--------------------- where rtt{m) denotes the return on asset i between month t - 1 and month t, and 7(m)indicates its sample mean. This estimator is easy to compute and update at the end of each month. Unfortunately, it also has a number of limitations. For example, it assigns the same weight to all the observations in the sample. This makes sense if the distribution that generates the monthly returns does not change over the 10-year period. However, if market volatility increased (decreased) significantly over the last part of the sample, this simple estimator would take a long time (often too long) to capture this change, because each new observation added to the sample has a small weight. In addition, the estimator uses only monthly data and, therefore, is not able to accommodate changes in market conditions that may be reflected in data at higher frequency, for example daily. The natural question to ask at this point is whether these limitations are relevant in practice. More specifically, are we likely to change our investment decisions due to the choice of a particular covariance matrix estimator? To answer this question, we present two scenarios in which the covariance matrix estimator plays an important role, and discuss the sensitivity of our conclusions to the use of two alternative estimators. In the first example, we consider two specifications of a $100 million portfolio invested in 18 developed equity markets: a market capitalization weighted portfolio, with the weights measured at the end of May 2002, and an equally weighted portfolio. For each portfolio, we want to estimate the risk contribution from each individual position, and the Value at Risk (VaR), which we identify with the amount of capital that would be expected to be lost once in 100 months. The two covariance matrix estimators that we use are both based on standard techniques followed by investment professionals.1 The first estimator (risk model A) uses 10 years of daily data and assigns a larger weight to more recent observations, starting from a weight of 1 and reducing it by approximately 25 percent on a monthly basis. The second estimator (risk model B) uses nine years of monthly data and assigns the same weight to all observations. The left part of Table 16.1 shows that the two estimators generate different values in the risk decomposition of the value-weighted portfolio. Not surprisingly, the differences are more pronounced for the largest positions in the portfolio (United States, United Kingdom, and Japan). The estimated VaR also increases by more than 7 percent when using estimator A instead of B. The right part of Table 16.1 contains similar statistics for the equally weighted portfolio. The effect on risk decomposition is even more striking. For example, Hong Kong and Singapore are among the bottom contributors to risk when using ]At this point, we do not discuss which estimator is more desirable. We leave that analysis for the main section of this chapter.