1. Beta Prediction: Optimal Number of Observations and Liquidity Angela Ryu Economics 201FS Honors Junior Workshop: Finance Duke University April 14, 2010

2. Data F, XRX, CPB, T, HNZ, IBM, XOM, WMT, JPM, BAC (10 stocks) SPY Data from Aug 20 2004 – Jan 7 2009 (1093 days) Sampling frequency (SF): 1 – 20 min. Beta calculation days (BI): 1 – 50 days

3. Summary Motivation: Beta is thought to be time varying (Huang and Litzenberger, 1988). Empirically find optimal (sampling frequency, Beta interval day) pair so that the realized Beta yield minimum MSE. Method: Calculate MSE with varying sampling frequency and Beta int. day; find the minimum point. TODAY Given sampling frequency, compare “optimal” Beta Interval Days of stocks of different liquidity – Hyp 1: for stocks with more liquidity, shorter sampling frequency and shorter Beta trailing window can be used for Beta calculation for least MSE Compare number of observations for each lowest MSE point – Hyp 2: There will be a number(or narrow range) of observations which will be optimal to calculate Beta with least MSE. Conclude with an explanation

4. Correction Correction in code: Comparing MSEs using 3-D plot: used same vector (10 min, 11 days) for each stock for comparison

5. Liquidity Measure Within the interval, find number of zero log returns. – The greater the number, the less liquid the stock is At 10 minute frequency, the percentage of zeros of each stock’s log return: Less liquidMore liquid Stock% of zero returnsStock% of zero returns F 19.7116 IBM 3.5224 XRX 15.32 XOM 3.5754 CPB 11.4196 WMT 6.0553 T 10.2567 JPM 6.6283 HNZ 10.0207 BAC 6.7053

6. 3-D plot Changes are more subtle, but tendencies are similar (compared to the previous result): – As sampling frequency grows from 1 to 20, MSE level increases – Given sampling frequency, MSE slightly increases again after 30 - 35 minutes but more or less remains constant

7. Global Minimum of MSE Less liquidMore liquid StockGlobal MinStockGlobal Min F (1, 23) IBM (1, 6) XRX (1, 10) XOM (1, 8) CPB (1, 6) WMT (1,7) T JPM (2, 20) HNZ (1, 4) BAC (1, 15) * Global Minimum Point in (sampling frequency, Beta interval day) Global mins have similar number of observations. Since the difference in minimum MSE for each sampling frequency is not big (< 10^-6), there may be an “optimal” number of observations needed for Beta calculation that yields the minimum MSE for given frequency.

8. Minimum MSE Points Hyp 1 is not necessarily true. Different levels of microstructure noise in two groups is negligible. Less Liquid More Liquid

9. Comparing Number of Observations Hyp 2 matches the result. Microstructure noise is a possible factor in stabilization of the number of observations necessary for calculating beta with the least MSE Again, there isn’t a significant difference between less liquid and more liquid stocks. Less Liquid More Liquid

10. Analysis When calculating beta, a certain threshold number of past data seems to be necessary for the least MSE. However, including more may distort the “prediction” because Beta is time varying. Microstructure noise within a stock affects the number of observations that yields minimum MSE. For shorter sampling frequencies ( =5 mins) the number remained relatively stable. Although 1 min sampling yields global minimum of MSE, since the level of MSE does not vary much among different sampling frequencies and 1 min requires significantly more obs., using “optimal” beta calculated in longer sampling frequency may be better in use. On the other hand, different level of noise among groups with varying liquidity did not affect the Beta and MSE (Some statistical method should be applied to verify). This may be so because the effect of number of observation overcomes the inter-difference of noise.

11. For the Final Report Find ways to statistically compare results from the two groups Compare the precision of Beta prediction (with optimal number of observations) to other classic models Literature to back up conceptual basis