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Knowledge of the efficient frontier is essential to understanding what the 'efficient surface' (and hence RiskWave) is all about..

Background

Harry Markowitz revolutionised investment theory with the introduction of mean-variance analysis in a 1952 essay entitled ‘Portfolio Selection’. Markowitz’s insight was to place emphasis on the correlations between securities and use diversification to minimise the risk of a portfolio for a given return through the careful selection of asset weights.

Diversification

By combining assets with less than perfect positive correlation, one can reduce the risk in the portfolio without sacrificing any of the portfolio’s expected return. This is known as diversification. In general, the lower the correlations of the assets in a portfolio, the less risky the resulting portfolio will be. This is true regardless of how risky the portfolio’s assets are when analysed in isolation.

Efficient Portfolio

A portfolio of assets (or contracts), is referred to as “efficient” if it has the lowest possible risk (represented by the variance of the portfolio’s return) for its level of expected return. Or equivalently, the portfolio has the highest level of expected return for its level of risk.

Efficient Frontier

The efficient frontier is the set of all efficient portfolios for the rates of return Mu-min 𝑡𝑜 Mu-max. Portfolios located above the efficient frontier are sub-optimal, as a lower level of risk can be achieved for the same rate of return. (Note that the axes are usually flipped the other way round; however this orientation is more useful for the purposes of the efficient surface)

The efficient frontier has two branches which are separated by the minimum variance point (MVP).

The efficient branch represents the best rate of return possible for a given risk level

The inefficient branch represents the worst rate of return possible for a given risk level.

Thus, in order for a portfolio to be considered optimal, we require that the portfolio be located on the efficient branch of the efficient frontier.

The Efficient Surface

Mean-variance analysis (MVA) can be applied to portfolios of catastrophe modelled (re)insurance contracts. The word 'asset' is therefore replaced by 'contract' - the principles remain essentially the same.

In applying MVA it becomes clear that the "classic" efficient frontier is insufficient for a portfolio manager's needs. The total written portfolio premium amount must be incorporated and this gives rise to the 'Efficient Surface'

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