it is estimated using a finite set of observations. The exercise also explores how a shrinkage technique of Ledoit and Wolf can mitigate this kind of distortion. (a) Assume n = 10 assets have returns that follow a multivariate normal distribution with expected returns equal to zero and true covariance matrix equal to the n x n diagonal matrix 0.8 0.85 V = 1.2 1.25 (The diagonal entries are equally spaced at 0.05 intervals.) Generate T = 120 samples rt, t = 1, ..., T, from this joint distribution. Each of these samples re E R10 is drawn from the ten-dimensional multivariate normal distribution N(0, V). (You may find the murnorm function in R useful for doing this.) This exercise is taken from Optimization Methods in Finance (2"d edition) by Cornuejols, Pena and Tutuncui and published by Cambridge University Press. 1 (i) Use the T samples to estimate the sample covariance matrix V as follows. Let r := (1/T) >tart, Z := 1t - r, t= 1,..., T, and V := Plot the eigenvalues both of the true covariance matrix V and of the estimated covariance matrix V on the same plot. Do you observe anything peculiar? (9 marks) (ii) Using the estimated covariance matrix V, find the estimated minimum-risk fully invested portfolio x. (Fully invested means that the portfolio weights should sum to 1.) Compute the estimated minimum variance x Vx, the actual minimum variance x Vx, and the true minimum variance (x*) V(x*), where x* is the true minimum-risk fully invested portfolio for V. (8 marks) (iii) Repeat parts (i) and (ii) several times (anywhere from a handful to a few thousand times). What do you observe? (8 marks) (b) We will next apply the shrinkage technique of Ledoit and Wolf. To that end, let di, i = 1, ...,n, denote the eigenvalues of the covariance matrix V and 1 := (1) Et-, di. Define C := AIn where In is the n x n identity matrix, and a := min 1 + Et- trace((ZAZ) - V)2), 1) trace((V - C)2) Finally, consider the shrunken matrix V := (1 - a) V + ac. (i) Plot the eigenvalues of the true covariance matrix V, of the sample covariance V, and of the shrunken covariance V on the same plot. What do you observe now? (9 marks) (ii) Using the shrunken covariance V find the estimated minimum-risk fully invested portfolio x. Compute the estimated minimum variance x Vx, the actual mini- mum variance x Vx, and the true minimum variance (x*) V(x*). What do you observe? Are the results any different from part (a) (ii)? (8 marks)