(16_2') +(P " 2) {cov[^+2 (i),r;-; (<i)] + cov[r(-; (rf),r/j(+2 (<*)]} + +{cov[^,i+p-i (d)>%t $)] + cov[rij( (i),f/i(+p_i (<*)]} The expression in equation (16.2') is more intuitive than it looks. To compute the monthly covariance between the two assets, one must estimate several covariances between daily returns, including the covariances between returns that occur on different days within the month. The covariances between returns that occur on the same day have a larger weight, because we observe p simultaneous daily returns each month. Returns that are farther apart within the month are observed less often, and therefore their covariances have a smaller weight. To use a slightly more technical terminology, equation (16.2') indicates that when dealing with high-frequency data (e.g., daily data), one must take into account the serial correlation between returns to construct a covariance estimator for a longer horizon (e.g., one month). This is an interesting result, because it warns us against the temptation to estimate the monthly covariance by simply multiplying the daily covariance between the two assets by the number of business days within a month. Such a procedure is correct only when daily returns are identically and independently distributed (iid) because, in this case, all the covariances between returns observed on different days are equal to zero. The natural question at this point is: What degree of serial correlation should one assume when dealing with daily data? Unfortunately, there is not a simple answer that fits all scenarios. If we had a very large sample of data, then we could simply apply equation (16.2'). For example, if the true covariance between returns with two or more day lags were zero, the sample covariances of those returns would probably be very close to zero as well. However, if the sample of available data is not sufficiently large, then the estimated sample covariances are likely to reflect noise (spurious correlation) rather than a real statistical link between returns. To get a sense of how serious the role of noise can be in small samples, we performed a simple experiment. We generated 1,000 observations from a bivariate distribution, assuming zero correlation between the two random variables. Next, we tested how the sample estimates of the correlation change when using only a subset of the observations. To do this, we constructed two different estimators: The first one used only 50 random observations from the sample; the second one used 100 random observations. We computed each estimator 100 times. Not surprisingly, both estimators were on average very close to zero. However, as documented in Table 16.5, the dispersion around the mean (standard deviation) for the first estimator was almost double the dispersion for the second estimator. The largest estimated correlation when using 50 observations was equal to 0.48, and the smallest was -0.36-quite a large variation when one is trying to estimate the risk of a portfolio. The extreme values were reduced to half the size when we used 100 observations.