volatility and correlation co-move a lot Long-run: market volatility and correlation co-move way less Implications for asset allocation at various horizons 9 / 10
31 Concluding remarks A very interesting paper! Suggestions: 1 economic significance? 2 robustness? 3 can methodology be extended to other applications? 4 more comments on the co-movement of correlations and volatilities at different frequencies 10 / 10
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