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1.2 Diversification

An investor can reduce portfolio risk simply by holding unrelated instruments. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk. For diversification to work the component assets must have unrelated risks.

Mathematically:

From the formulae above: if any two assets in the portfolio have a correlation of less than 1 (i.e. are not perfectly correlated) the portfolio variance and hence volatility will be less than the weighted average of the individual instruments' volatilities.

1.3 The efficient frontier

Every possible asset combination can be plotted in risk-return space. For every level of return there exists one portfolio with the lowest risk; conversely, for every level of risk there is one portfolio with the highest return. The combination of all such portfolios is called the efficient frontier (sometimes “the Markowitz frontier”.)

The efficient frontier is illustrated above, with return ' on the y axis, and risk ' on the x axis.

The efficient frontier will be concave – this is because the risk-return characteristics of a portfolio change in a non-linear fashion as its component weightings are changed. (As described above, portfolio risk is a function of the correlationIn probability theory and statistics, the correlation also called correlation coefficient between two random variables is found by dividing their covariance by the product of their standard deviations. It is defined only if these standard deviations are f of the component assets, and thus changes in a non-linear fashion as the weighting of component assets changes.)

The region above the frontier is unachievable by holding risky assets alone. No portfolios can be constructed corresponding to the points in this region. Points below the frontier are suboptimal. A rational investor will hold a portfolio only on the frontier.

2 The risk free asset

The risk free asset is the (hypothetical) asset which pays a risk free rate - it is usually proxied by an investment in short-dated Government bonds. The risk free asset has zero variance in returns (hence risk free); it is also uncorrelated with any other asset. As a result, when it is combined with any other asset, or portfolio of assets, the change in return and also in risk is linear.

Because both risk and return change linearly as the risk free asset is introduced into a portfolio, this combination will plot a straight line in risk return space. The line starts at 100% in cash and weight of the risky portfolio = 0 (i.e. intercepting the return axis at the risk free rate) and goes through the portfolio in question where cash holding = 0 and portfolio weight = 1.

Mathematically:

Return is the weighted average of the risk free asset, rf, and the risky portfolio, p, and is therefore linear:

Return =

Since the asset is risk free, portfolio standard deviation is simply a function of the weight of the risky portfolio in the position. This relationship is linear.

Standard deviation =

2.1 Portfolio leverage

An investor can add leverage to the portfolio by holding the risk free asset. The addition of the risk free asset allows for a position in the region above the efficient frontier. Thus, by combining a risk-free asset with risky assets, it is possible to construct portfolios whose risk-return profiles are superior to those on the efficient frontier.

An investor holding a portfolio of risky assets, with a holding in cash, has a positive risk-free weighting (a de-leveraged portfolio). The return and standard deviation will be lower than the portfolio alone, but since the efficient frontier is convex, this combination will sit above the efficient frontier – i.e. offering a higher return for the same risk as the point below it on the frontier.
The investor who borrows money to fund her purchase of the risky assets has a negative risk-free weighting -i.e a leveraged portfolio. Here the return is geared to the risky portfolio. This combination will again offer a return superior to those on the frontier.

2.2 The market portfolio

To ensure that the combination held is always above the efficient frontier, the line plotted must be tangential to the efficient frontier (as opposed to going through it). For a given risk free rate, there is thus only one portfolio on the efficient frontier which can be combined with cash efficiently; i.e. the tangent portfolio. This is the market portfolio . See the illustration at top.

All rational investors will hold some combination of the market portfolio and the risk free asset.

2.3 Capital Market Line

When the market portfolio is combined with the risk free asset, the result is the Capital Market Line. All points along the CML have superior risk-return profiles to any portfolio on the efficient frontier. (A position with zero cash weighting is on the efficient frontier - the market portfolio.)

The CML is illustrated above, with return ' on the y axis, and risk ' on the x axis.

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