basis and the half-life associated with it. The half-life is an interesting measure because it identifies how many months one must go back in the history of the data to find an observation with a weight equal to 0.5. For example, assume 21 business days in a month and a daily decay rate of 0.5 percent. Applying the recursive weighting formula, it is easy to verify that this corresponds to a monthly decay rate of approximately 10 percent and a half-life of 6.6 months. Since we are working with second moments, our weights will be assigned to squared returns and cross products between returns on different assets, so that the standard covariance formula is modified as:8 c6vT[r;fi),f;( %w\!\t(d)w\!\t(d) _ t=\ (16.4) I' More generally, to incorporate a weighting scheme into the monthly covariance estimator defined in equation (16.3'), we can proceed in steps: First, assign weights to the original return data; then apply the expression in (16.3') to the modified data. Formally, define the matrix of weighted daily returns as: .1/2 a. R(d) = wul*R(d) T-l T-l (1-5) 2 ru{d) (1-5) 2 Xl{d) I I (1-5)2 rT_u(d) (l-8)2rT_u(rf) T-l (1-5) 2 %N(d) (l-8)2fT_ljN(rf) rT,N ft) (16.5; where the symbol * indicates that each element in the vector of weights w must be multiplied by all the elements in the corresponding row of R(d). Once the daily returns have been adjusted by the weighting scheme, the modified formula for the co-variance matrix estimator is sw = p-s0(d)+^(p-kisk(d)+sk(d) i=l (16. whei 8Obviously assigning a weight wt to the cross product is equivalent to assigning the square root of that weight to each of the components of the cross product. As it will become clear later, the latter specification is easier to implement when working with matrices.