Question

8. (Tracking) Suppose that it is impractical to use all the assets that are incorporated into a specified portfolio (such as a given efficient portfolio). One alternative is to find the portfolio, made up of a given set of n stocks, that tracks the specified portfolio most closely--in the sense of minimizing the variance of the difference in returns Specifically, suppose that the target portfolio has (random) rate of return 71. Suppose that there are 11 assets with (random) rates of return 7, 12,...) We wish to find the portfolio rate of return r = air, twarz + + Onn (with Lima; = 1) minimizing var() - rm). (a) Find a set of equations for the ai's (b) Although this portfolio tracks the desired portfolio most closely in terms of variance, it may sacrifice the mean Hence a logical approach is to minimize the variance of the tracking error subject to achieving a given mean return. As the mean is varied, this results in a family of portfolios that are efficient in a new sense-say, tracking efficient. Find the equation for the x;'s that are tracking efficient

Answer 1

(a) Formula for the variance of a sum Var[ax + by] = a²Var[x] + 2abCov[x, y] + b²Var[y]

We can write Var[r − r_M] = Var[r] − 2Cov[r, r_M] + Var[r_M] =
=1
$\sum_{j=1}^{n}$ α_i α_j σ_ij − 2
=1
α_iσ_iM + σ²_M

where σ_ij is the covariance of stocks i and j

σ_iM is the covariance of stock i and the tracked portfolio

σ²_M is the variance of the return of the tracked portfolio.

Formula for the variance of the return of a portfolio Var[r] = Var[
=1
α_ir_i ] =
=1
$\sum_{j=1}^{n}$ α_i α_jCov[r_i , r_j ] and the linearity of covariance with respect to another of the variates Cov [
=1
a_i x,y] =
=1
=1 a_iCov[x, y].

Set up the Lagrangian to minimize Var[r − rM] subject to a_i = 1

=1

L =
=1
$\sum_{j=1}^{n}$ α_i α_j σ_ij − 2
=1
α_iσ_iM + σ²_M − $\lambda$ (
=1
a_i − 1 )

=
=1
α_i² αi_²+
=1
$\sum_{j=1,j\neq i}^{n}$ α_i α_j σ_ij− 2
=1
α_iσ_iM + σ²_M − $\lambda$ (
=1
a_i − 1 )

Differentiation with respect to α_is and $\lambda$ and setting the derivatives to zero yields

∂L/ ∂αi = 2α_iσ_i²+ 2 $\sum_{j=1,j\neq i}^{n}$ α_j σ_ij − 2σ_iM − $\lambda$ = 0, ∀i = 1, . . . , n

∂L ∂λ =
=1
α_i− 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
=1
α_i¯r_i = ¯r_M.

The Lagrangian is as below:

L =
=1
α_i² α_i²+
=1
$\sum_{j=1,j\neq i}^{n}$ α_i α_j σ_ij− 2
=1
α_iσ_iM + σ²_M − $\lambda$ (
=1
a_i − 1 ) -$\mu$(  
=1
α_i¯r_i = ¯r_M).

Again, we differentiate with respect to α's, λ and µ and set the derivatives to zero and get

∂L ∂αi = 2α_iα_i+ 2 $\sum_{j=1,j\neq i}^{n}$ α_j σ_ij − 2σ_iM − $\lambda$ − µ¯ri_i= 0 ∀i = 1, . . . , n,

∂L ∂λ ==
=1
α_i− 1 = 0,

∂L ∂µ =
=1
α_i¯r_i = ¯r_M = 0.

These n + 2 equations can be solved to and the tracking efficient αi 's.