The Divergence of High- and Low-Frequency Estimation: Causes and Consequences

1. The Divergence of High- and Low-Frequency Estimation: Causes and Consequences Will Kinlaw Mark Kritzman David Turkington

2. Motivation

3. The divergence of high- and low-frequency estimation

4. The divergence of high- and low-frequency estimation

5. U.S. and emerging markets stocks: monthly returns Correlation = 0.69 Source: State Street Associates

6. U.S. and emerging markets stocks: annual returns Correlation = 0.44 Source: State Street Associates

7. U.S. and emerging markets stocks: triennial returns Correlation = 0.04 Source: State Street Associates

8. The divergence of high- and low-frequency estimation • It is common practice to assume that correlations estimated from high-frequency returns are similar to correlations estimated from low-frequency returns. • It is also common practice to assume that volatility scales with the square root of time. • Neither of these assumptions is supported by evidence.

9. Emerging markets and U.S. equities: Relative return distribution Excessdispersion refers to the fraction of a distribution that falls outside the one-standard deviation tails of the distribution implied by monthly returns. Excess Dispersion Excess Dispersion Actual Implied Source: State Street Associates

10. Divergence is different from sampling error 1 2 Sampling error arises when we use parameters from one sample… 3 4 5 6 7 8 9 … to predict parameters in another sample. 10 11 12

11. Divergence is different from sampling error 1 1 2 3 4 2 Divergence arises when we extrapolate parameters derived from high-frequency observations… … to estimate low-frequency parameters within the same sample. 5 6 7 3 8 9 10 4 11 12

12. Mathematics of divergence

13. The relation of high- and low- frequency volatility The volatility of the cumulative continuous returns of x over q periods is given by:

14. The relation of high- and low- frequency volatility The volatility of the cumulative continuous returns of x over q periods is given by: This term reflects annualization in the absence of lagged effects

15. The relation of high- and low- frequency volatility The volatility of the cumulative continuous returns of x over q periods is given by: This term captures the impact of auto-correlation

16. The relation of high- and low- frequency volatility: Impact of first-order auto-correlation Notes: This example assumes x and y have a contemporaneous correlation of zero. Source: State Street Associates

17. The relation of high- and low- frequency correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:

18. The relation of high- and low- frequency correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by: This term captures the lagged cross-correlation between x and y

19. The relation of high- and low- frequency correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by: This term captures the auto-correlation of x

20. The relation of high- and low- frequency correlation The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by: This term captures the auto-correlation of y

21. The relation of high- and low- frequency correlation: Impact of first-order cross-correlation Notes: This example assumes x and y have a contemporaneous correlation of zero. Source: State Street Associates

22. The relation of high- and low- frequency: Shorter versus longer lags Notes: This example assumes x and y have a contemporaneous correlation of zero. Source: State Street Associates

23. Low-frequency tracking error We can now use the long-horizon standard deviations of x and y, together with their correlation, to compute the standard deviation of their relative performance after q periods, which we define as the tracking error between x and y:

24. Evidence and causes of divergence

25. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

26. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

27. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

28. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

29. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

30. Attribution of excess dispersion of triennial relative returns Source: State Street Associates

31. Excess dispersion of triennial returns: Major asset classes Source: State Street Associates

32. Implications for portfolio construction

33. Benchmark and portfolio weights Source: State Street Associates

34. Standard deviations and correlations Source: State Street Associates

35. Low-frequency underperformance risk Source: State Street Associates

36. Balancing high- and low-frequency optimality • If investors care only about performance over short horizons or within long horizons they could construct portfolios that reflect aversion to risk based on covariances of high-frequency returns. • Alternatively, if they are concerned only with performance at the conclusion of long horizons, they could estimate covariances of low-frequency returns. • Or, as is the most likely case, they care about performance over both short and long horizons, they could include separate estimates of risk; one based on covariances of high-frequency returns and one based on covariances of low-frequency returns.

37. Iso-expected return curve balancing high- and low-frequency tracking error Portfolio 1 Portfolio 2 Portfolio 3 Source: State Street Associates

38. Portfolio weights Portfolio 1: Monthly Portfolio 2: Blended Portfolio 3: Triennial Source: State Street Associates

39. Implications for currency hedging

40. Illustrative portfolio – USD based Currency Exposure (%) Portfolio Structure (%) Note: We assume a 60/40 allocation into equities and bonds. The country equity allocation is based on the MSCI DM Equity Index as of March 2013. The country bond allocation is based on the Citigroup Global Bond Indexas of March 2013. Note: The chart shows the currency exposure as of March 2013. We use the Euro to proxy for the Danish kroner and the U.S. dollar to proxy for both the Singapore dollar and Hong Kong dollar, as these currencies trade within tight bands around the currencies to which they are pegged. We ignore the Israeli Shekel because it comprises a very small portion of the portfolio and is relatively more costly to hedge than the other developed market currencies. Source: State Street Associates

41. Excess dispersion of triennial returns Source: State Street Associates

42. Optimal hedging amount – USD based Source: State Street Associates

43. Low-frequency underperformance risk Source: State Street Associates

44. Optimal hedging amount – JPY based Source: State Street Associates

45. Low-frequency underperformance risk Source: State Street Associates

46. Optimal hedging amount – AUD based Source: State Street Associates

47. Low-frequency underperformance risk Source: State Street Associates

48. Implications for performance measurement

49. The distortion of hedge fund performance • The conventional practice for evaluating the Sharpe ratios of hedge funds is to estimate their standard deviations from monthly returns and then to multiply these monthly standard deviations by the square root of 12. • This approach implicitly assumes that hedge fund returns are serially independent at all lags, which is not the case.

50. Percentage of hedge funds that change quantile Monthly versus triennial Sharpe ratio Source: State Street Associates