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Given a vector of attributes(eg.E/P ratios, betas) for N assets

$a^T = {a_1,a_2,...,a_N}$ The exposure of portfolio $h_P$ to attribute a is

$a = \sum_{n}a_n h_{P,n}$

Proposition: There is a unique portfolio $h_a$ that has minimum risk and unit exposure to a. The holdings(weights) of the characteristic portfolio $h_a$ are given by

$h_a = \frac{V^{-1}a}{a^TV^{-1}a}$

For the prrof we write:

Minimise $h^TVh$ subject to constriant : $h^Ta=1$

Using Langrange multiplier we get the equations:

a. $h^Ta = 1$

b. $Vh - \lambda a = 0$

Question: How does substituting a in b yields the result of the proposition ?

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From b. we get $Vh = \lambda a$, so $h=\lambda V^{-1}a$ (assuming V is invertible).

Using this to evaluate a. we get $h^Ta = \lambda a^T V^{-1}a=1$ (assuming $V^{-1}$ is symmetric). We can solve this for lambda: $\lambda=\frac{1}{a^T V^{-1}a}$

Now we can use this lambda in the previous expression for h to find the final explicit expression for h:

$$h=\frac{V^{-1}a}{a^T V^{-1}a}$$

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