for anybody interested in the estimation of variances and covariances. Under rather general conditions, the accuracy of second moment estimators improves with the ability to sample data at higher frequency within a given period, rather than by extending the sampling period while keeping the sampling frequency constant. The intuition behind this result, unlike its mathematical derivation, is rather simple. If market volatilities and correlations move over time, focusing on shorter horizons and high-frequency data increases the probability of using observations from the same volatility regime. Going too far back in history would contaminate the sample with data from a different regime, thus biasing the risk estimates.3 Using Daily Data to Estimate a Monthly Covariance Matrix In the discussion that follows, we assume that we are interested in estimating a covariance matrix to forecast risk with a one-month horizon, and we propose an estimator that uses daily returns. Obviously, our estimator can be generalized to any horizon (quarter, year, etc.), but we will focus on one month to keep the notation simple. Let r( t(d) be the daily return on asset i computed from the close of day t - 1 to the close of day t. If returns are continuously compounded, then time aggregation for any horizon can be performed by simply adding returns at higher frequency. For example, if a month contains p business days, then the monthly return on asset i, which we denote with r (m), can be computed by adding the daily returns for that month: P r;(m) = ^rht(d) (16.1) t=i Since the covariance between two sums of random variables is equal to the sum of the covariances between each pair of random variables in the sums, the co-variance between the monthly returns on two generic assets i and / can be computed as:4 P P cov[f;(ffl),f;.(ffl)] = ^^cov[r,.;(i),r;.E(<i)] (16.2) It is useful to rewrite equation (16.2) in a more disaggregate form, to better understand all the components involved in the calculation of the monthly covariance: 3See Merton (1980) for a formal discussion of this result. 4In our discussion, we focus on the covariance between two generic assets. However, the same arguments apply to variances. In fact, the variance of the return on any asset can be obtained as a special case in which i = /'.