Efficient frontier portfolio

The model is subject to the same set of constraints as the deterministic model, with 0i as the risk trade-off parameter (or simply termed the risk factor) associated with risk reduction for the expected profit. 0j is varied over the entire range of (0, oo) to generate a set of feasible decisions that have maximum return for a given level of risk, which is equivalent to the efficient frontier portfolios for investment applications. 

Markowitz was the first to propose an explicit quemtification of the asset-allocation problem (Markowitz 1959). Three categorical inputs are required the expected return for each asset in the portfolio, the risk or variance of each asset s return, and the correlation between asset returns. The objective is to select the optimal weights for each asset that meiximizes total portfolio return for a given level of portfoho risk. The set of optimum portfohos over the risk spectrum traces out what is called the efficient frontier. 

It should be obvious that the production of MV portfolios is not extraordinarily input intensive. Efficient frontiers for five asset portfolios require only five predicted returns, five standard deviations, and a five-by-five symmetrical matrix of correlation coefficients. Yet the process yields indispensable information that allows investors to select suitable portfolios. 

For example, consider a simple portfolio of U.S. equities and bonds. Normally managers with the best forecasts wiU achieve better performance than other managers investing in the same assets. But another manager, who may not forecast U.S. equity and bond returns extremely well, can outperform by allocating funds to assets such as international equities and bonds. These assets possess different returns, risks, and correlations with each other and U.S. assets. Their inclusion shifts the efficient frontier upward beyond that resulting when only U.S. stocks and bonds are considered. 

In general, new assets that effectively shift the efficient frontier outward must possess differential return and risk profiles, and have low correlation with the existing assets in the portfolio. Incorporating an asset that has similar returns and risk and is coUinear with an asset rdready in the portfolio is duplicative. One needs only one of the two. 

Synthetic rebalancing cannot always be done, however. This is likely to be the case for rttiquid assets and those with legal covenants limiting transfer. For example, an investor may own a partnership or hold a concentrated stock position in a trust whose position cannot be swapped away. In these situations, MV optimization must be amended to include these assets with their weights restricted to the prescribed levels. The returns, risk, and correlation forecasts for the restricted assets must then be incorporated exphcitly in the analysis to take account of their interaction with other assets. The resulting constrained optimum portfolios will comprise a second-best efficient frontier but may not be too far off the unconstrained version. 

Recently, Duarte (1999) has suggested that value-at-risk approaches be used to derive efficient frontiers. TTiis approach differs from MV analysis in that it uses Monte Carlo simulation to determine the dollar value of the portfolio that is at risk with a particular degree of statistical confidence. That... 

Inflation-linked bonds also have different behavioural characteristics to other assets. They form a distinct asset class offering portfolio diversification benefits. This can be demonstrated using efficient frontier analysis. However, we should bear in mind what has already been said in the... 

Efficient frontiers also invariably place Treasury bills as the risk-free asset. T-bills may be risk-free from a creditworthiness point of view, bnt it is not tenable that a three-month nominal asset is a risk-free instrn-ment for someone with, say, a 30-year savings horizon. If you are investing for 30 years, over which time you are interested in your prospective real returns, then a 30-year linker (to be held to maturity) is your riskfree asset, almost by definition. 100% invested in that bond becomes the lowest risk portfolio on yonr frontier. Efficient frontier analysis starts to lose its impact once this premise is accepted, not least because you do not have a large data sample of consecutive, nonoverlapping 30-year periods (for any asset) to produce robust analysis. ... 

In Exhibits 8.14 and 8.15, we use the same data to construct two efficient frontiers of portfolios—one without index-linked gilts as an available asset choice and one where index-linked can be selected. For the first frontier below, without linkers, gold is still selected for the lowest risk portfolios because of its diversifying characteristics, in spite of its dreadful risk-return trade-off over the 21 years. Flowever, it quickly disappears from optimal portfolios along the frontier if risk tolerance is raised a tiny bit. The asset mixes of a selection of portfolios along the frontier are also detailed. 

EXHIBIT 8.14 Efficient Frontier of Three-Asset Portfolios, When Equities, Gilts, and Gold Are Available... 

When you add index-linked gilts as an asset choice, this results in the deselection of gold in even the lowest risk portfolios and the extension of the efficient frontier into the low risk domain to the left, as can be seen in Exhibits 8.14 and 8.15. 

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