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**Portfolio optimisation.**

The expected return of a portfolio of k assets is given by

µp = W’R

where

W’ is the transpose of the k x 1 series of asset weights which results in a 1 x k matrix and

R is the k x 1 matrix of asset returns.

Typically when optimising a portfolio of assets we will want to either

a) Maximise the return of the portfolio or

b) Minimise the risk of the portfolio (often we use the Sharpe ratio as a proxy for the portfolio risk)

So for each asset we need to calculate the average return and the Sharpe ratio given by

(µp – rf)/σp

where

σp is the portfolio volatility

µp is the expected portfolio return and

rf is the risk free rate e.g LIBOR , USA T-bill

In order to optimise (i.e maximise return/minimise Sharpe ratio) we make the assumption in the first instance that all the assets in the portfolio are equally weighted. We use these initial weights to calculate the portfolio return µp and the portfoliovolatility σp.

Recall that σp = √X’∑X where ∑ is the variance covariance matrix and X is the asset weight matrix and X' its transpose.

Now that we have all of this data we can use excel’s solver to do our optimisation by

either maximising µp (or minimising the Sharpe ratio ) with the following constraints

a) Constraint 1 -> SUM(weights)= 100%

b) Constraint 2 -> Any individual weight > 0 (if we disallow short selling)

c) Constraint 3 -> σ <= smallest σ of any individual asset