Enterprise Risk Management Formula Book

10. Portfolio optimisation

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10.1        Mean-variance portfolio optimisation


If there are  asset categories then a (one period) mean-variance efficient portfolio,  where  is the amount (or weight) invested in the ’th asset, given a benchmark, , assumed future (one period) mean returns  on each asset, , and an assumed covariance matrix between the (one period) returns on different assets, , is a portfolio that maximises the utility function,  for some risk aversion parameter, , subject to relevant constraints on the , where:



The constraints that are applied are usually linear, i.e. of the form  which if  is an  dimensional vector is understood as meaning that there are  constraints each of the form . In such a formulation, equality constraints, including that the amounts invested add up to the total portfolio value (or the weights add to unity), can be written as two inequality constraints, e.g.  and  combined.


The implied alphas with a mean-variance risk-return model given a portfolio  and benchmark  are the mean returns that need to be assumed for the different asset categories for  to be mean-variance optimal for some value of . They can only meaningfully be determined for assets whose weights in the portfolio are not constrained (other than by the constraint that weights add to unity). They are then  where  where  and  are arbitrary scalar constants.


10.2        Capital Asset Pricing Model (CAPM)


The security market line is:





The capital market line (for efficient portfolios) is:



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