(a) Formula for the variance of a sum Var[ax + by] = a2Var[x] + 2abCov[x, y] + b2Var[y]
We can write Var[r − rM] = Var[r] − 2Cov[r, rM] + Var[rM] = αi αj σij − 2 αiσiM + σ2M
where σij is the covariance of stocks i and j
σiM is the covariance of stock i and the tracked portfolio
σ2M is the variance of the return of the tracked portfolio.
Formula for the variance of the return of a portfolio Var[r] = Var[ αiri ] = αi αjCov[ri , rj ] and the linearity of covariance with respect to another of the variates Cov [ ai x,y] = =1 aiCov[x, y].
Set up the Lagrangian to minimize Var[r − rM] subject to ai = 1
L = αi αj σij − 2 αiσiM + σ2M − ( ai − 1 )
= αi2 αi2+ αi αj σij− 2 αiσiM + σ2M − ( ai − 1 )
Differentiation with respect to αis and and setting the derivatives to zero yields
∂L/ ∂αi = 2αiσi2+ 2 αj σij − 2σiM − = 0, ∀i = 1, . . . , n
∂L ∂λ = αi− 1 = 0,
and we have n + 1 equations and n + 1 variables from which the αi 's can be solved.
b) Similarly with the added constraint αi¯ri = ¯rM.
The Lagrangian is as below:
L = αi2 αi2+ αi αj σij− 2 αiσiM + σ2M − ( ai − 1 ) -( αi¯ri = ¯rM).
Again, we differentiate with respect to α's, λ and µ and set the derivatives to zero and get
∂L ∂αi = 2αiαi+ 2 αj σij − 2σiM − − µ¯rii= 0 ∀i = 1, . . . , n,
∂L ∂λ == αi− 1 = 0,
∂L ∂µ = αi¯ri = ¯rM = 0.
These n + 2 equations can be solved to and the tracking efficient αi 's.
8. (Tracking) Suppose that it is impractical to use all the assets that are incorporated into...
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